Transition probabilities and measurement statistics of postselected ensembles

TL;DR

This paper investigates the effects of intermediate quantum measurements with postselection on transition probabilities, characterizing possible outcome distributions and trade-offs between randomness and transition enhancement.

Contribution

It provides a detailed analysis of measurement statistics in postselected ensembles, establishing constraints and quantifying the limits of transition probability enhancement.

Findings

Measurement can increase transition probability but with limits

Trade-off exists between measurement randomness and transition enhancement

Maximum transition probability increase factor is 2

Abstract

It is well-known that a quantum measurement can enhance the transition probability between two quantum states. Such a measurement operates after preparation of the initial state and before postselecting for the final state. Here we analyze this kind of scenario in detail and determine which probability distributions on a finite number of outcomes can occur for an intermediate measurement with postselection, for given values of the following two quantities: (i) the transition probability without measurement, (ii) the transition probability with measurement. This is done for both the cases of projective measurements and of generalized measurements. Among other constraints, this quantifies a trade-off between high randomness in a projective measurement and high measurement-modified transition probability. An intermediate projective measurement can enhance a transition probability such that the failure probability decreases by a factor of up to 2, but not by more.