Philosophy of Polarization-Path Entanglement in Quantum Optics

Abstract

This paper explores the formal structure and philosophical implications of polarization-path entanglement in quantum optics, where different degrees of freedom of a single photon become entangled. We examine the mathematical conditions under which coherence is preserved or lost, emphasizing the role of distinguishability and information flow. The analysis is situated within major interpretational frameworks (including Copenhagen, Many-Worlds, QBism, and Bohmian mechanics) to evaluate whether such entanglement reflects physical reality or epistemic constraints. Finally, we discuss experimental realizations, relevance to quantum information processing, and open conceptual questions regarding the ontological status of single-particle entanglement.

1. Introduction

Quantum entanglement, famously referred to as “spooky action at a distance” by Albert Einstein in his correspondence with Max Born [1], remains at the center of both foundational debates and practical applications in quantum optics [2]. In recent decades, entanglement has evolved from a conceptual challenge to a crucial resource in quantum information science, with applications ranging from quantum computation and quantum teleportation to quantum key distribution (QKD) [3].

A large body of research in quantum optics has focused on two-photon entanglement, particularly in polarization degrees of freedom. Such entangled photon pairs are routinely generated through spontaneous parametric down-conversion (SPDC) and have enabled pivotal experimental tests of quantum nonlocality, including violations of Bell inequalities [4,5]. Furthermore, two-photon entanglement serves as the basis of multiple quantum protocols, like quantum teleportation [6,7] or quantum cryptography, most notably the Ekert QKD protocol (E91) [8]. Recently, entangled photon pairs have been used to interconnect trapped-ion modules, enabling distributed quantum computing over an optical network [9]. These studies demonstrate both the non-classical features of entanglement and its utility in real-world applications.

Photons, however, are not limited to polarization alone. As quantum systems, they possess multiple degrees of freedom, including spatial mode (path), time-bin, orbital angular momentum, and frequency [10,11]. This internal richness allows for the generation of entanglement not just between separate photons but also between different degrees of freedom within a single photon [12]. Such intra-particle entanglement opens up new possibilities in encoding and processing quantum information, leading to protocols based on hyperentanglement, compact quantum gates, and high-dimensional quantum communication [13,14]. It also provides a powerful platform for investigating the very nature of quantum coherence and measurement, since entanglement is no longer limited to spatially separated particles.

Among the various forms of entanglement, polarization-path entanglement plays a particularly illustrative role [15,16]. It emerges naturally in optical systems such as polarizing beam splitters (PBSs), Mach–Zehnder interferometers, and SPDC setups. In such systems, a single photon’s polarization can become entangled with its spatial path, allowing us to investigate entanglement within a single particle using distinct degrees of freedom. This type of intra-particle entanglement has been exploited not only in foundational tests of quantum mechanics but also in demonstrations of quantum erasure, delayed-choice experiments, and hybrid quantum information protocols [10,17].

The study of polarization-path entanglement is not limited to experimental manipulation. It opens up deeper questions about the nature of quantum superposition, the observer effect, and the boundary between coherence and decoherence. In particular, tracing out one degree of freedom (such as the path) can result in a maximally mixed polarization state, even if the global system remains in a pure state. This simple mathematical operation carries significant philosophical implications regarding what it means for part of a system to be in a definite state and how information is distributed or lost through entanglement [18].

This paper explores polarization-path entanglement from both formal and philosophical perspectives. We begin by introducing the quantum formalism underlying separable and entangled states (Section 2). Next, we examine the structure of polarization-path-entangled states and analyze how partial tracing over inaccessible degrees of freedom leads to decoherence (Section 3). We argue that the significance of polarization-path entanglement lies in its conceptual transparency and accessibility, making it an ideal framework for investigating foundational issues such as the interpretation of mixed states, the role of measurement, and the question of objectivity in quantum theory.

