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Journal of the Optical Society of America A

Coherence, entanglement, and complementarity in mixed classical light

Asma Al-Qasimi

Published Aug 13, 2020• 7 citations
Publisher

Abstract

The classical analogues of quantum correlations have been the subject of considerable interest in the past few years; however, all of these investigations consider the classical analogue of only pure quantum mechanical states. In this work, we study mixed classical light states and derive relations between the polarization coherence of the field and the (classical) entanglement between its degrees of freedom. We show, for a specific model of mixed states, where the purity is determined by a single parameter, how the coherence shared between polarization and entanglement shrinks as the level of purity decreases. The sum of the square of polarization and entanglement remains constant, a behavior consistent with previous results [Phys. Rev. Lett.117, 153901 (2016)PRLTAO0031-900710.1103/PhysRevLett.117.153901] of pure states, even though the constant for the mixed-state case is now smaller in value.

DOI
10.1364/josaa.395172

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The classical analogues of quantum correlations have been the subject of considerable interest in the past few years; however, all of these investigations consider the classical analogue of only pure quantum mechanical states. In this work, we study mixed classical light states and derive relations between the polarization coherence of the field and the (classical) entanglement between its degrees of freedom. We show, for a specific model of mixed states, where the purity is determined by a single parameter, how the coherence shared between polarization and entanglement shrinks as the level of purity decreases. The sum of the square of polarization and entanglement remains constant, a behavior consistent with previous results [Phys. Rev. Lett.117, 153901 (2016)PRLTAO0031-900710.1103/PhysRevLett.117.153901] of pure states, even though the constant for the mixed-state case is now smaller in value.
Entanglement and polarization are mutually constrained by the relationship C2+P2=1{C}^{2}+{P}^{2}=1C2+P2=1, which engages the concurrence CCC of a pure, two-qubit state and the degree of polarization PPP of either of its two subsystems. How the above constraint generalizes for mixed states is an open question. We address mixed, two-qubit states of the XXX type, i.e., those whose density matrix has nonzero elements only in the two main diagonals. We focus on mixed states that arise out of a pure, two-qubit state that is subjected to either the amplitude damping channel or the depolarizing channel. We derive alternative constraints for concurrence and polarization and test them experimentally with polarization-entangled photons. We argue that our theoretical results hold also for classical light, whenever two binary degrees of freedom can be entangled.
We analyze features of mixed biphoton polarization states, which arise from pure states of polarization-frequency biphoton ququarts after averaging over frequencies of photons. For mixed states, we find their concurrence CCC, Schmidt parameter KKK, degree of polarization PPP, as well as the von Neumann mutual information III. In some simple cases, we also find the relative entropy Srel{S}_{\mathrm{rel}}Srel​ and the degree of classical correlations Ccl{C}_{\mathrm{cl}}Ccl​. In mixed states, the Schmidt parameter does not characterize the degree of entanglement anymore, as it does in pure states. Nevertheless, the Schmidt parameter remains useful even in the case of mixed states because it remains directly related to the degree of polarization. We compare results occurring in the cases of full pure polarization-frequency states of ququarts and mixed states (averaged over frequencies). Differences between these results can be seen in experiments with and without frequency filters in front of a detector.
Quantum polarization is an important property that has been extensively used for quantum communication purposes. In this work, the mixture of an unpolarized two-mode state and a two-mode pure state is analysed. This mixture is controlled by a single parameter, namely the degree of purity. The quantum polarization of the two-mode mixed state is discussed using the quantum Stokes parameters and the quantum degree of polarization. A relation between the degree of purity and the quantum degree of polarization is established. The mixture is used to model coherent light amplified by an erbium-doped fibre amplifier, which focuses the loss of polarization due to the unpolarized light resulting from spontaneous emission. Finally, the use of such a mixed state in the quantum key distribution using multiphoton pulses is analysed.
We analyze features of mixed biphoton polarization states which arise from pure states of polarization-frequency biphoton ququarts after averaging over frequencies of photons. For mixed states we find their concurrence C, Schmidt parameter K and degree of polarization P, as well as the von Neumann mutual information I. In some simple cases we find also the relative entropy S_{rel} and the degree of classical correlations C_{cl}. We show that in mixed states the Schmidt parameter does not characterize anymore the degree of entanglement, as it does in pure states. Nevertheless, the Schmidt parameter remains useful even in the case of mixed states because it remains directly related to the degree of polarization. We compare results occurring in the cases of full pure polarization-frequency states of ququarts, mixed states (averaged over frequencies) and states with separated high- and low-frequency parts. Differences between these results can be seen in experiments with and without a dichroic beam-splitter, as well as with and without frequency filters in front of detector.
The principle of complementarity relationship introduced by Niels Bohr in 1927 states that particle and wave behavior are two mutually exclusive properties for a quantum system.This relationship has taken the attention of researchers for decades due to its conceptual importance.This thesis contains a theoretical research work conducted on quantum complementarity of vectorial light fields and in the doublepinhole interference.Here we have adopted the classical Stokes parameters within a quantum framework and calculated the Stokes parameters in the observation plane in double-pinhole interference.Based on these calculations, we further evaluated visibility and distinguishability parameters for an arbitrary pure single-photon state and for a specific single-photon pure state, where the photon is x polarized at the first pinhole and y polarized at the second pinhole.We have then examined and interpreted the wave-particle duality for the mentioned two states.Finally, we performed mixed state calculation where fully unpolarized photon is incident on the two pinholes, but with the photon at one pinhole rotated by an angle of π/2.From the obtained results we concluded that a priori which path information (WPI) regarding the complementarity relations for the single photon pure vectorial quantum state couples not to the intensity visibility but to the total visibility.We further found that the vectorial quantum state gives rise to polarization modulation.From the results obtained for the mixed state calculations, we discovered that the last Stokes parameter has a non-zero value and it contributes to the polarization modulation.Thereby we further recommended to repeat the calculation for different mixed states other than what we have considered here to extend the exploration about the nature of polarization modulation.
The nonclassicality of a macroscopic single-mode optical superposition state is potentially convertible into entanglement, when the state is mixed with the vacuum on a beam splitter. Considering light beams with polarization degree of freedom in Euclidean space as coherent product states in a bipartite Hilbert space, we propose a method to convert the two orthogonal polarizations into simultaneous entanglement and classical nonseparability through nonclassicality in the superpositions of coherent and displaced Fock states. Equivalent Bell state emerges from the resulted superpositions and the proportion of mixed entanglement and nonseparablity is determined by the displacement amplitudes along the polarization directions. We characterize the state nonclassicality via features in Wigner distributions and propose an experimental method for generating these states and measuring them via homodyne tomography.