Emerging from the superposition principle, the resource theory of coherence plays a crucial role in many information-processing tasks. Recently, a generalization to this resource theory was investigated with respect to arbitrary positive operator valued measurement (POVM) based on Naimark's dilation theorem. Here, we introduce the notion of Block Incoherent Operations (BIO), Strictly Block Incoherent Operations (SBIO) and Physically Block Incoherent Operations (PBIO) and provide an analytical expression for Kraus operators of these operations to have a better understanding of the resource theory of block coherence which in turn gives a more clear picture of POVM based resource theory of coherence. A dilation theorem corresponding to SBIO has been introduced to enlighten the proper physical interpretation of this operation. These free operations will be helpful in finding out the conditions of state transformations and could be implemented in various protocols. For a transparent view of this resource theory, we have successfully introduced the concept of state transformation under SBIO.
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 …
Quantum coherence is an essential feature of quantum mechanics and is an important physical resource in quantum information. Recently, the resource theory of quantum coherence has been established parallel with that of entanglement. In the resource theory, a resource can be well defined if given three ingredients: the free states, the resource, the (restricted) free operations. In this paper, we study the resource theory of coherence in a different light, that is, we consider the total coherence defined by the basis-free coherence maximized among all potential basis. We define the distillable …
We demonstrate that quantum Fisher information and superradiance can be formulated as coherence measures in accordance with the resource theory of coherence, thus establishing a direct link between metrological information, superradiance and coherence. We also show that the trivial coherence measure has a metrological interpretation. The arguments are then generalized to show that coherence may be considered as the underlying fundamental resource for any functional of state that is first of all faithful, and second of all concave or linear. Arguments are then presented that show quantum Fisher…
Quantum discord and quantum entanglement are resources in some quantum information processing (QIP) models. However, in recent years, the evidence that separable states or classically correlated states can also accomplish QIP is demonstrated. It provides a useful tool since such states are easier to prepare. Quantum coherence is a measure of non-classical correlation, containing entanglement and discord as a subset. Nowadays, it is of interest whether quantum coherence can act as a resource in QIP independently or not, without the help from quantum discord or entanglement. In this paper, we sh…
The resource theory of quantum coherence studies the off-diagonal elements of a density matrix in a distinguished basis, whereas the resource theory of purity studies all deviations from the maximally mixed state. We establish a direct connection between the two resource theories, by identifying purity as the maximal coherence which is achievable by unitary operations. The states that saturate this maximum identify a universal family of maximally coherent mixed states. These states are optimal resources under maximally incoherent operations, and thus independent of the way coherence is quantif…
Quantum coherence is a fundamental feature of quantum mechanics and an underlying requirement for most quantum information tasks. In the resource theory of coherence, incoherent states are diagonal with respect to a fixed orthonormal basis, i.e., they can be seen as arising from a von Neumann measurement. Here, we introduce and study a generalization to a resource theory of coherence defined with respect to the most general quantum measurement, i.e., to an arbitrary positive-operator-valued measure (POVM). We establish POVM-based coherence measures and POVM-incoherent operations which coincide…
In this paper we study an error correcting protocol that specifically derives its error correcting properties from elementary units of coherence. The entire protocol from beginning to end is performed using non-coherence increasing operations, resulting in the consumption of the input coherence, thus necessitating further quantum resources if one wishes to perform the protocol again. We show that even when the input quantum resource is just 1 coherent qubit, one may acquire partial protection from phase flip errors, and that this can be scaled up to protect against arbitrary qubit errors with …
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 quantify the dynamical quantum resource in the resource non-increasing (RNI) framework, namely, the free dynamical resource is defined by the channels that cannot increase the static resourcefulness of any input state. We present two kinds of approaches to quantifying the dynamical resource, the distance measures and the maximal increasing static resource (MISR). As a demonstration, we quantify the dynamical total coherence with our presented measures. It is shown that the distance based measures have good operational interpretation through quantum processing tasks and can be numerically ca…
