Wave-particle duality of unpolarized photons

Abstract: Photons in a two-path interferometer best embody wave-particle duality (WPD), which is a core concept of quantum theory. So far, the WPD relation is commonly written as $V^2+D^2 \leq 1$, where $V$ is the interference fringe visibility and $D$ is path distinguishability, i.e., the distinguishability of which path a photon passed. This inequality is saturated only when the which-way marker (WWM), which embodies which-path information (WPI) via an internal degree of freedom of photons, such as polarization, is in a pure state. For mixed-state WWM, conventionally defined distinguishability underestimates the amount of WPI and thus does not saturate the WPD relation. Here, we introduce a generalized measure of distinguishability $D$ that properly quantifies the WPI and saturates the WPD relation for all pure- and mixed-state WWM within a purification-based framework. To this end, mixed-state WWM is treated as a result of entanglement formation between the WWM and an external degree of freedom, e.g., environment, and $D$ is defined so that it incorporates the total WPI shared between the WWM and the environment. We show that $D$ thus defined is experimentally quantifiable, independently of $V$, without access to the environment. We experimentally evaluate $V$ and $D$ using true single photons generated in the completely mixed (unpolarized) state, and thus verify the saturated WPD relation.

• The paper proposes a purification-based generalization of path distinguishability that accurately quantifies which-way information in mixed-state WWMs.
• It experimentally verifies a saturated duality relation (V² + D² = 1) using unpolarized photons in a Michelson interferometer setup.
• The study links the generalized distinguishability to entanglement metrics like concurrence, enhancing the understanding of quantum measurement in noisy environments.

Wave-Particle Duality for Unpolarized Photons: Purification-Based Quantification

Background and Motivation

This paper interrogates the validity and completeness of the widely used wave-particle duality (WPD) relation in quantum optics, which relates interference fringe visibility ($V$) and path distinguishability ($D$) through the inequality $V^2+D^2 \leq 1$. Conventionally, $D$ is quantified as half the trace distance between which-way marker (WWM) states—typically an internal degree of freedom such as polarization. However, this measure saturates the inequality only when the WWM is in a pure state and fails for mixed states, notably for completely unpolarized photons. The conventional $D$ thus underestimates the total accessible which-path information (WPI), yielding $D=0$ for the unpolarized case, despite demonstrable effects on interference. Addressing this deficiency, the authors seek a generalization of $D$ to correctly encapsulate WPI for photons with mixed-state WWMs, rooted in a purification framework.

Generalized Distinguishability via Purification

The authors extend the measurement of path distinguishability $D$ to include the entirety of WPI, both present within the WWM itself and that which is shared with any external degrees of freedom (such as the environment) via entanglement. Mixed-state WWMs are thus treated as reduced states from pure entangled states of a larger system. The general purified state for the two-path interferometer, entangled with system $R$ (e.g., environment), is expressed as:

$$\ket{\Psi}_{QR} = c_0 \ket{0}_Q\otimes\ket{\psi_0}_R + c_1 \ket{1}_Q\otimes\ket{\psi_1}_R,$$

with the corresponding reduced density matrices for the WWM obtained via partial trace.

For pure-state WWMs, $D$0 and $D$1 satisfy the saturated relation $D$2. In the mixed-state scenario, the conventional distinguishability $D$3 does not account for information distributed in the external system, resulting in $D$4 for mixed states. The authors propose a purification-based generalized $D$5 obtained via maximization over the possible decompositions and measurement bases, thereby restoring $D$6 universally. Notably, this $D$7 is experimentally accessible through optimal decomposition of the mixed state into pure components, circumventing the need for direct environmental access.

Entanglement, Concurrence, and Duality Relations

The paper rigorously explores the relationship between distinguishability, entanglement (concurrence $D$8), and predictability. The generalized duality and triality relations are formulated:

• $D$9 (visibility, path predictability, entanglement tripartite relation)
• In the case of equal path amplitudes (no predictability), $V^2+D^2 \leq 1$0 and $V^2+D^2 \leq 1$1, equating $V^2+D^2 \leq 1$2 with $V^2+D^2 \leq 1$3 in properly purified systems.

