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23 pages, 455 KB

Open AccessArticle

Multi-Output Nondestructive Controlled Quantum Teleportation Through a Partially Entangled Channel

by Miao Liu, Zhaoyuan Zhang, Jiayin Peng and Jiangang Tang

Mathematics 2026, 14(9), 1432; https://doi.org/10.3390/math14091432 - 24 Apr 2026

Cited by 1 | Viewed by 242

Through a detailed study of nondestructive controlled quantum teleportation (NCQT) with two outputs in three-dimensional and high-dimensional Hilbert spaces, we propose an NCQT scheme that synchronously and probabilistically teleports N arbitrary unknown d-dimensional single-particle states from one sender to N different receivers [...] Read more.

Through a detailed study of nondestructive controlled quantum teleportation (NCQT) with two outputs in three-dimensional and high-dimensional Hilbert spaces, we propose an NCQT scheme that synchronously and probabilistically teleports N arbitrary unknown d-dimensional single-particle states from one sender to N different receivers under the supervision of a controller, by using a d-dimensional partially entangled

(2 N + 1)

-particle state as the quantum channel. The protocol succeeds if and only if all receivers recover their respective target states, and the optimal success probability of the N-output NCQT is determined by the minimum superposition coefficient of each product two-qudit state in the partially entangled channel. In our scheme, each receiver introduces an auxiliary qubit to assist in the local recovery test. When the auxiliary-qubit measurement outcome is

| 0 ⟩

, the receiver can restore the target state; when the outcome is

| 1 ⟩

, the corresponding unknown original state is retained by the sender. Accordingly, the N-output controlled teleportation process can be repeated as many times as additional quantum channels are available after a failed attempt. The results show that weakly entangled channels can still realize N-output controlled teleportation through sufficiently many repetitions, whereas strongly entangled channels require only a small number of repetitions to achieve the same goal. Full article

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17 pages, 572 KB

Open AccessArticle

Superdense Coding Using Higher Dimensional Embedding

by Elio Thadhani, Sharjeel Ahmad, Hussain Ali Razvi, Facundo Martin Lopez and Eric Chitambar

Entropy 2026, 28(4), 387; https://doi.org/10.3390/e28040387 - 1 Apr 2026

Viewed by 887

Abstract

Quantum dense coding is a foundational protocol in quantum communication, allowing two classical bits to be transmitted by sending a single qubit when a maximally entangled pair is shared. In this work, we consider Embedded Dense Coding (EDC)—a generalization of deterministic dense coding [...] Read more.

Quantum dense coding is a foundational protocol in quantum communication, allowing two classical bits to be transmitted by sending a single qubit when a maximally entangled pair is shared. In this work, we consider Embedded Dense Coding (EDC)—a generalization of deterministic dense coding that embeds one subsystem into a higher-dimensional Hilbert space. To assess the operational advantage of EDC compared to standard dense coding, we consider the probability of transmission error when fixing the rate of entanglement consumed per classical message sent. We first demonstrate that EDC enables a smaller one-shot transmission error compared to standard dense coding when using quantum channels with nonzero rates of dephasing and loss. We then demonstrate that even with noiseless communication channels, EDC leads to smaller overall errors when the sender and receiver have noisy local processors. This advantage is shown through concrete implementations of EDC on IBM’s Heron processor. Full article

(This article belongs to the Section Quantum Information)

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16 pages, 526 KB

Open AccessArticle

Symmetric and Antisymmetric Quantum States from Graph Structure and Orientation

by Matheus R. de Jesus, Eduardo O. C. Hoefel and Renato M. Angelo

Entropy 2026, 28(4), 386; https://doi.org/10.3390/e28040386 - 1 Apr 2026

Viewed by 444

Abstract

Graph states provide a powerful framework for describing multipartite entanglement in quantum information science. In their standard formulation, graph states are generated by controlled-Z interactions and naturally encode symmetric exchange properties. Here we establish a precise correspondence between graph topology and exchange [...] Read more.

