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arXiv:2305.20092 · PRX Quantum

Probing Postmeasurement Entanglement without Postselection

Samuel J. Garratt, Ehud Altman

Published Jul 18, 2024• 51 citations
arXivHTMLPDF

Abstract

We study the problem of observing quantum collective phenomena emerging from large numbers of measurements. These phenomena are difficult to observe in conventional experiments because, in order to distinguish the effects of measurement from dephasing, it is necessary to postselect on sets of measurement outcomes with Born probabilities that are exponentially small in the number of measurements performed. An unconventional approach, which avoids this exponential “postselection problem”, is to construct cross-correlations between experimental data and the results of simulations on classical computers. However, these cross-correlations generally have no definite relation to physical quantities. We first show how to incorporate classical shadows into this framework, thereby allowing for the construction of quantum information-theoretic cross-correlations. We then identify cross-correlations that both upper and lower bound the measurement-averaged von Neumann entanglement entropy, as well as cross-correlations that lower bound the measurement-averaged purity and entanglement negativity. These bounds show that experiments can be performed to constrain postmeasurement entanglement without the need for postselection. To illustrate our technique, we consider how it could be used to observe the measurement-induced entanglement transition in Haar-random quantum circuits. We use exact numerical calculations as proxies for quantum simulations and, to highlight the fundamental limitations of classical memory, we construct cross-correlations with tensor-network calculations at finite bond dimension. Our results reveal a signature of measurement-induced criticality that can be observed using a quantum simulator in polynomial time and with polynomial classical memory. Published by the American Physical Society 2024

