Coherence distillation is one of the central problems in the resource theory of coherence. In this Letter, we complete the deterministic distillation of quantum coherence for a finite number of coherent states under strictly incoherent operations. Specifically, we find the necessary and sufficient condition for the transformation from a mixed coherent state into a pure state via strictly incoherent operations, which recovers a connection between the resource theory of coherence and the algebraic theory of majorization lattice. With the help of this condition, we present the deterministic coherence distillation scheme and derive the maximum number of maximally coherent states obtained via this scheme.
Coherence distillation is a basic information-theoretic task in the resource theory of coherence. In this paper, we develop the framework of the approximate coherence distillation under strictly incoherent operations. This protocol considers the situation that we cannot transform an initial state $ρ$ into a target state $ψ$ with certainty; instead, we aim to deterministically transform the initial state $ρ$ into an intermediate state $ϕ$ that is most approximate to the target state $ψ$ in terms of fidelity. We also present the explicit conversion strategy of the approximate coherence distillat…
We investigate the issue of assisted coherence distillation in the asymptotic limit (considering infinite copies of the resource states), by coordinately performing the identical local operations on the auxiliary systems of each copy. When we further restrict the coordinate operations to projective measurements, the distillation process is branched into many sub-processes. Finally, a simple formula is given that the assisted distillable coherence should be the maximal average coherence of the residual states. The formula makes the experimental research of assisted coherence distillation possib…
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 assis…
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 introdu…
We complete the task of optimal probabilistic coherence distillation protocol, whose aim is to transform a general state into a set of n-level maximally coherent states via strictly incoherent operations (SIO). Specifically, we present the necessary and sufficient condition for the transformation from a mixed state into a pure-state ensemble via SIO. With the aid of this condition and the simplex algorithm, we can accomplish the probabilistic distillation protocol by presenting an analytical expression of the maximal average distillable coherence of a general state and constructing the corresp…
We experimentally investigate problems of one-shot coherence distillation [Regula, Fang, Wang, and Adesso, Phys. Rev. Lett. 121, 010401 (2018)]. Based on a set of optical devices, we design a type of strictly incoherent operation (SIO), which is applicable in high-dimensional cases and can be applied to accomplish the transformations from higher-dimensional states to lower-dimensional states. Furthermore, a relatively complete process of the one-shot coherence distillation is experimentally demonstrated for three- and four-dimensional input states. Experimental data reveal an interesting resul…
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…
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…
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 dif…
We construct quantum coherence resource theories in symmetrized Fock space (QCRTF), thereby providing an information-theoretic framework that connects analyses of quantum coherence in discrete-variable (DV) and continuous variable (CV) bosonic systems. Unlike traditional quantum coherence resource theories, QCRTF can be made independent of the single-particle basis and allow to quantify coherence within and between particle number sectors. For example, QCRTF can be formulated in such a way that neither Bose-Einstein condensates nor Heisenberg-Weyl coherent states are considered as quantum many…
The remarkable phenomenon of catalyst tells us that adding a catalyst could help state transformation. In this paper, we consider the problem of catalyst-assisted probabilistic coherence distillation for mixed states under strictly incoherent operations. To this end, we first present the necessary and sufficient conditions for distilling a target pure coherent state from an initial mixed state via stochastic strictly incoherent operations and the maximal probability of obtaining the target pure state from the initial state. With the help of these results, we present the necessary and sufficien…
Finite-Blocklength and Exponential Analyses for Universal Entanglement and Coherence Distillation under Various Free Operations
Institute for Advanced Study in Mathematics, Harbin Institute of Technology（哈尔滨工业大学高等研究院）；School of Data Science, The Chinese University of Hong Kong, Shenzhen（香港中文大学（深圳）数据科学学院）
Blind quantum computation (BQC) provides an efficient method for the client who does not have enough sophisticated technology and knowledge to perform universal quantum computation. The single-server BQC protocol requires the client to have some minimum quantum ability, while the double-server BQC protocol makes the client's device completely classical, resorting to the pure and clean Bell-state shared by two servers. In this paper, we provide a deterministic entanglement distillation protocol in a practical noisy environment for the double-server BQC protocol. This protocol can obtain the pur…
We describe how to achieve optimal entanglement generation and one-way entanglement distillation rates by coherent implementation of a class of secret key generation and secret key distillation protocols, respectively. This short paper is a high-level descrioption of our detailed papers [8] and [10].