Graph states provide a powerful framework for describing multipartite entanglement in quantum information science. In their standard formulation, graph states are generated by controlled-Z interactions and naturally encode symmetric exchange properties. Here we establish a precise correspondence between graph topology and exchange symmetry by proving that a graph state is fully symmetric under particle permutations if and only if the underlying graph is complete. We then introduce a generalized graph-based construction using a non-commutative two-qudit gate, denoted

G R

, which requires directed edges and an explicit vertex ordering. We show that complete directed graphs generate fully antisymmetric multipartite states when endowed with appropriate orientations. Together, these results provide a unified graph-theoretic description of bosonic and fermionic exchange symmetry based on graph completeness and edge orientation. Full article

(This article belongs to the Special Issue Graph Theory and Its Applications in Quantum Mechanics)

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13 pages, 633 KB

Open AccessArticle

Qudit-Native Simulation of the Potts Model

by Maksim A. Gavreev, Evgeniy O. Kiktenko, Aleksey K. Fedorov and Anastasiia S. Nikolaeva

Entropy 2026, 28(2), 160; https://doi.org/10.3390/e28020160 - 31 Jan 2026

Viewed by 572

Abstract

Simulating entangled, many-body quantum systems is notoriously hard, especially in the case of the high-dimensional nature of the underlying physical objects. In this work, we propose an approach for simulating the Potts model based on the Suzuki–Trotter decomposition that we construct for qudit [...] Read more.

Simulating entangled, many-body quantum systems is notoriously hard, especially in the case of the high-dimensional nature of the underlying physical objects. In this work, we propose an approach for simulating the Potts model based on the Suzuki–Trotter decomposition that we construct for qudit systems. Specifically, we introduce two qudit-native decomposition schemes: (i) the first utilizes the Mølmer–Sørensen gate and additional local levels to encode the Potts interactions, while (ii) the second employs a light-shift gate that naturally fits qudit architectures. These decompositions enable a direct and efficient mapping of the Potts model dynamics into hardware-efficient qudit gate sequences for a trapped-ion platform. Furthermore, we demonstrate the use of a Suzuki–Trotter approximation with our evolution-into-gates framework for detecting the dynamical quantum phase transition. Our results establish a pathway toward qudit-based digital quantum simulation of many-body models and provide a new perspective on probing nonanalytic behavior in high-dimensional quantum many-body models. Full article

(This article belongs to the Special Issue Quantum Computing: From Basics to Advanced Algorithms)

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16 pages, 732 KB

Open AccessArticle

High-Dimensional Quantum Key Distribution with N-Qudits States in Optical Fibers

by Jesús Liñares, Xesús Prieto-Blanco and Alexandre Vázquez-Martínez

Appl. Sci. 2026, 16(3), 1396; https://doi.org/10.3390/app16031396 - 29 Jan 2026

Viewed by 515

Abstract

We present a high-dimensional quantum key distribution protocol by using N-qudits quantum light states—that is, product states with N photons, each of them in a quantum superposition of dimension d, which provides a high dimension

d^{N}

and, accordingly, a very [...] Read more.

We present a high-dimensional quantum key distribution protocol by using N-qudits quantum light states—that is, product states with N photons, each of them in a quantum superposition of dimension d, which provides a high dimension

d^{N}

and, accordingly, a very high security level. We present the implementation of this protocol in different types of optical fibers, where quantum states can undergo polarization and phase perturbations under propagation in optical fibers; however, polarization perturbations can be notably reduced in a passive or active way, and, more importantly, these states can become insensitive to phase perturbations. Thus, N-qubits are fully robust to relative phase perturbations between any pair of 1-qubits, and therefore do not require any phase compensation, which, on the contrary, is absolutely necessary in high-dimensional QKD with 1-qudits (one photon). Likewise, quantum states also undergo attenuation, that is, some photons are lost under propagation in the optical fibers and thus

