The resource theory of quantum coherence is an important topic in quantum information science. Standard coherence distillation and dilution problems have been thoroughly studied. In this paper, we introduce and study the problem of one-shot coherence distillation with catalysts. In order to distill more coherence from a state of interest, a catalytic system can be involved and a jointly free operation is applied to both systems. The joint output state should be a maximally coherent state in tensor product with the unchanged catalysts, with some allowable fidelity error. We consider several different definitions of this problem. First, with a small fidelity error in both systems, we show that, even via the smallest free operation class (physically implementable incoherent operations), the distillable coherence of any state with no restriction on the catalysts is infinite, which is a ``coherence embezzling phenomenon.'' We then define and calculate a lower bound for the distillable coherence when the dimension of catalysts is restricted. Finally, in consideration of physical relevance, we define the ``perfect catalysts'' scenario where the catalysts are required to be pure and precisely unchanged. Interestingly, we show that in this setting catalysts basically provide no advantages in pure state distillation via incoherent operations and strictly incoherent operations under certain smoothing restriction. Our work enhances the understanding of catalytic effect in quantum resource theory.
The resource theory of quantum coherence is an important topic in quantum information science. Standard coherence distillation and dilution problems have been thoroughly studied. In this paper, we introduce and study the problem of one-shot coherence distillation with catalysts. In order to distill more coherence from a state of interest, a catalytic system can be involved and a jointly free operation is applied to both systems. The joint output state should be a maximally coherent state in tensor product with the unchanged catalysts, with some allowable fidelity error. We consider several different definitions of this problem. First, with a small fidelity error in both systems, we show that, even via the smallest free operation class (physically implementable incoherent operations), the distillable coherence of any state with no restriction on the catalysts is infinite, which is a coherence embezzling phenomenon.'' We then define and calculate a lower bound for the distillable coherence when the dimension of catalysts is restricted. Finally, in consideration of physical relevance, we define the perfect catalysts'' scenario where the catalysts are required to be pure and precisely unchanged. Interestingly, we show that in this setting catalysts basically provide no advantages in pure state distillation via incoherent operations and strictly incoherent operations under certain smoothing restriction. Our work enhances the understanding of catalytic effect in quantum resource theory.
The ability to distill quantum coherence is pivotal for optimizing the performance of quantum technologies; however, such a task cannot always be accomplished with certainty. Here we develop a general framework of probabilistic distillation of quantum coherence in a one-shot setting, establishing fundamental limitations for different classes of free operations. We first provide a geometric interpretation for the maximal success probability, showing that under maximally incoherent operations (MIO) and dephasing-covariant incoherent operations (DIO) the problem can be simplified into efficiently computable semidefinite programs. Exploiting these results, we find that DIO and its subset of strictly incoherent operations have equal power in the probabilistic distillation of coherence from pure input states, while MIO are strictly stronger. We then prove a fundamental no-go result: Distilling coherence from any full-rank state is impossible even probabilistically. We further find that in some conditions the maximal success probability can vanish suddenly beyond a certain threshold in the distillation fidelity. Finally, we consider probabilistic coherence distillation assisted by a catalyst and demonstrate, with specific examples, its superiority to the unassisted and deterministic cases.
Quantum resource distillation is a fundamental task in quantum information science and technology. Minimizing the overhead of distillation is crucial for the realization of quantum computation and other technologies. Here we explicitly demonstrate how, for general quantum resources, suitably designed quantum catalysts (i.e., auxiliary systems that remain unchanged before and after the process) can lift the multi-shot or asymptotic average distillation overhead to the one-shot setting. This can enable one-shot distillation with constant overhead, thereby overcoming the established logarithmic lower bound. In particular, for magic state distillation, our catalysis method paves a path for tackling the diverging batch size problem associated with code-based low-overhead protocols by enabling arbitrary reduction of the protocol size for any desired accuracy. Notably, this yields constant-overhead magic state distillation with controllable protocol size. Furthermore, we demonstrate a tunable spacetime trade-off between overhead and success probability enabled by catalysts which offers significant versatility for practical implementation. Finally, we extend catalysis techniques to dynamical quantum resources and show that channel mutual information determines one-shot catalytic channel transformation, thereby advancing our understanding for both dynamical catalysis and information theory. In quantum information processing, distilling pure quantum states from noisy ones in the practical one-shot setting imposes a high overhead. Here, Fang and Liu propose a one-shot distillation scheme that exploits catalysis to arbitrarily reduce distillation overhead, and apply this to magic state distillation to improve over existing results.
