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Physical Review A

Entangled state for constructing a generalized phase-space representation and its statistical behavior

Liyun Hu, Hong-Yi Fan

Published Aug 19, 2009• 15 citations
Publisher

Abstract

We construct a generalized phase-space representation (GPSR) based on the idea of Einstein-Podolsky-Rosen quantum entanglement, i.e., we generalize the Torres-Vega-Frederick phase-space representation to the entangled case, which is characteristic of the features when two particles' relative coordinate, total momentum operators, and their conjugative variables, respectively, operate on the GPSR. This representation is complete and nonorthogonal. The Weyl-ordered form of the density operator of GPSR is derived, and its identification with the generalized Husimi operator is recognized, which clearly exhibit its statistical behavior. The minimum uncertainty relation obeyed by the GPSR is also demonstrated.

DOI
10.1103/physreva.80.022115

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We construct a generalized phase-space representation (GPSR) based on the idea of Einstein-Podolsky-Rosen quantum entanglement, i.e., we generalize the Torres-Vega-Frederick phase-space representation to the entangled case, which is characteristic of the features when two particles' relative coordinate, total momentum operators, and their conjugative variables, respectively, operate on the GPSR. This representation is complete and nonorthogonal. The Weyl-ordered form of the density operator of GPSR is derived, and its identification with the generalized Husimi operator is recognized, which clearly exhibit its statistical behavior. The minimum uncertainty relation obeyed by the GPSR is also demonstrated.
Our primary purpose of this work is to explicitly construct the general multipartite Einstein–Podolsky–Rosen (EPR) entangled state in multi-mode Fock space for a system with different masses of particles, which makes up a new quantum mechanical representation owing to completeness relation and orthogonal property. Its entanglement can be seen more clearly by analyzing its standard Schmidt decomposition. In addition, some applications of the multipartite entanglement are proposed including deriving the generalized Wigner operator and squeezing operator.
By virtue of the Einstein–Podolsky–Rosen entangled state, which is the common eigenvector of two particles' relative coordinate and total momentum, we establish the bosonic operator version of the Hamiltonian for a quantum point-mass pendulum. The Hamiltonian displays the correct Schrodinger equation in the entangled state representation. The corresponding Heisenberg operator equations which predict the angular momentum–angle uncertainty relation are derived. The quantum operator description of two quantum pendulums coupled by a spring is also derived.
By virtue of the Einstein-Podolsky-Rosen entangled state, which is the common eigenvector of two panicles' relative coordinate and total momentum, we establish the bosonic operator version of the Hamiltonian for a quantum point-mass pendulum. The Hamiltonian displays the correct Schroedlnger equation in the entangled state representation.The corresponding Heisenberg operator equations which predict the angular momentum-angle uncertainty relation are derived. The quantum operator description of two quantum pendulums coupled by a spring is also derived.
We find the explicit state vector for Torres-Vega-Frederick phase space representation [Go. Torres-Vega and J. H. Frederick, J. Chem. Phys. 98, 3103 (1993)], denoted by Gamma. This set of states make up a complete and nonorthogonal representation. The Weyl ordered form of Gamma Gamma [see text for the sign] is derived, which can clearly exhibit the statistical behavior of marginal distribution of Gamma Gamma [see text for the sign]. The minimum uncertainty relation for mid R:Gamma is demonstrated, which shows it being a coherent squeezed state.
The Einstein–Podolsky–Rosen (EPR) wavefunction is a two-particle wavefunction that is a simultaneous eigenfunction of the relative coordinate and the centre-of-mass momentum. The Fock space representation of this function is easily obtained as a direct product of an eigenfunction of the relative coordinate and an eigenfunction of the centre-of-mass momentum. Straightforward generalization to an arbitrary number of particles is shown to be feasible, and a classification of multiparticle entangled states is outlined. The relation between entanglement and inseparability is pointed out, and the status of the EPR states within the much broader family of entangled states is emphasized.