Many-Body Entropies and Entanglement from Polynomially Many Local Measurements,Physical Review X

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Many-Body Entropies and Entanglement from Polynomially Many Local Measurements
Physical Review X ( IF 11.6 ) Pub Date : 2024-08-26 , DOI: 10.1103/physrevx.14.031035
Benoît Vermersch _{1,

2,

3} , Marko Ljubotina ₄ , J. Ignacio Cirac _{5,

6} , Peter Zoller _{2,

3} , Maksym Serbyn ₄ , Lorenzo Piroli _{7,

8}

Affiliation

Estimating global properties of many-body quantum systems such as entropy or bipartite entanglement is a notoriously difficult task, typically requiring a number of measurements or classical postprocessing resources growing exponentially in the system size. In this work, we address the problem of estimating global entropies and mixed-state entanglement via partial-transposed (PT) moments and show that efficient estimation strategies exist under the assumption that all the spatial correlation lengths are finite. Focusing on one-dimensional systems, we identify a set of approximate factorization conditions (AFCs) on the system density matrix, which allow us to reconstruct entropies and PT moments from information on local subsystems. This identification yields a simple and efficient strategy for entropy and entanglement estimation. Our method could be implemented in different ways, depending on how information on local subsystems is extracted. Focusing on randomized measurements providing a practical and common measurement scheme, we prove that our protocol requires only polynomially many measurements and postprocessing operations, assuming that the state to be measured satisfies the AFCs. We prove that the AFCs hold for finite-depth quantum-circuit states and translation-invariant matrix-product density operators and provide numerical evidence that they are satisfied in more general, physically interesting cases, including thermal states of local Hamiltonians. We argue that our method could be practically useful to detect bipartite mixed-state entanglement for large numbers of qubits available in today’s quantum platforms.

Published by the American Physical Society

2024

中文翻译：

多项式多局部测量的多体熵和纠缠

众所周知，估计多体量子系统（如熵或二分纠缠）的全局属性是一项艰巨的任务，通常需要大量测量或经典后处理资源，使系统大小呈指数级增长。在这项工作中，我们解决了通过部分转置（PT）矩估计全局熵和混合状态纠缠的问题，并表明在所有空间相关长度都是有限的假设下存在有效的估计策略。专注于一维系统，我们在系统密度矩阵上确定了一组近似分解条件（AFC），这使我们能够从局部子系统的信息中重建熵和 PT 矩。这种识别产生了一种简单而有效的熵和纠缠估计策略。我们的方法可以以不同的方式实现，具体取决于如何提取本地子系统的信息。专注于随机测量，提供一种实用且通用的测量方案，我们证明我们的协议只需要多项式多测量和后处理操作，假设要测量的状态满足 AFC。我们证明了 AFC 适用于有限深度量子电路状态和平移不变矩阵乘积密度运算符，并提供数值证据证明它们在更一般的、物理上有趣的情况下得到满足，包括局部哈密顿量的热状态。我们认为，我们的方法对于检测当今量子平台中可用的大量量子比特的二分混合态纠缠可能非常有用。美国物理学会 2024 年出版

更新日期：2024-08-26

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