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arXiv:1404.1496 · Laser Physics

Three-photon polarization ququarts: polarization, entanglement and Schmidt decompositions

Mikhail Vladimirovich Fedorov, Nikolai Miklin

Published Feb 12, 2015• 6 citations
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Abstract

We consider polarization states of three photons, propagating collinearly and having equal given frequencies but with arbitrary distributed horizontal or vertical polarizations of photons. A general form of such states is a superposition of four basic three-photon polarization modes, to be referred to as the three-photon polarization ququarts (TPPQ). All such states can be considered as consisting of one- and two-photon parts, which can be entangled with each other. The degrees of entanglement and polarization, as well as the Schmidt decomposition and Stokes vectors of TPPQ are found and discussed.

DOI
10.1088/1054-660x/25/3/035204

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We consider polarization states of three photons, propagating collinearly and having equal given frequencies but with arbitrary distributed horizontal or vertical polarizations of photons. A general form of such states is a superposition of four basic three-photon polarization modes, to be referred to as the three-photon polarization ququarts (TPPQ). All such states can be considered as consisting of one- and two-photon parts, which can be entangled with each other. The degrees of entanglement and polarization, as well as the Schmidt decomposition and Stokes vectors of TPPQ are found and discussed.
In a general form, three-photon polarization states are superpositions of contributions from four three-photon polarization modes with four arbitrary complex coefficients. Entanglement of such states is defined as entanglement between their one-photon and two-photon parts. The degree of entanglement is found explicitly in terms of such entanglement quantifiers as the generalized concurrence, entropy of reduced states and the Schmidt parameter. Polarization Stokes vectors, as well as the Schmidt decompositions of three-photon states are found and discussed. Similarity and differences with respect to biphoton polarization states are discussed too.
For a general-form polarization biphoton qutrit, physically corresponding to a pair of arbitrarily polarized photons in a single frequency and wavevector mode, we explicitly find polarization Schmidt modes. A simple method is suggested for factorizing the state vector and the explicit expressions for the factorizing photon creation operators are found. The degrees of entanglement and polarization of a qutrit are shown to depend directly on the commutation features of the factorizing operators. Clear graphic representations for the Stokes vectors of the qutrit state as a whole, its Schmidt modes and factorizing single-photon creation operators are given based on the Poincaré sphere. An experimental scheme is proposed for measuring the parameters of the Schmidt decomposition as well as for demonstrating the operational meaning of qutrit entanglement.
For a general-form polarization biphoton qutrit, physically corresponding to a pair of arbitrarily polarized photons in a single frequency and wavevector mode, we explicitly find polarization Schmidt modes. A simple method is suggested for factorizing the state vector and the explicit expressions for the factorizing photon creation operators are found. The degrees of entanglement and polarization of a qutrit are shown to depend directly on the commutation features of the factorizing operators. Clear graphic representations for the Stokes vectors of the qutrit state as a whole, its Schmidt modes and factorizing single-photon creation operators are given based on the Poincare sphere. An experimental scheme is proposed for measuring the parameters of the Schmidt decomposition as well as for demonstrating the operational meaning of qutrit entanglement.
We investigate in a general form entanglement of biphoton qutrits and ququarts, i.e. states formed in the processes of collinear and, correspondingly, degenerate and non-degenerate Spontaneous Parametric Down-Conversion. Indistinguishability of photons and, for ququarts, joint presence of the frequency and and polarization entanglement are fully taken into account. In the case of qutrits the most general 3-parametric families of maximally entangled and non-entangled states are found, and anti-correlation of the degree of entanglement and polarization is shown to occur and to be characterized by a rather simple formula. Biphoton ququarts are shown to be two-qudits with the single-photon Hilbert space dimensionality d=4d=4d=4, which differs them significantly from the often used two-qubit model (d=2d=2d=2). New expressions for entanglement quantifiers of biphoton ququarts are derived and discussed. Rather simple procedures for a direct measurement of the degree of entanglement are described for both qutrits and ququarts.
We investigate, in a general form, entanglement of biphoton qutrits and ququarts, i.e. states formed in the processes of, correspondingly, degenerate and non-degenerate spontaneous parametric down-conversion. Indistinguishability of photons and, for ququarts, joint presence of the frequency and polarization entanglement are fully taken into account. In the case of qutrits, the most general three-parametric families of maximally entangled and non-entangled states are found, and anticorrelation of the degree of entanglement and polarization is shown to occur and to be characterized by a rather simple formula. Biphoton ququarts are shown to be two-qudits with the single-photon Hilbert space dimensionality d =4, which differentiates them significantly from the often used two-qubit model ( d =2). New expressions for entanglement quantifiers of biphoton ququarts are derived and discussed. Rather simple procedures for a direct measurement of the degree of entanglement are described for both qutrits and ququarts.
Motivated by their frequent use in quantum mechanical studies of entanglement, we give a brief overview of Schmidt modes and Schmidt decompositions of two-particle wave functions. We discuss methods of their derivation and include a little-known approach used in the original work by E. Schmidt [Math. Annalen, 63 (1906), 433]. This employs the bipartite wave function itself rather than the more complicated two-party reduced density matrix. As an illustration, Schmidt modes for two-photon polarisation qutrits are derived in a general form. The derivation is accompanied by a series of simple examples with special choices of parameters. Relationships between Schmidt modes, polarisation Stokes vectors and entanglement are also discussed.