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arXiv:2007.06305 · Physical Review Letters

Mixed-State Entanglement from Local Randomized Measurements

Andreas Elben, Richard Kueng, Hsin-Yuan Huang, Rick van Bijnen, Christian Kokail, Marcello Dalmonte, Pasquale Calabrese, Barbara Kraus, John P. Preskill, P. Zoller, Benoît Vermersch

Published Nov 11, 2020• 290 citations
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Abstract

We propose a method for detecting bipartite entanglement in a many-body mixed state based on estimating moments of the partially transposed density matrix. The estimates are obtained by performing local random measurements on the state, followed by postprocessing using the classical shadows framework. Our method can be applied to any quantum system with single-qubit control. We provide a detailed analysis of the required number of experimental runs, and demonstrate the protocol using existing experimental data [Brydges et al., Science 364, 260 (2019)SCIEAS0036-807510.1126/science.aau4963].

DOI
10.1103/physrevlett.125.200501

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We propose a method for detecting bipartite entanglement in a many-body mixed state based on estimating moments of the partially transposed density matrix. The estimates are obtained by performing local random measurements on the state, followed by postprocessing using the classical shadows framework. Our method can be applied to any quantum system with single-qubit control. We provide a detailed analysis of the required number of experimental runs, and demonstrate the protocol using existing experimental data [Brydges et al., Science 364, 260 (2019)SCIEAS0036-807510.1126/science.aau4963].
We propose a method for detecting bipartite entanglement in a many-body mixed state based on estimating moments of the partially transposed density matrix. The estimates are obtained by performing local random measurements on the state, followed by postprocessing using the classical shadows framework. Our method can be applied to any quantum system with single-qubit control. We provide a detailed analysis of the required number of experimental runs, and demonstrate the protocol using existing experimental data [Brydges et al., Science 364, 260 (2019)].
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
Quantum measurements are slow, while classical processors are fast, yet existing hybrid protocols never exploit this asymmetry. In this work, we propose an alternative formulation of classical estimators as online algorithms that are updated incrementally upon obtaining a new sample. Classical shadows are the natural fit for this approach: designed around the principle of measuring first and asking questions later, each snapshot is a self-contained classical description that can be processed immediately and independently. As a first demonstration, we focus on mixed state entanglement verification via PT-moments, moments of the partially transposed density matrix that provide experimentally accessible sufficient conditions for entanglement. We construct two unbiased online estimators that together characterize the fundamental challenge between memory footprint and per-shot computational cost: one scales to large systems at low moment order, the other handles high moment orders at the expense of memory exponential in system size. The online estimator certifies entanglement reliably and, by exploiting all (Tm)\binom{T}{m}(mT​) combinations of snapshots, requires fewer samples than state-of-the-art baselines, turning entanglement detection from a purely offline diagnostic into a protocol that runs concurrently with the experiment.
Abstract We propose an ordered set of experimentally accessible conditions for detecting entanglement in mixed states. The k -th condition involves comparing moments of the partially transposed density operator up to order k . Remarkably, the union of all moment inequalities reproduces the Peres-Horodecki criterion for detecting entanglement. Our empirical studies highlight that the first four conditions already detect mixed state entanglement reliably in a variety of quantum architectures. Exploiting symmetries can help to further improve their detection capabilities. We also show how to estimate moment inequalities based on local random measurements of single state copies (classical shadows) and derive statistically sound confidence intervals as a function of the number of performed measurements. Our analysis includes the experimentally relevant situation of drifting sources, i.e. non-identical, but independent, state copies.
For the certification and benchmarking of medium-size quantum devices, efficient methods to characterize entanglement are needed. In this context, it has been shown that locally randomized measurements on a multiparticle quantum system can be used to obtain valuable information on the so-called moments of the partially transposed quantum state. This allows one to infer some separability properties of a state, but how to use the given information in an optimal and systematic manner has yet to be determined. We propose two general entanglement detection methods based on the moments of the partially transposed density matrix. The first method is based on the Hankel matrices and provides a family of entanglement criteria, of which the lowest order reduces to the known p_{3}-positive-partial-transpose criterion proposed in A. Elben et al. [Phys. Rev. Lett. 125, 200501 (2020)PRLTAO0031-900710.1103/PhysRevLett.125.200501]. The second method is optimal and gives necessary and sufficient conditions for entanglement based on some moments of the partially transposed density matrix.
Moment-based relaxations of the positive partial transpose (PPT) criterion have been recently introduced, as a hierarchy of entanglement criteria involving only experimentally accessible quantities of a given bipartite state. The goal of this work is to study their typical detection performance on high-dimensional bipartite systems. Concretely, we investigate whether random bipartite mixed states on Cd⊗Cd\mathbb C^d\otimes\mathbb C^dCd⊗Cd, obtained as the marginal over an environment Cs\mathbb C^sCs of a uniformly distributed pure state, generically satisfy or violate them. For each fixed level m∈Nm\in\mathbb Nm∈N in this hierarchy of moment-based PPT criteria, we are able to identify a threshold environment dimension s=λmd2s=λ_md^2s=λm​d2 at which the behavior of the associated random state switches from violating to satisfying it, with probability going to 111 as ddd grows. The proof combines combinatorics of permutations techniques to estimate the average value of moments of partially transposed random states and concentration of measure arguments to bound the probability of deviating from such average, when the underlying local dimension ddd is large. We additionally need tools from the theory of Hankel determinant evaluation via orthogonal polynomials.