We give the first systematic study on the second order asymptotics of the operational task of coherence distillation with and without assistance. In the unassisted setting, we introduce a variant of randomness extraction framework where free incoherent operations are allowed before the incoherent measurement and the randomness extractors. We then show that the maximum number of random bits extractable from a given quantum state is precisely equal to the maximum number of coherent bits distillable from the same state. This relation enables us to derive tight second order expansions of both tasks in the independent and identically distributed setting. Remarkably, the incoherent operation classes that can empower coherence distillation for generic states all admit the same second order expansions, indicating their operational equivalence for coherence distillation in both asymptotic and large block length regimes. We then generalize the above line of research to the assisted setting, arising naturally in bipartite quantum systems where Bob distills coherence from the state at hand, aided by the benevolent Alice possessing the other system. More precisely, we introduce a new assisted incoherent randomness extraction task and establish an exact relation between this task and the assisted coherence distillation. It strengthens the one-shot relation in the unassisted setting and confirms that this cryptographic framework offers a new perspective to the study of quantum coherence distillation. Likewise, this relation yields second order characterizations to the assisted tasks. As by-products, we show the strong converse property of the tasks above from their second order expansions.
We give the first systematic study on the second order asymptotics of the operational task of coherence distillation with and without assistance. In the unassisted setting, we introduce a variant of randomness extraction framework where free incoherent operations are allowed before the incoherent measurement and the randomness extractors. We then show that the maximum number of random bits extractable from a given quantum state is precisely equal to the maximum number of coherent bits distillable from the same state. This relation enables us to derive tight second order expansions of both tasks in the independent and identically distributed setting. Remarkably, the incoherent operation classes that can empower coherence distillation for generic states all admit the same second order expansions, indicating their operational equivalence for coherence distillation in both asymptotic and large block length regimes. We then generalize the above line of research to the assisted setting, arising naturally in bipartite quantum systems where Bob distills coherence from the state at hand, aided by the benevolent Alice possessing the other system. More precisely, we introduce a new assisted incoherent randomness extraction task and establish an exact relation between this task and the assisted coherence distillation. It strengthens the one-shot relation in the unassisted setting and confirms that this cryptographic framework offers a new perspective to the study of quantum coherence distillation. Likewise, this relation yields second order characterizations to the assisted tasks. As by-products, we show the strong converse property of the tasks above from their second order expansions.
We introduce and study the task of assisted coherence distillation. This task arises naturally in bipartite systems where both parties work together to generate the maximal possible coherence on one of the subsystems. Only incoherent operations are allowed on the target system, while general local quantum operations are permitted on the other; this is an operational paradigm that we call local quantum-incoherent operations and classical communication. We show that the asymptotic rate of assisted coherence distillation for pure states is equal to the coherence of assistance, an analog of the entanglement of assistance, whose properties we characterize. Our findings imply a novel interpretation of the von Neumann entropy: it quantifies the maximum amount of extra quantum coherence a system can gain when receiving assistance from a collaborative party. Our results are generalized to coherence localization in a multipartite setting and possible applications are discussed.
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.
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.
In the task of assisted coherence distillation via the set of operations X, where X is either local incoherent operations and classical communication (LICC), local quantum-incoherent operations and classical communication (LQICC), separable incoherent operations (SI), or separable quantum incoherent operations (SQI), two parties, namely Alice and Bob, share many copies of a bipartite joint state. The aim of the process is to generate the maximal possible coherence on the subsystem of Bob. In this paper, we investigate the assisted coherence distillation of some special mixed states, the states with vanished basis-dependent discord and Werner states. We show that all the four sets of operations are equivalent for assisted coherence distillation, whenever Alice and Bob share one of those mixed quantum states. Moreover, we prove that the assisted coherence distillation of the former can reach the upper bound, namely QI relative entropy, while that of the latter can not. Meanwhile, we also present a sufficient condition such that the assistance of Alice via the set of operations X can not help Bob improve his distillable coherence, and this condition is that the state shared by Alice and Bob has vanished basis-dependent discord.
Historically, entanglement and coherence distillation focused on the distillable rate under full knowledge of the initial state. In this paper, we develop a unified finite-blocklength and asymptotic theory for universal distillation from convex compact sequences of possible states stable under tensor products. We first resolve an open question by showing that two regularized variants of the generalized quantum Hoeffding divergence need not coincide, and establish fundamental properties of the Hoeffding divergence and anti-divergence. For universal entanglement distillation, we connect finite-blocklength performance under non-entangling, PPT-preserving, and dually free operations to composite correlated hypothesis testing, yielding reliability functions for non-entangling and PPT-preserving operations characterized by regularized quantum Hoeffding divergences and strong converse bounds via the anti-divergence. We recover the entanglement concentration error exponent, zero-rate exponents under a continuity condition, and distillable entanglements, and characterize the dually free cases by regularized relative entropies under restricted measurements. Additionally, five concrete noise models justify the black-box assumptions for both resources. For coherence distillation under maximally incoherent and dephasing-covariant incoherent operations, we extend the one-shot semidefinite program characterization to universal inputs; relating a relaxed optimization to hypothesis testing yields the high-rate reliability function. A coherent-seed approach extends the single-state formula to the entire direct regime when the zero-error rate is positive. Finally, we completely characterize the states with positive zero-error coherence distillation rate, show that the rate equals the order-zero Petz Rényi relative entropy of coherence, and derive dimension-independent one-shot bounds ......
We compute on all quantum states several measures that characterise asymptotic quantum coherence manipulation under restricted classes of operations. We focus on the distillable coherence, i.e. the maximum rate of production of approximate pure bits of coherence starting from independent copies of an input state ρ, and on the coherence cost, i.e. the minimum rate of consumption of pure coherence bits that is needed to generate many copies of ρ with vanishing error. We obtain the first closed-form expression for the distillable coherence under strictly incoherent operations (SIO), proving that it coincides with that obtained via physically incoherent operations (PIO). This shows that SIO and PIO are equally weak at distilling coherence, sheds light on the recently discovered phenomenon of generic bound coherence, and provides us with an explicit optimal distillation protocol that is amenable to practical implementations. We give a single-letter formula for the coherence cost under PIO, showing that it is finite on a set of states with nonzero volume. Since PIO can be realised in a laboratory with incoherent ancillae, unitaries, and measurements, our result puts fundamental limitations on coherence manipulation in an experimentally relevant setting. We find examples of `abyssally bound' states with vanishing PIO distillable coherence yet infinite PIO coherence cost. Our findings complete the picture of asymptotic coherence manipulation under all the main classes of incoherent operations.