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.
Quantum computers promise transformative speedups, but environmental noise destroys their fragile states. Conventional quantum error correction (QEC) encodes information redundantly across physical qubits, yet fails above a threshold of about 1% and incurs polynomial qubit overhead. A recent theorem from the resource theory of coherence shows that catalytic covariant operations amplify coherence at an unbounded rate, but this result has never been cast as an operational protocol. The challenge is to turn an asymptotic theorem into a recovery scheme that works at any noise strength with realistic resources. Here we show that catalytic coherence amplification can be cast as an error-correction primitive, Catalytic Quantum Error Correction (CQEC), which recovers a known target state from noisy copies without any error magnitude threshold whenever the target's coherent modes are preserved. In an effective model of the recovery map, fidelity exceeds 0.99 across 200 noise configurations spanning d = 4-64; the catalyst cost drops from the constructive bound n* ~ d^4 e^(2 gamma) to 32 copies at matched fidelity (10^4- to 10^9-fold) via a pipeline of dynamical decoupling, Clifford twirling, and recursive swap-test purification. A first explicitly CPTP, exactly covariant joint-channel implementation validates genuine recovery under dephasing at small dimension (0.54 -> 0.77) while showing that shallow circuits consume the catalyst, quantifying the model-vs-channel gap. These results turn an abstract resource-theoretic statement into a concrete protocol candidate complementary to stabilizer- and purification-based QEC; an open-source package reproducing the benchmarks accompanies this work (arXiv:2603.25774, https://github.com/deeptell-inc/cqec).
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.
The no-cloning theorem is a cornerstone result in quantum mechanics that forbids copying of general quantum information. More quantitatively, even given
N copies of an unknown state, any channel producing an
(N+1)-copy state incurs a nonzero trace-distance error on some inputs. Turning this around, one can ask about the sample complexity of approximate
N→N+1 cloning: for a desired small error
ε, what
N suffices? For arbitrary pure states on a Hilbert space
H, the answer is
N≃dimH/ε - astronomically large for most
H of interest. What if the input is promised to lie in a structured family? We examine a range of families fundamental to many-body physics and quantum information:
n-mode fermionic Gaussian and Slater states,
n-mode bosonic Gaussian states, and qudit phase states. For fermions, we construct optimal cloning channels, reducing the complexity from exponential to polynomial in
n. For bosonic Gaussian states, we construct an explicit cloner that certifies polynomial complexity, without any constraint on the energy of the state; this channel, however, is not optimal in general. The fermionic and bosonic results follow from a single representation-theoretic framework, which generalizes previous work by Werner and by Chiribella and Yang; it also yields the cloners' unitary implementation. Furthermore, we explain why for many families the cloning complexity is linear in the dimension of the family manifold, via a controlled saddle-point evaluation of the frame potential at high
N. The same calculation accounts for the analogous scaling of tomography complexity. The above framework does not cover phase states; for these we give an independent construction of a channel with optimal cloning fidelity. Our work complements recent results on cloning stabilizer states.
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.
Quantum coherence is one of the fundamental aspects distinguishing classical and quantum theories. Coherence between different energy eigenstates is particularly important, as it serves as a valuable resource under the law of energy conservation. A fundamental question in this setting is how well one can prepare good coherent states from low coherent states and whether a given coherent state is convertible to another one. Here, we show that any low coherent state is convertible to any high coherent state arbitrarily well in two operational settings: asymptotic and catalytic transformations. For a variant of asymptotic coherence manipulation where one aims to prepare desired states in local subsystems, the rate of transformation becomes unbounded regardless of how weak the initial coherence is. In a non-asymptotic transformation with a catalyst, a helper state that locally remains in the original form after the transformation, we show that an arbitrary state can be obtained from any low coherent state. Applying this to the standard asymptotic setting, we find that a catalyst can increase the coherence distillation rate significantly-from zero to infinite rate. We also prove that such anomalous transformation requires small but nonzero coherence in relevant modes, establishing the condition under which a sharp transition of the operational capability occurs. Our results provide a general characterization of the coherence transformability in these operational settings and showcase their peculiar properties compared to other common resource theories such as entanglement and quantum thermodynamics.
What are the fundamental limits and advantages of using a catalyst to aid thermodynamic transformations between quantum systems? In this paper, we answer this question by focusing on transformations between energy-incoherent states under the most general energy-conserving interactions among the system, the catalyst, and a thermal environment. The sole constraint is that the catalyst must return unperturbed and uncorrelated with the other subsystems. More precisely, we first upper bound the set of states to which a given initial state can thermodynamically evolve (the catalyzable future) or from which it can evolve (the catalyzable past) with the help of a strict catalyst. Secondly, we derive lower bounds on the dimensionality required for the existence of catalysts under thermal process, along with bounds on the catalyst's state preparation. Finally, we quantify the catalytic advantage in terms of the volume of the catalyzable future and demonstrate its utility in an exemplary task of generating entanglement and cooling a quantum system using thermal resources. Published by the American Physical Society 2024