Hilbert's projective metric in quantum information theory

TY - JOUR

T1 - Hilbert's projective metric in quantum information theory

AU - Reeb, David Franz

AU - Kastoryano, Michael James

AU - Wolf, Michael Marc

PY - 2011/8/4

Y1 - 2011/8/4

N2 - We introduce and apply Hilbert's projective metric in the context of quantum information theory. The metric is induced by convex cones such as the sets of positive, separable or positive partial transpose operators. It provides bounds on measures for statistical distinguishability of quantum states and on the decrease of entanglement under protocols involving local quantum operations and classical communication or under other cone-preserving operations. The results are formulated in terms of general cones and base norms and lead to contractivity bounds for quantum channels, for instance, improving Ruskai's trace-norm contraction inequality. A new duality between distinguishability measures and base norms is provided. For two given pairs of quantum states we show that the contraction of Hilbert's projective metric is necessary and sufficient for the existence of a probabilistic quantum operation that maps one pair onto the other. Inequalities between Hilbert's projective metric and the Chernoff bound, the fidelity and various norms are proven.

AB - We introduce and apply Hilbert's projective metric in the context of quantum information theory. The metric is induced by convex cones such as the sets of positive, separable or positive partial transpose operators. It provides bounds on measures for statistical distinguishability of quantum states and on the decrease of entanglement under protocols involving local quantum operations and classical communication or under other cone-preserving operations. The results are formulated in terms of general cones and base norms and lead to contractivity bounds for quantum channels, for instance, improving Ruskai's trace-norm contraction inequality. A new duality between distinguishability measures and base norms is provided. For two given pairs of quantum states we show that the contraction of Hilbert's projective metric is necessary and sufficient for the existence of a probabilistic quantum operation that maps one pair onto the other. Inequalities between Hilbert's projective metric and the Chernoff bound, the fidelity and various norms are proven.

U2 - 10.1063/1.3615729

DO - 10.1063/1.3615729

M3 - Journal article

SN - 0022-2488

VL - 52

SP - 082201

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

IS - 8

ER -