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arXiv:1108.3537 · Physical Review B

Flows of Rényi entropies

Yuli V. Nazarov

Published Nov 17, 2011• 14 citations
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Abstract

We demonstrate that the condensed matter quantum systems encompassing two reservoirs connected by a junction permit a natural definition of flows of conserved measures, i.e., R\'enyi entropies. Such flows are similar to the flows of physical conserved quantities such as charge and energy. We develop a perturbation technique that permits efficient computation of R\'enyi entropy flows and analyze second- and fourth-order contributions. Second-order approximation was shown to correspond directly to the transition events in the system and thereby to possess a set of intuitive features. The analysis of fourth-order corrections reveals a more complicated picture: The intuitive relations do not hold anymore, and the corrections exhibit divergencies in low-temperature limit, manifesting an intriguing nonanalytical dependence of the flows on coupling strength in the limit of weak couplings and vanishing temperatures.

DOI
10.1103/physrevb.84.205437

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We demonstrate that the condensed matter quantum systems encompassing two reservoirs connected by a junction permit a natural definition of flows of conserved measures, i.e., R'enyi entropies. Such flows are similar to the flows of physical conserved quantities such as charge and energy. We develop a perturbation technique that permits efficient computation of R'enyi entropy flows and analyze second- and fourth-order contributions. Second-order approximation was shown to correspond directly to the transition events in the system and thereby to possess a set of intuitive features. The analysis of fourth-order corrections reveals a more complicated picture: The intuitive relations do not hold anymore, and the corrections exhibit divergencies in low-temperature limit, manifesting an intriguing nonanalytical dependence of the flows on coupling strength in the limit of weak couplings and vanishing temperatures.
Recently, an interesting quantity called the quantum Renyi divergence (or "sandwiched" Renyi relative entropy) was defined for pairs of positive semi-definite operators ρ\rhoρ and σ\sigmaσ. It depends on a parameter α\alphaα and acts as a parent quantity for other relative entropies which have important operational significances 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 Renyi 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 Renyi entropy is obtainable from the quantum Renyi divergence only if ρ\rhoρ and σ\sigmaσ 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 Renyi divergence with parameter α≥1/2\alpha \ge 1/2α≥1/2, and the α\alphaα-relative R\'enyi entropy with α∈[0,1)\alpha\in [0,1)α∈[0,1).
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).
At the interface of quantum information and condensed matter physics, the study of entanglement in quantum many-body systems requires a new toolset which combines concepts from each. This thesis introduces a set of computational methods to study phases and phase transitions in lattice models of quantum systems, using the Renyi entropies as a means of quantifying entanglement. The scaling of entanglement entropy can give valuable insight into the phase of a condensed matter system. It can be used to detect exotic types of phases, to pinpoint transitions between phases, and can give us universal information about a system. \nThe first approach in this thesis is a technique to measure entanglement in finite size \nlattice systems using zero-temperature quantum Monte Carlo simulations. The algorithm \nis developed, implemented, and used to explore anomalous entanglement scaling terms in \nthe spin-1/2 Heisenberg antiferromagnet. \nIn the second part of this thesis, a new and complementary numerical technique is introduced to study entanglement not just in finite size systems, but as we approach the thermodynamic limit. This “numerical linked-cluster expansion” is used to study two different systems at their quantum critical points — continuous phase transitions occurring at zero temperature, at which these systems exhibit universal properties. Remarkably, these universal properties can be reflected in the scaling of entanglement. \nEntanglement offers a new perspective on condensed matter systems, one which takes us closer to genuinely understanding what goes on in these materials at the quantum mechanical level. This thesis demonstrates the first steps in developing an extensive list of computational tools that can be used to study entanglement over a wide range of interacting quantum many-body systems. With the ever increasing computational power available, it may be only a matter of time before these tools are used to create a comprehensive framework for the characterization of condensed matter phases and phase transitions.
Concepts of information theory are increasingly used to characterize collective phenomena in condensed matter systems, such as the use of entanglement entropies to identify emergent topological order in interacting quantum many-body systems. Here, we employ classical variants of these concepts, in particular R\'enyi entropies and their associated mutual information, to identify topological order in classical systems. Like for their quantum counterparts, the presence of topological order can be identified in such classical systems via a universal, subleading contribution to the prevalent volume and boundary laws of the classical R\'enyi entropies. We demonstrate that an additional subleading O(1)O(1)O(1) contribution generically arises for all R\'enyi entropies S(n){S}^{(n)}S(n) with $n\ensuremath{\ge}2$ when driving the system towards a phase transition, e.g., into a conventionally ordered phase. This additional subleading term, which we dub connectivity contribution, tracks back to partial subsystem ordering and is proportional to the number of connected parts in a given bipartition. Notably, the Levin-Wen summation scheme, typically used to extract the topological contribution to the R\'enyi entropies, does not fully eliminate this additional connectivity contribution in this classical context. This indicates that the distillation of topological order from R\'enyi entropies requires an additional level of scrutiny to distinguish topological from nontopological O(1)O(1)O(1) contributions. This is also the case for quantum systems, for which we discuss which entropies are sensitive to these connectivity contributions. We showcase these findings by extensive numerical simulations of a classical variant of the toric code model, for which we study the stability of topological order in the presence of a magnetic field and at finite temperatures from a R\'enyi entropy perspective.
Quantum entanglement is one essential element to characterize many-body quantum systems. However, the entanglement measures are mostly discussed in Hermitian systems. Here, we propose a natural extension of entanglement and Rényi entropies to non-Hermitian quantum systems. There have been other proposals for the computation of these quantities, which are distinct from what is proposed in the current paper. We demonstrate the proposed entanglement quantities which are referred to as generic entanglement and Rényi entropies. These quantities capture the desired entanglement properties in non-Hermitian critical systems, where the low-energy properties are governed by the non-unitary conformal field theories (CFTs). We find excellent agreement between the numerical extrapolation of the negative central charges from the generic entanglement/Rényi entropy and the non-unitary CFT prediction. Furthermore, we apply the generic entanglement/Rényi entropy to symmetry-protected topological phases with non-Hermitian perturbations. We find the generic n-th Rényi entropy captures the expected entanglement property, whereas the traditional Rényi entropy can exhibit unnatural singularities due to its improper definition.