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arXiv:quant-ph/0703243 · Physical Review A

Ensemble-averaged entanglement of two-particle states in Fock space

Jan Naudts, Tobias G. W. Verhulst

Published Jun 12, 2007• 45 citations
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Abstract

Recent results, extending the Schmidt decomposition theorem to wave functions of pairs of identical particles, are reviewed. They are used to give a definition of reduced density operators in the case of two identical particles. Next, a method is discussed to calculate time averaged entanglement. It is applied to a pair of identical electrons in an otherwise empty band, and to a pair of bosons in a quadratic model with infinite range hopping. The effect of degeneracy of the spectrum of the Hamiltonian on the average entanglement is emphasized.

DOI
10.1103/physreva.75.062104

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We relate the notion of entanglement for quantum systems composed of two identical constituents to the impossibility of attributing a complete set of properties to both particles. This implies definite constraints on the mathematical form of the state vector associated with the whole system. We then analyze separately the cases of fermion and boson systems, and we show how the consideration of both the Slater-Schmidt number of the fermionic and bosonic analog of the Schmidt decomposition of the global state vector and the von Neumann entropy of the one-particle reduced density operators can supply us with a consistent criterion for detecting entanglement. In particular, the consideration of the von Neumann entropy is particularly useful in deciding whether the correlations of the considered states are simply due to the indistinguishability of the particles involved or are a genuine manifestation of the entanglement. The treatment leads to a full clarification of the subtle aspects of entanglement of two identical constituents which have been a source of embarrassment and of serious misunderstandings in the recent literature.
We review two general criteria for deciding whether a pure bipartite quantum state describing a system of two identical particles is entangled or not. The first one considers the possibility of attributing a complete set of objective properties to each particle belonging to the composed system, while the second is based both on the consideration of the Slater-Schmidt number of the fermionic and bosonic analog of the Schmidt decomposition and on the evaluation of the von Neumann entropy of the one-particle reduced statistical operators. 1
We review two general criteria for deciding whether a pure bipartite quantum state describing a system of two identical particles is entangled or not. The first one considers the possibility of attributing a complete set of objective properties to each particle belonging to the composed system, while the second is based both on the consideration of the Slater-Schmidt number of the fermionic and bosonic analogue of the Schmidt decomposition and on the evaluation of the von Neumann entropy of the one-particle reduced statistical operators.
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.