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Validity and failure of some entropy inequalities for CAR systems

Hajime Moriya

Published Sep 14, 2013• 5 citations
OpenAlex

Abstract

Basic properties of von Neumann entropy such as the triangle inequality and what we call MONO-SSA are studied for CAR systems. We show that both inequalities hold for every even state by using symmetric purification which is applicable to such a state. We construct a certain class of non-even states giving examples of the non-validity of those inequalities.

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Validity and failure of some entropy inequalities for CAR systems

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Basic properties of von Neumann entropy such as the triangle inequality and what we call MONO–SSA are studied for CAR systems. We show that both inequalities hold for every even state by using symmetric purification which is applicable to such a state. We construct a certain class of noneven states giving examples of the nonvalidity of those inequalities.

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Basic properties of von Neumann entropy such as the triangle inequality and what we call MONO-SSA are studied for CAR systems. We show that both inequalities hold for every even state by using symmetric purification which is applicable to such a state. We construct a certain class of non-even states giving examples of the non-validity of those inequalities.
We study quantum entanglement of bipartite CAR systems. We show that triangle inequality of von Neumann entropy does not hold in these systems. Meanwhile, we show non-separability in the sense of Florig-Summers for our CAR systems. Entanglement degrees by Narnhofer are generalized to an arbitrary, that is, not-necessarily commuting pair of general C ∗-systems. These new entanglement degrees are investigated with some explicit calculations for certain pairs of our CAR systems. We introduce a notion named half-sided entanglement via our generalized entanglement degree. By using this notion, we describe those certain notable features of quantum entanglement which originate from CAR. I
We exhibit infinitely many new, constrained inequalities for the von Neumann entropy, and show that they are independent of each other and the known inequalities obeyed by the von Neumann entropy (basically strong subadditivity). The new inequalities were proved originally by Makarychevfor the Shannon entropy, using properties of probability distributions. Our approach extends the proof of the inequalities to the quantum domain, and includes their independence for the quantum and also the classical cases.
The strong subadditivity inequality of von Neumann entropy relates the entropy of subsystems of a tripartite state ${\ensuremath{\rho}}_{ABC}$ to that of the composite system. Here, we define ${\mathbit{T}}^{(a)}({\ensuremath{\rho}}_{ABC})$ as the extent to which ${\ensuremath{\rho}}_{ABC}$ fails to satisfy the strong subadditivity inequality $S({\ensuremath{\rho}}_{B})+S({\ensuremath{\rho}}_{C})\ensuremath{\le}S({\ensuremath{\rho}}_{AB})+S({\ensuremath{\rho}}_{AC})$ with equality and investigate its properties. In particular, by introducing auxiliary subsystem EEE, we consider any purification $|{\ensuremath{\psi}}_{ABCE}\ensuremath{\rangle}$ of ${\ensuremath{\rho}}_{ABC}$ and formulate ${\mathbit{T}}^{(a)}({\ensuremath{\rho}}_{ABC})$ as the extent to which the bipartite quantum correlations of ${\ensuremath{\rho}}_{AB}$ and ${\ensuremath{\rho}}_{AC}$, measured by entanglement of formation and quantum discord, change under the transformation $B\ensuremath{\rightarrow}BE$ and $C\ensuremath{\rightarrow}CE$. Invariance of quantum correlations of ${\ensuremath{\rho}}_{AB}$ and ${\ensuremath{\rho}}_{AC}$ under such transformation is shown to be a necessary and sufficient condition for vanishing ${\mathbit{T}}^{(a)}({\ensuremath{\rho}}_{ABC})$. Our approach allows one to characterize, intuitively, the structure of states for which the strong subadditivity is saturated. Moreover, along with providing a conservation law for quantum correlations of states for which the strong subadditivity inequality is satisfied with equality, we find that such states coincide with those that the Koashi-Winter monogamy relation is saturated.
The Shannon entropy of a collection of random variables is subject to a number of constraints, the best-known examples being monotonicity and strong subadditivity. It remains an open question to decide which of these “laws of information theory” are also respected by the von Neumann entropy of many-body quantum states. In this article, we consider a toy version of this difficult problem by analyzing the von Neumann entropy of stabilizer states. We find that the von Neumann entropy of stabilizer states satisfies all balanced information inequalities that hold in the classical case. Our argument is built on the fact that stabilizer states have a classical model, provided by the discrete Wigner function: The phase-space entropy of the Wigner function corresponds directly to the von Neumann entropy of the state, which allows us to reduce to the classical case. Our result has a natural counterpart for multi-mode Gaussian states, which sheds some light on the general properties of the construction. We also discuss the relation of our results to recent work by Linden, Ruskai, and Winter [“The quantum entropy cone of stabiliser states,” e-print arXiv:1302.5453].