Catalysts are quantum systems that open up dynamical pathways between quantum states which are otherwise inaccessible under a given set of operational restrictions while, at the same time, they do not change their quantum state. We here consider the restrictions imposed by symmetries and conservation laws, where any quantum channel has to be covariant with respect to the unitary representation of a symmetry group, and present two results. First, for an exact catalyst to be useful, it has to build up correlations to either the system of interest or the degrees of freedom dilating the given process to covariant unitary dynamics. This explains why catalysts in pure states are useless. Second, if a quantum system ("reference frame") is used to simulate to high precision unitary dynamics (which possibly violates the conservation law) on another system via a global, covariant quantum channel, then this channel can be chosen so that the reference frame is approximately catalytic. In other words, a reference frame that simulates unitary dynamics to high precision degrades only very little.
Catalysts are quantum systems that open up dynamical pathways between quantum states which are otherwise inaccessible under a given set of operational restrictions while, at the same time, they do not change their quantum state. We here consider the restrictions imposed by symmetries and conservation laws, where any quantum channel has to be covariant with respect to the unitary representation of a symmetry group, and present two results. First, for an exact catalyst to be useful, it has to build up correlations to either the system of interest or the degrees of freedom dilating the given process to covariant unitary dynamics. This explains why catalysts in pure states are useless. Second, if a quantum system ("reference frame") is used to simulate to high precision unitary dynamics (which possibly violates the conservation law) on another system via a global, covariant quantum channel, then this channel can be chosen so that the reference frame is approximately catalytic. In other words, a reference frame that simulates unitary dynamics to high precision degrades only very little.
In the presence of conservation laws, superpositions of eigenstates of the corresponding conserved quantities cannot be generated by quantum dynamics. Thus, any such coherence represents a potentially valuable resource of asymmetry, which can be used, for example, to enhance the precision of quantum metrology or to enable state transitions in quantum thermodynamics. Here we ask if such superpositions, already present in a reference system, can be broadcast to other systems, thereby distributing asymmetry indefinitely at the expense of creating correlations. We prove a no-go theorem showing that this is forbidden by quantum mechanics in every finite-dimensional system. In doing so, we also answer some open questions in the quantum information literature concerning the sharing of timing information of a clock and the possibility of catalysis in quantum thermodynamics. We also prove that even weaker forms of broadcasting, of which Åberg's "catalytic coherence" is a particular example, can only occur in the presence of infinite-dimensional reference systems. Our results set fundamental limits on the creation and manipulation of quantum coherence and shed light on the possibilities and limitations of quantum reference frames to act catalytically without being degraded.
We investigate the fundamental constraint on amplifying the asymmetry in quantum states with correlating catalysts. Here a correlating catalyst is a finite-dimensional auxiliary, which exactly preserves its reduced state while allowed to become correlated with the quantum system. Interestingly, we prove that under translationally invariant operations, uncorrelating catalysts in pure states are useless in any state transformation, while with a correlating catalyst, one can extend the set of accessible states from an initially asymmetric state. Moreover, we show that the power of a catalyst increases with its dimension, and further, with a large enough catalyst, a qubit state with an arbitrarily small amount of asymmetry can be converted to any mixed qubit state. In doing so, we build a bridge between two important results concerning the restrictions on coherence conversion, the no-broadcasting theorem and the catalytic coherence. Our results may also apply to the constraints on coherence evolution in quantum thermodynamics and to the distribution of timing information between quantum clocks.
There are various physical scenarios in which one can only implement operations with a certain symmetry. Under such restriction, a system in a symmetry-breaking state can be used as a catalyst, e.g. to prepare another system in a desired symmetry-breaking state. This sort of (approximate) catalytic state transformations are relevant in the context of (i) state preparation using a bounded-size quantum clock or reference frame, where the clock or reference frame acts as a catalyst, (ii) quantum thermodynamics, where again a clock can be used as a catalyst to prepare states which contain coherence with respect to the system Hamiltonian, and (iii) cloning of unknown quantum states, where the given copies of state can be interpreted as a catalyst for preparing the new copies. Using a recent result of Fawzi and Renner on approximate recoverability, we show that the achievable accuracy in this kind of catalytic transformations can be determined by a single function, namely the relative entropy of asymmetry, which is equal to the difference between the entropy of state and its symmetrized version: if the desired state transition does not require a large increase of this quantity, then it can be implemented with high fidelity using only symmetric operations. Our lower bound on the achievable fidelity is tight in the case of cloners, and can be achieved using the Petz recovery map, which interestingly turns out to be the optimal cloning map found by Werner.
When reference frames are treated quantum mechanically, the subsystem structure of quantum systems is no longer absolute, but depends on the choice of the quantum reference frame (QRF). This raises a basic question: which dynamical properties are preserved across QRFs, and which depend on the physical reference used to define the system? We study this question in the general setting of open quantum systems. At the operational level, after a QRF transformation, the old reference frame and environmental degrees of freedom may be inaccessible and must therefore be traced out. This motivates the definition of reduced quantum-reference-frame channels: maps that connect the joint description in one frame to the accessible subsystem in another. We characterize their symmetry-constrained structure and define a regime in which a reduced entropy-coherence conservation law holds. We also identify when the induced reduced action on the open system admits a classical interpretation as random frame misalignment, and when it instead reflects quantum reduced-frame effects. We then apply the framework to pure-dephasing dynamics and derive a necessary and sufficient compatibility condition for population preservation. When the frame symmetry commutes with the open system's free Hamiltonian, coherences acquire a multiplicative frame factor, so that locally inferred decoherence rates split into environmental and reference-induced contributions. Ramsey interferometry gives this split a direct operational meaning. Finally, a gravity-motivated dephasing model illustrates how degradation of a phase reference can mimic signatures usually attributed to intrinsic decoherence mechanisms.
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.
Catalysis refers to the possibility of enabling otherwise inaccessible quantum state transitions by supplying an auxiliary system, provided that the auxiliary is returned to its initial state at the end of the protocol. We show that previous studies on catalysis are largely impractical, because even small errors in the system's initial state can irreversibly degrade the catalyst. To overcome this limitation, we introduce "robust catalytic transformations" and explore the fundamental extent of their capabilities. We demonstrate that robust catalysis is closely tied to the property of resource broadcasting. In particular, in completely resource nongenerating theories, robust catalysis is possible if and only if resource broadcasting is possible. We develop a no-go theorem under a set of general axioms, demonstrating that robust catalysis is unattainable for a broad class of quantum resource theories. However, surprisingly, we also identify thermodynamical scenarios where maximal robust catalytic advantage can be achieved. Our approach clarifies the practical prospects of catalytic advantage for a wide range of quantum resources, including entanglement, coherence, thermodynamics, nonstabilizerness, and imaginarity.