Statistical Framework for Quantum Teleportation: Fidelity Analysis and Resource Optimization

Abstract

This paper establishes a comprehensive statistical framework for analyzing quantum teleportation protocols under realistic noisy conditions. We develop novel mathematical tools to characterize the complete statistical distribution of teleportation fidelity, including both mean and variance, for systems experiencing decoherence and channel imperfections. Our analysis demonstrates that the teleportation fidelity follows a characteristic distribution $F\sim \mathcal{P}(F_{\mathrm{avg}},\mathrm{\Delta }{F}^2)$ where the variance $\mathrm{\Delta }{F}^2$ depends crucially on entanglement quality and channel noise. We derive the optimal resource allocation condition $\partial E_{\mathrm{ent}}/\partial F/\partial C_{\mathrm{classical}}/\partial F=\beta /\alpha$ that minimizes total resource consumption while achieving target fidelity. Furthermore, we introduce a Bayesian adaptive protocol that enhances robustness against noise through real-time statistical estimation. The theoretical framework is validated through numerical simulations and provides practical guidance for experimental implementations in quantum communication networks.

1. Introduction

Quantum teleportation, as one of the most remarkable phenomena in quantum information science, encapsulates the profound implications of quantum nonlocality. The concept was first proposed theoretically by Bennett et al. [1] and has since evolved into a cornerstone protocol in quantum information science. Its central idea is that an unknown quantum state can be perfectly transferred from a sender (commonly referred to as Alice) to a receiver (Bob) through the use of shared entanglement and classical communication, without any physical transmission of the quantum system itself. By ingeniously combining the nonclassical correlations embodied in entanglement, emphasized in the seminal work of Einstein, Podolsky, and Rosen [2], with classical communication mechanisms, quantum teleportation enables reliable quantum state transfer. From its theoretical conception to experimental verification, the development of quantum teleportation has witnessed the transition of quantum physics from abstract theory to operational technology. Early experimental efforts were primarily focused on photonic systems and were subsequently extended to a wide range of physical platforms, including atomic, ionic, and superconducting systems [3,4,5,6]. These experiments not only successfully validated the fundamental principles of the teleportation protocol but also systematically revealed the unavoidable challenges encountered in realistic physical implementations, such as decoherence, channel noise, and imperfections in measurement and control. In overcoming these challenges, quantum teleportation has continuously driven progress in frontier fields including quantum communication, quantum computation [7,8], and quantum networks [9,10,11], as well as photonic quantum technologies and scalable experimental platforms [12,13,14].

However, as the field moves from idealized theoretical protocols to practical, large-scale applications, it becomes increasingly clear that teleportation is subject to a wide range of noise sources and imperfections that can hinder performance. These include, but are not limited to (i) decoherence of the shared entangled resource (e.g., due to photon loss, spontaneous emission, or thermal noise), modeled by channels such as amplitude damping, phase damping, or depolarizing noise [15,16]; (ii) imperfect Bell-state measurements with limited efficiency or fidelity, which are particularly challenging in linear optics [17]; (iii) noise in the classical communication channel, leading to bit-flip errors in the correction message; (iv) and imperfect implementation of the final unitary correction operations. A comprehensive statistical framework must therefore account for these combined, non-ideal effects to predict and optimize real-world teleportation performance.

As research advances from idealized theoretical protocols toward scalable and engineering-oriented applications, the need for rigorous and systematic statistical analysis frameworks under realistic constraints becomes increasingly urgent. Due to the presence of stochastic processes such as decoherence, measurement and control errors, and channel noise, the fidelity and reliability of quantum teleportation in experimental realizations often exhibit pronounced statistical fluctuations. Consequently, statistical methods play an increasingly critical role in quantum information processing. Although the teleportation protocol is deterministic in principle under ideal conditions, in practical physical platforms, different noise mechanisms can substantially affect performance and alter the optimal allocation of entanglement and classical communication resources [3,4,5,6,18,19,20,21]. Related studies in quantum noise, noise-limited operation, and entanglement-assisted performance further indicate that quantitative conclusions depend sensitively on the physical context and underlying noise mechanisms. Motivated by these non-ideal effects, previous studies have investigated the statistical properties of quantum teleportation from multiple perspectives, including fidelity optimization [22,23], quantitative characterization of entanglement resources [24,25,26], systematic analysis of error sources [27,28,29,30,31], and the characterization of channel capacity and performance limits [32,33,34,35]. These works have provided valuable insights into teleportation behavior under specific noise models or within particular physical platforms. Nevertheless, most existing studies focus on isolated performance metrics or restricted scenarios, and a comprehensive statistical framework capable of unifying performance quantification, resource–performance trade-offs, and protocol reliability across multiple realistic noise mechanisms and operational constraints remains lacking.

In response to these challenges, this paper develops a comprehensive statistical framework for analyzing quantum teleportation protocols, with particular emphasis on fidelity optimization, resource allocation, and performance limits under realistic noisy conditions. Building upon the foundational work of Bennett et al. [1] and recent advances in quantum information theory [36,37], we introduce a unified statistical perspective to address the inherent randomness and uncertainty present in practical teleportation systems. Specifically, we first propose novel statistical metrics to characterize the overall performance of hybrid quantum–classical teleportation channels, employing unified measures that simultaneously quantify entanglement quality and classical communication reliability, thereby overcoming the limitations of traditional single-figure performance indicators. Second, by considering non-ideal EPR pairs and imperfect classical channels, we derive analytically tractable expressions for the complete statistical distribution of teleportation fidelity, explicitly including both the mean value and the variance, thus providing a rigorous mathematical foundation for assessing performance stability. Furthermore, we establish strict theoretical relationships between resources and performance by identifying sufficient conditions for achieving target fidelities with minimal resource overhead, quantitatively revealing the trade-offs among entanglement consumption, classical communication cost, and teleportation accuracy. Finally, we propose and analyze an adaptive teleportation strategy that incorporates real-time statistical estimation, significantly enhancing protocol robustness under diverse noise environments and operational imperfections. Our analysis uncovers intricate and fundamental statistical structures underlying the interplay among entanglement consumption, classical communication cost, and transmission accuracy, and explicitly identifies achievable fidelity limits as well as previously unknown performance boundaries and scaling behaviors under realistic channel imperfections. The resulting statistical framework is mathematically self-consistent and broadly applicable, naturally extending existing teleportation theory to address key challenges in practical scenarios such as long-distance quantum communication [38,39] and satellite-based quantum networks [40,41]. Beyond enriching the theoretical foundations of quantum teleportation, this work provides practically meaningful tools for the design, comparison, and optimization of emerging quantum technologies. As quantum communication networks continue to scale toward practical deployment [42], and as quantum computing architectures increasingly demand reliable state-transfer capabilities [43], a systematic and rigorous statistical characterization of the teleportation process will play an ever more central role in the future development of quantum information technologies.

The remainder of this paper is organized as follows. Section 2 presents a statistical analysis of teleportation outcomes, including the probability distribution of Bell measurement results and their statistical dependence and independence properties. Section 3 discusses channel-capacity and information-theoretic aspects relevant to teleportation, clarifying the roles and physical interpretations of classical and quantum resources. Section 4 introduces statistical verification and hypothesis-testing methods for demonstrating quantum advantage and experimental benchmarking. Section 5 explores applications and extensions, including resource-cost modeling, optimization principles, and adaptive and Bayesian strategies. Finally, Section 6 concludes the paper and outlines directions for future research.

2. Statistical Analysis of Teleportation Outcomes

This section analyzes the fundamental sources of randomness in the quantum teleportation protocol from a statistical perspective. We first investigate the intrinsic probabilistic structure introduced by Alice’s Bell-state measurement and rigorously demonstrate that, in the ideal teleportation protocol, the four Bell-measurement outcomes occur with equal probability and are statistically independent of the unknown input state. This uniformity and independence form the foundation of the statistical analysis of quantum teleportation. We then address the unavoidable non-idealities of entanglement resources in practical scenarios by introducing a resource-quality parameter, the singlet fraction $F_{\mathrm{EPR}}$, and derive the quantitative relationship between teleportation fidelity and entanglement quality under the depolarizing (Bell-diagonal) degradation model. These statistical links between the ideal and noisy regimes provide a unified starting point for the subsequent analysis of fidelity distributions, resource constraints, and experimental verification.

2.1. Probability Distributions in Measurement Outcomes

The Bell-state measurement constitutes a fundamental probabilistic element in the quantum teleportation protocol. Following the seminal formulation of quantum teleportation [1], the four mutually exclusive measurement outcomes correspond to the complete set of Bell states:

$$\begin{array}{cc}\hfill |{\mathrm{\Psi }}^{\pm }{\rangle }_{12}& ={\displaystyle \frac{1}{\sqrt{2}}}\left(\right|01\rangle \pm {|10\rangle )}_{12},\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill |{\mathrm{\Phi }}^{\pm }{\rangle }_{12}& ={\displaystyle \frac{1}{\sqrt{2}}}\left(\right|00\rangle \pm {|11\rangle )}_{12}.\hfill \end{array}$$

(2)

To simplify the presentation, we adopt a standard two-bit parametrization of the Bell basis [43]: we label the four Bell outcomes by $(m,n)\in {\{0,1\}}^2$ and write ${\left\{\right|B_{mn}\rangle }_{12}\}$ for the Bell basis. Here, $(m,n)$ are two binary labels that uniquely index the four Bell-state measurement outcomes (equivalently, two classical bits). One convenient identification is $|B_{00}\rangle \equiv |{\mathrm{\Phi }}^{+}\rangle$, $|B_{10}\rangle \equiv |{\mathrm{\Phi }}^{-}\rangle$, $|B_{01}\rangle \equiv |{\mathrm{\Psi }}^{+}\rangle$, and $|B_{11}\rangle \equiv |{\mathrm{\Psi }}^{-}\rangle$ (up to an overall phase under alternative conventions).

For an arbitrary unknown input state $|\varphi \rangle$ and ideal maximally entangled resource preparation, it is a standard result that the probability distribution over Alice’s Bell-measurement outcomes is perfectly uniform [44]. This equiprobability is independent of the specific input state and invariant under any unitary transformation on the input. While well established, it forms a statistical cornerstone for teleportation and underpins our subsequent analysis of fidelity distributions and verification procedures.

To make the statistical structure explicit without unnecessary intermediate calculations, Alice’s Bell-state measurement on particles 1 and 2 can be represented by the projectors onto the Bell basis of the $(1,2)$ subsystem:

$$\begin{array}{c}\hfill {\mathrm{\Pi }}_{12}^{mn}=|B_{mn}\rangle \langle B_{mn}{|}_{12}\otimes I_3,(m,n)\in {\{0,1\}}^2,\end{array}$$

(3)

where $I_3$ is the identity operator on particle 3 (Bob’s particle), and ${\mathrm{\Pi }}_{12}^{mn}$ is the Bell-basis projector on subsystems $(1,2)$ associated with outcome $(m,n)$.

The corresponding outcome probability is

$$\begin{array}{c}\hfill P(m,n)=\mathrm{Tr}\left[{\mathrm{\Pi }}_{12}^{mn}{\left(\right|\varphi \rangle }_1\langle \varphi |\otimes {\rho }_{23})\right],\end{array}$$

(4)

where $(m,n)$ label the classical outcomes of the Bell-state measurement, and ${\rho }_{23}$ represents the shared two-particle resource state between Alice and Bob. In the ideal case ${\rho }_{23}=|{\mathrm{\Psi }}^{-}{\rangle }_{23}\langle {\mathrm{\Psi }}^{-}|$, one obtains $P(m,n)=1/4$ for all $(m,n)$.

In practical implementations, however, deviations from this ideal uniformity may arise due to imperfections in various components of the protocol, e.g., (a) the entangled resource state ${\rho }_{23}$ may not be the ideal Bell state; (b) the Bell measurement may be incomplete or noisy (e.g., efficiency $\eta <1$); and (c) the classical communication channel may introduce errors in transmitting the outcome label.

Measurement Outcome Distribution. For an arbitrary unknown input state $|\varphi \rangle$ and ideal EPR pair preparation, the probability distribution over Alice’s measurement outcomes is perfectly uniform $P(m,n)={\displaystyle \frac{1}{4}},(m,n)\in {\{0,1\}}^2.$ Equivalently, $P\left(\right|{\mathrm{\Psi }}^{\pm }\rangle )=P\left(\right|{\mathrm{\Phi }}^{\pm }\rangle )=1/4$. This equiprobable distribution remains invariant under all unitary transformations of the input state $|\varphi \rangle$.

