State-Independent Error-Disturbance Tradeoff For Measurement Operators

TL;DR

This paper introduces a state-independent way to quantify measurement disturbance in quantum systems, establishing a trade-off relation between error and disturbance in finite-dimensional Hilbert spaces, with potential generalizations.

Contribution

It proposes a novel, state-independent disturbance measure based on classical distribution distinguishability and derives an error-disturbance trade-off relation for finite-dimensional quantum measurements.

Findings

Trade-off relation holds for 2D Hilbert space measurements.

Relation extends to measurements with mutually unbiased bases.

Error must be zero to minimize total error and disturbance.

Abstract

In general, classical measurement statistics of a quantum measurement is disturbed by performing an additional incompatible quantum measurement beforehand. Using this observation, we introduce a state-independent definition of disturbance by relating it to the distinguishability problem between two classical statistical distributions --- one resulting from a single quantum measurement and the other from a succession of two quantum measurements. Interestingly, we find an error-disturbance trade-off relation for any measurements in two-dimensional Hilbert space and for measurements with mutually unbiased bases in any finite-dimensional Hilbert space. This relation shows that error should be reduced to zero in order to minimize the sum of error and disturbance. We conjecture that a similar trade-off relation can be generalized to any measurements in an arbitrary finite-dimensional Hilbert space.