Test-measured Rényi divergences

Abstract

One possibility of defining a quantum Rényi $\alpha$-divergence of two quantum states is to optimize the classical Rényi $\alpha$-divergence of their post-measurement probability distributions over all possible measurements (measured Rényi divergence), and maybe regularize these quantities over multiple copies of the two states (regularized measured Rényi $\alpha$-divergence). A key observation behind the theorem for the strong converse exponent of asymptotic binary quantum state discrimination is that the regularized measured Rényi $\alpha$-divergence coincides with the sandwiched Rényi $\alpha$-divergence when $\alpha >1$. Moreover, it also follows from the same theorem that to achieve this, it is sufficient to consider $2$-outcome measurements (tests) for any number of copies (this is somewhat surprising, as achieving the measured Rényi $\alpha$-divergence for $n$ copies might require a number of measurement outcomes that diverges in $n$, in general). In view of this, it seems natural to expect the same when $\alpha <1$; however, we show that this is not the case. In fact, we show that even for commuting states (classical case) the regularized quantity attainable using $2$-outcome measurements is in general strictly smaller than the Rényi $\alpha$-divergence (which is unique in the classical case). In the general quantum case this shows that the above "regularized test-measured" Rényi $\alpha$-divergence is not even a quantum extension of the classical Rényi divergence when $\alpha <1$, in sharp contrast to the $\alpha >1$ case.

v3: 30 pages, minor improvements. Thanks to a comment by an anonymous reviewer, we can now show that the two different ways to regularize the test-measured Rényi $\alpha$-divergence lead to different quantities

References in corpus (6)

• The Quantum Chernoff Bound
• The Chernoff lower bound for symmetric quantum hypothesis testing
• Error Exponent in Asymmetric Quantum Hypothesis Testing and Its Application to Classical-Quantum Channel coding
• The Converse Part of The Theorem for Quantum Hoeffding Bound
• Error exponents in hypothesis testing for correlated states on a spin chain
• Pretty good measures in quantum information theory

Cited by in corpus (6)

• Some continuity properties of quantum Rényi divergences
• Geometric relative entropies and barycentric Rényi divergences
• Quantum Rényi divergences and the strong converse exponent of state discrimination in operator algebras
• Locally-Measured Rényi Divergences
• Multivariate Fidelities
• Super-exponential distinguishability of correlated quantum states