The Quantum Blackwell Theorem and Minimum Error State Discrimination

TL;DR

This paper extends the classical Blackwell Theorem to quantum channels, establishing a new way to compare their noisiness via state distinguishability and applying it to quantum cryptography security.

Contribution

It introduces a quantum analogue of the Blackwell Theorem linking channel noise to state distinguishability degradation and applies it to eavesdropper detection.

Findings

Quantum channel noisiness can be characterized by state distinguishability.

The method is equivalent to existing degradation measures of quantum channels.

Application demonstrated in quantum cryptography security analysis.

Abstract

A quantum analogue of the famous Blackwell Theorem in classical statistics has recently been proposed. Given two quantum channels A and B, a set of payoff functions have been proven to have values for B at least as high as they are for A if and only if there exists a quantum garbling channel E such that A=EB. When such a channel E exists, we can globally compare A and B in terms of their `noisiness'. We show that this method of channel noise comparison is equivalent to one obtained by considering the degradation of the distinguishability of states. Here, the channel A is said to be at least as noisy as the channel B if any ensemble of states, fed into each channel and possibly entangled with ancillae, emerges no more distinguishable from A than it does from channel B, where distinguishability is quantified by the minimum error discrimination probability. We also provide a novel application to eavesdropper detection in quantum cryptography.