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…
Quantum coherence with respect to orthonormal bases has been studied extensively in the past few years. Recently, Bischof, et al. [Phys. Rev. Lett. 123, 110402 (2019)] generalized it to the case of general positive operator-valued measure (POVM) measurements. Such POVM-based coherence, including the block coherence as a special case, have significant operational interpretations in quantifying the advantage of quantum states in quantum information processing. In this work we first establish an alternative framework for quantifying the block coherence and provide several block coherence measures…
Quantum coherence is one of the most important resources in quantum information. Indeed, preventing the loss of coherence is one of the most important technical challenges obstructing the development of large-scale quantum computers. Recently, there has been substantial progress in developing mathematical resource theories of coherence, paving the way towards its quantification and control. To date however, these resource theories have only been mathematically formalised within the realms of convex-geometry, information theory, and linear algebra. This approach is limited in scope, and makes i…
Imaginarity as a Resource within Quantum Coherence: Geometric Decomposition and Operational Conversion
School of Science, East China University of Technology（华东交通大学理学院）；School of Computer and Computing Science, Hangzhou City University（杭州城市大学计算机与计算科学学院）；School of Mathematical Sciences, Capital Normal University（首都师范大学数学科学学院）
Quantum coherence, like entanglement, is a fundamental resource in quantum information. In recent years, remarkable progress has been made in formulating resource theory of coherence from a broader perspective. The notions of block-coherence and POVM-based coherence have been established. Certain challenges, however, remain to be addressed. It is difficult to define incoherent operations directly, without requiring incoherent states, which proves a major obstacle in establishing the resource theory of dynamical coherence. In this paper, we overcome this limitation by introducing an alternate d…
In this paper, we detail an orthogonalization procedure that allows for the quantification of the amount of coherence present an arbitrary superposition of coherent states. The present construction is based on the quantum coherence resource theory introduced by Baumgratz et al., and the coherence resource monotone that we identify is found to characterize the nonclassicality traditionally analyzed via the Glauber-Sudarshan $P$ distribution. This suggests that identical quantum resources underlie both quantum coherence in the discrete finite dimensional case and the nonclassicality of quantum l…
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…
One of the fundamental features of quantum mechanics is the superposition principle, a manifestation of which is embodied in quantum coherence. Coherence of a quantum state is invariably defined with respect to a preferred set of pointer states, and there exist quantum coherence measures with respect to deterministically as well as probabilistically distinguishable sets of quantum state vectors. Here we study the resource theory of quantum coherence with respect to an arbitrary set of quantum state vectors, that may not even be probabilistically distinguishable. Geometrically, a probabilistica…
We define the quantum-incoherent relative entropy of coherence ($\mathcal{QI}$ REC) of quantum channels in the framework of the resource theory by using the Choi-Jamiolkowsky isomorphism. Coherence-breaking channels are introduced as free operations and their corresponding Choi states as free states. We also show the relationship between the coherence of channel and the quantum discord and find that basis-dependent quantum asymmetric discord can never be more than the $\mathcal{QI}$ REC for any quantum channels. {Also}, we prove the $\mathcal{QI}$ REC is decreasing for any divisible quantum in…
Measuring quantum states provides means to generate genuine random numbers. It has been shown that genuine randomness can be obtained even with an uncharacterized quantum source. In this work, we propose a framework that formalizes the idea of realizing source-independent quantum random number generation via measuring coherence. Without full state tomography, the coherence of the source can be estimated by coherence witnesses. The existing uncertainty-relation-based schemes can be treated as special cases under the coherence framework, as we design a nonlinear coherence witness that can essent…
Multiple quantum (MQ) NMR coherence spectra, which can be obtained experimentally in MQ NMR, can be transferred from the sender to the remote receiver without mixing the MQ-coherences of different orders and distortions. The only effect of such transfer is scaling of the certain blocks of sender's density matrix (matrices of MQ-coherences of different order). Such a block-scaled transfer is an alternative to the perfect state transfer. In particular, equal scaling of higher order MQ-coherences matrices is possible. Moreover, there are states which can be transferred to the receiver preserving …
