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Europhysics Letters (EPL)

Relative Rényi entropy and fidelity susceptibility

Ágnes Nagy, E. Romera

Published Mar 1, 2015• 37 citations
Publisher

Abstract

\n\nIt is shown that there is a linear relation between the relative Rényi entropy and the density functional fidelity susceptibility. The derivative of the relative Rényi entropy with respect to the Rényi parameter α is also proportional to the density functional fidelity susceptibility. The benefit of these relations is illustrated for the Dicke model (considering a numerical study and a variational analytical approximation). These relations are useful to detect the second-order QPT in this model.\n

DOI
10.1209/0295-5075/109/60002

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\n\nIt is shown that there is a linear relation between the relative Rényi entropy and the density functional fidelity susceptibility. The derivative of the relative Rényi entropy with respect to the Rényi parameter α is also proportional to the density functional fidelity susceptibility. The benefit of these relations is illustrated for the Dicke model (considering a numerical study and a variational analytical approximation). These relations are useful to detect the second-order QPT in this model.\n
Within density functional theory (DFT), where the density is the fundamental variable, quantum phase transitions (QPTs) can be formulated through a Hamiltonian H^=H^0+∑iξiA^i, such that the control parameters {ξi} are in bijective correspondence (in the nondegenerate case) with the “densities” ai=⟨A^i⟩, and the functional Q({ai}) acts as the Legendre transform of the energy; this structure even permits the use of Rényi entropy (for a given order) as an alternative control parameter, while degeneracy can be handled via a subspace density. On this foundation, information-theoretic measures provide sensitive diagnostics of criticality: fidelity and its susceptibility χ, Fisher information, relative Rényi entropy, and the Kullback–Leibler divergence are locally linked by Rq≈qIKL≈2qχ(δλ)2, revealing their proportionality in the small-parameter-shift regime. Applied to the Dicke model, numerical analyses show that fidelity exhibits pronounced curvature or divergence near λc=ωω0/2 and that the response sharpens with increasing j, corroborating that these information measures capture QPTs with precision within the DFT framework.
Recently, an interesting quantity called the quantum Rényi divergence (or 'sandwiched' Rényi relative entropy) was defined for pairs of positive semi-definite operators ρ and σ. It depends on a parameter α and acts as a parent quantity for other relative entropies which have important operational significance in quantum information theory: the quantum relative entropy and the min- and max-relative entropies. There is, however, another relative entropy, called the 0-relative Rényi entropy, which plays a key role in the analysis of various quantum information-processing tasks in the one-shot setting. We prove that the 0-relative Rényi entropy is obtainable from the quantum Rényi divergence only if ρ and σ have equal supports. This, along with existing results in the literature, suggests that it suffices to consider two essential parent quantities from which operationally relevant entropic quantities can be derived—the quantum Rényi divergence with parameter α ⩾ 1/2, and the α-relative Rényi entropy with α ∈ [0, 1).
We consider a two-parameter family of Rényi relative entropies Dα,z(ρ ∥ σ) that are quantum generalisations of the classical Rényi divergence Dα(p ∥ q). This family includes many known relative entropies (or divergences) such as the quantum relative entropy, the recently defined quantum Rényi divergences, as well as the quantum Rényi relative entropies. All its members satisfy the quantum generalizations of Rényi’s axioms for a divergence. We consider the range of the parameters α, z for which the data-processing inequality holds. We also investigate a variety of limiting cases for the two parameters, obtaining explicit formulas for each one of them.
According to the recently proposed density-functional analog of quantum phase transitions (QPTs) the ``density'' determines the ``control parameter'' (corresponding to the density-functional theory ``external potential''). It was also proved that in the nondegenerate case there is a one-to-one map between the ground-state wave function and the control parameter. It is now shown that there is a one-to-one map between the ground-state subspace and the control parameter. It is pointed out that there is a one-to one map between the subspace density and the control parameter. The R\'enyi entropies are proved to be strictly monotonic functions of the control parameter in the neighborhood of the transition point. These properties are illustrated with the Dicke model which exhibits a QPT from a normal phase (nondegenerate) to the superradiant case (degenerate) in the thermodynamic limit.
The fidelity-based smooth min-relative entropy is a distinguishability measure that has appeared in a variety of contexts in prior work on quantum information, including resource theories like thermodynamics and coherence. Here we provide a comprehensive study of this quantity. First we prove that it satisfies several basic properties, including the dataprocessing inequality. We also establish connections between the fidelity-based smooth min-relative entropy and other widely used information-theoretic quantities, including smooth min-relative entropy and smooth sandwiched Rényi relative entropy, of which the sandwiched Rényi relative entropy and smooth max-relative entropy are special cases. After that, we use these connections to establish the second-order asymptotics of the fidelity-based smooth min-relative entropy and all smooth sandwiched Rényi relative entropies, finding that the first-order term is the quantum relative entropy and the second-order term involves the quantum relative entropy variance. Utilizing the properties derived, we also show how the fidelity-based smooth min-relative entropy provides one-shot bounds for operational tasks in general resource theories in which the target state is mixed, with a particular example being randomness distillation. The above observations then lead to second-order expansions of the upper bounds on distillable randomness, as well as the precise second-order asymptotics of the distillable randomness of particular classical–quantum states. Finally, we establish semi-definite programs for smooth maxrelative entropy and smooth conditional min-entropy, as well as a bilinear program for the fidelity-based smooth min-relative entropy, which we subsequently use to explore the tightness of a bound relating the last to the first.