How many copies are needed for state discrimination?

Given a collection of states (rho_1, ..., rho_N) with pairwise fidelities F(rho_i, rho_j) <= F < 1, we show the existence of a POVM that, given rho_i^{otimes n}, will identify i with probability >= 1-epsilon, as long as n>=2(log N/eps)/log (1/F). This improves on previous results which were either dimension-dependent or required that i be drawn from a known distribution.Comment: 1 page, submitted to QCMC'06, answer is O(log # of states

A lower bound on the probability of error in quantum state discrimination

We give a lower bound on the probability of error in quantum state discrimination. The bound is a weighted sum of the pairwise fidelities of the states to be distinguished.Comment: 4 pages; v2 fixes typos and adds remarks; v3 adds a new referenc

The Optimal Single Copy Measurement for the Hidden Subgroup Problem

The optimization of measurements for the state distinction problem has recently been applied to the theory of quantum algorithms with considerable successes, including efficient new quantum algorithms for the non-abelian hidden subgroup problem. Previous work has identified the optimal single copy measurement for the hidden subgroup problem over abelian groups as well as for the non-abelian problem in the setting where the subgroups are restricted to be all conjugate to each other. Here we describe the optimal single copy measurement for the hidden subgroup problem when all of the subgroups of the group are given with equal a priori probability. The optimal measurement is seen to be a hybrid of the two previously discovered single copy optimal measurements for the hidden subgroup problem.Comment: 8 pages. Error in main proof fixe

Quantum algorithms for hidden nonlinear structures

Attempts to find new quantum algorithms that outperform classical computation have focused primarily on the nonabelian hidden subgroup problem, which generalizes the central problem solved by Shor's factoring algorithm. We suggest an alternative generalization, namely to problems of finding hidden nonlinear structures over finite fields. We give examples of two such problems that can be solved efficiently by a quantum computer, but not by a classical computer. We also give some positive results on the quantum query complexity of finding hidden nonlinear structures.Comment: 13 page