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Laser Physics

Entanglement of multiphoton states in polarization and quadrature variables

Mikhail Vladimirovich Fedorov

Published Oct 24, 2019• 6 citations
Publisher

Abstract

Abstract A brief overview of entanglement characterization is given in the cases of biphoton and multiphoton polarization states. The importance of the approach is stressed based on the Schmidt decomposition. This method is applied to the characterization of the quadrature entanglement in two-mode multiphoton states. The fruitfulness of this approach is illustrated by a series of examples.

DOI
10.1088/1555-6611/ab4bc5

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•Jan 1, 2013•10 citations
Abstract A brief overview of entanglement characterization is given in the cases of biphoton and multiphoton polarization states. The importance of the approach is stressed based on the Schmidt decomposition. This method is applied to the characterization of the quadrature entanglement in two-mode multiphoton states. The fruitfulness of this approach is illustrated by a series of examples.
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.
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.
The purpose of this paper is to study entanglement of quantum states by means of Schmidt decomposition. The notion of Schmidt information which characterizes the non-randomness of correlations between two observers that conduct measurements of EPR-states is proposed. In two important particular cases - a finite number of Schmidt modes with equal probabilities and Gaussian correlations- Schmidt information is equal to Shannon information. A universal measure of a dependence of two variables is proposed. It is based on Schmidt number and it generalizes the classical Pearson correlation coefficient. It is demonstrated that the analytical model obtained can be applied to testing the numerical algorithm of Schmidt modes extraction. A thermodynamic interpretation of Schmidt information is given. It describes the level of entanglement and correlations of micro-system with its environment
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.