TY - GEN

T1 - All inequalities for the relative entropy

AU - Ibinson, Ben

AU - Linden, Noah

AU - Winter, Andreas

PY - 2006

Y1 - 2006

N2 - The relative entropy of two distributions of n random variables, and more generally of two n-party quantum states, is an important quantity exhibiting, for example, the extent to which the two distributions/states are different. The relative entropy of the states formed by restricting to a smaller number m of parties is always less than or equal to the relative entropy of the two original n-party states. This is the monotonicity of relative entropy. Using techniques from convex geometry, we prove that monotonicity under restrictions is the only general inequality satisfied by relative entropies. In doing so we make a connection to secret sharing schemes with general access structures: indeed, it turns out that the extremal rays of the cone defined by monotonicity are populated by classical secret sharing schemes. A suprising outcome is that the structure of allowed relative entropy values of subsets of multiparty states is much simpler than the structure of allowed entropy values. And the structure of allowed relative entropy values (unlike that of entropies) is the same for classical probability distributions and quantum states.

AB - The relative entropy of two distributions of n random variables, and more generally of two n-party quantum states, is an important quantity exhibiting, for example, the extent to which the two distributions/states are different. The relative entropy of the states formed by restricting to a smaller number m of parties is always less than or equal to the relative entropy of the two original n-party states. This is the monotonicity of relative entropy. Using techniques from convex geometry, we prove that monotonicity under restrictions is the only general inequality satisfied by relative entropies. In doing so we make a connection to secret sharing schemes with general access structures: indeed, it turns out that the extremal rays of the cone defined by monotonicity are populated by classical secret sharing schemes. A suprising outcome is that the structure of allowed relative entropy values of subsets of multiparty states is much simpler than the structure of allowed entropy values. And the structure of allowed relative entropy values (unlike that of entropies) is the same for classical probability distributions and quantum states.

UR - http://www.scopus.com/inward/record.url?scp=39049132515&partnerID=8YFLogxK

U2 - 10.1109/ISIT.2006.261841

DO - 10.1109/ISIT.2006.261841

M3 - Conference contribution

AN - SCOPUS:39049132515

SN - 1424405041

SN - 9781424405046

T3 - IEEE International Symposium on Information Theory - Proceedings

SP - 237

EP - 241

BT - Proceedings - 2006 IEEE International Symposium on Information Theory, ISIT 2006

T2 - 2006 IEEE International Symposium on Information Theory, ISIT 2006

Y2 - 9 July 2006 through 14 July 2006

ER -