Abstract
This paper systematically develops the resource theory of asymmetric distinguishability, as initiated roughly a decade ago [K. Matsumoto, arXiv:1010.1030 (2010)]. The key constituents of this resource theory are quantum boxes, consisting of a pair of quantum states, which can be manipulated for free by means of an arbitrary quantum channel. We introduce bits of asymmetric distinguishability as the basic currency in this resource theory, and we prove that it is a reversible resource theory in the asymptotic limit, with the quantum relative entropy being the fundamental rate of resource interconversion. The distillable distinguishability is the optimal rate at which a quantum box consisting of independent and identically distributed (i.i.d.) states can be converted to bits of asymmetric distinguishability, and the distinguishability cost is the optimal rate for the reverse transformation. Both of these quantities are equal to the quantum relative entropy. The exact one-shot distillable distinguishability is equal to the min-relative entropy, and the exact one-shot distinguishability cost is equal to the max-relative entropy. Generalizing these results, the approximate one-shot distillable distinguishability is equal to the smooth min-relative entropy, and the approximate one-shot distinguishability cost is equal to the smooth max-relative entropy. As a notable application of the former results, we prove that the optimal rate of asymptotic conversion from a pair of i.i.d. quantum states to another pair of i.i.d. quantum states is fully characterized by the ratio of their quantum relative entropies.
Highlights
IntroductionThe ability to distinguish one possibility from another is what allows us to discover new scientific laws and make predictions of future possibilities
Distinguishability plays a central role in all sciences
The theory that we develop applies in the more general setting of quantum distinguishability, as it did in Refs. [5,6], in particular when the distributions p and q are replaced by quantum states ρ and σ, respectively, and the common transformations allowed on a quantum box (ρ, σ ) are quantum channels
Summary
The ability to distinguish one possibility from another is what allows us to discover new scientific laws and make predictions of future possibilities. In the process of scientific discovery, we form a hypothesis based on conjecture, which is to be tested against a conventional or null hypothesis by repeated trials or experiments. If the null hypothesis is accepted, one can form alternative hypotheses to test against the null hypothesis in future experiments. Repetition allows for increasing the distinguishability between the two hypotheses. If the two different hypotheses are relatively distinguishable, fewer trials are required to decide between the possibilities.
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