Universal Uniformity of Measurement Outcomes. For the quantum teleportation protocol employing ideal EPR pairs and perfect Bell-state measurements, the outcome probability distribution is uniform and independent of the specific form of the input quantum state.

Proof. 

Using the standard teleportation decomposition in the Bell basis, one may write

$${|\varphi \rangle }_1\otimes |{\mathrm{\Psi }}^{-}{\rangle }_{23}={\displaystyle \frac{1}{2}}\sum _{m,n\in \{0,1\}}{|B_{mn}\rangle }_{12}\otimes {\left(U_{mn}|\varphi \rangle \right)}_3,$$

(5)

where $U_{mn}$ are the Pauli correction operators associated with the Bell outcome $(m,n)$ (up to an overall phase under different labeling conventions). Since each Bell component appears with identical amplitude, the Born rule immediately yields a uniform outcome distribution independent of the input state $|\varphi \rangle$. □

Statistical Verification of the No-Cloning Theorem. The observed equiprobable distribution of measurement outcomes provides an operational consistency check of the no-cloning theorem within the quantum teleportation protocol.

• Derivation. The no-cloning theorem, a cornerstone of quantum information theory, prohibits the creation of perfect copies of an arbitrary unknown quantum state. Within the teleportation protocol framework, this fundamental limitation manifests through three interconnected observations:

• The original state $|\varphi \rangle$ undergoes irreversible destruction during Alice’s Bell measurement.
• The measurement outcomes exhibit perfect uniformity with probability ${\displaystyle \frac{1}{4}}$ for each Bell outcome, reflecting the complete absence of which-path information.
• Each distinct measurement outcome necessitates a corresponding unitary correction operation on Bob’s side to faithfully reconstruct the original state.

In ideal, noiseless theory, this uniformity is exact. In realistic experimental contexts, finite sampling and operational noise can lead to small but non-zero deviations from the ideal distribution. As a conceptual illustration (rather than a foundational claim), one may quantify such deviations using standard hypothesis testing tools.

For N independent teleportation trials, the expected frequency for each measurement outcome is $E_i=N/4$. Here, N denotes the total number of independent teleportation trials. A commonly used test statistic is the Chi-squared quantity

$${\chi }^2=\sum _{i=1}^4{\displaystyle \frac{{(O_i-E_i)}^2}{E_i}},$$

(6)

where $O_i$ denotes the observed frequency for outcome i.

Under the null hypothesis of an ideal uniform distribution, this statistic asymptotically follows a Chi-squared distribution with 3 degrees of freedom, yielding the associated p-value

$$p_{\mathrm{val}}=P({\chi }_3^2>{\chi }_{\mathrm{observed}}^2).$$

(7)

where $p_{\mathrm{val}}$ denotes the p-value associated with the Chi-squared goodness-of-fit test.

In this framework, statistically significant deviations (e.g., $p_{\mathrm{val}}<0.05$) should be interpreted as indications of experimental imperfections or operational noise, rather than a fundamental violation of the no-cloning theorem.

2.2. Realistic Noise Models in Physical Implementations

The phrase “realistic noisy conditions” does not refer to a single universal noise model. Instead, it denotes a family of platform-dependent decoherence channels and operational imperfections whose microscopic origins differ across experimental architectures (photonic, atomic/ionic, and solid-state). Our statistical framework is intentionally general; however, any concrete application requires specifying an effective noise model (or learning it statistically from data). In addition to the depolarizing entanglement-degradation model used in Equation (16), several physically motivated noise models commonly arise in practice.

Amplitude Damping (AD). AD models irreversible energy relaxation, such as spontaneous emission (atoms/ions) or photon loss processes (optics) in effective two-level encodings. For a single qubit, the AD channel is represented by Kraus operators $K_0=|0\rangle \langle 0|+\sqrt{1-\gamma }|1\rangle \langle 1|$ and $K_1=\sqrt{\gamma }|0\rangle \langle 1|$, where $\gamma \in [0,1]$ is the damping probability [44], where $\gamma$ denotes the probability of irreversible energy relaxation induced by the amplitude-damping process. When the shared entangled resource undergoes AD-type noise, the resulting two-qubit state becomes asymmetric and the functional relation between resource quality and teleportation fidelity generally deviates from the symmetric depolarizing-case expression [45].

Phase Damping/Dephasing (PD). PD captures loss of phase coherence without energy exchange (e.g., magnetic-field fluctuations, charge noise, or $1/f$ noise in solid-state devices), and can be modeled as random phase kicks or as a dephasing channel with standard Kraus representations [44]. Similar to AD, PD breaks the simple “single-parameter” mapping between an EPR-quality scalar and the output fidelity unless further symmetry assumptions are imposed [45].

Generalized Amplitude Damping (GAD). At finite temperature, relaxation and excitation processes coexist. The GAD channel provides a more complete model combining dissipation and thermal effects and reduces to AD in the zero-temperature limit [44]. This model is especially relevant for solid-state qubits and atomic systems where the environment is not effectively at zero temperature.

Non-Uniform Pauli Noise (anisotropic Pauli channels). In many devices, different error mechanisms occur with different rates, leading to Pauli channels of the form $\mathcal{N}\left(\rho \right)=(1-p_x-p_y-p_z)\rho +p_{x}X\rho X+p_{y}Y\rho Y+p_{z}Z\rho Z$, where $p_x$, $p_y$, and $p_z$ are the probabilities of the corresponding Pauli errors.

Such anisotropic Pauli noise models naturally encompass symmetric depolarizing noise as a special case and are widely used in contemporary error-mitigation and noise-learning contexts, where the Pauli–Lindblad structure is often statistically inferred from experiments [44,46].

Non-Markovian Noise (memory effects). In structured environments (e.g., strong coupling, narrow-band reservoirs, or engineered environments), noise can exhibit memory and information backflow, invalidating Markovian assumptions. Standard definitions and quantification of quantum non-Markovianity are reviewed in [47,48]. Non-Markovianity can modify both average performance and temporal fluctuations, thereby affecting not only mean teleportation fidelity but also higher-order statistics that our framework explicitly tracks (e.g., variance and time scales).

Beyond environment-induced decoherence, realistic conditions also include operational noise:imperfect Bell-state measurements (BSM), finite detector efficiency, mode mismatch, classical communication bit-flip errors, and imperfect unitary corrections. In particular, in polarization-encoded photonic implementations using only passive linear optics (plus vacuum ancillas), it is impossible to unambiguously discriminate all four Bell states with success probability exceeding 50%, which imposes a fundamental operational limitation on standard BSM-based teleportation schemes [49]. These operational effects motivate our emphasis on statistically characterizing teleportation not only by a single mean-fidelity number, but also by dispersion measures and platform-relevant coherence times.

Finally, we stress the scope of Equation (13) ($F_{\mathrm{avg}}=(2F_{\mathrm{EPR}}+1)/3$), and this closed-form linear relation holds under the symmetric depolarizing (Bell-diagonal) entanglement-degradation model. Under AD/PD/GAD, anisotropic Pauli noise, or non-Markovian dynamics, the mapping between a chosen resource-quality parameterization and the output fidelity is generally different and may not be known a priori. Our Bayesian adaptive protocol and resource-optimization principle remain applicable in such cases because they operate on statistically estimable quantities—such as $F_{\mathrm{avg}}$, the fidelity variance $\mathrm{\Delta }{F}^2$, and coherence-time parameters (e.g., $T_2$)—without requiring an exact closed-form noise-to-fidelity function.

Having clarified the relevant physical noise mechanisms, we now return to the protocol-level statistical structure and derive the independence relations that constrain what classical messages can reveal about the unknown input.

2.3. Statistical Dependence and Independence

Statistical Independence in Teleportation Protocols. The classical message $m\in \{00,01,10,11\}$ (encoding 2 classical bits) exhibits statistical independence from the unknown quantum state $|\varphi \rangle$ when their joint probability distribution factorizes as

$$P(m,|\varphi \rangle )=P(m\left)P\right(|\varphi \rangle )$$

(8)

for all possible quantum states $|\varphi \rangle$ and classical messages m.

Independence of Classical Communication and Quantum State. In the ideal quantum teleportation protocol implementation, the classical message transmitted by Alice maintains perfect statistical independence from the unknown quantum state $|\varphi \rangle$ being teleported.

• Derivation. From our preceding detailed analysis of the teleportation measurement statistics, the marginal probability distribution for each classical message (corresponding to specific measurement outcomes) demonstrates perfect uniformity:

$$P\left(m\right)={\displaystyle \frac{1}{4}}\mathrm{for}\mathrm{all}m\in \{00,01,10,11\}.$$

(9)

Moreover, the conditional probability $P\left(m\right||\varphi \rangle )$, i.e., the likelihood of obtaining message m when teleporting an arbitrary input state $|\varphi \rangle$, is independent of $|\varphi \rangle$:

$$P\left(m\right||\varphi \rangle )={\displaystyle \frac{1}{4}}\mathrm{for}\mathrm{all}|\varphi \rangle \mathrm{and}\mathrm{all}m.$$

(10)

Assuming a prior distribution $P\left(\right|\varphi \rangle )$ over possible input states (e.g., uniform in the absence of prior information), the joint distribution factorizes the following:

$$\begin{array}{cc}\hfill P(m,|\varphi \rangle )& =P\left(m\right||\varphi \rangle )P\left(\right|\varphi \rangle )={\displaystyle \frac{1}{4}}P\left(\right|\varphi \rangle )=P\left(m\right)P\left(\right|\varphi \rangle ).\hfill \end{array}$$

(11)

This establishes statistical independence between the classical communication and the unknown quantum state in the ideal teleportation protocol. Consequently, the classical message reveals no information about the teleported quantum state, a property relevant to quantum–classical information separation and secure quantum communication.

Quantification of EPR Correlation Strength.Building upon the foundational work of [2] and subsequent quantification frameworks by [23], the correlation strength in shared EPR pairs can be precisely characterized by the entanglement fidelity, also known as the singlet fraction [22,44].

$$F_{\mathrm{EPR}}=\langle {\mathrm{\Psi }}^{-}\left|\rho \right|{\mathrm{\Psi }}^{-}\rangle ,$$

(12)

where $\rho$ denotes the density matrix of the experimentally prepared entangled state. A value $F_{\mathrm{EPR}}=1$ corresponds to a perfect EPR pair, while $F_{\mathrm{EPR}}<1$ indicates imperfect entanglement due to noise or decoherence.

Functional Dependence of Teleportation Fidelity on EPR Correlation.Within the widely used depolarizing (Bell-diagonal) noise model for entanglement degradation (Refs. [22,44]), the achievable teleportation fidelity exhibits a simple linear dependence on the EPR correlation strength. Specifically, for qubit systems, the teleportation fidelity is given by

$$F_{\mathrm{avg}}={\displaystyle \frac{2F_{\mathrm{EPR}}+1}{3}}.$$

(13)

This relation is a standard result in quantum teleportation theory for Bell-diagonal (Werner-type) resources, and we adopt it as a fundamental theoretical foundation to underpin the statistical characterization and optimization analysis developed in the subsequent sections.

• Derivation. To model experimental imperfections in EPR pair preparation, we adopt a symmetric depolarizing channel acting on the ideal entangled state. Under this assumption, the noisy EPR resource can be represented by the Bell-diagonal density matrix [22,44]:

$$\rho =F_{\mathrm{EPR}}|{\mathrm{\Psi }}^{-}\rangle \langle {\mathrm{\Psi }}^{-}|+{\displaystyle \frac{1-F_{\mathrm{EPR}}}{3}}\sum _{i=1}^3|{\mathrm{\Psi }}_i\rangle \langle {\mathrm{\Psi }}_i|,$$

(14)

where $\left\{\right|{\mathrm{\Psi }}_i\rangle \}$ denotes the set of remaining Bell states $\left\{\right|{\mathrm{\Psi }}^{+}\rangle ,|{\mathrm{\Phi }}^{+}\rangle ,|{\mathrm{\Phi }}^{-}\rangle \}$.

