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Journal of Mathematical Analysis and Applications

Studies in the theory of the generalized density operators II. Systems of identical particles

Julius J. Brandstatter, C Peltzer

Published Jul 1, 1967• 3 citations
Publisher
DOI
10.1016/0022-247x(67)90031-5

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Quantum Entanglement for Systems of Identical Bosons. I General Theory

Bryan J. Dalton, John Goold, B. M. Garraway +1 more

These two accompanying papers treat two mode entanglement for systems of identical massive bosons and the relationship to spin squeezing and other quantum correlation effects. Entanglement is a key quantum feature of composite systems where the probabilities for joint measurements on the composite sub-systems are no longer determined from measurement probabilities on the separate sub-systems. We focus on the meaning of entanglement, the quantum paradoxes associated with entangled states, and important tests that allow an experimentalist to determine whether a quantum state - in particular, one for massive bosons is entangled. Our tests for entanglement fully utilise the symmetrisation principle and super-selection rules for bosonic massive particles. These papers provides detailed arguments necessary for the conclusions of a recent paper which presented results for rigorously demonstrating the entanglement of a two-mode Bose-Einstein condensate (BEC). In the first paper (I), we discuss the meaning of entanglement for systems of identical particles. For such systems, the relevant quantum density operators must satisfy the symmetrisation principle and global and local super-selection rules prohibiting states in which there are coherences between differing particle numbers. We fully justify these requirements. In the second quantisation approach, both the system and sub-systems are modes (or sets of modes) rather than particles, particles being associated with different mode occupancies. In the accompanying review paper (II), we consider spin squeezing and other tests for entanglement for two-mode bosonic systems. Starting from paper (I) we determine which tests are useful for detecting entanglement in massive bosonic (BEC), as opposed to photonic, systems. Several new inequalities are derived, and recent key experiments are analysed.

arXiv:1506.06906•Jun 23, 2015•1 citation
We derive the analogue of the Weyl formula for the average level density, in the case of a system of identical particles (taking fermions as an example). This is accomplished by introducing the projection operator onto states in Hilbert space which have the symmetry imposed by particle identity. The semiclassical result is obtained by using the Wigner representation for this operator and by letting h {r_arrow} 0. Numerical examples for few degrees of freedom show in general good agreement with exact diagonalization even near the ground state of the system.
For a system of identical particles (we take fermions as an example), we derive the analogue of the Weyl formula for the average level density. This is done by introducing the projection operator onto states in Hilbert space which have the symmetry imposed by particle identity. The semi-classical result is obtained by using the Wigner representation for this operator and by letting ℏ → 0. Numerical examples for few degrees of freedom show in general good agreement with exact diagonalization even near the ground state.
These two accompanying papers treat two mode entanglement for systems of identical massive bosons and the relationship to spin squeezing and other quantum correlation effects. Entanglement is a key quantum feature of composite systems where the probabilities for joint measurements on the composite sub-systems are no longer determined from measurement probabilities on the separate sub-systems. We focus on the meaning of entanglement, the quantum paradoxes associated with entangled states, and important tests that allow an experimentalist to determine whether a quantum state - in particular, one for massive bosons is entangled. Our tests for entanglement fully utilise the symmetrisation principle and super-selection rules for bosonic massive particles. These papers provides detailed arguments necessary for the conclusions of a recent paper which presented results for rigorously demonstrating the entanglement of a two-mode Bose-Einstein condensate (BEC). In the first paper (I), we discuss the meaning of entanglement for systems of identical particles. For such systems, the relevant quantum density operators must satisfy the symmetrisation principle and global and local super-selection rules prohibiting states in which there are coherences between differing particle numbers. We fully justify these requirements. In the second quantisation approach, both the system and sub-systems are modes (or sets of modes) rather than particles, particles being associated with different mode occupancies. In the accompanying review paper (II), we consider spin squeezing and other tests for entanglement for two-mode bosonic systems. Starting from paper (I) we determine which tests are useful for detecting entanglement in massive bosonic (BEC), as opposed to photonic, systems. Several new inequalities are derived, and recent key experiments are analysed.