Information-disturbance tradeoff in quantum measurement on the uniform ensemble and on the mutually unbiased bases

TL;DR

This paper analyzes the fundamental tradeoff between information gain and disturbance in quantum measurements, showing convexity of the tradeoff frontier, optimal measurement dynamics, and properties of specific quantum state ensembles.

Contribution

It introduces a convex structure for the information-disturbance tradeoff, identifies optimal measurement dynamics, and characterizes certain quantum ensembles as spherical 2-designs.

Findings

The information-disturbance frontier is convex for any initial state distribution.

Least-disturbing measurements follow the square-root dynamics.

Uniform ensembles on mutually unbiased bases form spherical 2-designs.

Abstract

I consider the tradeoff between the information gained about an initially unknown quantum state, and the disturbance caused to that state by the measurement process. I show that for any distribution of initial states, the information-disturbance frontier is convex, and disturbance is nondecreasing with information gain. I consider the most general model of quantum measurements, and all post-measurement dynamics compatible with a given measurement. For the uniform initial distribution over states, I show that the least-disturbing way of making any measurement is with conditional dynamics satisfying a generalization of the projection postulate, the ``square-root dynamics.'' Thus, procedures for achieving a point on the information-disturbance frontier may be assumed to involve such conditional dynamics. Also, the information-disturbance frontier for the uniform ensemble may be achieved with ``isotropic'' (unitarily covariant) dynamics. This results in a significant simplification of the optimization problem for calculating the tradeoff in this case, giving hope for a closed-form solution. I also show that the discrete ensembles uniform on the d(d+1) vectors of a certain set of d+1 ``mutually unbiased'' or conjugate bases in d dimensions form spherical 2-designs in CP_{d-1} when d is a power of an odd prime. This implies that many of the results of the paper apply also to these discrete ensembles.