This thesis deals with assessing optimal decision strategies and error rates for the task of binary quantum channel discrimination, where the main focus lies on proving optimal asymmetric error exponents (both asymptotically and non-asymptotically) for different variants of this problem. The task of binary quantum channel discrimination is to distinguish a quantum channel (a quantum device with a quantum input and a quantum output), of which n identical copies of are given, into one of two classes. In the simplest instance the task is just to recognize the channel as one of two possible options. Even when the n copies of the channel are assumed to be independent and identically distributed (i.i.d.), the optimal joint input states will often be entangled, and hence this becomes a highly non-trivial non-i.i.d. hypothesis testing problem, which in certain cases can also be shown to require adaptivity for optimal discrimination performance. This thesis works on exploring the landscape of when adaptivity is helpful, and what can be said about optimal asymmetric error rates in various regimes. Concretely, first, in the already mentioned simple case of distinguishing between two single options, and for arbitrary (finite) number of channel uses, we prove a relation between the error rates of optimal adaptive and non-adaptive (so called parallel) strategies, with a mostly explicit construction of a "good" parallel strategy (in a certain quantified sense) given an adaptive one. Additionally, we show how optimal parallel strategies and error rates can be computed as a semi-definite program (SDP) that grows only polynomially in n. Next, we study the problem of composite channel discrimination, where the two classes to distinguish between can now be very general sets of channels, and which to our knowledge has so far not been studied even classically, despite its broad applicability to real-world discrimination tasks. Also here, we establish asymptotic asymmetric error exponents in multiple scenarios, together with some conditions on when they are equal (or not equal) between adaptive and non-adaptive strategies, and hence are able to give conditions under which adaptivity is asymptotically (not) helpful. Finally, we lift many of these quantum channel discrimination results (and also the information theoretical toolbox they are based on), so far studied only in finite dimensions, to the infinite-dimensional setting of general separable Hilbert spaces, and establish a general theory of quantum channel discrimination also in this setting. For this, we prove (among many other things) the chain rule for the geometric Rényi divergences in infinite dimensions, and then establish relations between error rates of adaptive and parallel strategies similar to the finite-dimensional case. While in finite dimensions we can establish asymptotic equality of the asymmetric error exponent for adaptive and parallel strategies for all pairs of channels, in infinite dimensions we can do this only under an additional finiteness condition, and we discuss this condition in detail.
Hypothesis testing, Quantum channel discrimination, Quantum channels, Quantum information theory
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