We characterize minimal measurement setups for validating the quantum coherence of an unknown quantum state. We show that for a $d$-level system, the optimal strategy consists of measuring $d$ orthonormal bases such that each measured basis is mutually unbiased with respect to the reference basis, and together with the reference basis they form an informationally complete set of measurements. We show that, in general, any strategy capable of validating quantum coherence allows one to evaluate also the exact value of coherence. We then give an explicit construction of the optimal measurements f…
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…
Quantum resource theories provide a diverse and powerful framework for extensively studying the phenomena in quantum physics. Quantum coherence, a quantum resource, is the basic ingredient in many quantum information tasks. It is a subject of broad and current interest in quantum information, and many new concepts have been introduced and generalized since its establishment. Here we show that the block coherence can be transformed into entanglement via a block incoherent operation. Moreover, we find that the POVM-based coherence associated with block coherence through the Naimark extension act…
Connections between the resource theories of coherence and purity (or non-uniformity) are well known for discrete-variable, finite-dimensional, quantum systems. We establish analogous results for continuous-variable systems, in particular Gaussian systems. To this end, we define the concept of maximal coherence at fixed energy, which is achievable with energy-preserving unitaries. We show that the maximal Gaussian coherence (where states and operations are required to be Gaussian) can be quantified analytically by the relative entropy. We then propose a resource theory of non-uniformity, by co…
We generalize the recently proposed resource theory of coherence (or superposition) [Baumgratz, Cramer & Plenio, Phys. Rev. Lett. 113:140401; Winter & Yang, Phys. Rev. Lett. 116:120404] to the setting where not only the free ("incoherent") resources, but also the objects manipulated are quantum operations, rather than states. In particular, we discuss an information theoretic notion of coherence capacity of a quantum channel, and prove a single-letter formula for it in the case of unitaries. Then we move to the coherence cost of simulating a channel, and prove achievability results for unitari…
Quantum coherence is an essential feature of quantum mechanics which is responsible for the departure between classical and quantum world. The recently established resource theory of quantum coherence studies possible quantum technological applications of quantum coherence, and limitations which arise if one is lacking the ability to establish superpositions. An important open problem in this context is a simple characterization for incoherent operations, constituted by all possible transformations allowed within the resource theory of coherence. In this work, we contribute to such a character…
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 …
That superpositions of states can be useful for performing tasks in quantum systems has been known since the early days of quantum information, but only recently has quantitative theory of quantum coherence been proposed. Here we apply that theory to an analysis of the Deutsch-Jozsa algorithm, which depends on quantum coherence for its operation. The Deutsch-Jozsa algorithm solves a decision problem, and we focus on a probabilistic version of that problem, comparing probability of being correct for both classical and quantum procedures. In addition, we study a related decision problem in which…
As the precious resource for quantum information processing, quantum coherence can be created remotely if the involved two sites are quantum correlated. It can be expected that the amount of coherence created should depend on the quantity of the shared quantum correlation, which is also a resource. Here, we establish an operational connection between coherence induced by steering and the quantum correlation. We find that the steering-induced coherence quantified by such as relative entropy of coherence and trace-norm of coherence is bounded from above by a known quantum correlation measure def…
Quantum coherence is a fundamental manifestation of the quantum superposition principle. Recently, Baumgratz \emph{et al}. [Phys. Rev. Lett. \textbf{113}, 140401 (2014)] presented a rigorous framework to quantify coherence from the view of theory of physical resource. Here we propose a new valid quantum coherence measure which is a convex roof measure, for a quantum system of arbitrary dimension, essentially using the generalized Gell-Mann matrices. Rigorous proof shows that the proposed coherence measure fulfills all the requirements dictated by the resource theory of quantum coherence measur…
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 …
We derive a monogamy inequality for any local quantum resource and entanglement. It results from the fact that there is always a convex measure for a quantum resource, as shown here, and from the relation between entanglement and local entropy. One of its consequences is an entanglement monogamy different from that usually discussed. If the local resource is nonuniformity or coherence, it is satisfied by familiar resource and entanglement measures. The ensuing upper bound for the local coherence, determined by the entanglement, is independent of the basis used to define the coherence.
