Quantum observables over time for information recovery

TL;DR

This paper introduces quantum observables over time (QOOT) to analyze quantum information dynamics, establishing conditions for their definition, and developing a recovery protocol that outperforms existing error mitigation methods.

Contribution

It defines QOOT as a dual to quantum states over time, characterizes their conditions, and constructs a recovery protocol with optimal sampling overhead.

Findings

QOOTs can be used to define time-reversal for quantum channels.

The proposed recovery protocol retrieves noiseless expectation values.

Our method outperforms probabilistic error cancellation in efficiency.

Abstract

We introduce the concept of quantum observables over time (QOOT), an operator that jointly describes two observables at two distinct time points, as a dual of the quantum state over time formalism. We provide a full characterization of the conditions under which a QOOT can be properly defined, via a no-go theorem. We use QOOTs to establish a notion of time-reversal for generic quantum channels with respect to a reference observable, enabling the systematic construction of recovery maps that preserve the latter. These recovery maps, although generally non-physical, can be decomposed into realizable channels, enabling their application in noiseless expectation value estimation tasks. We provide explicit examples and compare our protocol with other error mitigation methods. We show that our protocol retrieves the noiseless expectation value of the reference observable and can achieve optimal sampling overhead, outperforming probabilistic error cancellation.