Abstract
In a recent paper (A. Fring and T. Frith, Phys. Rev A 100, 101102 (2019)), a Dyson scheme to deal with density matrix of non-Hermitian Hamiltonians has been used to investigate the entanglement of states of a PT-symmetric bosonic system. They found that von Neumann entropy can show a different behavior in the broken and unbroken regime. We show that their results can be recast in terms of an abstract model of pseudo-Hermitian random matrices. It is found however that although the formalism is practically the same, the entanglement is not of Fock states but of Bell states.
Highlights
The importance of the authors’ findings of Ref. [1] lies in the consequences it has to the question of time evolution of non-Hermitian Hamiltonians [2,3], or, more precisely, to the evolution of their associated density matrices
The non-Hermiticity addressed in Refs. [1,4,5] is that related to the PT-symmetry
By linearly superposing the operators Û, R, Ŝ and Û, T, Ŝ, two Hamiltonians are constructed in which the matrix Ŝ plays the role of the non-Hermitian term. In both cases, the time evolution leads to entanglement of, respectively, the chiral states of Rin the first Hamiltonian and the bipartite states of Tin the second one
Summary
The importance of the authors’ findings of Ref. [1] lies in the consequences it has to the question of time evolution of non-Hermitian Hamiltonians [2,3], or, more precisely, to the evolution of their associated density matrices. [1] lies in the consequences it has to the question of time evolution of non-Hermitian Hamiltonians [2,3], or, more precisely, to the evolution of their associated density matrices. It shows that by introducing an appropriate time-dependent metric, the density matrix of a non-Hermitian Hamiltonian can be linked to one of a Hermitian one by a similarity transformation Both share the same von Neumamm entropy. Since the beginning of the studies of PT-symmetric systems there was an interest in investigating random matrix ensemble to model properties of this kind of Hamiltonians. This comes naturally as time reversal symmetry plays an important role in RMT. In the conclusion section, the physical implications of the results are discussed
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