The manipulation of quantum coherence is one of the principal issues in the resource theory of coherence, with two critical topics being the purification and enhancement of coherence. Here, we present two no-go theorems for the deterministic purification of coherence and the probabilistic enhancement of coherence, respectively. Specifically, we prove that a quantum state cannot be deterministically purified if it can be expressed as a convex combination of an incoherent state and a coherent state. Besides, we give an easy-to-verified sufficient and necessary condition to determine whether a st…
Quantum coherence, which quantifies the superposition properties of a quantum state, plays an indispensable role in quantum resource theory. A recent theoretical work [Phys. Rev. Lett. \textbf{116}, 070402 (2016)] studied the manipulation of quantum coherence in bipartite or multipartite systems under the protocol Local Incoherent Operation and Classical Communication (LQICC). Here we present the first experimental realization of obtaining maximal coherence in assisted distillation protocol based on linear optical system. The results of our work show that the optimal distillable coherence rate…
In quantum mechanics, photonic de Broglie waves have been understood as a unique property of quantum mechanics satisfying the complementarity between particle and wave natures of light, where the photonic de Broglie wavelength is inversely proportional to the number of entangled photons acting on a beam splitter. Very recently, the heart of nonclassical feature of photon bunching on a beam splitter was newly interpreted using pure wave nature of coherence optics [arXiv:1911.07174v2], paving a road to unconditionally secured classical key distribution [arXiv:1807.04233v3]. Here, Mach-Zehnder in…
We study the one-shot distillation of general quantum resources, providing a unified quantitative description of the maximal fidelity achievable in this task, and revealing similarities shared by broad classes of resources. We establish fundamental quantitative and qualitative limitations on resource distillation applicable to all convex resource theories. We show that every convex quantum resource theory admits a meaningful notion of a pure maximally resourceful state which maximizes several monotones of operational relevance and finds use in distillation. We endow the generalized robustness …
We present an optimal probabilistic protocol to distill quantum coherence. Inspired by a specific entanglement distillation protocol, our main result yields a strictly incoherent operation that produces one of a family of maximally coherent states of variable dimension from any pure quantum state. We also expand this protocol to the case where it is possible, for some initial states, to avert any waste of resources as far as the output states are concerned, by exploiting an additional transformation into a suitable intermediate state. These results provide practical schemes for efficient quant…
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, an operational paradigm that we call local quantum-incoherent operations and classical communication (LQICC). We show that the asymptotic rate of assisted coherence distillation for pure states is equal to the coherence of assistance, an analog of the ent…
Quantum coherence, emerging from the 'superposition' of quantum states, is widely used in various information processing tasks. Recently, the resource theory of multilevel quantum coherence is attracting substantial attention. In this paper, we mainly study the deterministic transformations of resource pure states via free operations in the theoretical framework for multilevel coherence. We prove that any two multilevel coherent resource pure states can be interconverted with a nonzero probability via a completely positive and trace non-increasing $k$-coherence-preserving map. Meanwhile, we pr…
The role of coherence in quantum thermodynamics has been extensively studied in the recent years and it is now well-understood that coherence between different energy eigenstates is a resource independent of other thermodynamics resources, such as work. A fundamental remaining open question is whether the laws of quantum mechanics and thermodynamics allow the existence of a coherence distillation machine, i.e., a machine that, by possibly consuming work, obtains pure coherent states from mixed states, at a nonzero rate. This is related to another fundamental question: Starting from many copies…
We compute analytically the maximal rates of distillation of quantum coherence under strictly incoherent operations (SIO) and physically incoherent operations (PIO), showing that they coincide for all states, and providing a complete description of the phenomenon of bound coherence. In particular, we establish a simple, analytically computable necessary and sufficient criterion for the asymptotic distillability under SIO and PIO. We use this result to show that almost every quantum state is undistillable --- only pure states as well as states whose density matrix contains a rank-one submatrix …
Quantum coherence and quantum entanglement represent two fundamental features of non-classical systems that can each be characterized within an operational resource theory. In this paper, we unify the resource theories of entanglement and coherence by studying their combined behavior in the operational setting of local incoherent operations and classical communication (LIOCC). Specifically we analyze the coherence and entanglement trade-offs in the tasks of state formation and resource distillation. For pure states we identify the minimum coherence-entanglement resources needed to generate a g…
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 that can be distilled from the same state. This relation enables us to derive tight second order expansions of…
We propose the notion of faithful coherent states based on the fidelity-based coherence witness. The criterion for detecting faithful coherent states can be restricted to a subclass of fidelity-based criterion under unitary transformations for single and bipartite systems. We can realize these unitary transformations by using quantum gates and circuits, and establish the connection of faithful coherence states and coherence distillation, maximum relative entropy of coherence.