The equal weighting $(1-F_{\mathrm{EPR}})/3$ among the three non-target Bell states reflects the assumption of a symmetric depolarizing (isotropic) noise channel. In this model, all Pauli errors occur with equal probability, leading to a uniform mixture over the remaining Bell-state components. This choice represents a standard and analytically tractable description of isotropic decoherence rather than the most general noise scenario. More general noise models, such as asymmetric Pauli noise or amplitude-damping channels, would give rise to unequal coefficients among the Bell states. While our framework can be extended to such cases, the symmetric depolarizing model suffices to illustrate the core statistical relationship between entanglement quality and teleportation fidelity.

Physically, this symmetric noise model describes a situation in which the EPR pair is in the target Bell state with weight $F_{\mathrm{EPR}}$, while the remaining weight $(1-F_{\mathrm{EPR}})$ is distributed uniformly among the other Bell components due to isotropic decoherence. Note that the completely mixed two-qubit state corresponds to $F_{\mathrm{EPR}}=1/4$ (equal overlap with all Bell states), whereas $F_{\mathrm{EPR}}=0$ represents a uniform mixture over the three Bell states orthogonal to $|{\mathrm{\Psi }}^{-}\rangle$.

To maintain a self-contained presentation, we briefly recall the standard definition of teleportation fidelity under this model [22]. The teleportation fidelity for an arbitrary pure input state $|\varphi \rangle$ is defined as the average quantum fidelity over the uniform Haar measure [44]:

$$F_{\mathrm{avg}}=\int d\varphi \langle \varphi |{\rho }_{\mathrm{out}}|\varphi \rangle .$$

(15)

For a Bell-diagonal resource state of the above form, the induced teleportation channel is an isotropic depolarizing channel [22]. Accordingly, the output state can be expressed as

$${\rho }_{\mathrm{out}}=p|\varphi \rangle \langle \varphi |+(1-p){\displaystyle \frac{I}{2}},p={\displaystyle \frac{4F_{\mathrm{EPR}}-1}{3}}.$$

(16)

This representation admits a clear interpretation: p quantifies the weight of the correctly transmitted component, while $(1-p)$ corresponds to isotropic depolarization.

The teleportation fidelity therefore evaluates as

$$\begin{array}{cc}\hfill F_{\mathrm{avg}}& =\int d\varphi \left[p·1+(1-p){\displaystyle \frac{1}{2}}·1\right]={\displaystyle \frac{1+p}{2}}={\displaystyle \frac{2F_{\mathrm{EPR}}+1}{3}}.\hfill \end{array}$$

(17)

The normalization condition $\int d\varphi =1$ for the Haar measure ensures mathematical consistency. This derivation establishes the exact linear relationship between EPR correlation quality and achievable teleportation performance [22], providing a crucial quantitative tool for experimental design and resource optimization.

Minimum EPR Correlation Threshold for Quantum Advantage.For qubit teleportation to surpass the optimal classical measure-and-prepare strategies, the EPR correlation fidelity must satisfy $F_{\mathrm{EPR}}>{\displaystyle \frac{1}{2}},$ which is equivalent to exceeding the classical teleportation fidelity bound $F_{\mathrm{avg}}>F_{\mathrm{classical}}={\displaystyle \frac{2}{3}}$ under the Bell-diagonal depolarizing model. Here, $F_{\mathrm{classical}}$ denotes the optimal average fidelity achievable by classical measure-and-prepare strategies (i.e., without shared entanglement).

• Derivation. The threshold $F_{\mathrm{EPR}}=1/2$ admits a clear operational and physical interpretation. For qubit systems, the maximum average fidelity achievable by any classical measure-and-prepare strategy—i.e., protocols without entanglement resources—is bounded by $F_{\mathrm{classical}}={\displaystyle \frac{2}{3}}$, as established in Ref. [50]. Within the Bell-diagonal depolarizing model, Equation (13) produces $F_{\mathrm{avg}}=(2F_{\mathrm{EPR}}+1)/3$. Requiring the teleportation fidelity to exceed the classical bound $F_{\mathrm{avg}}>2/3$ directly implies the condition $F_{\mathrm{EPR}}>1/2.$ Beyond this algebraic relation, the value $F_{\mathrm{EPR}}=1/2$ marks the separability boundary for Bell-diagonal two-qubit states: states with $F_{\mathrm{EPR}}\le 1/2$ are separable and can be simulated by classical correlations alone, whereas $F_{\mathrm{EPR}}>1/2$ guarantees the presence of genuine entanglement. Consequently, only when $F_{\mathrm{EPR}}$ exceeds this threshold can the teleportation protocol achieve fidelities that are statistically unattainable by any classical strategy, thereby establishing a genuine quantum advantage.

In practice, robust experimental verification of quantum advantage must account for statistical fluctuations and systematic errors, motivating an operational margin condition

$$F_{\mathrm{avg}}>F_{\mathrm{classical}}+\delta ,$$

(18)

where $\delta$ summarizes uncertainties arising from finite sampling, measurement noise, and systematic imperfections. This requirement translates into a correspondingly stronger operational constraint on the entanglement quality, ensuring that the observed teleportation fidelity exceeds the classical limit unambiguously under finite-precision experimental conditions.

To summarize, this section systematically establishes the protocol-level statistical structure of quantum teleportation under both ideal and noisy conditions, including the probabilistic properties of Bell-state measurement outcomes and the quantitative dependence of the observable teleportation fidelity on the quality of shared entanglement resources (e.g., $F_{\mathrm{EPR}}$). To place these statistical performance characterizations on a rigorous information-theoretic basis, the next section analyzes the corresponding resource constraints, including the minimal classical communication cost required to complete the teleportation process and the fundamental limits imposed by channel capacity and mutual-information bounds on the effective information flow of the teleportation channel.

3. Channel Capacity and Information Theory

In this section, we formalize the resource requirements implied by Section 2 through information-theoretic benchmarks, focusing on the minimal classical-bit cost and capacity/mutual-information bounds for teleportation channels.

3.1. Classical Channel Capacity

Minimum Classical Bit Requirement for Qubit Teleportation. The teleportation of an arbitrary unknown quantum state of a two-level system requires exactly two classical bits, and this requirement is both necessary and sufficient for perfect state transfer.

• Derivation. We establish both sufficiency and necessity through rigorous information-theoretic arguments:

Sufficiency: Following the seminal protocol in Ref. [1], the Bell-state measurement has four possible outcomes forming a complete set of orthogonal projections onto the Bell basis. Each outcome can be encoded by two classical bits $(m,n)\in {\{0,1\}}^2$, which Bob uses to select the corresponding correction operation.

$$U_{mn}={\sigma }_z^m{\sigma }_x^n,m,n\in \{0,1\},$$

(19)

where, $(m,n)=(0,0),(0,1),(1,0),(1,1)$ correspond to $I,{\sigma }_x,{\sigma }_z,i{\sigma }_y$, respectively, up to a global phase. This compact expression unifies the four Pauli corrections and ensures perfect reconstruction of the original quantum state at Bob’s location when applied to the corresponding conditional state.

• Derivation. To establish the necessity of 2 classical bits, we employ a proof by contradiction. Suppose, contrary to our claim, that only 1 classical bit were sufficient for perfect teleportation. This would imply the existence of only 2 distinct correction operations. However, the teleportation protocol fundamentally requires discrimination among 4 distinct measurement scenarios corresponding to the 4 Bell states.

Formally, the number of required correction operations equals the cardinality of the Bell measurement outcome set, which is exactly 4 for qubit teleportation. Since 1 classical bit can encode at most 2 distinct messages, it is information-theoretically insufficient to specify all necessary correction operations.

This distinguishability requirement can be rigorously formalized through the accessible information framework. For an ensemble of quantum states $\{p_i,{\rho }_i\}$, the Holevo bound [33] provides the maximum classical information accessible through quantum measurements:

$$I_{\mathrm{acc}}\le \chi =S\left(\sum _i{p}_i{\rho }_i\right)-\sum _i{p}_{i}S\left({\rho }_i\right),$$

(20)

where $S\left(\rho \right)=-\mathrm{tr}\left(\rho {\log }_2\rho \right)$ denotes the von Neumann entropy.

In the teleportation context, the four possible measurement scenarios constitute an ensemble with uniform probabilities $p_i=1/4$. For ideal teleportation conditions, the Holevo quantity $\chi$ for this ensemble evaluates exactly to 2 bits, demonstrating that 2 classical bits are information-theoretically necessary to distinguish among the four cases.

More operationally, if only 1 classical bit were available, at least two of the four measurement outcomes would necessarily be mapped to the same classical message, thereby receiving identical correction operations. This degeneracy would inevitably result in imperfect teleportation fidelity for certain input states, violating the requirement for universal perfect teleportation.

Generalized Classical Requirement for d-dimensional Quantum Systems. The teleportation of an arbitrary unknown quantum state of a d-dimensional system requires at least $2{\log }_{2}d$ classical bits, establishing a fundamental lower bound on classical communication requirements.

• Derivation. Consider a general d-dimensional quantum system. The teleportation protocol necessitates the specification of a unitary correction operation that depends deterministically on the measurement outcome. For a complete Bell measurement in d dimensions, the number of possible outcomes scales as $d^2$, corresponding to the dimension of the operator basis required for state reconstruction [51].

The minimal classical information required to specify one among $d^2$ distinct possibilities is given by

$$C_{\min }={\log }_2\left(d^2\right)=2{\log }_{2}d\mathrm{bits}.$$

(21)

where $C_{\min }$ denotes the minimal required classical communication cost.

This bound emerges from fundamental information-theoretic considerations. While an arbitrary pure state in d dimensions is parameterized by $2d-2$ real parameters (accounting for normalization and global phase), the classical communication must specifically identify the necessary correction operation within the $d^2$-dimensional space of possible unitaries.

The minimal classical information requirement can be further substantiated through the resource theory of quantum communication [52]. Teleportation can be viewed as a quantum channel assisted by classical communication, where the classical capacity requirement is dictated by the dimensionality of the required correction operations.

For the special case of qubits ($d=2$), we recover the established result of 2 classical bits. For qutrits ($d=3$), this bound predicts a requirement of $2{\log }_{2}3\approx 3.17$ bits, which necessitates 4 bits in practical implementations due to the integer nature of bit counts.

This scaling relation reveals a fundamental aspect of quantum information: the classical communication cost grows linearly with the number of encoded qubits. Specifically, for a system comprising k qubits ($d=2^k$), the required classical information scales as

$$C=2{\log }_2\left(2^k\right)=2k\mathrm{bits}.$$

(22)

Tightness of the Classical Information Bound.

The bound $C_{\min }=2{\log }_{2}d$ is tight and asymptotically achievable with optimal encoding strategies in the limit of large d.

• Derivation. The tightness of this bound follows from the existence of teleportation protocols that asymptotically achieve this information-theoretic limit. For large-dimensional systems, the required classical bits can be encoded with near-optimal efficiency using advanced block coding techniques.

The achievability can be rigorously demonstrated through the theory of typical sequences in information theory [53]. Consider a sequence of L independent teleportation events. The total number of possible measurement outcome sequences is ${\left(d^2\right)}^L=d^{2L}$.

Employing optimal source coding (data compression), the average number of bits required to encode a typical sequence of length L is given by

$$L·H=L·\left(-\sum _{i=1}^{d^2}{p}_i{\log }_2{p}_i\right),$$

(23)

where H denotes the Shannon entropy of the measurement-outcome distribution. For uniformly distributed measurement outcomes, $p_i=1/d^2$ for all i, the entropy evaluates to $H=-{\sum }_{i=1}^{d^2}{\displaystyle \frac{1}{d^2}}{\log }_2{\displaystyle \frac{1}{d^2}}=2{\log }_{2}d.$ Consequently, in the asymptotic limit of many teleportation events ($L\to \infty$), the average classical communication cost per teleported state approaches $2{\log }_{2}d$ bits, confirming the tightness of the bound.

For finite block lengths, a small overhead arises due to the suboptimality of finite-length codes. However, this overhead diminishes as $O(\log L/L)$ and becomes negligible for practical implementations with sufficiently large L. This asymptotic achievability demonstrates that the bound $2{\log }_{2}d$ represents not merely a theoretical limit, but an operationally achievable benchmark for optimal teleportation protocols.

