The Operational Meaning of Min- and Max-Entropy

• Published in: IEEE transactions on information theory Vol. 55; no. 9; pp. 4337 - 4347
• Main Authors: Konig, R., Renner, R., Schaffner, C.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01-09-2009; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Applied sciences; Channel capacity; Classical and quantum physics: mechanics and fields; Communication channels; Cryptography; Entropy; Entropy measures; Error analysis; Exact sciences and technology; Information processing; Information security; Information theory; Information, signal and communications theory; max-entropy; Maximum entropy method; Merging; min-entropy; operational interpretations; Physics; Probability; Quantum entanglement; quantum hypothesis testing; Quantum information; quantum information theory; Quantum mechanics; Random variables; Signal and communications theory; single-shot information theory; singlet fraction; Telecommunications and information theory; Veins; single-shot information theory; Lower bound; quantum information theory; operational interpretations; Logarithmic function; min-entropy; Entropy; singlet fraction; Hypothesis test; Entropy measures; quantum hypothesis testing; Minimax method; Information processing; max-entropy; Secret key; Quantum information; Safety; Quantum theory; Information theory
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2009.2025545

In this paper, we show that the conditional min-entropy H min ( A | B ) of a bipartite state rhoAB is directly related to the maximum achievable overlap with a maximally entangled state if only local actions on the B -part of rhoAB are allowed. In the special case where A is classical, this overlap corresponds to the probability of guessing A given B . In a similar vein, we connect the conditional max-entropy H max ( A | B ) to the maximum fidelity of rhoAB with a product state that is completely mixed on A . In the case where A is classical, this corresponds to the security of A when used as a secret key in the presence of an adversary holding B . Because min- and max-entropies are known to characterize information-processing tasks such as randomness extraction and state merging, our results establish a direct connection between these tasks and basic operational problems. For example, they imply that the (logarithm of the) probability of guessing A given B is a lower bound on the number of uniform secret bits that can be extracted from A relative to an adversary holding B .