DOI
10.1103/prxquantum.5.030311

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We study the problem of observing quantum collective phenomena emerging from large numbers of measurements. These phenomena are difficult to observe in conventional experiments because, in order to distinguish the effects of measurement from dephasing, it is necessary to postselect on sets of measurement outcomes with Born probabilities that are exponentially small in the number of measurements performed. An unconventional approach, which avoids this exponential “postselection problem”, is to construct cross-correlations between experimental data and the results of simulations on classical computers. However, these cross-correlations generally have no definite relation to physical quantities. We first show how to incorporate classical shadows into this framework, thereby allowing for the construction of quantum information-theoretic cross-correlations. We then identify cross-correlations that both upper and lower bound the measurement-averaged von Neumann entanglement entropy, as well as cross-correlations that lower bound the measurement-averaged purity and entanglement negativity. These bounds show that experiments can be performed to constrain postmeasurement entanglement without the need for postselection. To illustrate our technique, we consider how it could be used to observe the measurement-induced entanglement transition in Haar-random quantum circuits. We use exact numerical calculations as proxies for quantum simulations and, to highlight the fundamental limitations of classical memory, we construct cross-correlations with tensor-network calculations at finite bond dimension. Our results reveal a signature of measurement-induced criticality that can be observed using a quantum simulator in polynomial time and with polynomial classical memory. Published by the American Physical Society 2024
We address how one can empirically infer properties of quantum states generated by dynamics involving measurements. Our focus is on many-body settings where the number of measurements is extensive, making brute-force approaches based on postselection intractable due to their exponential sample complexity. We introduce a general-purpose scheme that can be used to infer any property of the postmeasurement ensemble of states (e.g., the average entanglement entropy, or frame potential) using a scalable number of experimental repetitions. We first identify a general class of “estimable properties” that can be directly extracted from experimental data. Then, based on empirical observations of such quantities, we show how one can indirectly infer information about any particular given nonestimable quantity of interest through classical postprocessing. Our approach is based on an optimization task, where one asks what are the minimum and maximum values that the desired quantity could possibly take, while ensuring consistency with observations. The true value of this quantity must then lie within a feasible range between these extrema, resulting in two-sided bounds. Narrow feasible ranges can be obtained by using a classical simulation of the device to determine which estimable properties one should measure. Even in cases where this simulation is inaccurate, unambiguous information about the true value of a given quantity realized on the quantum device can be learned. As an immediate application, we show that our method can be used to verify the emergence of quantum state designs in experiments. We identify some fundamental obstructions that in some cases prevent sharp knowledge of a given quantity from being inferred, and discuss what can be learned in cases where classical simulation is too computationally demanding to be feasible. In particular, we prove that any observer who cannot perform a classical simulation cannot distinguish the output states from those sampled from a maximally structureless ensemble. Published by the American Physical Society 2024
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Information-theoretic phase transitions, such as the measurement-induced phase transition (MIPT), characterize the robustness of quantum dynamics to local monitoring and are naturally formulated in terms of trajectories conditioned on typical measurement outcomes, which are naively accessible only through postselection. Here, we implement forced measurements to investigate how explicit postselection alters the nature of the transition. We find that postselection fundamentally alters the universality class by reweighting trajectories that are otherwise rare. In particular, we obtain a correlation-length exponent ν≈2.1ν\approx 2.1ν≈2.1 larger than that of the standard MIPT and a negative effective central charge ceff≈−0.35c_\mathrm{eff}\approx -0.35ceff​≈−0.35. We also compare the postselected MIPT to the entanglement transition of random tensor networks, and demonstrate that their universality class is the same. This setup further allows time-periodic, translationally invariant circuits with postselected weak measurements. In both models, we find that an on-site dimension of at least 3 (qutrits but not qubits) is necessary to induce a transition.
We present a technique for enhancing the estimation of quantum state properties by incorporating approximate prior knowledge about the quantum state. This consists in performing randomized measurements on a quantum processor and comparing the results with those obtained from a classical computer that stores an approximation of the quantum state. We provide unbiased estimators for expectation values of multicopy observables and present performance guarantees in terms of variance bounds that depend on the prior knowledge accuracy. We demonstrate the effectiveness of our approach through experimental and numerical examples detecting mixed-state entanglement, and estimating polynomial approximations of the von Neumann entropy and state fidelities. Published by the American Physical Society 2024
Measurement-induced entanglement (MIE) captures how local measurements generate long-range quantum correlations and drive dynamical phase transitions in many-body systems. Yet estimating MIE experimentally remains challenging: direct evaluation requires extensive post-selection over measurement outcomes, raising the question of whether MIE is accessible with only polynomial resources. We address this challenge by reframing MIE detection as a data-driven learning problem that assumes no prior knowledge of state preparation. Using measurement records alone, we train a neural network in a self-supervised manner to predict the uncertainty metric for MIE--the gap between upper and lower bounds of the average post-measurement bipartite entanglement. Applied to random circuits with one-dimensional all-to-all connectivity, our method reveals a learnability transition with increasing circuit depth: below a threshold the MIE can be effectively learned with resources that grow only polynomially with system size, whereas above it the required resources grow exponentially. This computational phase transition coincides with the breakdown of efficient classical simulation of the underlying quantum state. We further observe signatures of this transition on current noisy quantum devices. These results highlight the power of data-driven approaches for learning MIE and delineate the practical limits of its classical learnability.
We investigate prospects of employing the linear cross entropy to experimentally access measurement-induced phase transitions without requiring any postselection of quantum trajectories. For two random circuits that are identical in the bulk but with different initial states, the linear cross entropy χ between the bulk measurement outcome distributions in the two circuits acts as an order parameter, and can be used to distinguish the volume law from area law phases. In the volume law phase (and in the thermodynamic limit) the bulk measurements cannot distinguish between the two different initial states, and χ=1. In the area law phase χ<1. For circuits with Clifford gates, we provide numerical evidence that χ can be sampled to accuracy ϵ from O(1/ϵ^{2}) trajectories, by running the first circuit on a quantum simulator without postselection, aided by a classical simulation of the second. We also find that for weak depolarizing noise the signature of the measurement-induced phase transitions is still present for intermediate system sizes. In our protocol we have the freedom of choosing initial states such that the "classical" side can be simulated efficiently, while simulating the "quantum" side is still classically hard.