Past observable dynamics of a continuously monitored qubit

TL;DR

This paper reveals that for a continuously monitored qubit, the past measurement record better approximates a shifted, smoothed estimate of the observable, converging to a generalized weak value without postselection, and can outperform full quantum state knowledge.

Contribution

It introduces a novel interpretation of the past measurement record as a smoothed estimate converging to a weak value, challenging traditional views on quantum measurement records.

Findings

The past record approximates a shifted Gaussian process centered at a smoothed observable.

The smoothed estimate converges to the real part of a generalized weak value in the continuous limit.

Smoothed estimates from incomplete information can outperform full quantum state knowledge.

Abstract

Monitoring a quantum observable continuously in time produces a stochastic measurement record that noisily tracks the observable. For a classical process such noise may be reduced to recover an average signal by minimizing the mean squared error between the noisy record and a smooth dynamical estimate. We show that for a monitored qubit this usual procedure returns unusual results. While the record seems centered on the expectation value of the observable during causal generation, examining the collected past record reveals that it better approximates a moving-mean Gaussian stochastic process centered at a distinct (smoothed) observable estimate. We show that this shifted mean converges to the real part of a generalized weak value in the time-continuous limit without additional postselection. We verify that this smoothed estimate minimizes the mean squared error even for individual measurement realizations. We go on to show that if a second observable is weakly monitored concurrently, then that second record is consistent with the smoothed estimate of the second observable based solely on the information contained in the first observable record. Moreover, we show that such a smoothed estimate made from incomplete information can still outperform estimates made using full knowledge of the causal quantum state.