Cryptographic distinguishability measures for quantum-mechanical states

• Published in: IEEE transactions on information theory Vol. 45; no. 4; pp. 1216 - 1227
• Main Authors: Fuchs, C.A., van de Graaf, J.
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.05.1999; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Computer information security; Cryptographic protocols; Cryptography; Electrical engineering; Error analysis; History; Inequalities; Information security; Joining; Joining processes; Mutual information; Pattern recognition; Probability distribution; Quantum computing; Quantum cryptography; Quantum theory; Veins
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/18.761271

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 (1948) 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.