This connects WPD directly to measures of entanglement, providing a unified approach to quantifying all relevant which-way information, even in realistic experimental conditions with uncontrolled environmental degrees of freedom.

Experimental Demonstration with Unpolarized Photons

The authors implement a Michelson interferometer using true single photons emitted from a nitrogen-vacancy (NV) center in diamond, naturally unpolarized due to thermal mixing of excited states. Polarization acts as the WWM, with WPI manipulated by quarter-wave plates in each path.

Key Experimental Steps:

• Characterization: The photon source demonstrates nearly perfect unpolarized behavior in both static (density matrix, Stokes vector) and dynamic (second-order correlation, polarization correlation function) terms.
• Interference and Quantum Erasure: In absence of WPI, interference fringes appear with maximal visibility. Introduction of maximal WPI (orthogonal polarization rotations) destroys interference, even though the polarization density matrix remains unchanged. Quantum erasure—detection in an appropriate circular polarization basis—restores interference, underlining the operational reality of WPI in mixed-state WWMs.
• Measurement of Generalized $V^2+D^2 \leq 1$4: The experiment splits the unpolarized photon ensemble into pure linear polarizations ($V^2+D^2 \leq 1$5, $V^2+D^2 \leq 1$6), quantifies distinguishability per decomposition, and averages weighted by the decomposition coefficients, as prescribed by the generalized measure. Optimal measurement bases are selected to maximize the likelihood for which-way determination, compatible with Helstrom bounds for state discrimination.

Numerical Results:

• Visibility $V^2+D^2 \leq 1$7 varies with WPI manipulation, decreasing as distinguishability increases.
• Generalized $V^2+D^2 \leq 1$8 achieves maximum at full WPI, even for completely unpolarized input.
• Saturated Relation: $V^2+D^2 \leq 1$9 remains close to unity across all tested settings, both pure and mixed-state WWMs, in accordance with theoretical predictions and distinct from the conventional $D$0.
• Quantum Erasure: Experimental data confirm revival and destruction of interference solely through measurement basis changes, establishing equivalence with quantum erasure protocols for pure-state WWMs.

Theoretical and Practical Implications

Theoretical Implications

This work establishes that the wave-particle duality relation can be universally formulated (and experimentally verified) for both pure and mixed-state WWMs by adopting a purification-based distinguishability measure. It confirms that conventional measures fail to account for entanglement-induced WPI shared with the environment, which is crucial in realistic quantum systems. The generalized $D$1 is shown to be equivalent to concurrence in the purified setting, linking duality directly with entanglement theory in a quantum optical context.

Practical Implications

The generalized $D$2 is experimentally accessible without direct environmental probing, simply requiring optimal decomposition and measurement basis selection—a significant result for practical quantum optics, quantum information, and foundational interference studies. Wave-particle duality tests and complementarity analyses can now be performed in settings where WWMs are unavoidably mixed, i.e., in true single-photon sources, noisy or decohered environments, and when polarization is thermally randomized or otherwise uncontrolled.

Speculation on Future Developments

The purification-based approach and quantification of WPI in mixed-state WWMs may be extended to higher-dimensional systems, multipartite entanglement, and more complex interferometric setups. The framework may inform studies of multi-path duality, generalized coherence theorems, and investigations into decoherence, environmental entanglement, and quantum measurement theory. Further research may explore computational algorithms for finding optimal decompositions in high-dimensional spaces, and implications for quantum technologies where mixed-state markers are ubiquitous.

Conclusion

The paper rigorously addresses the deficiency in conventional path distinguishability measures for mixed-state WWMs (notably unpolarized photons), proposing and experimentally validating a purification-based generalization. This enables a universal and saturated wave-particle duality relation, $D$3, irrespective of the purity of the WWM and provides experimental procedures for its quantification. Its findings integrate duality, entanglement, and quantum measurement, setting the stage for further theoretical exploration and practical implementation in quantum optics and quantum information science.

Reference: "Wave-particle duality of unpolarized photons" (2605.25738)