One-Shot Yield-Cost Relations in General Quantum Resource Theories

Abstract

Although it is well known that the amount of resources that can be asymptotically distilled from a quantum state or channel does not exceed the resource cost needed to produce it, the corresponding relation in the non-asymptotic regime is not well understood. Here, we establish a quantitative relation between the one-shot distillable resource yield and dilution cost in terms of transformation errors involved in these processes. Notably, our bound is applicable to quantum state and channel manipulation with respect to any type of quantum resource and any class of free transformations thereof, encompassing broad types of settings, including entanglement, quantum thermodynamics, quantum communication, and Bell non-locality. We also show that our techniques provide strong converse bounds relating the distillable resource and resource dilution cost in the asymptotic regime. Moreover, we introduce a class of channels that generalize twirling maps encountered in many resource theories, and by directly connecting it with resource quantification, we compute analytically several smoothed resource measures and improve our one-shot yield-cost bound in relevant theories. We use these operational insights to exactly evaluate important measures for various resource states in the resource theory of magic states.

Quantifying dynamical magic with completely stabilizer preserving operations as free

“…Let b be the (3×1) Bloch vector corresponding to the state σ. Then, the state ρ can be converted to the state σ using CSPOs if there exists an x ∈ R 31 + such that Ax = b.…”

Tight constraints on probabilistic convertibility of quantum states

“…(We refer the interested reader to Ref. [78] where the properties of such states are explained from the perspective of a class of twirling maps applicable to general resource theories.) We note, however, a difference between Theorem 19 and the finding of Section IV where the states also appear -in Theorem 13, we only needed the resource theory to be affine to ensure the optimality of , while (ii) here is a more demanding criterion.…”