Quantum information theory
Quantum hypothesis testing is instrumental in analyzing various problems in classical-quantum communication. In this line of research, we explore the existing connections between quantum and classical hypothesis testing, and how this connection refines our understanding of converse bounds in classical-quantum channels, and even enables the definition (and sometimes the construction) of optimal codes.
Multiple quantum hypothesis testing and classical-quantum channel converse bounds
In [1], we derive alternative exact expressions for the minimum error probability of a hypothesis test discriminating among M quantum states. The first expression corresponds to the error probability of a binary hypothesis test with certain parameters; the second involves the optimization of a given information-spectrum measure. Particularized in the setting of classical-quantum channel coding, this characterization implies the tightness of two existing converse bounds; one derived by Matthews and Wehner using hypothesis-testing, and one obtained by Hayashi and Nagaoka via an information-spectrum approach.
Generalized perfect codes for symmetric classical-quantum channels
In [2], we define a new family of codes for symmetric classical-quantum channels and establish their optimality. To this end, we extend the classical notion of generalized perfect and quasi-perfect codes to channels defined over some finite dimensional complex Hilbert output space. The resulting optimality conditions depend on the channel considered and on an auxiliary state defined on the output space of the channel. For certain N-qubit classical-quantum channels, we show that codes based on a generalization of Bell states are quasi-perfect and, therefore, they feature the smallest error probability among all codes of the same blocklength and cardinality.
Error Probability Trade-off in Quantum Hypothesis Testing via the Nussbaum-Szkoła Mapping
The error probability trade-off of quantum hypothesis testing is related to that of a certain surrogate classical hypothesis test via the Nussbaum-Szkoła mapping. This connection was used in the information-theoretic literature to establish the asymptotic error exponent of Bayesian quantum hypothesis testing and asymmetric quantum hypothesis testing (Hoeffding bound). In [3], we analyze the non-asymptotic gap between the error probability of a quantum test and the corresponding classical test via the Nussbaum-Szkoła mapping, showing that in certain scenarios the multiplicative gap can be as large as a factor of two in both types of error probabilities.