N^{'}

(<N)-qudits are used; however, even for standard optical fiber attenuation values, high secret key rates are still obtained. Finally, we analyse the security of this high-dimensional protocol under an intercept and resend attack performed by Eve, and the resulting secure key rates are calculated, showing a significant increase with the dimension provided by number N of photons. Full article

(This article belongs to the Section Optics and Lasers)

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19 pages, 1436 KB

Open AccessArticle

The Geometry of Qubit Decoherence: Linear Versus Nonlinear Dynamics in the Bloch Ball

by Alan C. Maioli, Evaldo M. F. Curado, Jean-Pierre Gazeau and Tomoi Koide

Physics 2026, 8(1), 8; https://doi.org/10.3390/physics8010008 - 14 Jan 2026

Viewed by 916

Abstract

We present two complementary approaches to the Gorini–Kossakowski–Sudarshan–Lindblad equation for an open qubit. First, based on linearity, yields solutions illustrated by mixed-state trajectories in the Bloch ball, including non-random asymptotic fixed points and exceptional points. Second, exploiting the SU

(2)

symmetry, [...] Read more.

We present two complementary approaches to the Gorini–Kossakowski–Sudarshan–Lindblad equation for an open qubit. First, based on linearity, yields solutions illustrated by mixed-state trajectories in the Bloch ball, including non-random asymptotic fixed points and exceptional points. Second, exploiting the SU

(2)

symmetry, leads to a nonlinear dynamical system that separates angular dynamics from radial dissipation. This symmetry-based perspective presents a promising route toward generalization to open qudits. Full article

(This article belongs to the Special Issue A Themed Issue in Honor of Professor Viktor Dodonov—55 Years in Quantum Physics)

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20 pages, 585 KB

Open AccessArticle

Transition-Aware Decomposition of Single-Qudit Gates

by Denis A. Drozhzhin, Evgeniy O. Kiktenko, Aleksey K. Fedorov and Anastasiia S. Nikolaeva

Entropy 2026, 28(1), 56; https://doi.org/10.3390/e28010056 - 31 Dec 2025

Cited by 2 | Viewed by 765

Abstract

Quantum computation with d-level quantum systems, also known as qudits, benefits from the possibility to use a richer computational space compared to qubits. However, for an arbitrary qudit-based hardware platform, the issue is that a generic qudit operation has to be decomposed [...] Read more.

Quantum computation with d-level quantum systems, also known as qudits, benefits from the possibility to use a richer computational space compared to qubits. However, for an arbitrary qudit-based hardware platform, the issue is that a generic qudit operation has to be decomposed into the sequence of native operations—pulses that are adjusted to the transitions between two levels in a qudit. Typically, not all levels in a qudit are simply connected to each other due to specific selection rules. Moreover, the number of pulses plays a significant role, since each pulse takes a certain execution time and may introduce error. In this paper, we propose a resource-efficient algorithm to decompose single-qudit operations into the sequence of pulses that are allowed by qudit selection rules. Using the developed algorithm, the number of pulses is at most

d (d - 1) / 2

for an arbitrary single-qudit operation. For specific operations, the algorithm could produce even fewer pulses. We provide a comparison of qudit decompositions for several types of trapped ions, specifically

{}^{171}{Yb}^{+}

,

{}^{137}{Ba}^{+}

and

{}^{40}{Ca}^{+}

with different selection rules, and also decomposition for superconducting qudits. Although our approach deals with single-qudit operations, the proposed approach is important for realizing two-qudit operations since they can be implemented as a standard two-qubit gate that is surrounded by efficiently implemented single-qudit gates. Full article

(This article belongs to the Special Issue Quantum Computing: From Basics to Advanced Algorithms)

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27 pages, 407 KB

Open AccessFeature PaperArticle

Equivalence Relations Between Conical 2-Designs and Mutually Unbiased Generalized Equiangular Tight Frames

by Katarzyna Siudzińska

Mathematics 2026, 14(1), 128; https://doi.org/10.3390/math14010128 - 29 Dec 2025

Cited by 1 | Viewed by 692 | Correction

Abstract

Quantum measurements play a fundamental role in quantum information. Therefore, increasing efforts are being made to construct symmetric measurement operators for qudit systems. A wide class of projective measurements corresponds to complex projective 2-designs, which include symmetric, informationally complete (SIC) POVMs and mutually [...] Read more.

