barriga2424

What I really want to discuss in this talk is hypothesis testing in quantum information theory. I will mostly restrict myself to the case of two simple hypotheses, each of which is represented by a quantum state (density matrix). The plan is to first review the basics and the difficulties of quantum hypothesis testing, covering both one-shot (Neyman- Pearson-Helstrom) and asymptotic (Stein, Chernoff) settings. Furthermore, I will indicate some applications in quantum Shannon theory, from coding theorems to strong converses and finite block-length analysis.

The main focus of the talk, however, will be on the peculiarity of quantum mechanics that we need to specify a measurement to learn about the quantum state, and that these measurements are often subject to physically motivated restrictions. Thus, a tension arises between a 'strong' set of measurements (decision rules) with a 'weak' set, i.e. all theoretically allowed measurements vs the restricted ones. This is actually a phenomenon arising generically in statistics; the classical analogue would be secret sharing, in which two perfectly distinguishable multi-partite hypotheses appear to be indistinguishable when accessing only a marginal. The quantum versions are richer in that, for instance, local operations and classical communication (LOCC) allow for state tomography, so the states cannot become perfectly indistinguishable but only nearly so, and hence the question is one of efficiency. We will focus on a couple of concrete examples and associated sets of ideas, including open problems:

1. Local operation and classical communication (LOCC): I will show simple arguments to bound the minimum bias of any LOCC test in a bipartite setting of two d-dimensional systems, some of which can even be extended to probabilistic theories beyond quantum mechanics. While these results are in the one-shot setting, in certain cases we also have a good understanding of the asymptotic case (Chernoff exponent).

2. Gaussian operations and classical computation (GOCC): These are motivated by the so-called linear optics describing most of quantum optics, but which cannot distinguish optimally even two coherent states of a single mode of light. Recently, we found states which are almost perfectly distinguishable by suitable measurements, but when restricted to GOCC, i.e. linear optics and post-processing, the states appear almost identical. The construction is probabilistic and relies on coding arguments; one of its interesting features is that the states can be prepared by GOCC, though as we show, they cannot be distinguished by GOCC. Open questions include whether one can give a constructive version of the argument, and whether even thermal states can be used, and how efficient the hiding is.

3. Returning to LOCC: It was known for a while that, asymptotically, n bits can be hidden in a bipartite system of n qubits each. Only recently it was shown that this is asymptotically optimal, by using the calculus of min-entropies. This is reminiscent of the fact that in secret sharing, each relevant share has to be at least as large as the hidden message. Indeed, we get bounds on the so-called data hiding capacity of any preparation system; these are, however, not always tight. While it is known that data hiding by separable states is possible (i.e., the state preparation can be done by LOCC), it is open whether the optimal information efficiency of one bit per local qubit can be achieved by separable hiding states.