One physical system always evolves from a state of low disorder to another of high disorder. This phenomenon can be identified by increased entropy. As we know, quantum resource theories provide a mathematical framework for examining diverse quantum resources in states, with the primary problem being whether quantum states can be transformed into each other by free operations. Unlike natural transformations, free operations in quantum resource theories never increase the resource measure of a primary system. Generally, the standard approach to this central problem is to study approximate transformations and apply catalysts to broaden the set of possible state transformations. By combining approximate transformations and catalytic transformations, we show that the quantum correlation between the primary system and the catalytic system after approximate catalytic transformations determines quantum resource changes in the primary system as well as the distribution of quantum correlations among different primary systems. We first find and establish that the resource measure of the primary system does not decrease under global free operations on the primary system and the corresponding catalytic system using the universal freezing technology and the catalytic transformation methods. A resource measure is monotonic under free operations, additive on tensor products, and continuous. Using these features, we then access relations between variations in the resource measure of the primary system under global free operations and quantum correlations between the primary and catalytic systems. Therefore, quantum correlations exist between the primary and catalytic systems after catalytic transformations, which are essential for nondecreasing resource measures of the primary system. Finally, we establish relationships among different primary systems after repeatedly applying the same catalyst. The correlation is closely associated with the reuse times of the same catalyst.
One physical system always evolves from a state of low disorder to another of high disorder. This phenomenon can be identified by increased entropy. As we know, quantum resource theories provide a mathematical framework for examining diverse quantum resources in states, with the primary problem being whether quantum states can be transformed into each other by free operations. Unlike natural transformations, free operations in quantum resource theories never increase the resource measure of a primary system. Generally, the standard approach to this central problem is to study approximate transformations and apply catalysts to broaden the set of possible state transformations. By combining approximate transformations and catalytic transformations, we show that the quantum correlation between the primary system and the catalytic system after approximate catalytic transformations determines quantum resource changes in the primary system as well as the distribution of quantum correlations among different primary systems. We first find and establish that the resource measure of the primary system does not decrease under global free operations on the primary system and the corresponding catalytic system using the universal freezing technology and the catalytic transformation methods. A resource measure is monotonic under free operations, additive on tensor products, and continuous. Using these features, we then access relations between variations in the resource measure of the primary system under global free operations and quantum correlations between the primary and catalytic systems. Therefore, quantum correlations exist between the primary and catalytic systems after catalytic transformations, which are essential for nondecreasing resource measures of the primary system. Finally, we establish relationships among different primary systems after repeatedly applying the same catalyst. The correlation is closely associated with the reuse times of the same catalyst.
Quantum catalysis, the ability to enable previously impossible transformations by using auxiliary systems without degrading them, has emerged as a powerful tool in various resource theories. Although catalytically enabled state transformations have been formally characterized by the monotonic behavior of entropic quantifiers (e.g., the von Neumann entropy or nonequilibrium free energy), such characterizations often rely on unphysical assumptions, namely, the ability of using catalysts of infinitely large dimension. This approach offers very limited insights into the practical significance of using catalysis for quantum information processing. Here, we address this problem across a broad class of quantum resource theories. Leveraging quantum information tools beyond the asymptotic regime, we establish sufficient conditions for the existence of catalytic transformations with finite-size catalysts. We further unveil connections between finite-size catalysis and multicopy transformations. Notably, we discover a phenomenon of catalytic resonance: by carefully tailoring the catalysts's state, one can drastically reduce the required dimension of the catalyst, thus enabling efficient catalytic transformations with minimal resources. Finally, we illustrate our findings with examples from the resource theories of entanglement and thermodynamics, as well in the context of catalytic unitary transformations.
Determining whether a given state can be transformed into a target state using free operations is one of the fundamental questions in the study of resource theories. Free operations in resource theories can be enhanced by allowing for a catalyst system that assists the transformation and is returned unchanged, but potentially correlated, with the target state. While this has been an active area of recent research, very little is known about the necessary properties of such catalysts. Here, we prove fundamental limits applicable to a large class of correlated catalytic transformations by showing that a small residual correlation between a catalyst and target state implies that the catalyst needs to be highly resourceful. In fact, the resource required diverge in the limit of vanishing residual correlation. In addition, we establish that in imperfect catalysis a small error generally implies a highly resourceful embezzling catalyst. We develop our results in a general resource theory framework and discuss its implications for the resource theory of athermality, the resource theory of coherence, and entanglement theory.
