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arXiv:1804.09698 · Entropy

The von Neumann Entropy for Mixed States

Jorge Andres Anaya-Contreras, Héctor M. Moya-Cessa, Arturo Zúñiga-Segundo

Published Jan 10, 2019• 15 citations
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Abstract

The Araki-Lieb inequality is commonly used to calculate the entropy of subsystems when they are initially in pure states, as this forces the entropy of the two subsystems to be equal after the complete system evolves. Then, it is easy to calculate the entropy of a large subsystem by finding the entropy of the small one. To the best of our knowledge, there does not exist a way of calculating the entropy when one of the subsystems is initially in a mixed state. For the case of a two-level atom interacting with a quantized field, we show that it is possible to use the Araki-Lieb inequality and find the von Neumann entropy for the large (infinite) system. We show this in the two-level atom-field interaction.

DOI
10.3390/e21010049

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The Araki-Lieb inequality is commonly used to calculate the entropy of subsystems when they are initially in pure states, as this forces the entropy of the two subsystems to be equal after the complete system evolves. Then, it is easy to calculate the entropy of a large subsystem by finding the entropy of the small one. To the best of our knowledge, there does not exist a way of calculating the entropy when one of the subsystems is initially in a mixed state. For the case of a two-level atom interacting with a quantized field, we show that it is possible to use the Araki-Lieb inequality and find the von Neumann entropy for the large (infinite) system. We show this in the two-level atom-field interaction.
We study the Jaynes-Cummings model with initial mixed states, for the atom or the field, are considered. The evolved mixed field density matrix is purified to a wavefunction that describes the interaction between a quantised field and an artificial four-level atom. This allows us to use the Araki-Lieb inequality to calculate the field entropy from the atomic entropy. We then suggest Hamiltonians that reproduce the field entropy dynamics. We finally show that more realistic Hamiltonians may be considered by using two entangled two-level atoms.
We study the Jaynes-Cummings interaction when initial mixed states, for the atom or the field, are considered. The evolved mixed field density matrix is purified to a wavefunction that describes the interaction between a quantised field and an artificial four-level atom. This allows us to use the Araki-Lieb inequality to calculate the field entropy from the atomic entropy. We then generate {\it artificial} Hamiltonians that reproduce the field entropy dynamics. We finally show that more realistic Hamiltonians may be considered by using two entangled two-level atoms.
This article presents self-contained proofs of the strong subadditivity inequality for von Neumann’s quantum entropy, S(ρ), and some related inequalities for the quantum relative entropy, most notably its convexity and its monotonicity under stochastic maps. Moreover, the approach presented here, which is based on Klein’s inequality and Lieb’s theorem that the function A→Tr eK+log A is concave, allows one to obtain conditions for equality. In the case of strong subadditivity, which states that S(ρ123)+S(ρ2)⩽S(ρ12)+S(ρ23) where the subscripts denote subsystems of a composite system, equality holds if and only if log ρ123=log ρ12−log ρ2+log ρ23. Using the fact that the Holevo bound on the accessible information in a quantum ensemble can be obtained as a consequence of the monotonicity of relative entropy, we show that equality can be attained for that bound only when the states in the ensemble commute. The article concludes with an Appendix giving a short description of Epstein’s elegant proof of Lieb’s theorem.
We present two results derived from the sharp log-Sobolev inequality for the uniform measure on a complete graph which concern the von Neumann entropy and the Entanglement of Formation of a state of finite and infinite-dimensional quantum systems. The first result is a sharp Lipschitz lower semicontinuity bound for the von Neumann entropy at any mixed state ρρρ with uniform positive spectrum (i.e. a state proportional to a projector) w.r.t. the fidelity deficit: the inequality  S(ρ)−S(σ)≤Cρ(1−F(ρ,σ)) \,S(ρ)-S(σ)\leq C_ρ(1-F(ρ,σ))\,S(ρ)−S(σ)≤Cρ​(1−F(ρ,σ)) valid for any state σσσ, where CρC_ρCρ​ is a constant depending on the rank of ρρρ. The second result is a sharp Lipschitz lower semicontinuity bound for the Entanglement of Formation at any pure state ρρρ with uniform positive spectrum of marginal states w.r.t. the fidelity deficit: the inequality  EF(ρ)−EF(σ)≤Cρ(1−Trρσ) \,E_F(ρ)-E_F(σ)\leq C_ρ(1-\mathrm{Tr}ρσ)\,EF​(ρ)−EF​(σ)≤Cρ​(1−Trρσ) valid for any state σσσ, where CρC_ρCρ​ is a constant depending on the Schmidt rank of ρρρ. In both cases the optimal constant CρC_ρCρ​ is equal to the optimal constant KdK_{d}Kd​ in the log-Sobolev inequality for the complete graph with ddd vertices: in the first case d=rankρd=\mathrm{rank}ρd=rankρ, in the second one d=rankρA=rankρBd=\mathrm{rank}ρ_A=\mathrm{rank}ρ_Bd=rankρA​=rankρB​. The authors are grateful to GPT 5.6 for valuable discussion and technical help in preparing this note.
We construct a complete set of Wannier functions that are localized at both given positions and momenta. This allows us to introduce the quantum phase space, onto which a quantum pure state can be mapped unitarily. Using its probability distribution in quantum phase space, we define an entropy for a quantum pure state. We prove an inequality regarding the long-time behavior of our entropy's fluctuation. For a typical initial state, this inequality indicates that our entropy can relax dynamically to a maximized value and stay there most of time with small fluctuations. This result echoes the quantum H theorem proved by von Neumann [Zeitschrift für Physik 57, 30 (1929)]. Our entropy is different from the standard von Neumann entropy, which is always zero for quantum pure states. According to our definition, a system always has bigger entropy than its subsystem even when the system is described by a pure state. As the construction of the Wannier basis can be implemented numerically, the dynamical evolution of our entropy is illustrated with an example.