On the central limit theorem for unsharp quantum random variables

TL;DR

This paper investigates the convergence properties of unsharp quantum measurement-generated random variables, demonstrating that their distributions approximate mixtures of normal distributions and showing entanglement can improve convergence rates.

Contribution

It provides a de Finetti-type theorem for separable states and shows how entanglement can enhance convergence rates in quantum measurements.

Findings

Distribution approximates mixture of normal distributions

Convergence rate is at best 1/√N for separable states

Entangled inputs can improve convergence rates

Abstract

In this letter we study the weak-convergence properties of random variables generated by unsharp quantum measurements. More precisely, for a sequence of random variables generated by repeated unsharp quantum measurements, we study the limit distribution of relative frequency. We provide the de Finetti-type representation theorem for all separable states, showing that the distribution can be well approximated by mixture of normal distributions. No symmetry restrictions, such as the permutational invariance were needed. Furthermore, we investigate the convergence rates and show that the relative frequency can stabilize to some constant at best at the rate of order $1/\sqrt{N}$ for all separable inputs. On the other hand, we provide an example of a strictly unsharp quantum measurement where the better rates are achieved by using entangled inputs. This means that in certain cases the noise generated by the measurement process can be suppressed by using entanglement. We deliver our result in the form of quantum information task where the player achieves the goal with certainty in the limiting case by using entangled inputs or fails with certainty by using separable inputs.