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arXiv:2207.13805 · Physical review. B./Physical review. B

Squashed entanglement in one-dimensional quantum matter

Alfonso Maiellaro, Francesco Romeo, R. Citro, Fabrizio Illuminati

Published Mar 28, 2023• 7 citations
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Abstract

Squashed entanglement and its universal upper bound, the quantum conditional mutual information, are faithful measures of bipartite quantum correlations defined in terms of multipartitions. As such, they are sensitive to the fine-grain structure of quantum systems. Building on this observation, we introduce the concept of quantum conditional mutual information between the edges of quantum many-body systems. We show that this quantity characterizes unambiguously one-dimensional topological insulators and superconductors, being equal to Bell-state entanglement in the former and to half Bell-state entanglement in the latter, mirroring the different statistics of the edge modes in the two systems. The edge-to-edge quantum conditional mutual information is robust in the presence of disorder or local perturbations, converges exponentially with the system size to a quantized topological invariant, even in the presence of interactions, and vanishes in the trivial phase. We thus conjecture that it coincides with the edge-to-edge squashed entanglement in the entire ground-state phase diagram of symmetry-protected topological systems, and we provide some analytical evidence supporting the claim. By comparing them with the entanglement negativity, we collect further indications that the quantum conditional mutual information and the squashed entanglement provide a very accurate characterization of nonlocal correlation patterns in one-dimensional quantum matter.