Quantum measurements play a fundamental role in quantum information. Therefore, increasing efforts are being made to construct symmetric measurement operators for qudit systems. A wide class of projective measurements corresponds to complex projective 2-designs, which include symmetric, informationally complete (SIC) POVMs and mutually unbiased bases (MUBs). In this paper, we establish a one-to-one correspondence between conical 2-designs and mutually unbiased generalized equiangular tight frames, both of which are common generalizations of SIC POVMs and MUBs to operators of arbitrary rank. It turns out that there exist rich families of operators that belong to only one of those two classes. This raises important questions about which symmetries have to be preserved for applicational prominence. Full article

(This article belongs to the Section E4: Mathematical Physics)

38 pages, 4891 KB

Open AccessArticle

Thermonuclear Fusion Based Quantum-Inspired Algorithm for Solving Multiobjective Optimization Problems

by Liliya Demidova and Vladimir Maslennikov

Algorithms 2025, 18(12), 793; https://doi.org/10.3390/a18120793 - 15 Dec 2025

Viewed by 856

Abstract

This paper introduces a novel quantum-inspired algorithm for numerical multiobjective optimization, uniquely integrating the multilevel structure of qudits with principles of controlled thermonuclear fusion. Moving beyond conventional qubit-based approaches, the algorithm leverages the qudit’s higher-dimensional state space to enhance search capabilities. Fusion-inspired dynamics—modeling [...] Read more.

This paper introduces a novel quantum-inspired algorithm for numerical multiobjective optimization, uniquely integrating the multilevel structure of qudits with principles of controlled thermonuclear fusion. Moving beyond conventional qubit-based approaches, the algorithm leverages the qudit’s higher-dimensional state space to enhance search capabilities. Fusion-inspired dynamics—modeling particle interaction, energy release, and plasma cooling—provide a powerful metaheuristic framework for navigating complex, high-dimensional Pareto fronts. A hybrid quantum-classical version of the algorithm is presented, designed to exploit the complementary strengths of both computational paradigms for improved efficiency in solving dynamic multiobjective problems. Experimental evaluation on standard dynamic multiobjective benchmarks demonstrates clear performance advantages. Both the quantum-inspired and hybrid variants consistently outperform leading classical algorithms such as NSGA-III, MOEA/D and GDE3, as well as the quantum-inspired NSGA-III, in key metrics: identifying a greater number of unique non-dominated solutions, ensuring superior uniformity along the Pareto front, maintaining stable convergence across generations, and achieving higher accuracy in approximating the ideal solution. Full article

(This article belongs to the Special Issue Numerical Optimization and Algorithms: 4th Edition)

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15 pages, 17666 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Multi-Dimensional Quantum-like Resources from Complex Synchronized Networks

by Debadrita Saha and Gregory D. Scholes

Entropy 2025, 27(9), 963; https://doi.org/10.3390/e27090963 - 16 Sep 2025

Viewed by 1046

Abstract

Recent publications have introduced the concept of quantum-like (QL) bits, along with their associated QL states and QL gate operations, which emerge from the dynamics of complex, synchronized networks. The present work extends these ideas to multi-level QL resources, referred to as QL [...] Read more.

Recent publications have introduced the concept of quantum-like (QL) bits, along with their associated QL states and QL gate operations, which emerge from the dynamics of complex, synchronized networks. The present work extends these ideas to multi-level QL resources, referred to as QL dits, as higher-dimensional analogs of QL bits. We employ systems of k-regular graphs to construct QL-dits for arbitrary dimensions, where the emergent eigenspectrum of their adjacency matrices defines the QL-state space. The tensor product structure of multi-QL dit systems is realized through the Cartesian product of graphs. Furthermore, we examine the potential computational advantages of employing d-nary QL systems over two-level QL bit systems, particularly in terms of classical resource efficiency. Overall, this study generalizes the paradigm of using synchronized network dynamics for QL information processing to include higher-dimensional QL resources. Full article

