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.
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.
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.
We generalize the theory of catalytic quantum randomness to distributed and dynamical settings. First, we expand the theory of catalytic quantum randomness by calculating the amount of (R\'enyi) entropy catalytically extractable from a distributed or dynamical randomness source. We show that no entropy can be catalytically extracted when one cannot implement local projective measurement on randomness source without altering its state. As an application, we prove that quantum operation cannot be hidden in correlation between two parties without using randomness, which is the dynamical generalization of the no-hiding theorem. Moreover, the formalism of distributed catalysis is applied to develop a formal definition of semantic quantum information and it follows that utilizing semantic information is equivalent to catalysis using a catalyst already correlated with the transforming system. By doing so, we unify the utilization of semantic and nonsemantic quantum information and conclude that one can always extract more information from an incompletely depleted classical randomness source, but it is not possible for quantum randomness sources.
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.
Randomness can help one to implement quantum maps that cannot be realized in a deterministic fashion.Recently, it was discovered that explicitly treating a randomness source as a quantum system could double the efficiency as a catalyst for some tasks.In this work, we first show that every quantum channel that can be implemented with a randomness source without leaking information to it must be a catalysis.For that purpose, we prove a new no-go theorem that generalizes the no-hiding theorem, the no-secret theorem that states no quantum information can be shared with other systems as a secret without leaking some information.Second, we show that nondegenerate catalysts should be used classically when no extra dimension is allowed, which leads to the fact that the quantum advantage of a catalytic process strictly comes from the uniformness of the randomness source.Finally, we discuss a method to circumvent the previous result that achieves quantum advantage with nondegenerate catalyst uniformized by employing extra work space.
Quantum catalysts can overcome otherwise impossible quantum state transformations without being consumed, and allowing them to become correlated with the output makes this assistance substantially more powerful. This raises a fundamental question for entanglement theory: which limitations on state manipulation remain when such correlated catalysts are freely available? We answer this question in the positive-partial-transpose (PPT) resource theory, which allows a substantially broader class of operations than local operations and classical communication (LOCC). We identify general conditions under which regularized relative-entropy measures become strongly superadditive, and use them to construct monotones that constrain correlated catalytic PPT transformations without any knowledge of the catalyst. In particular, we prove that the regularized PPT relative entropy is fully additive and strongly superadditive, resolving an open problem in entanglement theory. Most importantly, these constraints show that even arbitrary correlated catalysts cannot restore asymptotic reversibility: for an explicit state, the optimal entanglement distillation rate remains strictly smaller than the entanglement cost. Thus, substantial catalytic assistance does not remove some of the fundamental limitations of mixed-state entanglement manipulation.
Randomness can help one to implement quantum maps that cannot be realized in a deterministic fashion. Recently, it was discovered that explicitly treating a randomness source as a quantum system could double the efficiency as a catalyst for some tasks. In this work, we generalize this result to general randomness-utilizing processes and show that the gap between the efficiencies of classical and quantum catalytic randomness is generic. To explore the achievability of catalysis of quantum randomness, we first show that the second law of thermodynamics holds for general randomness-utilizing processes: the amount of randomness never decreases. Especially, for two important randomization processes, dephasing and erasure maps, randomness sources are forced to be used catalytically, i.e., in such a way that the randomness is preserved exactly. Second, we show that non-degenerate catalysts should be used classically, which leads to the fact that the quantum advantage of a catalytic process strictly comes from the uniformness of the randomness source. Finally, we discuss a method to circumvent the previous result that achieves quantum advantage with non-degenerate catalyst uniformized by employing extra work space.