On repeated (continuous) weak measurements of a single copy of an unknown quantum state

TL;DR

This paper studies repeated weak measurements on a single unknown quantum state, showing they cannot fully determine the state without ensemble measurements, and explores the tradeoff between measurement error and invasiveness.

Contribution

It provides a detailed analysis of repeated weak measurements on a single quantum state, including error quantification, the error-invasiveness tradeoff, and the distribution of measurement outcomes.

Findings

Repeated weak measurements do not suffice to determine the unknown state.

A precise error-invasiveness tradeoff is established.

Analytical expressions relate outcome distributions to strong measurements.

Abstract

In this paper we investigate repeated weak measurements,without post-selection, on a \emph{single copy} of an \emph{unknown} quantum state. The resulting random walk in state space is precisely characterised in terms of joint probabilities for outcomes. We conclusively answer, in the negative, the very important question whether the statistics of such repeated measurements can determine the unknown state. We quantify the notion of error in this context as the departure of a suitably averaged density matrix from the initial state. When the number of weak measurements is small the original state is preserved to a great degree, but only an ensemble of such measurements, of a complete set of observables, can determine the unknown state. By a careful analysis of errors, it is shown that there is a precise tradeoff between errors and \emph{invasiveness}. Lower the errors, greater the invasiveness. Though the outcomes are not independently distributed, an analytical expression is obtained for how averages are distributed, which is shown to be the way outcomes are distributed in a \emph{strong measurement}. An \emph{error-disturbance} relation, though not of the Ozawa-type, is also derived. In the limit of vanishing errors, the invasiveness approaches what would obtain from strong measurements.