In Section 4, we contrast how major interpretations of quantum mechanics, such as Copenhagen, Many-Worlds, QBism, Bohmian mechanics, and consistent quantum mechanics, account for polarization-path entanglement. Section 5 investigates broader philosophical questions, including the ontological status of such entanglement, the role of distinguishability, and the epistemic versus ontic nature of quantum states. Section 6 discusses the practical relevance of these ideas, highlighting experimental realizations and their role in quantum information processing. Finally, in Section 7, we present a generalization of polarization-path entanglement by including the temporal degree of freedom. This paper concludes in Section 8 with a summary of key results and a discussion of open conceptual tensions.

2. Quantum States, Tensor Products, and Entanglement

In quantum mechanics, the formalism of tensor products provides the foundation for describing composite systems. Whether one is dealing with multiple particles or distinct degrees of freedom within a single particle, the combined state space is constructed as the tensor product of individual Hilbert spaces. If ${\mathcal{H}}_A$ and ${\mathcal{H}}_B$ denote the Hilbert spaces associated with systems A and B, respectively, then the joint system $A+B$ is described within the space ${\mathcal{H}}_A\otimes {\mathcal{H}}_B$.

A state ${|\mathrm{\Psi }\rangle }_{AB}\in {\mathcal{H}}_A\otimes {\mathcal{H}}_B$ is called separable (or product) if it can be written as the tensor product of individual states [3], as follows:

$${|\mathrm{\Psi }\rangle }_{AB}={|\psi \rangle }_A\otimes {|\varphi \rangle }_B.$$

(1)

This means that the state of the composite system is entirely determined by the states of its constituents. No quantum correlations exist between A and B in this case, and measurements on one subsystem do not affect the state of the other.

In contrast, a state that cannot be written in this form is called entangled. A simple and well-known example is the Bell state [3], as follows:

$$|{\mathrm{\Phi }}^{+}\rangle ={\displaystyle \frac{1}{\sqrt{2}}}\left({|0\rangle }_A\otimes {|0\rangle }_B+{|1\rangle }_A\otimes {|1\rangle }_B\right),$$

(2)

which exhibits maximal entanglement. In such a state, neither subsystem A nor B has an independent state; their properties are fully correlated. Measurement outcomes on one subsystem are perfectly predictive of outcomes on the other, regardless of the spatial separation between them.

Entanglement is not limited to spatially separated particles. It can also occur between different degrees of freedom of a single system. In the case of polarization-path entanglement, the polarization and path of a single photon are described in a joint Hilbert space ${\mathcal{H}}_{\mathrm{pol}}\otimes {\mathcal{H}}_{\mathrm{path}}$, and the state of the photon may be entangled across these degrees of freedom. Such intra-particle entanglement serves as a practical and conceptually rich platform for exploring foundational aspects of quantum theory.

The distinction between separable and entangled states is crucial for understanding quantum coherence, measurement, and information flow. In particular, entangled states form the backbone of non-classical correlations observed in quantum experiments and are indispensable resources in quantum information processing.

3. Polarization-Path Entanglement in Quantum Optics

3.1. Formal Structure of Polarization-Path Entangled States

In quantum optics, it is possible to entangle different degrees of freedom of a single photon, such as its polarization and spatial path. This form of intra-particle entanglement is conceptually rich because it involves correlations between internal modes of a single quantum object rather than spatially separated particles. Such entanglement can be realized experimentally using beam splitters, wave plates, and interferometers.

A prototypical example involves a photon initially prepared in diagonal polarization, denoted as

$$|D\rangle ={\displaystyle \frac{1}{\sqrt{2}}}(|H\rangle +|V\rangle ),$$

(3)

where $|H\rangle$ and $|V\rangle$ refer to horizontal and vertical polarization states, respectively.

When this photon passes through a PBS, which transmits $|H\rangle$ into one spatial mode $|0\rangle$ and reflects $|V\rangle$ into another mode $|1\rangle$, the spatial and polarization degrees of freedom become correlated. The input state is

$$|{\psi }_{\mathrm{in}}\rangle =|D\rangle \otimes |\mathrm{in}\rangle ={\displaystyle \frac{1}{\sqrt{2}}}(|H\rangle +|V\rangle )\otimes |\mathrm{in}\rangle ,$$

(4)

where $|\mathrm{in}\rangle$ denotes the initial path of the photon.

