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A Note on Pseudo-Hermitian Systems with Point Interactions and Quantum Separability

Shao-Ming Fei

Published Jan 1, 2006• 0 citations
OpenAlex

Abstract

We study the quantum entanglement and separability of Hermitian and pseudo-Hermitian systems of identical bosonic or fermionic particles with point interactions. The separability conditions are investigated in detail. PACS: 02.30.Ik, 11.30.Er, 03.65.Fd Key words: Point interactions, PT-symmetry, Separability Pseudo-Hermitian quantum mechanical systems have been extensively investi-gated due to some mathematical and physical considerations [1]. For PT-symmetric systems with singular contact interactions, we have studied the classification, spec-tra and integrability problem in [2]. The results are generalized to the case of identical bosonic and fermionic many-body systems with PT-symmetric contact interactions and spin-coupling interactions [3]. On the other hand, as one of the most striking features of quantum phenom-ena [4], quantum entanglement has been identified as a key non-local resource in quantum information processing such as quantum computation [5], quantum tele-portation, dense coding, quantum cryptographic schemes, entanglement swapping and remote state preparation (RSP). Nevertheless the theory of quantum entan-glement is far from being satisfied. One of main problem in quantum entanglement is to judge a quantum state is separable or entangled, i.e. the separability. In this paper we study separability of Hermitian and pseudo-Hermitian systems of many body systems with contact interactions. We first consider contact interactions without spin coupling. A particle moving in one dimension with point interactions at the origin can be characterized by the non-separated1) boundary conditions imposed on the wave function ϕ at x = 0, ϕ

