Jon Tyson, Institute for Quantum Computing
Abstract
The problem of estimating the distinguishability of a collection of quantum states is one of the oldest in quantum information theory, dating back to theoretical studies of optimal detectors from the 1960s and 70s. Furthermore, as observed by Barnum and Knill (2002), studies of state distinguishability may be seen as a "simple" testing ground for the more general problem of estimating the reversibility of an arbitrary quantum channel in the sense of entanglement fidelity. (This generalization is sometimes is called "approximate quantum error recovery.")
This talk will exploit a natural inner product structure on the space of generalized measurements to derive explicit and provably tight two-sided bounds on the minimum-error distinguishability of arbitrary ensembles of mixed quantum states. This framework ties together the early results of Holevo (1978) and Curlander (1979) on the pure-state case with the work of Jezek-Fiurasek-Rehacek-Hradil (2002, 2004) and Ogawa-Nagaoka (1999). Furthermore, a mixed-state generalization of Holevo's asymptotically optimal measurement is derived. (Holevo's measurement has often been confused with the so-called "pretty good" measurement of Belavkin (1975) and Hausladen-Wootters (1994).) A conjecture of Schumacher and Westmoreland (1997) is resolved.
The methods introduced have further application to estimating the reversibility of quantum channels and the conditional min-entropy of bipartite quantum states. A technical discussion of these generalizations is left for future discussion.