The generalized second laws of quantum thermodynamics are usually formulated in terms of Rényi divergences and the associated family of generalized free energies. In catalytic thermal transformations, this framework typically certifies the existence of a suitable catalyst but does not make the catalytic contribution explicit in the resulting system-level inequalities. Here we develop a complementary formulation based on non-additive divergences, whose pseudo-additive structure yields a family of generalized free energies with an explicit catalyst-dependent correction term. For uncorrelated catalytic thermal transformations, we show that this leads to non-additive second-law relations providing nontrivial constraints on admissible catalysts when the catalyst is returned only approximately. We also analyze correlated catalytic thermal transformations and show, through explicit finite-dimensional examples, that reduced-state data are generally insufficient to characterize thermodynamic accessibility: the thermo-majorization behavior of the joint transformation can change while the system and catalyst marginals remain fixed, and even states with identical marginals and the same mutual information can exhibit different thermo-majorization accessibility.
Catalysts are substances that assist transformation of other resourceful objects without being consumed in the process. However, the fact that their `catalytic power' is limited and can be depleted is often overlooked, especially in the recently developing theories on catalysis of quantum randomness utilizing building correlation with catalyst. In this work, we establish a resource theory of one-shot catalytic randomness in which uncorrelatedness is consumed in catalysis of randomness. We do so by completely characterizing bipartite unitary operators that can be used to implement catalysis of randomness using partial transpose. By doing so, we find that every catalytic channel is factorizable, and therefore there exists a unital channel that is not catalytic. We define a family of catalytic entropies that quantifies catalytically extractable entropy within a quantum state and show how much degeneracy of quantum state can boost the catalytic entropy beyond its ordinary entropy. Based on this, we demonstrate that a randomness source can be actually exhausted after a certain amount of randomness is extracted. We apply this theory to systems under conservation law that forbids superposition of certain quantum states and find that non-maximally mixed states can yield the maximal catalytic entropy. We discuss implications of this theory to various topics including catalytic randomness absorption, the no-secret theorem and the possibility of multi-party infinite catalysis.
Quantum theory provides us with a mathematical framework to understand nature at the microscopic level.It explains phenomena that classical physics could not account for.A deeper understanding of these non-classical phenomena (e.g., entanglement and quantum coherence) is not only of a foundational interest, but also paves the way towards potential technological advancements.Developments in the field of quantum information showed that such quantum features help us to perform operational tasks which would be impossible in the framework of classical physics.Therefore, these quantum phenomena can be viewed as resources which help us to go beyond the restrictions imposed by classical physics.Inspired by this, the mathematical framework of quantum resource theories was developed.Each resource theory singles out a particular quantum resource by setting the states which contain no resource as free states.Furthermore, the set of free operations are defined such that they do not generate resource states out of free states.An important problem in any resource theory is to understand how different resources transform into each other under the action of free operations.This problem of state transformation will be the main focus of this thesis.We will start by studying the notion of deterministic transformations, where one aims to achieve a target state without a chance of failure.We will explore the connection between deterministic transformations and the problem of quantification of resources.We will then go further and concentrate on transformations which allow for a probability of failure, and study the fundamental restrictions imposed on the achievable error and the probability of success.Finally we will explore the idea of catalytic transformations, where we allow for an ancillary quantum system which increases the transformation power while remaining invariant in the process.For the resource theory of entanglement, we will give a full solution for this problem when the target state is pure and the initial state is distillable.Finally, we discuss potential applications of catalysis to the problem of sending quantum information through noisy channels.
Catalysts are substances that assist transformation of other resourceful objects without being consumed in the process. However, the fact that their ``catalytic power'' is limited and can be depleted is often overlooked, especially in the recently developing theories on catalysis of quantum randomness utilizing building correlation with catalyst. In this work, we establish a resource theory of one-shot catalytic randomness in which uncorrelatedness is consumed in catalysis of randomness. We do so by completely characterizing bipartite unitary operators that can be used to implement catalysis of randomness using partial transpose. By doing so, we find that every catalytic channel is factorizable, and therefore there exists a unital channel that is not catalytic. We define a family of catalytic entropies that quantifies catalytically extractable R\'enyi entropies from a quantum state and show how much the degeneracy of a quantum state can boost the catalytic entropy beyond its ordinary entropy. Based on this, we demonstrate that a randomness source can be actually exhausted after a certain amount of randomness is extracted. We apply this theory to systems under superselection rules that forbids superposition of certain quantum states and find that nonmaximally mixed states can yield the maximal catalytic entropy. We discuss implications of this theory to various topics, including catalytic randomness absorption, the no-secret theorem, and the possibility of multiparty infinite catalysis.