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We study the quantum entanglement and separability of Hermitian and pseudo-Hermitian systems of identical bosonic or fermionic particles with point interactions. The separability conditions are investigated in detail. PACS: 02.30.Ik, 11.30.Er, 03.65.Fd Key words: Point interactions, PT-symmetry, Separability Pseudo-Hermitian quantum mechanical systems have been extensively investi-gated due to some mathematical and physical considerations [1]. For PT-symmetric systems with singular contact interactions, we have studied the classification, spec-tra and integrability problem in [2]. The results are generalized to the case of identical bosonic and fermionic many-body systems with PT-symmetric contact interactions and spin-coupling interactions [3]. On the other hand, as one of the most striking features of quantum phenom-ena [4], quantum entanglement has been identified as a key non-local resource in quantum information processing such as quantum computation [5], quantum tele-portation, dense coding, quantum cryptographic schemes, entanglement swapping and remote state preparation (RSP). Nevertheless the theory of quantum entan-glement is far from being satisfied. One of main problem in quantum entanglement is to judge a quantum state is separable or entangled, i.e. the separability. In this paper we study separability of Hermitian and pseudo-Hermitian systems of many body systems with contact interactions. We first consider contact interactions without spin coupling. A particle moving in one dimension with point interactions at the origin can be characterized by the non-separated1) boundary conditions imposed on the wave function ϕ at x = 0, ϕ
In this final chapter we discuss composite systems in Quantum Mechanics. First, we lay out the necessary conditions that the state space of a composite system must satisfy by considering the trivial case of two juxtaposed systems. Then we point out the features that the interesting case of a composite system, whose parts interact with each other, must have. We then move to demonstrate how such a system can be naturally described (quantized) by employing tensor product spaces. The definition and the most relevant properties of tensor product spaces are introduced. Then the meaning of “interactions” in quantum mechanics is explored and the most nontrivial aspects of correlation and entanglement are discussed. Finally, in the last section, we introduce systems of identical particles. We describe how to represent the state space of such systems in terms of one-particle states, number operators and occupation number representations. Classification of identical particles into Bosons and Fermions is explained. Finally, introducing symmetric and antisymmetric states, the traditional quantization scheme using the symmetrization postulate is described. The permutation operator is introduced which is the main tool for dealing with identical particle systems in this language. We wind up by indicating how exchange degeneracy is removed which embodies the signature of indistinguishability.
Abstract These two accompanying papers are concerned with 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 in which the probabilities for joint measurements on the composite sub-systems are no longer determined from measurement probabilities on the separate sub-systems. There are many aspects of entanglement that can be studied. This two-part review focuses on the meaning of entanglement, the quantum paradoxes associated with entangled states, and the important tests that allow an experimentalist to determine whether a quantum state—in particular, one for massive bosons is entangled. An overall outcome of the review is to distinguish criteria (and hence experiments) for entanglement that fully utilize the symmetrization principle and the super-selection rules that can be applied to bosonic massive particles. In the first paper (I), the background is given for the meaning of entanglement in the context of systems of identical particles. For such systems, the requirement is that the relevant quantum density operators must satisfy the symmetrization principle and that global and local super-selection rules prohibit states in which there are coherences between differing particle numbers. The justification for these requirements is fully discussed. In the second quantization approach that is used, both the system and the sub-systems are modes (or sets of modes) rather than particles, particles being associated with different occupancies of the modes. The definition of entangled states is based on first defining the non-entangled states—after specifying which modes constitute the sub-systems. This work mainly focuses on the two mode entanglement for massive bosons, but is put in the context of tests of local hidden variable theories, where one may not be able to make the above restrictions. The review provides the detailed arguments necessary for the conclusions of a recent paper, where the question of how to rigorously demonstrate the entanglement of a two-mode Bose–Einstein condensate (BEC) has been examined. In the accompanying review paper (II), we consider spin squeezing and other tests for entanglement that have been proposed for two-mode bosonic systems. We apply the approach of review (I) to determine which tests, and which modifications of the tests, are useful for detecting entanglement in massive bosonic (BEC), as opposed to photonic, systems. Several new inequalities are derived, a theory for the required two-mode interferometry is presented, and key experiments to date are analyzed.
We study the properties of the entanglement spectrum in gapped non-interacting non-Hermitian systems, and its relation to the topological properties of the system Hamiltonian. Two different families of entanglement Hamiltonians can be defined in non-Hermitian systems, depending on whether we consider only right (or equivalently only left) eigenstates or a combination of both left and right eigenstates. We show that their entanglement spectra can still be computed efficiently, as in the Hermitian limit. We discuss how symmetries of the Hamiltonian map into symmetries of the entanglement spectrum depending on the choice of the many-body state. Through several examples in one and two dimensions, we show that the biorthogonal entanglement Hamiltonian directly inherits the topological properties of the Hamiltonian for line gapped phases, with characteristic singular and energy zero modes. The right (left) density matrix carries distinct information on the topological properties of the many-body right (left) eigenstates themselves. In purely point gapped phases, when the energy bands are not separable, the relation between the entanglement Hamiltonian and the system Hamiltonian breaks down.
We introduce the notion of pseudo-Hermiticity and show that every Hamiltonian with a real spectrum is pseudo-Hermitian. We point out that all the PT-symmetric non-Hermitian Hamiltonians studied in the literature belong to the class of pseudo-Hermitian Hamiltonians, and argue that the basic structure responsible for the particular spectral properties of these Hamiltonians is their pseudo-Hermiticity. We explore the basic properties of general pseudo-Hermitian Hamiltonians, develop pseudosupersymmetric quantum mechanics, and study some concrete examples, namely the Hamiltonian of the two-component Wheeler–DeWitt equation for the FRW-models coupled to a real massive scalar field and a class of pseudo-Hermitian Hamiltonians with a real spectrum.
Entanglement is known to be a relative notion, defined with respect to the choice of physical observables to be measured (i.e., the measurement setup used). This implies that, in general, the same state can be both separable and entangled for different measurement setups, but this does not exclude the existence of states which are separable (or entangled) for all possible setups. We show that for systems of bosonic particles there indeed exist such universally separable states: They are independently and identically distributed (i.i.d.) pure states. In contrast, there is no such state for fermionic systems with a few exceptional cases. We also find that none of the fermionic and bosonic systems admits universally entangled states.