We revisit the problem of switching off unwanted phase evolution and decoherence in a single two-state quantum system in the light of recent results on random dynamical decoupling methods [L. Viola and E. Knill, Phys. Rev. Lett. {\bf 94}, 060502 (2005)]. A systematic comparison with standard cyclic decoupling is effected for a variety of dynamical regimes, including the case of both semiclassical and fully quantum decoherence models. In particular, exact analytical expressions are derived for randomized control of decoherence from a bosonic environment. We investigate quantitatively control pr…
The resource framework of quantum coherence was introduced by Baumgratz, Cramer and Plenio [PRL 113, 140401 (2014)] and further developed by Winter and Yang [PRL 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 generalisation of the Winter-Yang protocol. This is compared to the distillable coherence under maximal incoherent operations (MIO) and dephasing-covariant incoherent operations (DIO), wh…
Deterministic quantum computation with one qubit (DQC1) is iconic in highlighting that exponential quantum speedup may be achieved with negligible entanglement. Its discovery catalyzed heated study of general quantum resources, and various conjectures regarding their role in DQC1's performance advantage. Coherence and discord are prominent candidates, respectively characterizing non-classicality within localized and correlated systems. Here we realize DQC1 within a superconducting system, engineered such that the dynamics of coherence and discord can be tracked throughout its execution. We exp…
Catalytic coherence transformations allow the otherwise impossible state transformations using only incoherent operations with the aid of an auxiliary system with finite coherence which is not being consumed in anyway. Here we find the necessary and sufficient conditions for the deterministic and stochastic catalytic coherence transformations between pair of pure quantum states. In particular, we show that the simultaneous decrease of a family of Rényi entropies of the diagonal parts of the states under consideration are necessary and sufficient conditions for the deterministic catalytic coher…
Based on the theory of quantum mechanics, intrinsic randomness in measurement distinguishes quantum effects from classical ones. From the perspective of states, this quantum feature can be summarized as coherence or superposition in a specific (classical) computational basis. Recently, by regarding coherence as a physical resource, Baumgratz et al.~present a comprehensive framework for coherence measures. Here, we propose a quantum coherence measure essentially using the intrinsic randomness of measurement. The proposed coherence measure provides an answer to the open question in completing th…
We present bounds for distilling many copies of a pure state from an arbitrary initial state in a general quantum resource theory. Our bounds apply to operations that are able to generate no more than a $δ$ amount of resource, where $δ\geq 0$ is a given parameter. To maximize applicability of our upper bound, we assume little structure on the set of free states under consideration besides a weak form of superadditivity of the function $G_{min}(ρ)$, which measures the overlap between $ρ$ and the set of free states. Our bounds are given in terms of this function and the robustness of resource. K…
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 non-asymptotic 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…
Coherence and entanglement are fundamental concepts in resource theory. The coherence (entanglement) of assistance is the coherence (entanglement) that can be extracted assisted by another party with local measurement and classical communication. We introduce and study the general coherence of assistance. First, in terms of real symmetric concave functions on the probability simplex, the coherence of assistance and the entanglement of assistance are shown to be in one-to-one correspondence. We then introduce two classes of quantum states: the assisted maximally coherent states and the assisted…
We establish a set of resource conversion relationships between quantum coherence and multipartite entanglement in many-body systems, where the operational measures of resource cost and distillation are focused. Under the multipartite incoherent operation, the initial coherence of single-party system bounds the maximal amounts of corresponding operational entanglement in an arbitrary bipartite partition as well as the genuine multipartite entanglement in many-body systems. Furthermore, the converted multipartite entanglement can be transferred to its subsystems and restored to coherence of a s…
We establish an operational theory of coherence (or of superposition) in quantum systems, by focusing on the optimal rate of performance of certain tasks. Namely, we introduce the two basic concepts - "coherence distillation" and "coherence cost" in the processing quantum states under so-called incoherent operations [Baumgratz/Cramer/Plenio, Phys. Rev. Lett. 113:140401 (2014)]. We then show that in the asymptotic limit of many copies of a state, both are given by simple single-letter formulas: the distillable coherence is given by the relative entropy of coherence (in other words, we give the …
Quantum resource theories have been widely studied to systematically characterize the non-classicality of quantum systems. Most resource theories focus on quantum states and study their interconversions. Although quantum channels are generally used as the tool for state manipulation, such a manipulation capability can be naturally regarded as a generalized quantum resource, leading to an open research direction in the resource theories of quantum channels. Various resource-theoretic properties of channels have been investigated, however, without treating channels themselves as operational reso…
We mainly study the block-coherence measures based on resource theory of block-coherence and the coherence measures based on positive-operator-valued measures (POVM). Several block-coherence measures including a block-coherence measure based on maximum relative entropy, the one-shot block coherence cost under the maximally block-incoherent operations, and a coherence measure based on coherent rank have been introduced and the relationships between these block-coherence measures have been obtained. We also give the definition of the maximally block-coherent state and describe the deterministic …
Any quantum resource theory is based on free states and free operations, i.e., states and operations which can be created and performed at no cost. In the resource theory of coherence free states are diagonal in some fixed basis, and free operations are those which cannot create coherence for some particular experimental realization. Recently, some problems of this approach have been discussed, and new sets of operations have been proposed to resolve these problems. As an example for such a problem, a quantum operation can be incoherent in one particular experimental realization, but it still …
Quantum resource theory under different classes of quantum operations advances multiperspective understandings of inherent quantum-mechanical properties, such as quantum coherence and quantum entanglement. We establish hierarchies of different operations for manipulating coherence and entanglement in distributed settings, where at least one of the two spatially separated parties are restricted from generating coherence. In these settings, we introduce new classes of operations and also characterize those maximal, i.e., the resource-non-generating operations, progressing beyond existing studies…