DOI
10.1103/physrevb.107.115160

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Squashed entanglement and its universal upper bound, the quantum conditional mutual information, are faithful measures of bipartite quantum correlations defined in terms of multipartitions. As such, they are sensitive to the fine-grain structure of quantum systems. Building on this observation, we introduce the concept of quantum conditional mutual information between the edges of quantum many-body systems. We show that this quantity characterizes unambiguously one-dimensional topological insulators and superconductors, being equal to Bell-state entanglement in the former and to half Bell-state entanglement in the latter, mirroring the different statistics of the edge modes in the two systems. The edge-to-edge quantum conditional mutual information is robust in the presence of disorder or local perturbations, converges exponentially with the system size to a quantized topological invariant, even in the presence of interactions, and vanishes in the trivial phase. We thus conjecture that it coincides with the edge-to-edge squashed entanglement in the entire ground-state phase diagram of symmetry-protected topological systems, and we provide some analytical evidence supporting the claim. By comparing them with the entanglement negativity, we collect further indications that the quantum conditional mutual information and the squashed entanglement provide a very accurate characterization of nonlocal correlation patterns in one-dimensional quantum matter.
Identifying entanglement-based order parameters characterizing topological systems, in particular topological superconductors and topological insulators, has remained a major challenge for the physics of quantum matter in the last two decades. Here we show that the end-to-end, long-distance, bipartite squashed entanglement between the edges of a many-body system, defined in terms of the edge-to-edge quantum conditional mutual information, is the natural nonlocal order parameter for topological superconductors in one dimension as well as in quasi-one-dimensional geometries. For the Kitaev chain in the entire topological phase, the edge squashed entanglement is quantized to ln(2)/2ln(2)/2ln(2)/2, half the maximal Bell-state entanglement, and vanishes in the trivial phase. Such a topological squashed entanglement exhibits the correct scaling at the quantum phase transition, is stable in the presence of interactions, and is robust against disorder and local perturbations. Edge quantum conditional mutual information and edge squashed entanglement defined with respect to different multipartitions discriminate topological superconductors from symmetry breaking magnets, as shown by comparing the fermionic Kitaev chain and the spin-1/2 Ising model in transverse field. For systems featuring multiple topological phases with different numbers of edge modes, like the quasi-1D Kitaev ladder, topological squashed entanglement counts the number of Majorana excitations and distinguishes the different topological phases of the system. In fact, we show that the edge quantum conditional mutual information and the edge squashed entanglement remain valid detectors of topological superconductivity even for systems, like the Kitaev tie with long-range hopping, featuring geometrical frustration and a suppressed bulk-edge correspondence.
Entanglement in many-body quantum systems is distributed across spatial regions, where its structure often dictates the information-processing capabilities of the state. Yet, characterizing the entanglement structure, especially for mixed states, remains a challenge. In this work, we propose hysteretic squashed entanglement TsqT_{sq}Tsq​, a conditional entanglement monotone that measures the genuine quantum correlations between two subregions, conditioned on a third region, in a many-body quantum state. TsqT_{sq}Tsq​ is upper bounded by the convex-roof extension of quantum conditional mutual information and exhibits several desirable properties like monogamy, convexity, asymptotic continuity, faithfulness, and additivity for tensor-product states. We study the conditional entanglement generation in a one-dimensional transverse-field Ising model under quench, where we show that TsqT_{sq}Tsq​ effectively squashes classical contributions and can detect genuine quantum correlations across both adjacent and long-range subsystems. We elucidate the utility of this measure as a robust quantifier of topological entanglement entropy for mixed states. This opens new operational resource-theoretic avenues for probing topological order and criticality.
Abstract It is shown that the recently established lower semicontinuity of the quantum conditional mutual information implies (in fact, is equivalent to) the lower semicontinuity of the loss of the quantum (conditional) mutual information under local channels considered as a function on the Cartesian product of the set of all states of a composite system and the sets of all local channels (equipped with the strong convergence topology). Some applications of this property are considered. New continuity conditions for the quantum mutual information and for the squashed entanglement in both bipartite and multipartite infinite-dimensional systems are obtained. It is proved, in particular, that the multipartite squashed entanglement of any countably indecomposable separable state with finite marginal entropies is equal to zero. Special continuity properties of the information gain of a quantum measurement with and without quantum side information are established that can be treated as robustness (stability) of these quantities with respect to perturbation of the measurement and the measured state.
The conditional quantum mutual information I(A; B|C) of a tripartite state ρABC is an information quantity which lies at the center of many problems in quantum information theory. Three of its main properties are that it is non-negative for any tripartite state, that it decreases under local operations applied to systems A and B, and that it obeys the duality relation I(A; B|C) = I(A; B|D) for a four-party pure state on systems ABCD. The conditional mutual information also underlies the squashed entanglement, an entanglement measure that satisfies all of the axioms desired for an entanglement measure. As such, it has been an open question to find Rényi generalizations of the conditional mutual information, that would allow for a deeper understanding of the original quantity and find applications beyond the traditional memoryless setting of quantum information theory. The present paper addresses this question, by defining different α-Rényi generalizations Iα(A; B|C) of the conditional mutual information, some of which we can prove converge to the conditional mutual information in the limit α → 1. Furthermore, we prove that many of these generalizations satisfy non-negativity, duality, and monotonicity with respect to local operations on one of the systems A or B (with it being left as an open question to prove that monotonicity holds with respect to local operations on both systems). The quantities defined here should find applications in quantum information theory and perhaps even in other areas of physics, but we leave this for future work. We also state a conjecture regarding the monotonicity of the Rényi conditional mutual informations defined here with respect to the Rényi parameter α. We prove that this conjecture is true in some special cases and when α is in a neighborhood of one.
We analyse two possible definitions of the squashed entanglement in an infinite-dimensional bipartite system: direct translation of the finite-dimensional definition and its universal extension. It is shown that the both definitions produce the same lower semicontinuous entanglement measure possessing all basis properties of the squashed entanglement on the set of states having at least one finite marginal entropy. It is also shown that the second definition gives an adequate lower semicontinuous extension of this measure to all states of the infinite-dimensional bipartite system. A general condition relating continuity of the squashed entanglement to continuity of the quantum mutual information is proved and its corollaries are considered. Continuity bound for the squashed entanglement under the energy constraint on one subsystem is obtained by using the tight continuity bound for quantum conditional mutual information (proved in the Appendix by using Winter’s technique). It is shown that the same continuity bound is valid for the entanglement of formation. As a result the asymptotic continuity of the both entanglement measures under the energy constraint on one subsystem is proved.
In this paper, we present a new entanglement monotone for bipartite quantum states. Its definition is inspired by the so-called intrinsic information of classical cryptography and is given by the halved minimum quantum conditional mutual information over all tripartite state extensions. We derive certain properties of the new measure which we call “squashed entanglement”: it is a lower bound on entanglement of formation and an upper bound on distillable entanglement. Furthermore, it is convex, additive on tensor products, and superadditive in general. Continuity in the state is the only property of our entanglement measure which we cannot provide a proof for. We present some evidence, however, that our quantity has this property, the strongest indication being a conjectured Fannes-type inequality for the conditional von Neumann entropy. This inequality is proved in the classical case.