After interaction with the PBS, this transforms into the entangled state [15,16], as follows:

$${|\psi \rangle }_{\mathrm{pol}-\mathrm{path}}={\displaystyle \frac{1}{\sqrt{2}}}\left(|H\rangle \otimes |0\rangle +|V\rangle \otimes |1\rangle \right).$$

(5)

This state resides in the composite Hilbert space ${\mathcal{H}}_{\mathrm{pol}}\otimes {\mathcal{H}}_{\mathrm{path}}$. It is maximally entangled in the sense that any attempt to describe one degree of freedom independently results in a mixed state. It also satisfies the Schmidt decomposition with equal Schmidt coefficients, confirming maximal entanglement.

We can also define a corresponding density matrix [19], as follows:

$${\rho }_{\mathrm{pol}-\mathrm{path}}={\left(|\psi \rangle \langle \psi |\right)}_{\mathrm{pol}-\mathrm{path}}={\displaystyle \frac{1}{2}}\left(|H0\rangle \langle H0|+|H0\rangle \langle V1|+|V1\rangle \langle H0|+|V1\rangle \langle V1|\right),$$

(6)

where $|H0\rangle \equiv |H\rangle \otimes |0\rangle$, and likewise for other terms. This joint state exhibits quantum coherence between the two degrees of freedom.

3.2. Tracing Out Path Information: Loss of Coherence

The coherence in Equation (5) depends critically on the indistinguishability of the path states. As long as the spatial modes $|0\rangle$ and $|1\rangle$ are not measured or otherwise accessible, interference effects can be observed in polarization measurements. However, if the path information is leaked to an environment or is made accessible through measurement, then the coherence between polarization components is lost.

To model this, we trace over the path subsystem to obtain the reduced density matrix for the polarization degree of freedom [3], as follows:

$${\rho }_{\mathrm{pol}}={\mathrm{Tr}}_{\mathrm{path}}\left[{\rho }_{\mathrm{pol}-\mathrm{path}}\right].$$

(7)

Evaluating this trace yields

$$\begin{array}{cc}\hfill {\rho }_{\mathrm{pol}}& ={\displaystyle \frac{1}{2}}\left(|H\rangle \langle H|\langle 0|0\rangle +|H\rangle \langle V|\langle 0|1\rangle +|V\rangle \langle H|\langle 1|0\rangle +|V\rangle \langle V|\langle 1|1\rangle \right)\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}\left(|H\rangle \langle H|+|V\rangle \langle V|\right)={\displaystyle \frac{1}{2}}{\mathbb{I}}_2,\hfill \end{array}$$

(8)

where ${\mathbb{I}}_2$ denotes an identity matrix and $\langle 0|1\rangle =\langle 1|0\rangle =0$ due to the orthogonality of the basis states. This describes a fully mixed state with no coherence (no off-diagonal terms), and it leads to classical probabilities for polarization measurements, rather than quantum interference.

This mathematical operation models the physical process of decoherence. The disappearance of off-diagonal elements in the reduced density matrix reflects the loss of quantum coherence due to accessible which-path information. If an environment or a measurement device becomes entangled with the path degree of freedom, the observer is effectively tracing out that information, leading to a classical probabilistic mixture.

It is also worth noting that the purity of the reduced density matrix, defined as $\mathrm{Tr}\left({\rho }_{\mathrm{pol}}^2\right)$, decreases as a result of this tracing. For the original pure entangled state, the purity is $\mathrm{Tr}\left({\rho }_{\mathrm{pol}-\mathrm{path}}^2\right)=1$, while for the mixed polarization state, we obtain

$$\mathrm{Tr}\left({\rho }_{\mathrm{pol}}^2\right)={\displaystyle \frac{1}{2}}.$$

(9)

This quantifies the loss of coherence and confirms that tracing over a subsystem reduces the purity of the system of interest.

Thus, the transition from a coherent entangled state to an incoherent mixed state underscores the central role of information accessibility in quantum mechanics. Coherence and interference depend not merely on the presence of entanglement but on the inaccessibility of which-path information. This interplay is fundamental to understanding not only quantum optics experiments but also the broader philosophical discussions surrounding measurement and reality in quantum theory.

