We present a universal technique for quantum-state estimation based on the maximum-likelihood method. This approach provides a positive-definite estimate for the density matrix from a sequence of measurements performed on identically prepared copies of the system. The method is versatile and can be applied to multimode radiation fields as well as to spin systems. The incorporation of physical constraints, which is natural in the maximum-likelihood strategy, leads to a substantial reduction of statistical errors. Numerical implementation of the method is based on a particular form of the Gauss decomposition for positive-definite Hermitian matrices.

We address the problem of discriminating with minimal error probability two given quantum operations. We show that the use of entangled input states generally improves the discrimination. For Pauli channels we provide a complete comparison of the optimal strategies where either entangled or unentangled input states are used.

In a recent paper, Walgate et. al. demonstrated that any two orthogonal multipartite pure states can be optimally distinguished using only local operations. We utilise their result to show that this is true for any two multiparty pure states, in the sense of inconclusive discrimination. There are also certain regimes of conclusive discrimination for which the same also applies, although we can only conjecture that the result is true for all conclusive regimes. We also discuss a class of states that can be distinguished locally according to any discrimination measure, as they can be locally recreated in the hands of one party. A consequence of this is that any two maximally entangled states can always be optimally discriminated locally, according to any figure of merit.

We show the rather counterintuitive result that entangled input states can strictly enhance the distinguishability of two entanglement-breaking channels.

We propose a covariant protocol for transmitting reference frames encoded on $N$ spins, achieving sensitivity $N^{-2}$ without the need of a preestablished reference frame and without using entanglement between sender and receiver. The protocol exploits the use of equivalent representations that were overlooked in the previous literature.