3.2. Quantum vs. Classical Information

Quantification of Information Duplication Capability. The fundamental limitation on information duplication in quantum mechanics is characterized by the optimal cloning fidelity [54]:

$$F_{\mathrm{clone}}=\underset{\mathcal{E}}{max}\underset{|\psi \rangle }{min}\langle \psi \left|\mathcal{E}\left(\right|\psi \rangle \langle \psi \left|\right)\right|\psi \rangle ,$$

(24)

where $\mathcal{E}$ represents the quantum operation implementing the cloning process.

Quantum Advantage in Information Transmission Protocols. Quantum teleportation provides a demonstrable advantage over purely classical communication strategies when the quantum mutual information between input and output states exceeds the fundamental classical capacity limits.

• Derivation. Let X represent the random variable encoding the input state information and Y the corresponding output state. The classical mutual information $I_{\mathrm{classical}}(X;Y)$ is fundamentally bounded by the Holevo bound as established by [33]:

$$I_{\mathrm{classical}}(X;Y)\le {\log }_{2}d$$

(25)

for d-dimensional quantum systems.

For quantum teleportation, the correlation between input and output quantum states can be quantified by the quantum mutual information, defined via the von Neumann entropy [44]:

$$I_{\mathrm{quantum}}({\rho }_{\mathrm{in}};{\rho }_{\mathrm{out}})=S\left({\rho }_{\mathrm{in}}\right)+S\left({\rho }_{\mathrm{out}}\right)-S({\rho }_{\mathrm{in}},{\rho }_{\mathrm{out}}),$$

(26)

where $S({\rho }_{\mathrm{in}},{\rho }_{\mathrm{out}})$ denotes the joint entropy of the input–output system [44].

In the ideal teleportation protocol, where Alice and Bob share a maximally entangled d-dimensional Bell state and all operations are noiseless, the input and output systems become perfectly correlated. In this limit, the quantum mutual information attains its maximal value:

$$I_{\mathrm{quantum}}=2{\log }_{2}d,$$

(27)

which is twice the classical information capacity of a d-dimensional system and therefore saturates the fundamental quantum limit.

We emphasize that the ideal teleportation performance relies critically on the quality of the shared entangled resource. In realistic implementations, the entangled pair is generally imperfect and must therefore be characterized explicitly. As discussed in the preceding sections, this imperfection is conveniently quantified by the entanglement fidelity $F_{\mathrm{EPR}}$, defined as the overlap between the experimentally prepared two-qubit state and the ideal Bell state. While $F_{\mathrm{EPR}}=1$ corresponds to perfect entanglement, values $F_{\mathrm{EPR}}<1$ model noisy or partially depolarized entanglement resources. Under the standard depolarizing noise model [22], the teleportation fidelity for qubit systems satisfies Equation (13), $F_{\mathrm{avg}}=(2F_{\mathrm{EPR}}+1)/3$, which explicitly captures the degradation of teleportation performance induced by imperfect entanglement. In particular, whenever $F_{\mathrm{EPR}}>\frac{1}{2}$, the resulting teleportation fidelity exceeds the optimal classical limit, $F_{\mathrm{avg}}>F_{\mathrm{classical}}=\frac{2}{3}$, which cannot be surpassed by any classical measure-and-prepare strategy.

This improvement in state-transfer fidelity is accompanied by a corresponding advantage in information transmission. For classical measure-and-prepare protocols achieving an average fidelity F, the accessible classical mutual information is bounded by [44]

$$I_{\mathrm{acc}}\le 1+F{\log }_{2}F+(1-F){\log }_2(1-F),$$

(28)

for qubit systems. In contrast, quantum teleportation preserves genuine quantum correlations between input and output states, enabling higher information transfer capability.

Consequently, even in the presence of imperfect entanglement, quantum teleportation can retain a mutual-information advantage over classical communication protocols whenever the entanglement fidelity surpasses the classical threshold. In this qualitative information-theoretic sense, one may state that

$$I_{\mathrm{quantum}}>{\log }_{2}d$$

(29)

although the exact value of $I_{\mathrm{quantum}}$ is reduced from its ideal maximum by noise.

Therefore, the role of $F_{\mathrm{EPR}}$ is not to modify the principle of teleportation itself, but to establish a quantitative link between the quality of the entanglement resource and the resulting information-theoretic performance. This link forms the foundation for the optimization framework developed in the subsequent sections.

Mutual Information in Quantum Teleportation Protocols.The mutual information between Alice’s measurement outcome M and Bob’s reconstructed state ${\rho }_{\mathrm{out}}$ is characterized by

$$I(M;{\rho }_{\mathrm{out}})=H\left(M\right)-H\left(M\right|{\rho }_{\mathrm{out}})=2{\log }_{2}d-\delta ,$$

(30)

where $\delta \ge 0$ quantifies the total imperfection in the teleportation protocol.

• Derivation. The mutual information can be rigorously computed using the quantum–classical information measure [55]. Consider the ensemble $\{p_{mn},{\rho }_{mn}\}$ of possible output states conditioned on measurement outcomes $(m,n)$.

The accessible information is bounded by the Holevo quantity:

$$\chi =S\left(\sum _{m,n}{p}_{mn}{\rho }_{mn}\right)-\sum _{m,n}{p}_{mn}S\left({\rho }_{mn}\right).$$

(31)

For ideal teleportation conditions, the average output state becomes $\overline{\rho }={\displaystyle \frac{I}{d}},$ with corresponding von Neumann entropy $S\left(\overline{\rho }\right)={\log }_{2}d$. Each conditional output state ${\rho }_{mn}$ is pure (representing a perfect replica of the input state up to a known unitary transformation), yielding $S\left({\rho }_{mn}\right)=0$. Consequently, ${\chi }_{\mathrm{ideal}}={\log }_{2}d-0={\log }_{2}d.$ However, this quantity captures only the classical information transmitted through the measurement outcomes. The complete mutual information must incorporate both the classical information and the quantum correlations preserved via entanglement.

The total correlation can be expressed as the sum of classical and quantum contributions:

$$I_{\mathrm{total}}=I_{\mathrm{classical}}+I_{\mathrm{quantum}}={\log }_{2}d+{\log }_{2}d=2{\log }_{2}d.$$

(32)

The imperfection deficit $\delta$ arises from various sources of protocol non-ideality:

$$\delta ={\delta }_{\mathrm{entanglement}}+{\delta }_{\mathrm{measurement}}+{\delta }_{\mathrm{channel}}+{\delta }_{\mathrm{operations}},$$

(33)

where

• ${\delta }_{\mathrm{entanglement}}$ quantifies information loss due to imperfect entanglement resources,
• ${\delta }_{\mathrm{measurement}}$ accounts for incomplete or noisy Bell measurements,
• ${\delta }_{\mathrm{channel}}$ represents degradation from classical channel noise,
• ${\delta }_{\mathrm{operations}}$ captures imperfections in unitary correction implementations.

Each contribution admits quantitative characterization through corresponding fidelity reductions and can be systematically minimized through error correction techniques and protocol optimization.

Optimality Conditions for Perfect Quantum Teleportation.Perfect teleportation ($\delta =0$) is achieved if and only if the conditions are simultaneously satisfied as follows:

• The shared entanglement resource is maximal ($F_{\mathrm{EPR}}=1$),
• The Bell measurement is perfect and informationally complete,
• The classical communication channel is noiseless and possesses sufficient capacity,
• The correction operations are implemented with perfect fidelity.

• Derivation. Each condition contributes specifically to the total imperfection deficit $\delta$:

• Imperfect entanglement: ${\delta }_{\mathrm{entanglement}}=2{\log }_{2}d(1-F_{\mathrm{EPR}})$
  Submaximal entanglement results in output state mixtures, reducing mutual information proportionally to the entanglement infidelity.
• Incomplete measurement: ${\delta }_{\mathrm{measurement}}=2{\log }_{2}d(1-\eta )$
  For measurement efficiency $\eta <1$, information loss scales with the inefficiency, particularly through non-zero overlap between undistinguished measurement outcomes.
• Classical channel noise: ${\delta }_{\mathrm{channel}}=2{\log }_{2}d-C$
  For a classical channel with capacity $C<2{\log }_{2}d$, the mutual information deficit equals the capacity shortfall.
• Imperfect operations: ${\delta }_{\mathrm{operations}}=2{\log }_{2}d(1-F_{\mathrm{op}})$
  Unitary correction operations implemented with average fidelity $F_{\mathrm{op}}<1$ introduce additional information loss proportional to the implementation infidelity.

The conjunction of all ideal conditions ($F_{\mathrm{EPR}}=1$, $\eta =1$, $C=2{\log }_{2}d$, $F_{\mathrm{op}}=1$) yields $\delta =0$, achieving the maximum possible mutual information of $2{\log }_{2}d$. Conversely, any deviation from these ideal conditions necessarily introduces $\delta >0$, resulting in suboptimal teleportation performance.

This optimality condition provides a comprehensive framework for designing high-performance teleportation systems and identifies critical components for optimization in practical implementations, enabling systematic performance enhancement through targeted improvements.

The information-theoretic analysis in Section 3 provides principled benchmarks for teleportation performance and clarifies how non-idealities contribute to the total imperfection budget $\delta$. In practice, however, these quantities are not accessed directly—they must be inferred from finite experimental data and are inevitably affected by statistical fluctuations. This motivates the next section, where we develop statistical procedures that turn the above theoretical benchmarks into experimentally implementable certification criteria.

4. Statistical Verification of Teleportation

In this section, we construct a complete statistical verification toolkit for quantum teleportation, including confidence-interval estimation, hypothesis testing, and benchmarking procedures. These tools translate theoretical fidelity relations into operational decision rules with controlled error probabilities, thereby enabling finite-data certification of quantum advantage in realistic experiments. We begin by specifying the fidelity metrics and averaging conventions (Uhlmann fidelity and Haar-averaged $F_{\mathrm{avg}}$), which serve as the basic observables for the subsequent statistical tests.

4.1. Fidelity Metrics

Quantum State Fidelity Measure. The fidelity between two quantum states $\rho$ and $\sigma$ is defined through the Uhlmann fidelity formula as introduced by [44,51,56]:

$$F(\rho ,\sigma )={\left(\mathrm{tr}\sqrt{\sqrt{\rho }\sigma \sqrt{\rho }}\right)}^2.$$

(34)

For pure states $|\psi \rangle$ and $|\varphi \rangle$, this expression simplifies to the overlap probability $F\left(\right|\psi \rangle ,|\varphi \rangle )={\left|\langle \psi |\varphi \rangle \right|}^2.$

Teleportation Fidelity Definition. The average teleportation fidelity for an ensemble of input states $\{p_i,|{\psi }_i\rangle \}$ is given by the weighted average:

$$F_{\mathrm{ens}}=\sum _i{p}_{i}F(|{\psi }_i\rangle \langle {\psi }_i|,{\rho }_{\mathrm{out}}^{\left(i\right)}),$$

(35)

where ${\rho }_{\mathrm{out}}^{\left(i\right)}$ denotes the output density matrix corresponding to the input state $|{\psi }_i\rangle$.

Fidelity as Complete Statistical Characterizer. The teleportation fidelity provides a comprehensive statistical characterization of protocol accuracy when evaluated over Haar-random distributed input states.

• Derivation. Consider an ensemble of input states distributed according to the Haar measure on $\mathbb{C}{P}^1$, corresponding to uniform sampling over the Bloch sphere for qubit systems. The average fidelity admits the integral representation:

$$F_{\mathrm{avg}}=\int d\psi \langle \psi |{\rho }_{\mathrm{out}}\left(\psi \right)|\psi \rangle ,$$

(36)

where $d\psi$ denotes the normalized uniform measure over pure states.

For a teleportation protocol described by a completely positive trace-preserving (CPTP) map $\mathcal{E}$, this becomes

$$F_{\mathrm{avg}}=\int d\psi \langle \psi \left|\mathcal{E}\left(\right|\psi \rangle \langle \psi \left|\right)\right|\psi \rangle .$$

(37)

This integral can be systematically evaluated using the moment operator formalism [22]. For qubit systems, any quantum channel $\mathcal{E}$ admits a representation in the Pauli operator basis. Exploiting the invariance properties of the Haar measure, we obtain the explicit expression $F_{\mathrm{avg}}={\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{6}}{\sum }_{i=1}^3\mathrm{tr}\left({\sigma }_i\mathcal{E}\left({\sigma }_i\right)\right),$ where ${\left\{{\sigma }_i\right\}}_{i=1}^3$ represent the Pauli matrices ${\sigma }_x,{\sigma }_y,$ and ${\sigma }_z$.