4. Interpretational Frameworks

Polarization-path entanglement, as a specific instance of quantum entanglement between internal degrees of freedom of a single particle, provides a valuable testbed for examining quantum interpretations. Although the phenomenon is well defined mathematically, its ontological status depends heavily on the interpretive framework.

4.1. Copenhagen Interpretation

In the Copenhagen view, the wavefunction represents potential outcomes of measurement. Polarization-path entanglement describes possible correlations, and upon measurement of one degree of freedom (typically polarization), the wavefunction “collapses” to a definite outcome. The path degree of freedom, if not measured, remains an abstract tool for computing outcome probabilities. This view treats the entanglement as epistemic: a predictive device rather than a physical entity [17].

4.2. Many-Worlds Interpretation

The Everett or Many-Worlds Interpretation (MWI) posits that all components of the superposition are realized in branching worlds. In this framework, a measurement of polarization leads to branching not only in the polarization basis but also in the spatial path, since both degrees of freedom are entangled. The polarization-path-entangled state reflects the structure of the universal wavefunction, and no collapse occurs [20]. The entanglement is ontologically real and corresponds to a high-dimensional structure of quantum reality.

4.3. QBism

QBism interprets the quantum state as an agent’s personal degree of belief about the outcomes of measurements. Entanglement, in this view, is not a physical linkage between degrees of freedom but a reflection of the agent’s expectations about measurement correlations [21]. The polarization-path-entangled state thus encodes the betting odds an agent assigns to different joint measurement outcomes, and not a physical superposition in space-time.

4.4. Bohmian Mechanics

In Bohmian mechanics, the ontology includes both a guiding wavefunction and well-defined particle positions. For a polarization-path-entangled photon, the actual spatial trajectory (path) is determined by a hidden variable, while the polarization evolves under the influence of the wavefunction. The total wavefunction remains entangled, but an effective wavefunction for polarization can be defined as conditional on the actual path taken [22]. This interpretation supports a realist view, but with a subtle layering between the actual and effective dynamics.

4.5. Consistent Quantum Mechanics

Consistent quantum mechanics (CQM) offers a framework in which quantum events are described by “histories”, which are sequences of projectors at successive times (subject to a consistency condition that ensures that classical probability rules apply) [23,24]. In this approach, there is no universal wavefunction collapse; instead, different consistent sets (or frameworks) provide mutually exclusive but internally coherent descriptions of a system’s evolution. A polarization-path-entangled photon traversing a beam splitter can be analyzed within a framework that includes projectors onto both polarization and path states at various stages. If the chosen framework does not resolve interference between paths, coherence is preserved; if it does, interference terms vanish, and the resulting description resembles classical particle behavior. In CQM, the appearance or disappearance of entanglement across the beam splitter is not a dynamical “loss” but a matter of which consistent set is used, underscoring the interpretive relativity of statements about quantum correlations. This allows polarization-path entanglement to be discussed in terms of well-defined probabilistic histories without invoking measurement as a special process.

4.6. Epistemic vs. Ontic Entanglement

The central question these interpretations address differently is whether polarization-path entanglement is a real feature of nature (ontic), or merely a formal expression of limited knowledge (epistemic). MWI and Bohmian mechanics lean toward ontic interpretations, while Copenhagen and QBism treat entanglement as fundamentally epistemic.

5. Philosophical Implications

The polarization-path setup sharpens foundational tensions regarding distinguishability, information, and quantum ontology.

5.1. Distinguishability and the Flow of Information

As discussed earlier, tracing out the path degree of freedom leads to decoherence in the polarization subsystem, cf. Equation (8). This mathematical operation presupposes distinguishability: the ability, even in principle, to determine the path of the photon. Thus, the mere availability of which-path information, regardless of whether it is accessed, can destroy observable interference patterns [25]. This raises the question of whether quantum coherence is fundamentally relational and contingent on the structure of accessible information.