Manipulation and quantification of quantum resources are fundamental problems in quantum physics. In the asymptotic limit, coherence distillation and dilution have been proposed by manipulating infinite identical copies of states. In the nonasymptotic setting, finite data-size effects emerge, and the practically relevant problem of coherence manipulation using finite resources has been left open. This Letter establishes the one-shot theory of coherence dilution, which involves converting maximally coherent states into an arbitrary quantum state using maximally incoherent operations, dephasing-covariant incoherent operations, incoherent operations, or strictly incoherent operations. We introduce several coherence monotones with concrete operational interpretations that estimate the one-shot coherence cost-the minimum amount of maximally coherent states needed for faithful coherence dilution. Furthermore, we derive the asymptotic coherence dilution results with maximally incoherent operations, incoherent operations, and strictly incoherent operations as special cases. Our result can be applied in the analyses of quantum information processing tasks that exploit coherence as resources, such as quantum key distribution and random number generation.
We characterize the distillation of quantum coherence in the one-shot setting, that is, the conversion of general quantum states into maximally coherent states under different classes of quantum operations. We show that the maximally incoherent operations (MIO) and the dephasing-covariant incoherent operations (DIO) have the same power in the task of one-shot coherence distillation. We establish that the one-shot distillable coherence under MIO and DIO is efficiently computable with a semidefinite program, which we show to correspond to a quantum hypothesis testing problem. Further, we introduce a family of coherence monotones generalizing the robustness of coherence as well as the modified trace distance of coherence, and show that they admit an operational interpretation in characterizing the fidelity of distillation under different classes of operations. By providing an explicit formula for these quantities for pure states, we show that the one-shot distillable coherence under MIO, DIO, strictly incoherent operations, and incoherent operations is equal for all pure states.
The resource framework of quantum coherence was introduced by Baumgratz, Cramer, and Plenio [Phys. Rev. Lett. 113, 140401 (2014)] and further developed by Winter and Yang [Phys. Rev. Lett. 116, 120404 (2016)]. We consider the one-shot problem of distilling pure coherence from a single instance of a given resource state. Specifically, we determine the distillable coherence with a given fidelity under incoherent operations (IO) through a generalization of the Winter-Yang protocol. This is compared to the distillable coherence under maximal incoherent operations (MIO) and dephasing-covariant incoherent operations (DIO), which can be cast as a semidefinite programme, that has been presented previously by Regula et al. [Phys. Rev. Lett. 121, 010401 (2018)]. Our results are given in terms of a smoothed min-relative entropy distance from the incoherent set of states, and a variant of the hypothesis-testing relative entropy distance, respectively. The one-shot distillable coherence is also related to one-shot randomness extraction. Moreover, from the one-shot formulas under IO, MIO, and DIO, we can recover the optimal distillable rate in the many-copy asymptotics, yielding the relative entropy of coherence. These results can be compared with previous work by some of the present authors [Zhao et al., Phys. Rev. Lett. 120, 070403 (2018)] on one-shot coherence formation under IO, MIO, DIO and also SIO. This shows that the amount of distillable coherence is essentially the same for IO, DIO, and MIO, despite the fact that the three classes of operations are very different. We also relate the distillable coherence under strictly incoherent operations (SIO) to a constrained hypothesis testing problem and explicitly show the existence of bound coherence under SIO in the asymptotic regime.
We characterize the operational task of environment-assisted distillation of quantum coherence under different sets of free operations when only a finite supply of copies of a given state is available. We first evaluate the one-shot assisted distillable coherence exactly, and introduce a semidefinite programming bound on it in terms of a smooth entropic quantity. We prove the bound to be tight for all systems in dimensions 2 and 3, which allows us to obtain computable expressions for the one-shot rate of distillation, establish an analytical expression for the best achievable fidelity of assisted distillation for any finite number of copies, and fully solve the problem of asymptotic zero-error assisted distillation for qubit and qutrit systems. Our characterization shows that all relevant sets of free operations in the resource theory of coherence have exactly the same power in the task of one-shot assisted coherence distillation, and furthermore resolves a conjecture regarding the additivity of coherence of assistance in dimension 3.