Cryptographic Distinguishability Measures for Quantum Mechanical States

TL;DR

This paper surveys quantum distinguishability measures rooted in classical concepts, explores their interrelations, and introduces a unified notion of cryptographic exponential indistinguishability relevant for quantum cryptography security analysis.

Contribution

It provides a comprehensive survey of quantum distinguishability measures, establishes inequalities among them, and defines a unified cryptographic indistinguishability concept for quantum states.

Findings

Derived inequalities relating quantum distinguishability measures

Introduced a unified notion of cryptographic exponential indistinguishability

Potential application in analyzing quantum cryptographic protocol security

Abstract

This paper, mostly expository in nature, surveys four measures of distinguishability for quantum-mechanical states. This is done from the point of view of the cryptographer with a particular eye on applications in quantum cryptography. Each of the measures considered is rooted in an analogous classical measure of distinguishability for probability distributions: namely, the probability of an identification error, the Kolmogorov distance, the Bhattacharyya coefficient, and the Shannon distinguishability (as defined through mutual information). These measures have a long history of use in statistical pattern recognition and classical cryptography. We obtain several inequalities that relate the quantum distinguishability measures to each other, one of which may be crucial for proving the security of quantum cryptographic key distribution. In another vein, these measures and their connecting inequalities are used to define a single notion of cryptographic exponential indistinguishability for two families of quantum states. This is a tool that may prove useful in the analysis of various quantum cryptographic protocols.