Operator algebra of the information-disturbance tradeoff in quantum measurements

TL;DR

The paper addresses how to quantify the information-disturbance tradeoff in quantum measurements within an operator-algebra framework. It introduces an explicit decomposition of a minimally disturbing measurement via $M_m = \sum_{a=0}^{d-1} \sqrt{p(m|a)} P_a$ into a superposition of orthogonal unitaries with coefficients $C_k = \frac{1}{d} \sum_{a=0}^{d-1} e^{-i 2\pi k a/d} \sqrt{p(m|a)}$, linking information gain to the associated disturbance through a discrete Fourier transform. The authors derive a tight bound on $p(a|m) \le \frac{1}{d} \left( \sum_k \sqrt{p(b+k|b,m)} \right)^2$ with $p(b+k|b,m) = p(m,b+k|b)/p(m|b)$ and show it is saturable by a uniform disturbance pattern, revealing the fundamental tradeoff. The framework clarifies how information leaks in quantum channels and provides tools for quick eavesdropping assessments in BB84-type protocols.

Abstract

Quantum measurements can be described by operators that assign conditional probabilities to different outcomes while also describing unavoidable physical changes to the system. Here, we point out that operators describing information gain at minimal disturbance can be expanded into a set of unitary operators representing experimentally distinguishable patterns of disturbance. The observable statistics of disturbance defines a tight upper bound on the information gain of the measurement.