Resource Theory Of Quantum Coherence Research Articles

Interpreting quantum coherence through a quantum measurement process

Recently, there has been a renewed interest in the quantification of coherence or other coherence-like concepts within the framework of quantum resource theory. However, rigorously defined or not, the notion of coherence or decoherence has already been used by the community for decades since the advent of quantum theory. Intuitively, the definitions of coherence and decoherence should be the two sides of the same coin. Therefore, a natural question is raised: how can the conventional decoherence processes, such as the von Neumann-L\"{u}ders (projective) measurement postulation or partially dephasing channels, fit into the bigger picture of the recently established theoretic framework? Here we show that the state collapse rules of the von Neumann or L\"{u}ders-type measurements, as special cases of genuinely incoherent operations (GIO), are consistent with the resource theories of quantum coherence. New hierarchical measures of coherence are proposed for the L\"{u}ders-type measurement and their relationship with measurement-dependent discord is addressed. Moreover, utilizing the fixed point theory for $C^\ast$-algebra, we prove that GIO indeed represent a particular type of partially dephasing (phase-damping) channels which have a matrix representation based on the Schur product. By virtue of the Stinespring's dilation theorem, the physical realizations of incoherent operations are investigated in detail and we find that GIO in fact constitute the core of strictly incoherent operations (SIO) and generally incoherent operations (IO) and the unspeakable notion of coherence induced by GIO can be transferred to the theories of speakable coherence by the corresponding permutation or relabeling operators.

The role of quantum coherence in dimer and trimer excitation energy transfer

Recent progress in resource theory of quantum coherence has resulted in measures to quantify coherence in quantum systems. Especially, the l1-norm and relative entropy of coherence have been shown to be proper quantifiers of coherence and have been used to investigate coherence properties in different operational tasks. Since long-lasting quantum coherence has been experimentally confirmed in a number of photosynthetic complexes, it has been debated if and how coherence is connected to the known efficiency of population transfer in such systems. In this study, we investigate quantitatively the relationship between coherence, as quantified by l1-norm and relative entropy of coherence, and efficiency, as quantified by fidelity, for population transfer between end-sites in a network of two-level quantum systems. In particular, we use the coherence averaged over the duration of the population transfer in order to carry out a quantitative comparison between coherence and fidelity. Our results show that although coherence is a necessary requirement for population transfer, there is no unique relation between coherence and the efficiency of the transfer process.

Structure of the Resource Theory of Quantum Coherence.

Quantum coherence is an essential feature of quantum mechanics which is responsible for the departure between the classical and quantum world. The recently established resource theory of quantum coherence studies possible quantum technological applications of quantum coherence, and limitations that 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 Letter, we contribute to such a characterization by proving several upper bounds on the maximum number of incoherent Kraus operators in a general incoherent operation. For a single qubit, we show that the number of incoherent Kraus operators is not more than 5, and it remains an open question if this number can be reduced to 4. The presented results are also relevant for quantum thermodynamics, as we demonstrate by introducing the class of Gibbs-preserving strictly incoherent operations, and solving the corresponding mixed-state conversion problem for a single qubit.

Quantum Coherence Quantifiers Based on Rényi α-Relative Entropy**Supported by by National Natural Science Foundation of China under Grant Nos. 11271237, 11671244, 61671280, and the Higher School Doctoral Subject Foundation of Ministry of Education of China under Grant No. 20130202110001, and Fundamental Research Funds for the Central Universities (GK201502004 and 2016CBY003), and the Academic Leaders and Academic Backbones, Shaanxi Normal University

The resource theories of quantum coherence attract a lot of attention in recent years. Especially, the monotonicity property plays a crucial role here. In this paper we investigate the monotonicity property for the coherence measures induced by the Rényi α-relative entropy, which present in [Phys. Rev. A 94 (2016) 052336]. We show that the Rényi α-relative entropy of coherence does not in general satisfy the monotonicity requirement under the subselection of measurements condition and it also does not satisfy the extension of monotonicity requirement, which presents in [Phys. Rev. A 93 (2016) 032136]. Due to the Rényi α-relative entropy of coherence can act as a coherence monotone quantifier, we examine the trade-off relations between coherence and mixedness. Finally, some properties for the single qubit of Rényi 2-relative entropy of coherence are derived.