When we perform the partial trace over the path degree of freedom, the reduced state of the polarization changes from pure to mixed: its purity drops from 1 to $1/2$, and all off-diagonal coherence terms vanish. In terms of quantum information theory, this implies a loss of coherence and an increase in entropy within the polarization subsystem. However, this does not imply a fundamental loss of information in the full quantum system. Rather, the information initially present in the polarization-path correlations is now inaccessible when we consider only polarization. The coherence is not destroyed globally—it is simply no longer present in the reduced description.

In the theory of open quantum systems [26,27,28], such behavior is interpreted as a flow of quantum information from the system to an external environment, which stores the coherence and renders the subsystem effectively classical. In the present case, however, the environment is not an external reservoir but an internal degree of freedom—the path of the photon. The path and polarization together still form a pure entangled state, and the full system retains its total information content. The loss of coherence in the polarization arises only because we choose to ignore (or cannot access) the path.

This internal redistribution of information reveals that decoherence is not necessarily tied to dissipation or physical leakage of energy, but can also result from the internal structure of entanglement. The information lost from the polarization subsystem flows into the path degree of freedom in the form of quantum correlations. If access to the path is restored, the lost coherence can in principle be recovered, e.g., by performing quantum erasure [10,17]. This reinforces the idea that decoherence is observer-dependent and contingent on which parts of the system are treated as relevant or observable.

In this sense, quantum information is never destroyed but merely redistributed across the system’s degrees of freedom. The apparent loss is a result of coarse graining: a choice in how we partition the system and what we choose to trace out. Polarization-path entanglement thus serves as a vivid model for understanding how quantum information behaves under restricted access to degrees of freedom and offers insight into the nature of decoherence without invoking an external environment.

5.2. Reality of Entanglement

Is polarization-path entanglement a real, physical phenomenon or merely a bookkeeping tool? The answer depends largely on one’s interpretation of the quantum formalism—particularly whether the wavefunction is taken to represent an element of physical reality or simply a computational device for predicting measurement outcomes. In polarization-path entanglement, a single photon appears to be “spread” across two degrees of freedom, raising the question of whether this constitutes genuine entanglement or an artifact of labeling.

Arguments against the “reality” of such entanglement often stem from the intuition that genuine entanglement requires multiple particles. However, as emphasized by van Enk [12], this intuition is misleading. The formal structure of quantum entanglement pertains to modes or subsystems—whether spatially distinct or internally encoded—not to particle count. A single photon entangled between two spatial modes (or between path and polarization) can, under appropriate interactions (e.g., absorption into atoms located in separate cavities), give rise to entanglement between truly separate physical systems. This operational pathway supports the claim that such intra-particle entanglement is physically meaningful.

Moreover, polarization-path entanglement has observable consequences. When the paths are recombined in a quantum eraser setup, interference can be restored, implying that the coherence (and thus the entanglement) was never truly lost, only rendered inaccessible due to path distinguishability. This reversibility challenges the view that polarization-path entanglement is a mere artifact of description. Instead, it suggests that entanglement, even in single-particle contexts, manifests in the physical correlations between subsystems and is subject to the same resource-theoretic and operational criteria as multi-particle entanglement.

Van Enk also highlights an important caveat: entanglement between degrees of freedom can be operationally inaccessible if the necessary reference frames (e.g., shared phases or coordinate systems) are not aligned or well defined [12]. This reinforces the relational nature of quantum entanglement—its observability depends not only on the structure of the state, but also on the framework used by agents to extract information from it.

In sum, polarization-path entanglement should not be dismissed as a formal artifact. It meets all the formal criteria for entanglement, can be operationally transferred to independent systems, and gives rise to observable quantum effects. Its “reality” may depend on the interpretational lens through which it is viewed, but its physical relevance and utility are well established.

5.3. Ontic vs. Epistemic Mixed States

The reduced polarization state after tracing out the path is a mixed state. However, is this mixture the result of ignorance about the path (epistemic), or an actual physical decoherence (ontic)? The answer lies at the heart of the measurement problem. Epistemic interpretations struggle to explain why interference disappears, while ontic approaches (e.g., environmental decoherence) offer a mechanism for how coherent superpositions become classical mixtures [17].

5.4. Quantum Holism

Polarization-path entanglement exemplifies quantum holism: the total state is well defined and pure, but its subsystems lack individual definiteness. This suggests that subsystems of a quantum object may not possess local properties independently of the global state, challenging classical intuitions about separability and objectivity.

