Quantum Doeblin Coefficients: Interpretations and Applications

Christoph Hirche, Ian George, Theshani Nuradha, Mark Wilde

Abstract:

In classical information theory, the Doeblin coefficient of a classical channel
provides an efficiently computable upper bound on the total-variation contraction
coefficient of the channel, leading to what is known as a strong data-processing
inequality. Here, we investigate quantum Doeblin coefficients as a generalization
of the classical concept. In particular, we define various new quantum Doeblin
coefficients, one of which has several desirable properties, including concatenation
and multiplicativity, in addition to being efficiently computable. We also develop
various interpretations of two of the quantum Doeblin coefficients, including representations
as minimal singlet fractions, exclusion values, reverse max-mutual and
oveloH informations, reverse robustnesses, and hypothesis testing reverse mutual
and oveloH informations. Our interpretations of quantum Doeblin coefficients as
either entanglement-assisted or unassisted exclusion values are particularly appealing,
indicating that they are proportional to the best possible error probabilities
one could achieve in state-exclusion tasks by making use of the channel. We also
outline various applications of quantum Doeblin coefficients, ranging from limitations
on quantum machine learning algorithms that use parameterized quantum
circuits (noise-induced barren plateaus), on error mitigation protocols, on
the sample complexity of noisy quantum hypothesis testing, on the fairness of
noisy quantum models, and on mixing, indistinguishability, and decoupling times
of time-varying channels. All of these applications make use of the fact that
quantum Doeblin coefficients appear in upper bounds on various trace-distance
contraction coefficients of a quantum channel. Furthermore, in all of these applications,
our analysis using quantum Doeblin coefficients provides improvements of
various kinds over contributions from prior literature, both in terms of generality
and being efficiently computable.