(This article belongs to the Special Issue Quantum Information and Probability: From Foundations to Engineering IV)

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69 pages, 1603 KB

Open AccessFeature PaperArticle

Intrinsic and Measured Information in Separable Quantum Processes

by David Gier and James P. Crutchfield

Entropy 2025, 27(6), 599; https://doi.org/10.3390/e27060599 - 3 Jun 2025

Cited by 1 | Viewed by 2089

Abstract

Stationary quantum information sources emit sequences of correlated qudits—that is, structured quantum stochastic processes. If an observer performs identical measurements on a qudit sequence, the outcomes are a realization of a classical stochastic process. We introduce quantum-information-theoretic properties for separable qudit sequences that [...] Read more.

Stationary quantum information sources emit sequences of correlated qudits—that is, structured quantum stochastic processes. If an observer performs identical measurements on a qudit sequence, the outcomes are a realization of a classical stochastic process. We introduce quantum-information-theoretic properties for separable qudit sequences that serve as bounds on the classical information properties of subsequent measured processes. For sources driven by hidden Markov dynamics, we describe how an observer can temporarily or permanently synchronize to the source’s internal state using specific positive operator-valued measures or adaptive measurement protocols. We introduce a method for approximating an information source with an independent and identically distributed, Markov, or larger memory model through tomographic reconstruction. We identify broad classes of separable processes based on their quantum information properties and the complexity of measurements required to synchronize to and accurately reconstruct them. Full article

(This article belongs to the Special Issue Quantum Probability and Randomness V)

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19 pages, 3197 KB

Open AccessArticle

Towards a Multiqudit Quantum Processor Based on a ¹⁷¹Yb⁺ Ion String: Realizing Basic Quantum Algorithms

by Ilia V. Zalivako, Anastasiia S. Nikolaeva, Alexander S. Borisenko, Andrei E. Korolkov, Pavel L. Sidorov, Kristina P. Galstyan, Nikita V. Semenin, Vasilii N. Smirnov, Mikhail A. Aksenov, Konstantin M. Makushin, Evgeniy O. Kiktenko, Aleksey K. Fedorov, Ilya A. Semerikov, Ksenia Yu. Khabarova and Nikolay N. Kolachevsky

Quantum Rep. 2025, 7(2), 19; https://doi.org/10.3390/quantum7020019 - 12 Apr 2025

Cited by 15 | Viewed by 5237

Abstract

We demonstrate a quantum processor based on a 3D linear Paul trap that uses

{}^{171}{Yb}^{+}

ions with eight individually controllable four-level qudits (ququarts), which is computationally equivalent to a sixteen-qubit quantum processor. The design of the developed ion trap provides high [...] Read more.

We demonstrate a quantum processor based on a 3D linear Paul trap that uses

{}^{171}{Yb}^{+}

ions with eight individually controllable four-level qudits (ququarts), which is computationally equivalent to a sixteen-qubit quantum processor. The design of the developed ion trap provides high secular frequencies and a low heating rate, which, together with individual addressing and readout optical systems, allows executing quantum algorithms. In each of the eight ions, we use four electronic levels coupled by E2 optical transition at 435 nm for qudit encoding. We present the results of single- and two-qubit operations benchmarking and realizing basic quantum algorithms, including the Bernstein–Vazirani and Grover’s search algorithms as well as H₂ and LiH molecular simulations. Our results pave the way to scalable qudit-based quantum processors using trapped ions. Full article

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21 pages, 926 KB

Open AccessArticle

Qutrit Control for Bucket Brigade RAM Using Transmon Systems

by Lazaros Spyridopoulos, Dimitris Ntalaperas and Nikos Konofaos

Appl. Sci. 2025, 15(7), 3950; https://doi.org/10.3390/app15073950 - 3 Apr 2025

Viewed by 1202

Abstract

Qudits allow the encoding and manipulation of additional quantum information compared to that stored to a two-level qubit system. Although manipulations of qudit states are generally more complex and can introduce extra sources of noise, qudits can still be used in a number [...] Read more.