The statistical variance of fidelity is defined through

$$\mathrm{\Delta }{F}^2=\mathrm{Var}\left(F\right)=\int d\psi [\langle \psi |\mathcal{E}\left(\right|\psi \rangle \langle \psi \left|\right)|\psi \rangle -F_{\mathrm{avg}}{]}^2.$$

(38)

This second moment can be computed using advanced Haar integration techniques, expressing the variance in terms of Pauli matrix components and higher-order correlation functions. While the complete statistical description requires knowledge of all moments of the fidelity distribution, in practical scenarios, the mean and variance often provide sufficient characterization, particularly for protocols exhibiting approximately Gaussian fidelity distributions. This statistical framework enables complete characterization of teleportation performance through the fidelity distribution over randomly sampled input states, establishing a powerful methodology for protocol optimization and experimental verification.

Necessity of Maximal Entanglement for Perfect Teleportation.Perfect teleportation, characterized by $F_{\mathrm{avg}}=1$, necessitates the utilization of maximally entangled resource states.

• Derivation. Consider a shared two-qubit entangled resource ${|\mathrm{\Psi }\rangle }_{23}$ with Schmidt decomposition

$${|\mathrm{\Psi }\rangle }_{23}=\sqrt{{\lambda }_0}|00\rangle +\sqrt{{\lambda }_1}|11\rangle ,$$

(39)

where ${\lambda }_0+{\lambda }_1=1$ and, without loss of generality, ${\lambda }_0\ge {\lambda }_1\ge 0$.

The teleportation protocol induces a completely positive trace-preserving map on the input state, which can be written in Kraus form as

$$\mathcal{E}\left(\rho \right)=\sum _{k=1}^4{K}_k\rho K_k^{†},$$

(40)

where the Kraus operators $K_k$ are determined by the Bell-basis measurement on subsystems $(1,2)$ together with the shared resource on $(2,3)$; one convenient representation (before Bob’s conditional unitary correction) is $K_k={}_{12}\langle {\mathrm{\Phi }}_k\left|{(I_1\otimes |\mathrm{\Psi }\rangle }_{23}\right)$, with $\left\{\right|{\mathrm{\Phi }}_k\rangle \}$ denoting the Bell basis.

A standard figure of merit for assessing the usefulness of a given resource state for teleportation is its (maximal) singlet fraction f, defined as the maximal overlap with a maximally entangled state under local unitary optimization on Bob’s side, $f={max}_{U\in U\left(2\right)}\langle {\mathrm{\Phi }}^{+}\left|(I\otimes U)\right|\mathrm{\Psi }\rangle \langle \mathrm{\Psi }|(I\otimes U^{†})|{\mathrm{\Phi }}^{+}\rangle$. For the Schmidt-decomposed pure state above, this maximization can be carried out analytically, yielding $f={(\sqrt{{\lambda }_0}+\sqrt{{\lambda }_1})}^2/2=(1+2\sqrt{{\lambda }_0{\lambda }_1})/2$.

It is well known that the optimal average teleportation fidelity achievable with a two-qubit resource of singlet fraction f is given by $F_{\mathrm{avg}}^{max}=(2f+1)/3$ [22]. Substituting the expression of f for the present pure state leads to

$$F_{\mathrm{avg}}^{max}={\displaystyle \frac{2}{3}}\left(1+\sqrt{{\lambda }_0{\lambda }_1}\right).$$

(41)

Therefore, perfect teleportation, characterized by $F_{\mathrm{avg}}^{max}=1$, is achieved if and only if $\sqrt{{\lambda }_0{\lambda }_1}=1/2$, i.e., ${\lambda }_0={\lambda }_1=1/2$, which is precisely the condition of maximal entanglement.

This conclusion can also be understood from an information-theoretic perspective. The entanglement entropy of the resource state,

$$S\left({\mathrm{tr}}_2\left(\right|\mathrm{\Psi }\rangle \langle \mathrm{\Psi }\left|\right)\right)=-{\lambda }_0{\log }_2{\lambda }_0-{\lambda }_1{\log }_2{\lambda }_1,$$

(42)

quantifies the amount of entanglement and attains its maximum value of one ebit exactly when ${\lambda }_0={\lambda }_1=1/2$. Hence, perfect teleportation fidelity necessarily requires maximally entangled resources.

Fidelity Deviation Measure. 上The fidelity deviation quantifies the variability of teleportation performance across different input states and is defined as

$$\mathrm{\Delta }F=\sqrt{\mathrm{Var}\left(F\right)}=\sqrt{\int d\psi {\left[F\left(\psi \right)-F_{\mathrm{avg}}\right]}^2}.$$

(43)

Universal Lower Bound on Fidelity Deviation.For any teleportation protocol employing non-maximally entangled resources, the fidelity deviation obeys the universal inequality

$$\mathrm{\Delta }F\ge {\displaystyle \frac{1}{2\sqrt{5}}}(1-F_{\mathrm{avg}}).$$

(44)

• Derivation. The variance of the teleportation fidelity can be written in terms of its second moment as

$$\mathrm{Var}\left(F\right)=\int d\psi F{\left(\psi \right)}^2-F_{\mathrm{avg}}^2.$$

(45)

For a general teleportation channel described by a completely positive trace-preserving map $\mathcal{E}$, the second moment of the teleportation fidelity admits a closed-form expression obtained via Haar integration over the Bloch sphere,

$$\begin{array}{cc}\int d\psi F{\left(\psi \right)}^2& =\int d\psi {\langle \psi \left|\mathcal{E}\left(\right|\psi \rangle \langle \psi \left|\right)\right|\psi \rangle }^2\hfill \\ \hfill & ={\displaystyle \frac{1}{5}}+{\displaystyle \frac{4}{5}}{F}_{\mathrm{avg}}-{\displaystyle \frac{1}{15}}\sum _{i<j}\mathrm{tr}\left({\sigma }_i\mathcal{E}\left({\sigma }_i\right)\right)\mathrm{tr}\left({\sigma }_j\mathcal{E}\left({\sigma }_j\right)\right).\hfill \end{array}$$

(46)

The numerical coefficients arise from the geometry of the Haar measure and the second-order moment structure of the Bloch sphere.

The cross-term contributions in the above expression are fundamentally constrained by the quality of the entanglement resource. For teleportation protocols employing non-maximally entangled states, these constraints imply a universal lower bound on the fidelity variance, $\mathrm{Var}\left(F\right)\ge {\displaystyle \frac{1}{20}}{(1-F_{\mathrm{avg}})}^2$, and hence a corresponding bound on the fidelity deviation, $\mathrm{\Delta }F\ge {\left(2\sqrt{5}\right)}^{-1}(1-F_{\mathrm{avg}})$. This bound reveals an intrinsic trade-off in teleportation protocols: as the average fidelity decreases from the ideal value of unity, performance fluctuations across different input states necessarily increase. Consequently, protocols based on non-maximally entangled resources cannot simultaneously achieve high average fidelity and low fidelity variation. From an experimental perspective, this result provides a practical diagnostic criterion: if the measured fidelity variation falls below the universal bound, it indicates either near-optimal teleportation performance or the presence of advanced error suppression mechanisms beyond the standard teleportation framework.

Statistical Test for Maximal Entanglement Verification.In the ideal, noise-free and asymptotic limit, maximal entanglement can be operationally certified through the simultaneous satisfaction of

$$F_{\mathrm{avg}}=1\mathrm{and}\mathrm{\Delta }F=0.$$

(47)

• Derivation. From our previous analysis, $F_{\mathrm{avg}}=1$ necessitates maximal entanglement resources. Furthermore, under maximal entanglement conditions with perfect operations, the teleportation fidelity becomes state-independent, namely,

$$F\left(\right|\psi \rangle ,{\rho }_{\mathrm{out}}\left(\psi \right))=1\forall |\psi \rangle .$$

(48)

This uniformity follows from the protocol’s symmetric structure when employing maximal entanglement, ensuring perfect teleportation regardless of the specific input state. Consequently, the fidelity variance vanishes identically, as follows:

$$\mathrm{Var}\left(F\right)=\int d\psi {[1-1]}^2=0.$$

(49)

Conversely, if both $F_{\mathrm{avg}}=1$ and $\mathrm{\Delta }F=0$ are observed (within statistical uncertainty), then within the standard teleportation framework, this implies perfect teleportation and hence maximal entanglement of the resource state.

This establishes an operational methodology for verifying maximal entanglement through statistical testing of the teleportation protocol, circumventing the need for complete quantum state tomography of the entangled resource. The test exhibits particular sensitivity because it can detect subtle deviations from maximal entanglement even when the average fidelity remains high but displays non-uniformity across different input states.

• Experimental verification. Experimental verification of teleportation fidelity proceeds through the following protocol:

• Prepare a comprehensive set of Haar-random input states.
• Execute teleportation for each input state.
• Measure the corresponding output state fidelity.
• Compute sample statistics:

  $$\begin{array}{ccc}\hfill {\widehat{F}}_{\mathrm{avg}}& ={\displaystyle \frac{1}{N}}\sum _{i=1}^N{F}_i,\widehat{\mathrm{\Delta }F}\hfill & \hfill =\sqrt{{\displaystyle \frac{1}{N-1}}\sum _{i=1}^N{(F_i-{\widehat{F}}_{\mathrm{avg}})}^2},\end{array}$$

  (50)
• Compare empirical results with theoretical predictions to assess entanglement quality.

The sample statistics converge to their true values as $N\to \infty$ by the strong law of large numbers. The estimation precision of the average fidelity is quantified by

$$\mathrm{Var}\left({\widehat{F}}_{\mathrm{avg}}\right)={\displaystyle \frac{\mathrm{Var}\left(F\right)}{N}},$$

(51)

while for the variance estimator one obtains $\mathrm{Var}\left(\widehat{\mathrm{\Delta }{F}^2}\right)={\displaystyle \frac{{\mu }_4-\mathrm{Var}{\left(F\right)}^2}{N}}$, where $\widehat{\mathrm{\Delta }{F}^2}$ denotes the sample estimator of the fidelity variance $\mathrm{\Delta }{F}^2$, and ${\mu }_4$ is the fourth central moment of the fidelity distribution. These results establish that both the mean fidelity and its fluctuations can be estimated with controlled precision, and that the requisite sample size for a desired statistical accuracy can be determined directly from the above variance expressions.

This experimental protocol establishes reliable statistical verification of teleportation performance and entanglement quality. The requisite sample size for a desired precision can be determined directly from the above variance expressions.

In practical implementations, instead of generating truly Haar-random input states—which would require arbitrary unitary operations—one can employ states uniformly distributed over the Bloch sphere, such as vertices of regular polyhedra or other symmetry-related state sets. This approximation preserves the essential statistical properties required for fidelity estimation while remaining experimentally feasible.

4.2. Experimental Validation

Teleportation Witness Operator FormulationA teleportation witness operator $\mathcal{W}$ is a Hermitian operator satisfying $\mathrm{tr}\left(\mathcal{W}\rho \right)\ge 0$ for all quantum states $\rho$ incapable of achieving teleportation fidelity beyond the classical limit, while $\mathrm{tr}\left(\mathcal{W}\rho \right)<0$ indicates the presence of useful entanglement for teleportation [23].

Bell Inequality Violation as Teleportation Capability Witness.For any two-qubit quantum state $\rho$ employed as a teleportation resource in the standard protocol, the maximal violation of the Clauser–Horne–Shimony–Holt (CHSH) inequality [57] provides a sufficient lower bound on the achievable average teleportation fidelity:

$$F_{\mathrm{avg}}\ge {\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{12}}{S}_{max}\left(\rho \right).$$

(52)

• Derivation. The CHSH operator is defined as

$$B=A_1\otimes B_1+A_1\otimes B_2+A_2\otimes B_1-A_2\otimes B_2,$$

(53)

where $A_i$ and $B_j$ represent local dichotomic observables with eigenvalues $\pm 1$ acting on the two subsystems.

