TY - JOUR

T1 - The operational meaning of min- and max-entropy

AU - König, Robert

AU - Renner, Renato

AU - Schaffner, Christian

N1 - Funding Information:
Manuscript received August 15, 2008; revised May 04, 2009. Current version published August 19, 2009. The work of R. König was supported by the National Science Foundation (NSF) under Grants PHY-0456720 and PHY-0803371. The work of C. Schaffner was supported by the European Union fifth framework project QAP IST 015848 and the NWO VICI project 2004–2009. The work of R. Renner was supported by the Swiss National Science Foundation under Grant 200021-119868.

PY - 2009

Y1 - 2009

N2 - In this paper, we show that the conditional min-entropy Hmin (A\B) of a bipartite state ρ AB is directly related to the maximum achievable overlap with a maximally entangled state if only local actions on the B-part of ρ AB 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 Hmax(A\B) to the maximum fidelity of ρ AB 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.

AB - In this paper, we show that the conditional min-entropy Hmin (A\B) of a bipartite state ρ AB is directly related to the maximum achievable overlap with a maximally entangled state if only local actions on the B-part of ρ AB 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 Hmax(A\B) to the maximum fidelity of ρ AB 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.

KW - Entropy measures

KW - Max-entropy

KW - Min-entropy

KW - Operational interpretations

KW - Quantum hypothesis testing

KW - Quantum information theory

KW - Single-shot information theory

KW - Singlet fraction

UR - http://www.scopus.com/inward/record.url?scp=69449101829&partnerID=8YFLogxK

U2 - 10.1109/TIT.2009.2025545

DO - 10.1109/TIT.2009.2025545

M3 - Article

AN - SCOPUS:69449101829

SN - 0018-9448

VL - 55

SP - 4337

EP - 4347

JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

IS - 9

ER -