Cryptographic Distinguishability Measures for Quantum Mechanical States

TL;DR

The paper studies how to quantify how close or far quantum states are from one another in cryptographic contexts, by surveying four measures (PE, K, B, SD) that generalize classical distribution-distinguishability notions. It frames these measures within density matrices and POVMs, derives explicit quantum forms for PE, K, and B, and discusses the (often intractable) SD optimization, along with a set of inequalities that relate the measures. A key result is that exponential indistinguishability across any one of the measures implies the same for the others, providing a canonical quantum notion of indistinguishability for state families. The work highlights applications to quantum cryptographic security analyses and protocol design, including parity-related arguments in quantum key distribution. Overall, it provides a unified framework and useful bounds for analyzing how distinguishable quantum state ensembles are in cryptographic settings.

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.