For an arbitrary two-qubit state $\rho$, the maximum quantum expectation value of the CHSH operator is given by [58]

$$S_{max}\left(\rho \right):=\underset{\mathcal{B}}{max}\mathrm{tr}\left(\rho \mathcal{B}\right)=2\sqrt{{\lambda }_1+{\lambda }_2},$$

(54)

where the maximization is taken over all CHSH Bell operators $\mathcal{B}$ constructed from local dichotomic observables on the two subsystems, and ${\lambda }_1,{\lambda }_2$ are the two largest eigenvalues of the matrix $T^{†}T$ associated with $\rho$.

For the standard teleportation protocol, the average fidelity can be expressed in terms of the same correlation matrix as [23]

$$F_{\mathrm{avg}}={\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{6}}\mathrm{tr}\left(\sqrt{T^{†}T}\right).$$

(55)

Using the eigenvalue decomposition of $T^{†}T$ and the identity $\mathrm{tr}\left(\sqrt{T^{†}T}\right)={\sum }_{k=1}^3\sqrt{{\mu }_k}$, with ${\mu }_k$ denoting its eigenvalues, we obtain the inequality $\mathrm{tr}\left(\sqrt{T^{†}T}\right)\ge \sqrt{{\lambda }_1}+\sqrt{{\lambda }_2}\ge \sqrt{{\lambda }_1+{\lambda }_2}.$ Substituting this bound into the expression for $F_{\mathrm{avg}}$ directly yields $F_{\mathrm{avg}}\ge {\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{6}}\sqrt{{\lambda }_1+{\lambda }_2}={\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{12}}{S}_{max}\left(\rho \right),$ which establishes the stated lower bound.

This provides an operational link between Bell nonlocality and teleportation performance: any CHSH violation $S_{max}\left(\rho \right)>2$ implies $F_{\mathrm{avg}}>{\displaystyle \frac{2}{3}},$ thereby certifying quantum teleportation fidelity strictly above the classical benchmark.

This connection is particularly valuable experimentally, since Bell tests can be simpler to implement than full teleportation benchmarking, enabling a practical witness of teleportation capability without executing the complete protocol.

Comprehensive Experimental Validation Protocol.A complete experimental validation of quantum teleportation requires the following complementary measurements:

• Quantum state tomography of the entangled resource state.
• Measurement of Bell inequality violation.
• Quantum process tomography of the teleportation protocol.
• Statistical analysis of the fidelity distribution.

• Derivation. The completeness of this protocol follows from information-theoretic requirements.

• State tomography: Complete characterization of a two-qubit state necessitates 15 independent measurements [59]. The resource state quality is quantified by the entanglement fidelity:

  $$F_{\mathrm{EPR}}=\langle {\mathrm{\Psi }}^{-}\left|\rho \right|{\mathrm{\Psi }}^{-}\rangle .$$

  (56)
• Bell measurement: The experimentally observed CHSH value is obtained from correlation measurements at optimized settings:

  $$S_{\exp }=\left|E({\theta }_1,{\varphi }_1)+E({\theta }_1,{\varphi }_2)+E({\theta }_2,{\varphi }_1)-E({\theta }_2,{\varphi }_2)\right|,$$

  (57)

  where $E(\theta ,\varphi )$ denotes the experimentally measured two-body correlation for local analyzer settings parametrized by angles $\theta$ and $\varphi$ (e.g., polarization or Bloch-sphere measurement directions). In practice, $S_{\exp }$ provides a directly measurable lower bound on $S_{max}\left(\rho \right)$ (up to finite-setting optimization).
• Process tomography: The teleportation process is fully characterized by the $\chi$-matrix representation [60]:

  $$\mathcal{E}\left(\rho \right)=\sum _{mn}{\chi }_{mn}{\sigma }_m\rho {\sigma }_n.$$

  (58)

  where $\left\{{\sigma }_m\right\}$ denotes the operator basis introduced previously. The process fidelity relative to the identity operation is given by $F_{\mathrm{process}}={\chi }_{00}$.
• Statistical analysis: For N independent runs, the standard error of the estimated mean fidelity satisfies

  $${\sigma }_F\approx {\displaystyle \frac{{\sigma }_F}{\sqrt{N}}},{{\sigma }_F}^2=\mathrm{Var}\left(F_i\right).$$

  (59)

When a Bernoulli-type approximation is appropriate, one may use ${{\sigma }_F}^2\approx F(1-F)$; otherwise, ${{\sigma }_F}^2$ is estimated empirically from the sample.

Each component provides complementary information: (i) tomography quantifies the resource quality, (ii) Bell tests offer an efficient witness of useful nonclassical correlations, (iii) process tomography identifies the implemented map, and (iv) statistical analysis quantifies reliability and precision. Collectively, these measurements establish a comprehensive validation framework for teleportation performance under realistic experimental conditions.

Hypothesis Testing Framework for Quantum Advantage.For teleportation experiments operating under noisy conditions, quantum advantage is established through the one-sided hypothesis test:

$$\begin{array}{cc}\hfill H_0:& F_{\mathrm{avg}}\le F_{\mathrm{classical}}(\mathrm{null}\mathrm{hypothesis}),\hfill \\ \hfill H_1:& F_{\mathrm{avg}}>F_{\mathrm{classical}}(\mathrm{alternative}\mathrm{hypothesis}).\hfill \end{array}$$

(60)

With test statistic

$$Z={\displaystyle \frac{\widehat{F}-F_{\mathrm{classical}}}{\widehat{{\sigma }_F}/\sqrt{N}}},$$

(61)

where $\widehat{F}={\displaystyle \frac{1}{N}}{\sum }_{i=1}^N{F}_i$ is the sample mean fidelity and ${\widehat{{\sigma }_F}}^2={\displaystyle \frac{1}{N-1}}{\sum }_{i=1}^N{(F_i-\widehat{F})}^2$ is the corresponding unbiased sample variance, computed from the measured fidelity samples ${\left\{F_i\right\}}_{i=1}^N$.

We emphasize that the individual fidelity outcomes $F_i$ are not assumed to follow a Gaussian distribution; rather, each $F_i$ is a bounded random variable taking values in $[0,1]$. Nevertheless, under the null hypothesis $H_0$ and for sufficiently large sample sizes N, the test statistic Z is approximately standard normal by virtue of the Central Limit Theorem, since $\widehat{F}$ is the mean of N independent and identically distributed random variables with finite variance. For smaller sample sizes, the distribution of Z is more accurately approximated by a Student’s t-distribution with $N-1$ degrees of freedom, while for typical experimental regimes ($N\gtrsim 30$), the normal approximation is well justified.

Accordingly, the corresponding one-sided p-value is given by

$$p=1-\mathrm{\Phi }\left(Z\right),$$

(62)

where $\mathrm{\Phi }$ denotes the cumulative distribution function of the standard normal distribution.

For a true average fidelity $F_1>F_{\mathrm{classical}}$, the (approximate) statistical power at significance level $\alpha$ is

$$1-\beta =\mathbb{P}\left(Z>z_{1-\alpha }|F_{\mathrm{avg}}=F_1\right)\approx 1-\mathrm{\Phi }\left(z_{1-\alpha }-{\displaystyle \frac{F_1-F_{\mathrm{classical}}}{{\sigma }_F/\sqrt{N}}}\right),$$

(63)

where $z_{1-\alpha }$ is the $(1-\alpha )$ quantile of the standard normal distribution and $\beta$ denotes the type-II error probability. Accordingly, the sample size required to achieve a target significance level $\alpha$ and statistical power $1-\beta$ is

$$N={\left({\displaystyle \frac{{\sigma }_F(z_{1-\alpha }+z_{1-\beta })}{F_1-F_{\mathrm{classical}}}}\right)}^2.$$

(64)

For qubit teleportation experiments, the classical benchmark fidelity is $F_{\mathrm{classical}}=2/3$. Here, ${{\sigma }_F}^2$ denotes the population variance relevant for experimental design and power analysis in Equation (64), whereas ${\widehat{{\sigma }_F}}^2$ is its empirical estimator obtained from the finite sample $\left\{F_i\right\}$ and used in the test statistic in Equation (61). This hypothesis testing framework therefore provides a statistically rigorous and experimentally practical methodology for claiming quantum advantage in teleportation experiments, while explicitly justifying the use of the normal approximation and accounting for both statistical and systematic uncertainties.

Decoherence-Resilient Teleportation Quality Measure.The teleportation quality under decoherence effects is quantified by the composite measure

$$Q=F_{\mathrm{avg}}·\exp \left(-{\displaystyle \frac{t}{T_2}}\right)·\left(1-{\displaystyle \frac{\mathrm{\Delta }F}{F_{\mathrm{avg}}}}\right).$$

(65)

which is introduced as a heuristic single-figure indicator combining mean performance, coherence-time decay, and fidelity uniformity.

Noise Threshold for Demonstrable Quantum Advantage.Quantum teleportation provides operational advantage over classical protocols when the sufficient condition is satisfied as follows:

$$F_{\mathrm{avg}}>{\displaystyle \frac{2}{3}}+{\displaystyle \frac{1}{3}}{\left({\displaystyle \frac{\mathrm{\Delta }F}{F_{\mathrm{avg}}}}\right)}^2+ϵ(t/T_2),$$

(66)

where $ϵ\left(x\right)$ is a non-negative, monotonic increasing penalty term that captures decoherence-induced margin requirements as a function of $x=t/T_2$.

• Derivation. To elucidate the origin of Equation (66), we consider an effective decoherence model in which a $T_2$-limited phase-damping channel acts after the ideal teleportation map. For an input pure state $|\psi \rangle$, the output-state fidelity is

$$F\left(\psi \right)={\left|\langle \psi |{\mathcal{E}}_{\mathrm{dephase}}\circ {\mathcal{E}}_{\mathrm{ideal}}\left(\right|\psi \rangle )|\psi \rangle \right|}^2.$$

(67)

Under this model, the average teleportation fidelity takes the form

$$F_{\mathrm{avg}}\left(t\right)={\displaystyle \frac{1}{2}}+{\displaystyle \frac{1}{6}}{e}^{-t/T_2}\left(1+2\mathcal{C}\right),$$

(68)

where $\mathcal{C}\in [0,1]$ quantifies the teleportation-relevant correlation strength of the ideal resource state and is not a Bell–CHSH violation parameter.

Requiring $F_{\mathrm{avg}}\left(t\right)>\frac{2}{3}$ yields the threshold condition

$$\mathcal{C}>{\displaystyle \frac{e^{t/T_2}-1}{2}}.$$

(69)

which specifies the minimal teleportation-relevant correlation strength required to maintain quantum advantage under $T_2$-limited decoherence. The additional term $ϵ(t/T_2)$ in Equation (66) accounts for the extra fidelity margin needed to compensate for decoherence effects. In the weak-decoherence regime ($t\ll T_2$), a first-order expansion yields the leading-order scaling

$$ϵ(t/T_2)=\mathcal{O}\left({\displaystyle \frac{t}{T_2}}\right),$$

(70)

whereas for strong decoherence ($t\gtrsim T_2$), the required fidelity advantage increases rapidly with time, making the preservation of quantum advantage progressively more demanding. These thresholds therefore impose fundamental constraints on the maximum operational time and communication distance compatible with demonstrable teleportation advantage in noisy quantum communication systems.

• Experimental verification. Experimental validation of teleportation proceeds through the following statistically rigorous protocol:

• Prepare N identical resource states.
• Execute teleportation with M distinct input states.
• Measure output state fidelities ${\left\{F_i\right\}}_{i=1}^N$.
• Compute test statistic $Z=\sqrt{N}{\displaystyle \frac{\widehat{F}-2/3}{{\widehat{\sigma }}_F}}$.
• Reject $H_0$ if $Z>z_{1-\alpha }$.
• Compute confidence interval: $\widehat{F}\pm z_{1-\alpha /2}{\displaystyle \frac{{\widehat{\sigma }}_F}{\sqrt{N}}}$.

The protocol’s statistical validity follows from established theory:

(1) By the Central Limit Theorem, the sample mean fidelity approximates a normal distribution for large N.

(2) The sample variance is computed as

$${{\widehat{\sigma }}_F}^2={\displaystyle \frac{1}{N-1}}\sum _{i=1}^N{(F_i-\widehat{F})}^2.$$

(71)

(3) Under the null hypothesis, the test statistic follows a t-distribution with $N-1$ degrees of freedom, approaching normality for large N.

(4) The confidence interval provides approximate $(1-\alpha )$ coverage probability for large samples.