6. Practical Relevance and Experimental Realizations

Polarization-path entanglement is not only of philosophical interest; it has practical utility in quantum optics and quantum information.

6.1. Experimental Demonstrations

Experiments employing beam displacers, interferometers, and entanglement-preserving elements have demonstrated polarization-path entanglement with high visibility [15]. These setups often serve as analogues to Bell-type experiments, where entanglement between different degrees of freedom is verified through violation of inequalities [29].

A particularly compelling recent demonstration of the utility of polarization-path entanglement was reported in the context of quantum illumination [15]. In this experiment, single photons generated via SPDC were entangled in polarization and path degrees of freedom and used as probes in a target detection protocol. The protocol relied on heralded detection: one photon of the SPDC pair was retained as a timing reference (idler), while the signal photon, entangled internally, was directed along two distinct paths, one of which encountered a weakly reflecting object submerged in thermal noise. By evaluating the Clauser–Horne–Shimony–Holt (CHSH) inequality [5] for triple-path coincidence measurements, the presence of the object could be inferred even at very low signal-to-noise ratios (SNRs). Notably, the protocol operated effectively without requiring full interferometric reconstruction, highlighting the practical advantage of internal entanglement in noisy environments. Although some issues regarding the consistency of the theoretical framework have been noted [30], this experiment confirms that polarization-path-entangled states not only are of foundational interest but also hold promise for real-world quantum sensing applications [15].

6.2. Hyperentanglement and Quantum Information

By entangling photons simultaneously in multiple degrees of freedom (e.g., polarization, path, time-bin, orbital angular momentum), researchers achieve hyperentangled states. These states enable advanced protocols such as complete Bell-state analysis, dense coding, and enhanced quantum teleportation [31]. Polarization-path entanglement can be used as a component within such hybrid systems.

Quantum state tomography (QST) is a crucial experimental tool in quantum information theory [32,33,34]. It enables the reconstruction of a system’s density matrix through a series of measurements on identically prepared quantum systems. By combining projections in multiple bases and applying linear inversion or maximum likelihood estimation techniques, QST allows one to fully characterize both pure and mixed quantum states. This makes it possible to verify the presence of entanglement, quantify decoherence, and benchmark the performance of quantum devices.

A particularly relevant advance is presented in the recent work on “Quantum tomography for arbitrary single-photon polarization-path states” [16]. This study introduces a compact and experimentally viable method for reconstructing the full density matrix of polarization-path-entangled single-photon states. The method exploits standard linear optics and Stokes parameter measurements on both path modes, supplemented with a novel concept of quantum two-point Stokes parameters. Importantly, it demonstrates that such intra-photon-entangled states are not just abstract mathematical constructions—they can be fully reconstructed and verified within the framework of quantum theory. This supports the view that polarization-path entanglement is a physically accessible and experimentally controllable resource, with direct implications for the study of open-system dynamics and photonic quantum technologies.

6.3. Toward Higher-Dimensional Systems

Polarization-path schemes help bridge the gap between qubit-based and qudit-based quantum architectures. By using path as an ancillary system, one can simulate higher-dimensional entanglement, offering insights into the structure of Hilbert space and experimental access to more complex entangled states [35].

Recent research has explored the application of such higher-dimensional encoding schemes in QKD protocols. By employing multiple degrees of freedom simultaneously, such as polarization, path, time-bin, or orbital angular momentum, quantum information can be encoded into high-dimensional quantum states, or qudits. This multidimensional encoding not only enhances the information capacity per photon but also improves robustness against certain types of noise and eavesdropping strategies. High-dimensional QKD protocols promise increased secret key rates and better tolerance to loss and detector imperfections, offering a promising direction for practical and scalable quantum communication networks [36,37].

7. Polarization–Path–Temporal Entanglement

7.1. Conceptual Framework and GHZ Analogy

Beyond two-degrees-of-freedom entanglement, a single photon can exhibit entanglement across three distinct degrees of freedom: polarization, spatial path, and temporal mode. This gives rise to a richer structure of intra-particle correlations, formally analogous to Greenberger–Horne–Zeilinger (GHZ) states [38,39].