Quantum coherence via skew information and its polygamy

Quantifying coherence is a key task in both quantum mechanical theory and practical applications. Here, a reliable quantum coherence measure is presented by utilizing the quantum skew information of the state of interest subject to a certain broken observable. This coherence measure is proven to fulfill all the criteria (especially the strong monotonicity) recently introduced in the resource theories of quantum coherence. The coherence measure has an analytic expression and an obvious operational meaning related to quantum metrology. In terms of this coherence measure, the distribution of the quantum coherence, i.e., how the quantum coherence is distributed among the multiple parties is studied and a corresponding polygamy relation is proposed. As a further application, it is found that the coherence measure forms the natural upper bounds for quantum correlations prepared by incoherent operations. The experimental measurement of our coherence measure as well as the relative-entropy coherence, $l_p$-norm coherence is studied finally.

Towards Resource Theory of Coherence in Distributed Scenarios

The search for a simple description of fundamental physical processes is an important part of quantum theory. One example for such an abstraction can be found in the distance lab paradigm: if two separated parties are connected via a classical channel, it is notoriously difficult to characterize all possible operations these parties can perform. This class of operations is widely known as local operations and classical communication (LOCC). Surprisingly, the situation becomes comparably simple if the more general class of separable operations is considered, a finding which has been extensively used in quantum information theory for many years. Here, we propose a related approach for the resource theory of quantum coherence, where two distant parties can only perform measurements which do not create coherence and can communicate their outcomes via a classical channel. We call this class local incoherent operations and classical communication (LICC). While the characterization of this class is also difficult in general, we show that the larger class of separable incoherent operations (SI) has a simple mathematical form, yet still preserving the main features of LICC. We demonstrate the relevance of our approach by applying it to three different tasks: assisted coherence distillation, quantum teleportation, and single-shot quantum state merging. We expect that the results obtained in this work also transfer to other concepts of coherence which are discussed in recent literature. The approach presented here opens new ways to study the resource theory of coherence in distributed scenarios.

Maximal coherence in a generic basis

Since quantum coherence is an undoubted characteristic trait of quantum physics, the quantification and application of quantum coherence has been one of the long-standing central topics in quantum information science. Within the framework of a resource theory of quantum coherence proposed recently, a \textit{fiducial basis} should be pre-selected for characterizing the quantum coherence in specific circumstances, namely, the quantum coherence is a \textit{basis-dependent} quantity. Therefore, a natural question is raised: what are the maximum and minimum coherences contained in a certain quantum state with respect to a generic basis? While the minimum case is trivial, it is not so intuitive to verify in which basis the quantum coherence is maximal. Based on the coherence measure of relative entropy, we indicate the particular basis in which the quantum coherence is maximal for a given state, where the Fourier matrix (or more generally, \textit{complex Hadamard matrices}) plays a critical role in determining the basis. Intriguingly, though we can prove that the basis associated with the Fourier matrix is a stationary point for optimizing the $l_1$ norm of coherence, numerical simulation shows that it is not a global optimal choice.

Comparison of incoherent operations and measures of coherence

Considerable work has recently been directed toward developing resource theories of quantum coherence. In most approaches, a state is said to possess quantum coherence if it is not diagonal in some specified basis. In this letter we establish a criterion of physical consistency for any resource theory in terms of physical implementation of the free operations, and we show that all currently proposed basis-dependent theories of coherence fail to satisfy this criterion. We further characterize the physically consistent resource theory of coherence and find its operational power to be quite limited. After relaxing the condition of physical consistency, we introduce the class of dephasing-covariant incoherent operations, present a number of new coherent monotones based on relative R\'{e}nyi entropies, and study incoherent state transformations under different operational classes. In particular, we derive necessary and sufficient conditions for qubit state transformations and show these conditions hold for all classes of incoherent operations.

