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arXiv:1808.01646 · Modern Physics Letters A

Induced entanglement entropy of harmonic oscillators in non-commutative phase space

Bingsheng Lin, Jian Xu, Taihua Heng

Published Jul 30, 2019• 9 citations
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Abstract

We study the entanglement entropy of harmonic oscillators in non-commutative phase space (NCPS). We propose a new definition of quantum Rényi entropy based on Wigner functions in NCPS. Using the Rényi entropy, we calculate the entanglement entropy of the ground state of the 2D isotropic harmonic oscillators. We find that for some values of the non-commutative parameters, the harmonic oscillators can be entangled in NCPS. This is a new entanglement-like effect caused by the non-commutativity of the phase space.

DOI
10.1142/s0217732319502699

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We study the entanglement entropy of harmonic oscillators in non-commutative phase space (NCPS). We propose a new definition of quantum Rényi entropy based on Wigner functions in NCPS. Using the Rényi entropy, we calculate the entanglement entropy of the ground state of the 2D isotropic harmonic oscillators. We find that for some values of the non-commutative parameters, the harmonic oscillators can be entangled in NCPS. This is a new entanglement-like effect caused by the non-commutativity of the phase space.
The quantum harmonic oscillator is a basic model for many complex ones.After integration,the Wigner function of quantum harmonic oscillator can be expressed by a simple form,which can be used to solve many related problems.This paper has placed the harmonic oscillator in non-commutative phase space,and studied the Wigner function of two-dimensional charged harmonic oscillator in electric field.The results show that when the non-commutative parameter is zero,the Wigner function of two-dimensional charged harmonic oscillator is equivalent to its form in commutative space.
Two isotropic harmonic oscillators in a magnetic field on the non-commutative phase space (NCPS) are studied in this paper.We derive the corresponding entanglement entropies of the vacuum state and we find analytically purity function, Hall conductivity and the energy density.Transition-state theory is also developed to examine the purity function and the energy density.By considering the action of the non-commutativity parameters, entanglement and Hall conductivity present a similar behavior.A large magnetic field acts as a disturbance of the accurate system information and the non-localization of the two harmonic oscillators in the atom by increasing its stored energy.The application of the transition-state theory increases entanglement and accelerates the non-localization phenomenon.
Mode entanglement in a two-mode squeezed state is studied. The entanglement spectrum is derived and interpreted as Boltzmann coefficients of harmonic oscillator eigenstates at thermal equilibrium. The effective temperature is expressed in terms of squeezing parameters and is equal to zero when the state is unsqueezed and increases without bound as the state is infinitely squeezed. It is demonstrated that the reduced density operator can be identified as a canonical ensemble of single harmonic oscillator at some effective temperature Te. The Renyi entanglement entropy is calculated and found to increase without bound as the squeezing parameter increases. The single-copy entanglement, which is a limiting value of the Renyi entropy, is interpreted as a partition function.