Qudits allow the encoding and manipulation of additional quantum information compared to that stored to a two-level qubit system. Although manipulations of qudit states are generally more complex and can introduce extra sources of noise, qudits can still be used in a number of applications when this error can be kept sufficiently low. One such application is the case of the Bucket Brigade Algorithm for realizing a Quantum RAM (QRAM), which inherently uses qutrits for encoding the state of address switches. In this paper, we study a methodology for qutrit manipulation that leverages efficient encoding techniques and pulse calibration methods for the case of transmon systems. The methodology employs an encoding scheme that allows the execution of controlled operations, using the subspace spanned by the two lowest levels of the transmon; we show how this scheme can be used for generating one- and two-qutrit gates by leveraging the Qiskit and Boulder Opal frameworks to compute the parameters of pulses that implement the quantum gates that are used by the BBA. For this type of gate, simulations show that the pulses perform the required operations with a low infidelity when errors introduced by the qutrit Hamiltonian dynamics are considered. Full article

(This article belongs to the Special Issue Novel Approaches for Quantum Computing and Quantum Information Processing)

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20 pages, 549 KB

Open AccessArticle

Transpiling Quantum Assembly Language Circuits to a Qudit Form

by Denis A. Drozhzhin, Anastasiia S. Nikolaeva, Evgeniy O. Kiktenko and Aleksey K. Fedorov

Entropy 2024, 26(12), 1129; https://doi.org/10.3390/e26121129 - 23 Dec 2024

Cited by 3 | Viewed by 2212

Abstract

In this paper, we introduce the workflow for converting qubit circuits represented by Open Quantum Assembly format (OpenQASM, also known as QASM) into the qudit form for execution on qudit hardware and provide a method for translating qudit experiment results back into qubit [...] Read more.

In this paper, we introduce the workflow for converting qubit circuits represented by Open Quantum Assembly format (OpenQASM, also known as QASM) into the qudit form for execution on qudit hardware and provide a method for translating qudit experiment results back into qubit results. We present the comparison of several qudit transpilation regimes, which differ in decomposition of multicontrolled gates: qubit as ordinary qubit transpilation and execution, qutrit with

d = 3

levels and single qubit in qudit, and ququart with

d = 4

levels and 2 qubits per ququart. We provide several examples of transpiling circuits for trapped ion qudit processors, which demonstrate potential advantages of qudits. Full article

(This article belongs to the Special Issue Quantum Computing with Trapped Ions)

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16 pages, 311 KB

Open AccessArticle

Communication Complexity of Entanglement-Assisted Multi-Party Computation

by Ruoyu Meng and Aditya Ramamoorthy

Entropy 2024, 26(11), 896; https://doi.org/10.3390/e26110896 - 23 Oct 2024

Cited by 1 | Viewed by 1606

Abstract

We consider a quantum and a classical version of a multi-party function computation problem with n players, where players

2, \dots, n

need to communicate appropriate information to player 1 so that a “generalized” inner product function with an appropriate promise [...] Read more.

We consider a quantum and a classical version of a multi-party function computation problem with n players, where players

2, \dots, n

need to communicate appropriate information to player 1 so that a “generalized” inner product function with an appropriate promise can be calculated. In the quantum version of the protocol, the players have access to entangled qudits but the communication is still classical. The communication complexity of a given protocol is the total number of classical bits that need to be communicated. When n is prime, and for our chosen function, we exhibit a quantum protocol (with complexity

(n - 1) (log n)

bits) and a classical protocol (with complexity

({(n - 1)}^{2} (log n^{2})

bits)). Furthermore, we present an integer linear programming formulation for determining a lower bound on the classical communication complexity. This demonstrates that our quantum protocol is strictly better than classical protocols. Full article

(This article belongs to the Special Issue Entropy, Quantum Information and Entanglement)

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