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Hysteretic squashed entanglement in many-body quantum systems

Siddhartha Das, Alexander Yosifov, Jinzhao Sun

Abstract

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 $T_{sq}$, a conditional entanglement monotone that measures the genuine quantum correlations between two subregions, conditioned on a third region, in a many-body quantum state. $T_{sq}$ 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 $T_{sq}$ 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.

Hysteretic squashed entanglement in many-body quantum systems

Abstract

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 , a conditional entanglement monotone that measures the genuine quantum correlations between two subregions, conditioned on a third region, in a many-body quantum state. 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 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.
Paper Structure (2 sections, 5 theorems, 36 equations, 2 figures, 1 table)

This paper contains 2 sections, 5 theorems, 36 equations, 2 figures, 1 table.

Table of Contents

  1. Appendix
  2. Supplemental Materials

Key Result

Proposition 1

For a many-body quantum system $ABCD$ composite of four regions, $T_{\mathop{\mathrm{\mathrm{sq}}}\nolimits}$ satisfies the following properties: 1) Convexity: For a state $\rho_{ABCD}$ expressed as a convex mixture $\sum_{x}p_X(x)\rho^x_{ABCD}$ of quantum states 2) Faithfulness: For a state $\rho_{ABCD}$, $T_{\mathop{\mathrm{\mathrm{sq}}}\nolimits}(A;C|B)_{\rho}=0$ iff there exists a state exten

Figures (2)

  • Figure 1: (a) Generation of tripartite correlations for the initial state $\ket{\psi_{0}}$ quenched under Eq. \ref{['eq:TFIM']} with $\gamma=0.5$. (b) As $h(t)$ ramps up, the system crosses the critical point, evident from the rapid suppression of the longitudinal magnetization order $\langle Z\rangle = N^{-1}\sum_i\langle Z\rangle$, hence marking the transition to a paramagnetic regime. (c) As purity decreases under $\gamma$, the gap widens, showing $T_{\text{sq}}$ effectively squashes classical correlations. (d) The average NN correlation function over time.
  • Figure 2: (a) For adjacent qubits, the quench induces local GHZ-type GME that quickly collapses due to $\gamma$ and monogamy as correlations spread. In contrast, $T_{\mathop{\mathrm{\mathrm{sq}}}\nolimits}$ remains positive, witnessing bipartite and GME. (b) For distant qubits, $\tau_{3} = 0$ as the correlations, mediated by quasiparticles, are bipartite. Under the quench, $T_{\mathop{\mathrm{\mathrm{sq}}}\nolimits}$ oscillates due to coherent recurrences in the non-local regime. In both scenarios, $T_{\mathop{\mathrm{\mathrm{sq}}}\nolimits}$ captures genuinely quantum correlations that $\tau_{3}$ misses.

Theorems & Definitions (9)

  • Definition 1
  • Proposition 1
  • Lemma 1
  • Theorem 1
  • Theorem 2
  • proof
  • proof
  • proof
  • Corollary 1