Probing post-measurement entanglement without post-selection

TL;DR

This paper introduces a method to observe post-measurement entanglement in quantum systems without relying on post-selection, using cross-correlations with classical simulations and shadow tomography, enabling feasible experimental detection of measurement-induced phenomena.

Contribution

It develops a framework combining shadow tomography and cross-correlations to bound entanglement measures, avoiding exponential post-selection and enabling practical observation of measurement-induced entanglement transitions.

Findings

Bounded measurement-averaged entanglement entropy and negativity via cross-correlations.

Demonstrated feasibility of observing measurement-induced criticality with polynomial resources.

Identified fundamental limitations of classical memory in simulating quantum measurement effects.

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 post-select on sets of measurement outcomes whose Born probabilities are exponentially small in the number of measurements performed. An unconventional approach, which avoids this exponential `post-selection 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 shadow tomography into this framework, thereby allowing for the construction of quantum information-theoretic cross-correlations. We then identify cross-correlations which both upper and lower bound the measurement-averaged von Neumann entanglement entropy, and additional cross-correlations which lower bound the measurement-averaged purity and entanglement negativity. These bounds show that experiments can be performed to constrain post-measurement entanglement without the need for post-selection. 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.