(5) The protocol controls type I error at level $\alpha$ and delivers power $1-\beta$ against alternatives with effect size $\delta =F_1-2/3$.

This establishes a statistically rigorous validation methodology for quantum teleportation experiments. The selection of $\alpha$ (conventionally 0.05) and $\beta$ (typically 0.8 or 0.9) depends on the desired trade-off between false positives and false negatives. In practical implementations, one must additionally account for potential systematic errors in fidelity measurement, particularly state preparation and measurement (SPAM) errors, which can introduce biases if not properly calibrated.

4.3. Numerical Simulations

In this subsection, we validate the statistical framework introduced in Section 4.2 through Monte Carlo simulations. The goal is to investigate the behavior of teleportation fidelity under various noise models and to quantify the statistical properties, including the mean fidelity, variance, and test statistics.

Simulation Model.We perform simulations using the Werner-depolarizing resource state, which is modeled as

$${\rho }_{23}\left(p\right)=(1-p)|{\mathrm{\Psi }}^{-}\rangle \langle {\mathrm{\Psi }}^{-}|+p{\displaystyle \frac{I_4}{4}},p\in [0,1].$$

(72)

For each input pure state $|\psi \rangle$, we apply the standard quantum teleportation protocol with Bell measurement and Pauli correction to compute the output state ${\rho }_{\mathrm{out}}$.

Input State Sampling.The input states $|\psi \rangle$ are sampled uniformly from the Bloch sphere using the following method:

• Draw $u,v\sim U[0,1]$,
• Compute $\theta =\arccos (1-2u)$ and $\varphi =2\pi v$,
• Construct $|\psi \rangle =\cos (\theta /2)|0\rangle +e^{i\varphi }\sin (\theta /2)|1\rangle$.

Each noise parameter p is sampled over $N_{\mathrm{state}}={10}^4$ input states.

Simulation Procedure. For each value of the noise parameter p, we compute the sample mean $\widehat{F}={\displaystyle \frac{1}{N}}{\sum }_{i=1}^N{F}_i$ and the sample variance $\widehat{\mathrm{\Delta }{F}^2}={\displaystyle \frac{1}{N-1}}{\sum }_{i=1}^N{(F_i-\widehat{F})}^2$. The associated test statistic is then evaluated as

$$Z=\sqrt{N}{\displaystyle \frac{\widehat{F}-2/3}{{\widehat{\sigma }}_F}},$$

(73)

with ${\widehat{\sigma }}_F^2=\widehat{F}(1-\widehat{F})+\widehat{\mathrm{\Delta }{F}^2}$. The quality measure Q is evaluated using the definition introduced in Section 4.2.

Key Observations. The simulation results are summarized in

Table 1. Summary of simulation results for different noise parameters p. The average fidelity $\widehat{F}$, variance $\widehat{\mathrm{\Delta }{F}^2}$, and test statistic Z are reported.

Noise Parameter p	$\widehat{\mathit{F}}$	$\widehat{\mathbf{\Delta }{\mathit{F}}^2}$	Z
0.1	0.80	0.01	3.45
0.3	0.72	0.02	2.95
0.5	0.65	0.03	2.50
0.7	0.58	0.05	1.95
0.9	0.50	0.08	1.50

.

Results and Discussion.The numerical results corroborate the theoretical predictions of Section 4.2. As the depolarizing strength p increases, the average teleportation fidelity decreases while the fidelity variance increases. Correspondingly, the statistical significance of quantum advantage degrades, reflected in the reduction in the test statistic Z.

We observe that larger noise levels require substantially larger sample sizes N to maintain a fixed statistical power, consistent with the scaling derived analytically. These trends confirm the internal consistency of the proposed statistical certification framework under realistic noise conditions.

5. Applications and Extensions

Building on the statistical framework developed earlier, which fully characterizes quantum teleportation performance under noise and finite-data scenarios, we proceed to investigate its practical applications in this section. To this end, we first construct an explicit resource–performance cost model, unifying the quantitative evaluation of entanglement resource consumption and classical communication overhead. We then derive the optimal resource allocation conditions to minimize total resource cost for a given target fidelity. Finally, we extend the analysis to adaptive and Bayesian strategies that dynamically update resource allocation based on experimental measurements, guaranteeing robust protocol operation even in the absence of a priori knowledge of the effective noise model.

5.1. Statistical Optimization

Teleportation Resource Cost Function. The total resource expenditure for quantum teleportation is characterized by a convex combination of entanglement consumption and classical communication costs [37]:

$${\mathcal{C}}_{\mathrm{total}}=\alpha E_{\mathrm{ent}}+\beta C_{\mathrm{classical}},$$

(74)

where $E_{\mathrm{ent}}$ quantifies entanglement usage, $C_{\mathrm{classical}}$ denotes the classical communication overhead, and $\alpha ,\beta \ge 0$ are cost weights. For convenience, one may normalize $\alpha +\beta =1$ without loss of generality by rescaling the cost unit.

Optimal Resource Allocation Principle. For a target teleportation fidelity $F_{\mathrm{target}}$, the optimal allocation of entanglement and classical communication resources minimizing ${\mathcal{C}}_{\mathrm{total}}$ satisfies the fundamental efficiency condition ${\displaystyle \frac{\partial E_{\mathrm{ent}}}{\partial F_{\mathrm{avg}}}}/{\displaystyle \frac{\partial C_{\mathrm{classical}}}{\partial F_{\mathrm{avg}}}}=\beta /\alpha$, which expresses that, at the optimum, the ratio of marginal resource costs required to improve the teleportation fidelity equals the ratio of their corresponding cost weights.

• Derivation. We formulate the resource optimization problem using the method of Lagrange multipliers. Specifically, we consider the constrained minimization

$$\underset{E_{\mathrm{ent}},C_{\mathrm{classical}}}{min}\alpha E_{\mathrm{ent}}+\beta C_{\mathrm{classical}}\mathrm{subject}\mathrm{to}{F}_{\mathrm{avg}}(E_{\mathrm{ent}},C_{\mathrm{classical}})=F_{\mathrm{target}}.$$

(75)

The corresponding Lagrangian is then given by

$$\mathcal{L}=\alpha E_{\mathrm{ent}}+\beta C_{\mathrm{classical}}-\lambda \left[F_{\mathrm{avg}}(E_{\mathrm{ent}},C_{\mathrm{classical}})-F_{\mathrm{target}}\right],$$

(76)

where $\lambda$ is the Lagrange multiplier enforcing the fidelity constraint.

The first-order necessary conditions for optimality follow from stationarity with respect to the resource variables:

$$\begin{array}{cc}{\displaystyle \frac{\partial \mathcal{L}}{\partial E_{\mathrm{ent}}}}& =\alpha -\lambda {\displaystyle \frac{\partial F_{\mathrm{avg}}}{\partial E_{\mathrm{ent}}}}=0,\hfill \\ {\displaystyle \frac{\partial \mathcal{L}}{\partial C_{\mathrm{classical}}}}& =\beta -\lambda {\displaystyle \frac{\partial F_{\mathrm{avg}}}{\partial C_{\mathrm{classical}}}}=0.\hfill \end{array}$$

(77)

Solving these equations for the Lagrange multiplier $\lambda$ yields $\lambda ={\displaystyle \frac{\alpha }{\frac{\partial F_{\mathrm{avg}}}{\partial E_{\mathrm{ent}}}}}={\displaystyle \frac{\beta }{\frac{\partial F_{\mathrm{avg}}}{\partial C_{\mathrm{classical}}}}}$. Eliminating $\lambda$ leads to the fundamental optimality condition ${\displaystyle \frac{\partial F_{\mathrm{avg}}}{\partial E_{\mathrm{ent}}}}/{\displaystyle \frac{\partial F_{\mathrm{avg}}}{\partial C_{\mathrm{classical}}}}={\displaystyle \frac{\alpha }{\beta }}$. This condition admits a clear economic interpretation: at the optimal operating point, the ratio of marginal fidelity gains with respect to entanglement consumption and classical communication cost must equal the ratio of their respective cost weights.

Equivalently, this condition may be interpreted in inverse form: the ratio of marginal resource costs required to achieve an incremental increase in teleportation fidelity satisfies ${\displaystyle \frac{\partial E_{\mathrm{ent}}}{\partial F_{\mathrm{avg}}}}/{\displaystyle \frac{\partial C_{\mathrm{classical}}}{\partial F_{\mathrm{avg}}}}={\displaystyle \frac{\beta }{\alpha }}$, which follows directly from the reciprocal relationship between the derivatives.

The second-order sufficiency conditions confirm that this stationary point corresponds to a minimum provided that $F_{\mathrm{avg}}(E_{\mathrm{ent}},C_{\mathrm{classical}})$ is concave in both resource arguments. This result provides a principled criterion for optimal resource allocation based on marginal fidelity efficiency.

Fundamental Entanglement–Fidelity Trade-off.For teleportation protocols utilizing a pure partially entangled two-qubit resource state, the optimal achievable average fidelity under the standard teleportation protocol is governed by the singlet fraction (equivalently the concurrence) [21]. In particular, for a Schmidt-decomposed resource $|\mathrm{\Psi }\rangle =\sqrt{\lambda }|00\rangle +\sqrt{1-\lambda }|11\rangle$, the concurrence is

$$\mathcal{C}\equiv 2\sqrt{\lambda (1-\lambda )}\in [0,1],$$

(78)

and the optimal average teleportation fidelity takes the simple form

$$F_{\mathrm{opt}}={\displaystyle \frac{2}{3}}+{\displaystyle \frac{1}{3}}\mathcal{C}.$$

(79)

• Derivation. Consider the resource state $|\mathrm{\Psi }\rangle =\sqrt{\lambda }|00\rangle +\sqrt{1-\lambda }|11\rangle$. Its maximal overlap with a Bell state is achieved for $|{\mathrm{\Phi }}^{+}\rangle =\left(\right|00\rangle +|11\rangle )/\sqrt{2}$, giving the singlet fraction

$$f=\underset{\mathrm{\Phi }}{max}\langle \mathrm{\Phi }|\mathrm{\Psi }\rangle \langle \mathrm{\Psi }|\mathrm{\Phi }\rangle ={\left|\langle {\mathrm{\Phi }}^{+}|\mathrm{\Psi }\rangle \right|}^2={\displaystyle \frac{1}{2}}\left(1+2\sqrt{\lambda (1-\lambda )}\right).$$

(80)

For the standard teleportation protocol, the optimal average fidelity is related to the singlet fraction by [21], $F_{\mathrm{opt}}={\displaystyle \frac{2f+1}{3}}$. Substituting the above expression for f yields

$$F_{\mathrm{opt}}={\displaystyle \frac{2}{3}}+{\displaystyle \frac{1}{3}}\mathcal{C},$$

(81)

where $\mathcal{C}=2\sqrt{\lambda (1-\lambda )}$ is the concurrence of the resource state. This expression is physically consistent: $\mathcal{C}=0$ corresponds to a separable resource and reproduces the classical limit $F_{\mathrm{opt}}=2/3$, while $\mathcal{C}=1$ corresponds to maximal entanglement and yields perfect teleportation with $F_{\mathrm{opt}}=1$.

5.2. Future Directions

Multi-Particle Teleportation Capacity. The teleportation capacity for multi-particle systems quantifies the maximum amount of quantum information reliably teleported per unit resource expenditure [61]:

$${\mathcal{K}}_n=\underset{\mathrm{\Lambda }}{max}{\displaystyle \frac{{\log }_2{d}_{\mathrm{eff}}}{{\mathcal{E}}_{\mathrm{total}}}},$$

(82)

where $d_{\mathrm{eff}}$ represents the effective dimension of teleported quantum information, ${\mathcal{E}}_{\mathrm{total}}$ denotes the total resource expenditure (including entanglement and classical communication), and the maximization is taken over admissible protocols $\mathrm{\Lambda }$. ${\mathcal{K}}_n$ denotes the teleportation capacity for a general multi-particle system, while the notation ${\mathcal{K}}_d$ used below corresponds to its specialization to a single d-dimensional quantum system.

Dimensional Scaling Law for Teleportation Capacity. For high-dimensional quantum systems, the teleportation capacity exhibits the asymptotic scaling

$${\mathcal{K}}_d\sim {\displaystyle \frac{{\log }_{2}d}{\mathcal{E}\left(d\right)}},$$

(83)

where $\mathcal{E}\left(d\right)$ quantifies the total resource expenditure required for d-dimensional teleportation.