To engineer such a state, one can begin with a photon in a diagonal polarization state $|D\rangle =(|H\rangle +|V\rangle )/\sqrt{2}$ and send it through a PBS. The PBS transmits horizontal polarization into one path ($|0\rangle$) and reflects vertical polarization into another path ($|1\rangle$). This generates polarization-path entanglement ${|\psi \rangle }_{\mathrm{pol}-\mathrm{path}}$ as in Equation (5).

Next, a temporal delay is introduced between the two spatial paths, for example, by extending the optical path length in one arm using a delay line. Assuming that the delay is greater than the photon’s coherence time, this introduces orthogonal time-bin states $|t_0\rangle$ and $|t_1\rangle$ corresponding to early and late arrival times. The final state becomes

$${|\psi \rangle }_{\mathrm{GHZ}-\mathrm{like}}={\displaystyle \frac{1}{\sqrt{2}}}\left(|H\rangle \otimes |0\rangle \otimes |t_0\rangle +|V\rangle \otimes |1\rangle \otimes |t_1\rangle \right).$$

(10)

In Figure 1, we present a conceptual scheme that shows the generation method for polarization–path–temporal entanglement. A more formal description can be given by associating each degree of freedom with a separate Hilbert space, as follows: ${\mathcal{H}}_{\mathrm{pol}}\otimes {\mathcal{H}}_{\mathrm{path}}\otimes {\mathcal{H}}_{\mathrm{time}}$, where each subspace is spanned by an orthonormal qubit basis, as follows:

$$\begin{array}{cc}\hfill & {\mathcal{H}}_{\mathrm{pol}}=\mathrm{span}\{|H\rangle ,|V\rangle \},\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill & {\mathcal{H}}_{\mathrm{path}}=\mathrm{span}\{|0\rangle ,|1\rangle \},\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill & {\mathcal{H}}_{\mathrm{time}}=\mathrm{span}\{|t_0\rangle ,|t_1\rangle \}.\hfill \end{array}$$

(13)

The PBS operation can be described by the mapping:

$$\begin{array}{cc}\hfill & |H\rangle \otimes |\mathrm{in}\rangle \mapsto |H\rangle \otimes |0\rangle ,\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill & |V\rangle \otimes |\mathrm{in}\rangle \mapsto |V\rangle \otimes |1\rangle .\hfill \end{array}$$

(15)

The delay line acts only on the ${\mathcal{H}}_{\mathrm{path}}\otimes {\mathcal{H}}_{\mathrm{time}}$ subsystem, applying the transformation, as follows:

$$\begin{array}{cc}\hfill & |0\rangle \otimes |t_0\rangle \mapsto |0\rangle \otimes |t_0\rangle ,\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill & |1\rangle \otimes |t_0\rangle \mapsto |1\rangle \otimes |t_1\rangle .\hfill \end{array}$$

(17)

Applying these operations to the initial state $|D\rangle \otimes |t_0\rangle$ yields the GHZ-like state in Equation (10). This representation makes it explicit that the final state is maximally entangled across all three subsystems, with reduced density matrices of each qubit being completely mixed, i.e., ${\rho }_{\mathrm{pol}}={\rho }_{\mathrm{path}}={\rho }_{\mathrm{time}}={\mathbb{I}}_2/2$.

The state Equation (10) is a genuine three-way entangled state in a single particle and structurally resembles the canonical GHZ state, as follows:

$$|\mathrm{GHZ}\rangle ={\displaystyle \frac{1}{\sqrt{2}}}\left(|000\rangle +|111\rangle \right).$$

(18)

Each “qubit” in Equation (10) corresponds to a different degree of freedom: polarization, path, and time. While such states do not involve multipartite entanglement across distinct particles, they realize a multipartite structure within the tensor product of internal Hilbert spaces of a single photon.

