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arXiv:0907.0386

Effective separability of typical entangled many-body states

S. Camalet

Published Jul 2, 2009• 0 citations
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Abstract

We consider two systems of harmonically trapped particles in a typical pure state of the Hilbert space defined by given values of the particle numbers and energies of the two gases. Such a state is entangled but we show that, for large systems, the resulting correlations between the two gases are identical to those of a separable mixture. This result can be generalized to other physical systems. We discuss the relation of this effective separability to the well-known existence of quantum correlations in any entangled state. We study in detail a small bipartite system and find that its correlations are well explained by the large systems results.

DOI
10.48550/arxiv.0907.0386

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We consider two systems of harmonically trapped particles in a typical pure state of the Hilbert space defined by given values of the particle numbers and energies of the two gases. Such a state is entangled but we show that, for large systems, the resulting correlations between the two gases are identical to those of a separable mixture. This result can be generalized to other physical systems. We discuss the relation of this effective separability to the well-known existence of quantum correlations in any entangled state. We study in detail a small bipartite system and find that its correlations are well explained by the large systems results.
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We discuss the question of entanglement versus separability of pure quantum states in direct product Hilbert spaces, and the relevance of the issue to physics. Different types of separability may be possible, depending on the particular factorization of the Hilbert space. A given orthonormal basis set for a Hilbert space is defined to be of type ( p , q ) if p elements of the basis are entangled and q are separable, relative to a given bi-partite factorization of that space. We conjecture that not all basis types exist for a given Hilbert space.
Unlike for bipartite states consisting of distinguishable particles, in the case of identical parties the notion of entanglement is still under debate. In the following, we review two different approaches to the entanglement of systems consisting of two bosons or fermions; the first approach is based on the particle aspect typical of first quantization and identifies separable pure states as those that allow to assign two orthogonal single particle vector states to both parties. The second approach makes full use of the mode aspect of second quantization whereby separability can be formulated as absence of nonlocal correlation among two different sets of modes. While the first approach applies to pure states only, the second one is more general and characterizes generic entangled states. In the following, we shall show that the mode-based approach indeed contains the particle-based one.
Our goal is to clarify the relation between entanglement and correlation energy in a bipartite system with infinite dimensional Hilbert space. To this aim we consider the completely solvable Moshinsky's model of two linearly coupled harmonic oscillators. Also for small values of the couplings the entanglement of the ground state is nonlinearly related to the correlation energy, involving logarithmic or algebraic corrections. Then, looking for witness observables of the entanglement, we show how to give a physical interpretation of the correlation energy. In particular, we have proven that there exists a set of separable states, continuously connected with the Hartree-Fock state, which may have a larger overlap with the exact ground state, but also a larger energy expectation value. In this sense, the correlation energy provides an entanglement gap, i.e. an energy scale, under which measurements performed on the 1-particle harmonic sub-system can discriminate the ground state from any other separated state of the system. However, in order to verify the generality of the procedure, we have compared the energy distribution cumulants for the 1-particle harmonic sub-system of the Moshinsky's model with the case of a coupling with a damping Ohmic bath at 0 temperature.