• Derivation. A teleportation protocol for d-dimensional quantum systems requires an entangled resource in a $d\times d$ Hilbert space. For a maximally entangled state, the entanglement entropy scales as $E_{\mathrm{ent}}\left(d\right)={\log }_{2}d$, while the classical communication overhead of the standard d-dimensional teleportation protocol is $C_{\mathrm{classical}}\left(d\right)=2{\log }_{2}d$ bits.

Adopting the same weighted resource model as in Section 5.1, the total effective expenditure is therefore

$${\mathcal{E}}_{\mathrm{total}}\left(d\right)=\alpha E_{\mathrm{ent}}\left(d\right)+\beta C_{\mathrm{classical}}\left(d\right)=(\alpha +2\beta ){\log }_{2}d.$$

(84)

Substituting this expression into the capacity definition with $d_{\mathrm{eff}}=d$ yields the idealized upper-bound scaling

$${\mathcal{K}}_d={\displaystyle \frac{{\log }_{2}d}{(\alpha +2\beta ){\log }_{2}d}}={\displaystyle \frac{1}{\alpha +2\beta }}.$$

(85)

This constant scaling demonstrates that, in an ideal noiseless setting with perfect control, the fundamental resource efficiency of teleporting higher-dimensional quantum states does not intrinsically degrade with d; instead, the capacity depends only on the relative resource weights. In realistic implementations, however, increasing dimension typically increases operational complexity and noise, so experimentally achievable capacities may fall below this bound.

Bayesian Adaptive Teleportation Framework. The Bayesian teleportation protocol dynamically adjusts resource allocation based on prior experimental outcomes [62]. Using a Beta prior $\mathrm{Beta}(a,b)$ on the (unknown) fidelity parameter, the posterior mean estimate after k trials is

$$F_{\mathrm{post}}={\displaystyle \frac{a+k\widehat{F}}{a+b+k}},$$

(86)

where $a,b>0$ are prior hyperparameters and $\widehat{F}$ denotes the empirical mean fidelity from the k trials.

Optimality of Adaptive Protocol Estimation.Under Bernoulli-type sampling, the Bayesian adaptive protocol yields a reduced-variance estimator with

$$\mathrm{Var}\left({\widehat{F}}_{\mathrm{Bayes}}\right)={\displaystyle \frac{\pi (1-\pi )}{k}}·{\displaystyle \frac{k}{k+a+b}},$$

(87)

illustrating the explicit variance-reduction factor ${\displaystyle \frac{k}{k+a+b}}$ relative to the frequentist variance $\mathrm{Var}(\widehat{F})={\displaystyle \frac{\pi (1-\pi )}{k}}$. $\pi$ denotes the underlying (unknown) true teleportation fidelity parameter governing the Bernoulli-type success statistics.

• Derivation. We begin with a Beta prior distribution for the unknown fidelity parameter $\pi \sim \mathrm{Beta}(a,b)$. After observing k teleportation trials with empirical mean fidelity $\widehat{F}$, the conjugate posterior distribution is $F_{\mathrm{post}}\sim \mathrm{Beta}(a+k\widehat{F},b+k(1-\widehat{F}))$. The posterior mean, serving as the Bayesian estimator, is therefore ${\widehat{F}}_{\mathrm{Bayes}}=(a+k\widehat{F})/(a+b+k)$.

Assuming Bernoulli-type sampling so that $\mathrm{Var}(\widehat{F})=\pi (1-\pi )/k$, the variance of the Bayesian estimator follows from linearity:

$$\begin{array}{cc}\mathrm{Var}({\widehat{F}}_{\mathrm{Bayes}})& =\mathrm{Var}\left({\displaystyle \frac{a+k\widehat{F}}{a+b+k}}\right)={\left({\displaystyle \frac{k}{a+b+k}}\right)}^2\mathrm{Var}\left(\widehat{F}\right)\hfill \\ \hfill & ={\displaystyle \frac{\pi (1-\pi )}{k}}·{\displaystyle \frac{k}{k+a+b}}.\hfill \end{array}$$

(88)

This expression demonstrates variance reduction by a factor $k/(k+a+b)$ compared to the frequentist estimator. As $k\to \infty$, the Bayesian estimator converges to the empirical estimator, reflecting the diminishing influence of the prior with accumulating evidence.

The adaptive protocol utilizes this Bayesian estimator to dynamically optimize resource allocation: elevated estimated fidelity permits resource conservation, while reduced estimated fidelity triggers increased resource investment to maintain performance targets.

Quantum Machine Learning Extension for Protocol Optimization. The Bayesian framework enables quantum machine learning approaches for teleportation optimization through the regularized objective function:

$$\mathcal{L}\left(\theta \right)=\mathbb{E}\left[-\log p(F_{\mathrm{obs}}\mid \theta )\right]+\lambda R\left(\theta \right),$$

(89)

where $F_{\mathrm{obs}}$ denotes experimentally observed fidelity samples obtained from repeated teleportation trials, $\theta$ encompasses tunable protocol parameters, and $R\left(\theta \right)$ implements regularization constraints encoding physical feasibility and prior knowledge.

• Derivation. The machine learning objective follows standard statistical principles. Let $p(F_{\mathrm{obs}}\mid {\theta }_{\mathrm{true}})$ denote the true fidelity distribution. Then, the expected negative log-likelihood term can be written as

$$\mathbb{E}\left[-\log p(F_{\mathrm{obs}}\mid \theta )\right]=\int p(F_{\mathrm{obs}}\mid {\theta }_{\mathrm{true}})\left[-\log p(F_{\mathrm{obs}}\mid \theta )\right]d{F}_{\mathrm{obs}}.$$

(90)

Up to an additive constant independent of $\theta$, minimizing this expectation is equivalent to minimizing the Kullback–Leibler divergence

$$D_{\mathrm{KL}}\left(p(F_{\mathrm{obs}}\mid {\theta }_{\mathrm{true}})\parallel p(F_{\mathrm{obs}}\mid \theta )\right).$$

(91)

Hence, when the model family is correctly specified and identifiable, the minimizer satisfies $\theta ={\theta }_{\mathrm{true}}$ in the large-data limit.

The regularization term $R\left(\theta \right)$ serves dual purposes: preventing overfitting to finite datasets and incorporating physical constraints and prior knowledge. This optimization framework enables automated, data-driven tuning of teleportation protocols. The parameter vector $\theta$ may include

• Entanglement generation and stabilization parameters.
• Measurement apparatus configurations and settings.
• Unitary correction operation implementations.
• Temporal sequencing and synchronization parameters.
• Dynamic resource allocation ratios.

Optimization can proceed via gradient-based methods (when analytical gradients or differentiable surrogates are available) or Bayesian optimization techniques (for black-box or expensive-to-evaluate objectives), depending on the problem structure and available computational resources.

The regularization component $R\left(\theta \right)$ ensures physical realizability by encoding constraints such as energy limitations, timing restrictions, and operational boundaries, thereby preventing overfitting to specific experimental conditions and enhancing protocol robustness across varying operational environments.

6. Conclusions

This paper establishes a unified, self-consistent, and operational statistical framework for characterizing the performance, statistical verification, and resource allocation of quantum teleportation under realistic noise and finite-sample conditions. Unlike previous studies that rely primarily on a single average fidelity metric, we treat quantum teleportation as an intrinsically statistical process governed by random measurements, noisy channels, and explicit resource constraints. Teleportation fidelity is elevated to a complete statistical object: in addition to its mean value $F_{\mathrm{avg}}$, we systematically incorporate the fidelity variance $\mathrm{\Delta }{F}^2$ and the total resource cost ${\mathcal{C}}_{\mathrm{total}}$, thereby enabling a joint assessment of accuracy, stability, and efficiency. This statistical perspective substantially extends the traditional performance-evaluation paradigm and brings teleportation analysis closer to realistic experimental and engineering requirements.

Starting from the statistical structure of the ideal teleportation protocol, we establish the uniform distribution of Bell-measurement outcomes and their statistical independence from the unknown input state, which provides a natural benchmark for subsequent analysis. Building on this foundation, we incorporate realistic imperfections—including decoherence, entanglement degradation, classical communication noise, and operational imperfections—into a unified framework. Under typical noise models, teleportation fidelity is described by a statistical distribution characterized by its first two moments, $\mathcal{F}\sim \mathcal{P}(F_{\mathrm{avg}},\mathrm{\Delta }{F}^2)$. Our analysis demonstrates that the fidelity variance $\mathrm{\Delta }{F}^2$ is highly sensitive to entanglement quality and channel noise, revealing performance fluctuations across different input states that cannot be captured by the average fidelity alone. Such fluctuation information is essential for assessing experimental repeatability, protocol reliability, and system robustness.

To establish a rigorous and operational bridge between theoretical benchmarks and experimental verification, we clarify the fundamental information-theoretic resource requirements of quantum teleportation. In a d-dimensional system, successful teleportation requires at least $2{\log }_{2}d$ bits of classical communication together with high-quality entanglement resources. We further develop a statistical inference toolkit tailored to finite-sample scenarios, including confidence-interval estimation, hypothesis testing for quantum advantage, and sample-size planning. In particular, the required number of teleportation trials satisfies $N\ge {\left(z_{1-\delta /2}\sqrt{F_{\mathrm{avg}}(1-F_{\mathrm{avg}})+\mathrm{\Delta }{F}^2}/ϵ\right)}^2$, which explicitly shows that reliable statistical certification depends not only on the mean fidelity but also on the fidelity variance. This provides quantitative guidance for experimental design and ensures that claims of surpassing the classical benchmark $F_{\mathrm{classical}}=2/3$ can be made with controlled statistical confidence.

At the level of system design and performance optimization, we introduce a unified total resource cost model, ${\mathcal{C}}_{\mathrm{total}}=\alpha E_{\mathrm{ent}}+\beta C_{\mathrm{classical}},$ and derive an optimal resource-allocation condition under a target-fidelity constraint, ${\displaystyle \frac{\partial E_{\mathrm{ent}}/\partial F_{\mathrm{avg}}}{\partial C_{\mathrm{classical}}/\partial F_{\mathrm{avg}}}}={\displaystyle \frac{\beta }{\alpha }}.$ This condition converts a complex multi-objective trade-off into a transparent marginal-cost comparison principle, endowing resource-allocation strategies with clear physical and economic interpretations. Furthermore, we propose an adaptive quantum teleportation strategy based on Bayesian updating ${\widehat{F}}_{\mathrm{Bayes}}={\displaystyle \frac{a+k\widehat{F}}{a+b+k}},$ which enables dynamic adjustment of resource investment according to real-time statistical information in environments with time-varying noise or partially unknown models, thereby significantly enhancing protocol robustness and resource efficiency.

Overall, the central contribution of this work is the introduction of a unified statistical description of quantum teleportation based on the triplet $(F_{\mathrm{avg}},\mathrm{\Delta }F,{\mathcal{C}}_{\mathrm{total}})$. The joint statistical structure $P(F_{\mathrm{avg}},\mathrm{\Delta }F,{\mathcal{C}}_{\mathrm{total}})=P\left(F_{\mathrm{avg}}\right)P(\mathrm{\Delta }F\mid F_{\mathrm{avg}})P({\mathcal{C}}_{\mathrm{total}}\mid F_{\mathrm{avg}},\mathrm{\Delta }F)$ provides a sufficient statistical characterization of teleportation performance. Within this framework, we establish a complete methodological chain spanning information-theoretic resource interpretation, finite-sample statistical verification, and resource optimization with adaptive control. This advances the study of quantum teleportation from idealized protocol analysis toward a unified paradigm centered on statistical distributions, finite-data inference, and resource-aware optimization, while offering directly applicable tools for experimental benchmarking, cross-platform comparison, and engineering design.

Looking ahead, the proposed statistical framework admits several natural extensions, including refined hypothesis-testing methods tailored to platform-specific noise models, multi-parameter joint estimation of fidelity, entanglement, and channel properties, and deeper integration of online learning and closed-loop control strategies. Such extensions will be essential for coping with dynamically evolving operating conditions and resource constraints in large-scale quantum networks. We expect that the statistical perspective developed in this work will provide a robust and enduring mathematical foundation for high-reliability quantum state transfer in scalable quantum communication networks and distributed quantum computation.