Uncertainty Relations in Pre- and Post-Selected Systems

TL;DR

This paper derives a new uncertainty relation for pre- and post-selected quantum systems, revealing unusual properties like non-zero bounds for commuting observables and the ability to prepare sharply defined states for non-commuting observables.

Contribution

It introduces a Robertson-Heisenberg like uncertainty relation tailored for PPS systems with physical interpretations and demonstrates novel properties contrasting standard quantum mechanics.

Findings

Lower bound of uncertainty relation does not vanish for commuting observables in PPS.

PPS systems allow sharp pre-selection for non-commuting observables.

Applications include improved detection of mixed states and tighter bounds on correlation functions.

Abstract

In this work, we derive Robertson-Heisenberg like uncertainty relation for two incompatible observables in a pre- and post-selected (PPS) system. The newly defined standard deviation and the uncertainty relation in the PPS system have physical meanings which we present here. We demonstrate two unusual properties in the PPS system using our uncertainty relation. First, for commuting observables, the lower bound of the uncertainty relation in the PPS system does not become zero even if the initially prepared state i.e., pre-selection is the eigenstate of both the observables when specific post-selections are considered. This implies that for such case, two commuting observables can disturb each other's measurement results which is in fully contrast with the Robertson-Heisenberg uncertainty relation. Secondly, unlike the standard quantum system, the PPS system makes it feasible to prepare sharply a quantum state (pre-selection) for non-commuting observables {(to be detailed in the main text)}. Some applications of uncertainty and uncertainty relation in the PPS system are provided: $(i)$ detection of mixedness of an unknown state, $(ii)$ stronger uncertainty relation in the standard quantum system, ($iii$) ``purely quantum uncertainty relation" that is, the uncertainty relation which is not affected (i.e., neither increasing nor decreasing) under the classical mixing of quantum states, $(iv)$ state dependent tighter uncertainty relation in the standard quantum system, and $(v)$ tighter upper bound for the out-of-time-order correlation function.