General Quantum Resource Theories: Distillation, Formation and Consistent Resource Measures

Abstract

Quantum resource theories (QRTs) provide a unified theoretical framework for understanding inherent quantum-mechanical properties that serve as resources in quantum information processing, but resources motivated by physics may possess structure whose analysis is mathematically intractable, such as non-uniqueness of maximally resourceful states, lack of convexity, and infinite dimension. We investigate state conversion and resource measures in general QRTs under minimal assumptions to figure out universal properties of physically motivated quantum resources that may have such mathematical structure whose analysis is intractable. In the general setting, we prove the existence of maximally resourceful states in one-shot state conversion. Also analyzing asymptotic state conversion, we discover catalytic replication of quantum resources, where a resource state is infinitely replicable by free operations. In QRTs without assuming the uniqueness of maximally resourceful states, we formulate the tasks of distillation and formation of quantum resources, and introduce distillable resource and resource cost based on the distillation and the formation, respectively. Furthermore, we introduce consistent resource measures that quantify the amount of quantum resources without contradicting the rate of state conversion even in QRTs with non-unique maximally resourceful states. Progressing beyond the previous work showing a uniqueness theorem for additive resource measures, we prove the corresponding uniqueness inequality for the consistent resource measures; that is, consistent resource measures of a quantum state take values between the distillable resource and the resource cost of the state. These formulations and results establish a foundation of QRTs applicable in a unified way to physically motivated quantum resources whose analysis can be mathematically intractable.

Quantifying Dynamical Total Coherence in a Resource Non-increasing Framework

“…Dynamical systems carry higher dimensional information, which implies that quantum features in quantum chan- * ycs@dlut.edu.cn nels can also be taken regard as resources. Therefore, the dynamical QRT [49][50][51][52][53][54][55][56][57][58][59][60][61][62][63]could also been established similar to but quite different from the QRT in static systems. Ref.…”

Tight constraints on probabilistic convertibility of quantum states

“…Many problems of this type can be understood as instances of the question of resource convertibility within the formalism of quantum resource theories [20,21]. However, most works that dealt with resource transformations, in particular in the context of general quantum resources, focused only on deterministic transformations [12,14,15,18,19,[22][23][24][25][26][27][28][29]. This limits the practical utility of such approaches, and in particular means that most previous results cannot be used to understand the ultimate limits of resource conversion -it is known that probabilistic transformations can be significantly more powerful than deterministic ones [4,[30][31][32], so constraining only deterministic protocols does not provide complete information about our capability to manipulate a given resource.…”

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

“…There have been many efforts to investigate the optimal rates of these tasks, known respectively as distillable resource yield and resource dilution cost, under various settings. In the asymptotic regime, an operational argument paralleling the second law of thermodynamics implies that the distillable resource is bounded from above by the dilution cost, because otherwise one could produce an unbounded amount of resources by repeating the distillationdilution cycle ad infinitum [1][2][3][4][5][6][7][8][9]. (See also Refs.…”