7.2. Implications and Applications

Entanglement across multiple internal degrees of freedom opens up new directions for compact quantum information encoding. Polarization–path–temporal states provide three qubits of logical encoding per photon, making them a promising resource for the following:

• Quantum communication with reduced photon flux;
• Multi-qubit quantum gates and quantum logic with minimal resources;
• Tests of contextuality and the compatibility of local measurements across different degrees of freedom;
• Advanced quantum sensing protocols using entanglement of multiple degrees of freedom to enhance information capacity and noise resilience.

Furthermore, such states can serve as testbeds for exploring entanglement monogamy and generalized entropic inequalities in composite quantum systems.

7.3. Interference Visibility and Temporal Indistinguishability

The observation of coherence in polarization–path–temporal-entangled states depends critically on maintaining indistinguishability across all three degrees of freedom. The superposition in Equation (10) is coherent only if the following are true:

• The spatial modes $|0\rangle$ and $|1\rangle$ are indistinguishable aside from path labels.
• The temporal modes $|t_0\rangle$ and $|t_1\rangle$ are well separated yet well defined.
• Polarization states are preserved throughout the interferometric delay lines.

Temporal jitter (random fluctuations in the photon’s detection time) acts as a source of decoherence. If the jitter is on the order of the time-bin separation $\mathrm{\Delta }t=|t_1-t_0|$, then time bins are no longer distinguishable, and interference visibility is lost. This degrades the entangled state’s purity and can make tomographic reconstruction inaccurate.

In practical implementations, achieving full state tomography of polarization–path–temporal states requires ultrafast detection systems with sub-nanosecond timing resolution and stable interferometric setups. Nevertheless, recent advances in quantum tomography techniques, such as the results on time-resolved measurements [40,41] or models for polarization-path tomography [16], demonstrate that these states not merely are theoretical constructs but are experimentally accessible and measurable with high fidelity.

8. Conclusions

Polarization-path entanglement serves as a particularly accessible and instructive case study for exploring deep questions in quantum theory. While often overshadowed by two-particle entanglement, intra-particle entanglement between internal degrees of freedom reveals the same formal richness and interpretational ambiguity. We have examined the mathematical structure of such states, including the role of the tensor product, entanglement criteria, and consequences of tracing out one subsystem, which leads to decoherence and information loss. These formal tools clarify the mechanisms through which coherence is destroyed—not through active measurement, but merely through the availability of distinguishing information.

From an interpretational standpoint, polarization–path entanglement highlights the persistent tensions in quantum foundations. The Copenhagen, MWI, QBism, Bohmian interpretations and CQM offer diverging ontological accounts of what it means for a photon to be in an entangled state across internal degrees of freedom. Whether entanglement reflects physical reality or a knowledge-based abstraction remains an open question, especially in single-photon contexts where the very notion of nonlocality becomes subtle. The analysis was enriched by insights from van Enk’s argument that such single-particle entanglement is operationally meaningful when it can be transferred to systems like atoms via local interactions [12].

On the practical side, polarization-path-entangled states have moved well beyond abstraction. They have been demonstrated experimentally using beam displacers, interferometers, and entanglement-preserving elements. Notably, they have been employed in quantum illumination protocols for object detection in the presence of strong thermal noise, showing resilience where traditional entangled photon-pair strategies would fail [15]. Their role in quantum technologies extends further: they can serve as building blocks in hyperentangled systems, enabling advanced protocols like dense coding, full Bell-state discrimination, and high-dimensional QKD protocols.

Importantly, QST techniques now allow for complete reconstruction of the density matrix of polarization-path states, affirming that such states are not merely theoretical constructs but physically well-defined objects within quantum theory [16]. This aligns with ongoing efforts to harness multiple-degrees-of-freedom encoding to increase efficiency in quantum communication and metrology.

In conclusion, polarization-path entanglement is both a powerful resource and a conceptual probe. It illuminates the boundaries between coherence and decoherence, objectivity and relationality, and operational tools and ontological commitments. Open questions remain—about the ultimate status of entanglement, the relationship between distinguishability and measurement, and the extent to which internal entanglement can reveal truly nonclassical features of quantum systems. As both a laboratory for ideas and a platform for innovation, polarization-path entanglement occupies a central place in the study of quantum foundations and emerging quantum technologies.