In quantum information and computation, a central challenge is to determine which quantum states can be transformed into which others under restricted sets of free operations. While many transformations are impossible directly, catalytic processes can enable otherwise forbidden conversions: an auxiliary quantum state (the catalyst) facilitates the transformation while remaining unchanged. In this work, we introduce flexible catalysis, a generalization in which the catalyst is allowed to transform into a different auxiliary state, provided it remains a valid catalyst. We show that this framework subsumes both standard catalytic and multicopy transformations, and we analyse its advantages across several classes of free operations. In particular, we prove that when the free operations are local unitaries or permutation matrices, flexible catalysis enables state extractions that are unattainable with standard catalysis alone.
In quantum information and computation, a central challenge is to determine which quantum states can be transformed into which others under restricted sets of free operations. While many transformations are impossible directly, catalytic processes can enable otherwise forbidden conversions: an auxiliary quantum state (the catalyst) facilitates the transformation while remaining unchanged. In this work, we introduce flexible catalysis, a generalization in which the catalyst is allowed to transform into a different auxiliary state, provided it remains a valid catalyst. We show that this framework subsumes both standard catalytic and multicopy transformations, and we analyse its advantages across several classes of free operations. In particular, we prove that when the free operations are local unitaries or permutation matrices, flexible catalysis enables state extractions that are unattainable with standard catalysis alone.
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.
Understanding the fundamental limits of state convertibility is crucial for establishing the boundaries of quantum information processing and thermodynamic efficiency. While auxiliary systems, catalysts, can facilitate otherwise impossible transformations, standard catalysis rigidly requires the auxiliary system to return to its exact initial state. In this work, we investigate the power of flexible catalysis, where the catalyst evolves through a cycle of states, restoring its initial configuration only after a finite number of steps. Focusing on the regime of fixed, finite dimensions, we analyze the capabilities of flexible catalysis within the resource theories of entanglement and quantum thermodynamics. In the context of entanglement, we derive conditions limiting flexible catalysts, yet show that flexible catalysis can be strictly more powerful than same-dimensional standard catalysis: it enables deterministic transformations achievable by no standard catalyst of the same dimension, and it strictly increases the success probability of stochastic local operations and classical communication. A similar deterministic advantage arises in quantum thermodynamics, where flexible catalysis enables state transformations that are impossible with any standard catalyst of fixed dimension and Hamiltonian but become achievable via flexible catalysis.
Catalysis refers to the possibility of performing an otherwise impossible local state transformation by sharing an additional state, i.e. a catalyst, which is returned at the end of the protocol. There is a stronger version, known as supercatalysis, in which the borrowed catalyst is returned in an enhanced form, i.e. more entangled. However, this phenomenon has remained little explored. In this work we introduce the supercatalytic entanglement gain as a figure of merit taking values in [0,1] that quantifies the performance of the protocol (with 0 corresponding to the standard case of catalysis and 1 representing the maximal possible gain) and we study in which cases it can be greater than zero and which strategies can maximize it. While it turns out that every catalytic transformation can be implemented in a supercatalytic fashion with entanglement gain equal to 1 if the state that is borrowed is chosen appropriately, other choices can make the gain strictly less than 1 and even 0. In fact, we prove that a large class of catalytic transformations are not fully supercatalyzable, i.e. there is at least one choice of catalyst for which the entanglement gain vanishes. On the other hand, the construction that shows that supercatalysis is always possible with maximal gain uses artificially highly entangled catalysts. For this reason, we also study minimal supercatalysis, where the entanglement content of the borrowed state is constrained in a precise and natural way. While we consider a scenario where we prove it is impossible to have entanglement gain equal to 1 in this case, we show that there exist minimal supercatalytic transformations with gain as close to 1 as desired. We also explore several examples and observe that, although choosing a catalyst with the least possible entanglement is often an optimal strategy for minimal supercatalysis, this is not necessarily always the case.
Abstract In chemistry, a catalyst is a substance which enables a chemical reaction or increases its rate, while remaining unchanged in the process. Instead of chemical reactions, quantum catalysis enhances our ability to convert quantum states into each other under physical constraints. The nature of the constraints depends on the problem under study and can arise, e.g. from energy preservation. This article reviews the most recent developments in quantum catalysis and gives a historical overview of this research direction. We focus on the catalysis of quantum entanglement and coherence, and also discuss this phenomenon in quantum thermodynamics and general quantum resource theories. We review applications of quantum catalysis and also discuss the recent efforts on universal catalysis, where the quantum state of the catalyst does not depend on the states to be transformed. Catalytic embezzling is also considered, a phenomenon that occurs if the catalyst’s state can change in the transition.
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 class of possible thermodynamic conversions can be extended by introducing an auxiliary system called catalyst, which assists in state conversion while its own state remains unchanged. We reveal a complete characterization of catalytic state conversion in quantum and single-shot thermodynamics by allowing an infinitesimal correlation between the system and the catalyst. Specifically, we prove that a single thermodynamic potential, which provides the necessary and sufficient condition for the correlated-catalytic state conversion, is given by the standard nonequilibrium free energy defined with the Kullback-Leibler divergence. This resolves the conjecture raised by Wilming, Gallego, and Eisert [Entropy 19, 241 (2017)ENTRFG1099-430010.3390/e19060241] and by Lostaglio and Müller [Phys. Rev. Lett. 123, 020403 (2019)PRLTAO0031-900710.1103/PhysRevLett.123.020403] in the positive. Moreover, we show that, with the aid of the work storage, any quantum state can be converted into another by paying the work cost equal to the nonequilibrium free energy difference. Our result would serve as a step towards establishing resource theories of catalytic state conversion in the fully quantum regime.