Quantum measurement is a class of quantum channels that sends quantum states to classical states. We set up resource theories of quantum coherence and quantum entanglement for quantum measurements and find relations between them. For this, we conceive a relative entropy type quantity to account for the quantum resources of quantum measurements. The quantum coherence of a quantum measurement can be converted into the entanglement in a bipartite quantum measurement through coherence-non generating transformations. Conversely, a quantum entanglement monotone of quantum measurements induces a quan…
We investigate the behavior of coherence in scattering quantum walk search on complete graph under the condition that the total number of vertices of the graph is greatly larger than the marked number of vertices we are searching, $N \gg v$. We find that the consumption of coherence represents the increase of the success probability for the searching,also the consumption of coherence is related to the efficiency of the algorithm represented by oracle queries.If no coherence is consumed, the efficiency of the algorithm will be the same as the classical blind search, implying that coherence is r…
A fundamental task in any physical theory is to quantify certain physical quantity in a meaningful way. In this paper we show that both fidelity distance and affinity distance satisfy the strong contractibility, and the corresponding resource quantifiers can be used to characterize a large class of resource theories. Under two assumptions, namely, convexity of "free states" and closure of free states under "selective free operations", our general framework of resource theory includes quantum resource theories of entanglement, coherence, partial coherence and superposition. In partial coherence…
Shor's factoring algorithm provides a super-polynomial speed-up over all known classical factoring algorithms. Here, we address the question of which quantum properties fuel this advantage. We investigate a sequential variant of Shor's algorithm with a fixed overall structure and identify the role of coherence for this algorithm quantitatively. We analyze this protocol in the framework of dynamical resource theories, which capture the resource character of operations that can create and detect coherence. This allows us to derive a lower and an upper bound on the success probability of the prot…
We introduce a rigorous framework for the quantification of coherence and identify intuitive and easily computable measures of coherence. We achieve this by adopting the viewpoint of coherence as a physical resource. By determining defining conditions for measures of coherence we identify classes of functionals that satisfy these conditions and other, at first glance natural quantities, that do not qualify as coherence measure. We conclude with an outline of the questions that remain to be answered to complete the theory of coherence as a resource.
Coherence is a cornerstone of quantum theory and a prerequisite for the advantage of quantum technologies. In recent work, the notion of coherence with respect to a general quantum measurement (POVM) was introduced and embedded into a resource-theoretic framework that generalizes the standard resource theory of coherence. In particular, POVM-incoherent (free) states and operations were established. In this work, we explore features of this framework which arise due to the rich structure of POVMs compared to projective measurements. Moreover, we introduce a rigorous, probabilisitic framework fo…
Quantum resources, such as coherence, discord, and entanglement, play as a key role for demonstrating advantage in many computation and communication tasks. In order to find the nature behind these resources, tremendous efforts have been made to explore the connections between them. In this work, we extend the single party coherence resource framework to the distributed scenario and relate it to basis-dependent discord. We show the operational meaning of basis-dependent discord in quantum key distribution. By formulating a framework of basis-dependent discord, we connect these quantum resource…
We define the coherence non-activating measurement as the positive operator-valued measurement which gives the same result whether or not the coherence in a quantum state is destroyed. A connection is built between the coherence activating ability of a measurement and its ability to steer quantum states when coherence non-activating measurement is free. Then we study the quantum discord based on coherence non-activating measurement and its behavior under local incoherent operations. Our results contribute to the study of resource non-activating condition, which is a complementary to the well-s…
We analyze the optimal basis for generating the maximum relative entropy of quantum coherence by an arbitrary gate on a two-qubit system. The optimal basis is not unique, and the high quantum coherence generating gates are also typically high entanglement generating ones and vice versa. However, the profile of the relative frequencies of Haar random unitaries generating different amounts of entanglement for a fixed amount of quantum coherence is different from the one in which the roles of entanglement and quantum coherence are reversed, although both follow the beta distribution.
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…
One of the main problems in any quantum resource theory is the characterization of the conversions between resources by means of the free operations of the theory. In this work, we advance on this characterization within the quantum coherence resource theory by introducing the generalized coherence vector of an arbitrary quantum state. The generalized coherence vector is a probability vector that can be interpreted as a concave roof extension of the pure states coherence vector. We show that it completely characterizes the notions of being incoherent, as well as being maximally coherent. Moreo…