Quantum Processes Which Do Not Use Coherence

A major signature of quantum mechanics beyond classical physics is coherence, the existence of superposition states. The recently developed resource theory of quantum coherence allows the formalisation of incoherent operations -- those operations which cannot create coherence. We identify the set of operations which additionally do not use coherence. These are such that coherence cannot be exploited by a classical observer, who measures incoherent properties of the system, to go beyond classical dynamics. We give a physical interpretation in terms of interferometry and prove a dilation theorem, showing how these operations can always be constructed by interacting the system in an incoherent way with an ancilla. Such a physical justification is not known for the incoherent operations, thus our results lead to a physically well-motivated resource theory of coherence. Next, we investigate the implications for coherence in multipartite systems. We show that quantum correlations can be defined naturally with respect to a fixed basis, providing a link between coherence and quantum discord. We demonstrate the interplay between these two quantities under our studied operations, and suggest implications for the theory of quantum discord by relating the studied operations to those which cannot create discord.

Channels that do not generate coherence

We define the coherence non-generating channel as the completely positive trace-preserving map which does not generate quantum coherence from an incoherent state. The incoherent operations are the strict subset of the coherence non-generating channels. We prove that the relative entropy of coherence is monotonically decreasing under the coherence non-generating channels, while the coherence of formation may increase under such channels. Interestingly, by combining mathematically the coherence of formation to the entanglement of formation, we show that the coherence of formation of a single-qubit is never increased by a coherence non-generating channel. This leads to the superadditivity property for the coherence increasing power of quantum channels, namely, while two channels can not increase coherence individually, they may increase the quantum coherence of a composed system. Further, we derive the general form of the rank-2 coherence non-generating qubit channels. Our results contribute to the resource theory of quantum coherence.

Critical Examination of Incoherent Operations and a Physically Consistent Resource Theory of Quantum Coherence.

Considerable work has recently been directed toward developing resource theories of quantum coherence. In this Letter, we establish a criterion of physical consistency for any resource theory. This criterion requires that all free operations in a given resource theory be implementable by a unitary evolution and projective measurement that are both free operations in an extended resource theory. We show that all currently proposed basis-dependent theories of coherence fail to satisfy this criterion. We further characterize the physically consistent resource theory of coherence and find its operational power to be quite limited. After relaxing the condition of physical consistency, we introduce the class of dephasing-covariant incoherent operations as a natural generalization of the physically consistent operations. Necessary and sufficient conditions are derived for the convertibility of qubit states using dephasing-covariant operations, and we show that these conditions also hold for other well-known classes of incoherent operations.

Robustness of Coherence: An Operational and Observable Measure of Quantum Coherence.

Quantifying coherence is an essential endeavor for both quantum foundations and quantum technologies. Here, the robustness of coherence is defined and proven to be a full monotone in the context of the recently introduced resource theories of quantum coherence. The measure is shown to be observable, as it can be recast as the expectation value of a coherence witness operator for any quantum state. The robustness of coherence is evaluated analytically on relevant classes of states, and an efficient semidefinite program that computes it on general states is given. An operational interpretation is finally provided: the robustness of coherence quantifies the advantage enabled by a quantum state in a phase discrimination task.

Measuring Quantum Coherence with Entanglement.

Quantum coherence is an essential ingredient in quantum information processing and plays a central role in emergent fields such as nanoscale thermodynamics and quantum biology. However, our understanding and quantitative characterization of coherence as an operational resource are still very limited. Here we show that any degree of coherence with respect to some reference basis can be converted to entanglement via incoherent operations. This finding allows us to define a novel general class of measures of coherence for a quantum system of arbitrary dimension, in terms of the maximum bipartite entanglement that can be generated via incoherent operations applied to the system and an incoherent ancilla. The resulting measures are proven to be valid coherence monotones satisfying all the requirements dictated by the resource theory of quantum coherence. We demonstrate the usefulness of our approach by proving that the fidelity-based geometric measure of coherence is a full convex coherence monotone, and deriving a closed formula for it on arbitrary single-qubit states. Our work provides a clear quantitative and operational connection between coherence and entanglement, two landmark manifestations of quantum theory and both key enablers for quantum technologies.