The concept of quantum coherence and its possible use as a resource are currently the subject of active researches. Uncertainty and complementarity relations for quantum coherence allow one to study its changes with respect to other characteristics of the process of interest. Protocols of quantum information processing often use measurements that have a special inner structure. Quantum designs are considered as tools with a perspective of fruitful applications in quantum information processing. We obtain uncertainty and certainty relations for coherence averaged with respect to a set of design-structured POVMs of rank one. To characterize the degree of coherence, the relative entropy of coherence is utilized. The derived relations are exemplified with quantum designs in the two-dimensional Hilbert space.
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 …
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 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…
Complementarity relations between various characterizations of a probability distribution are at the core of information theory. In particular, lower and upper bounds for the entropic function are of great importance. In applied topics, we often deal with situations, where the sums of certain powers of probabilities are known. The main question is how to convert the imposed restrictions into two-sided estimates on the Shannon entropy. It is addressed in two different ways. The more intuitive of them is based on truncated expansions of the Taylor type. Another method is based on the use of coef…
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…
Quantum measurement not only can destroy coherence but also can create it. Here, we estimate the maximum amount of coherence, one can create under a complete non-selective measurement process. For our analysis, we consider projective as well as POVM measurements. Based on our observations, we characterize the measurement processes into two categories, namely, the measurements with the ability to induce coherence and the ones without this ability. Our findings also suggest that the more POVM elements present in a measurement that acts on the quantum system, the less will be its coherence creati…
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…
Various forms of optimality for quantum observables described as normalized positive operator valued measures (POVMs) are studied in this paper. We give characterizations for observables that determine the values of the measured quantity with probabilistic certainty or a state of the system before or after the measurement. We investigate observables which are free from noise caused by classical post-processing, mixing, or pre-processing of quantum nature. Especially, a complete characterization of pre-processing and post-processing clean observables is given, and necessary and sufficient condi…
Maximal coherence of quantum measurement and the resource theory of sharpness
Institute for Convergence Research and Education in Advanced Technology, Yonsei University（延世大学先进技术与教育融合研究所）；Department of Quantum Information, Graduate School, Yonsei University（延世大学研究生院量子信息系）；Agency for Defense Development（国防发展局）；Department of Physics, University of Texas at El Paso（德克萨斯大学埃尔帕索分校物理系）
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…
Quantum coherence is a fundamental resource that quantum technologies exploit to achieve performance beyond that of classical devices. A necessary prerequisite to achieve this advantage is the ability of measurement devices to detect coherence from the measurement statistics. Based on a recently developed resource theory of quantum operations, here we quantify experimentally the ability of a typical quantum-optical detector, the weak-field homodyne detector, to detect coherence. We derive an improved algorithm for quantum detector tomography and apply it to reconstruct the positive-operator-va…
The conventional coherence is defined with respect to a fixed orthonormal basis, i.e., to a von Neumann measurement. Recently, generalized quantum coherence with respect to general positive operator-valued measurements (POVMs) has been presented. Several well-defined coherence measures, such as the relative entropy of coherence $C_{r}$, the $l_{1}$ norm of coherence $C_{l_{1}}$ and the coherence $C_{T,α}$ based on Tsallis relative entropy with respect to general POVMs have been obtained. In this work, we investigate the properties of $C_{r}$, $l_{1}$ and $C_{T,α}$. We estimate the upper bounds…
Unsharp POVM measurements allow the estimation and tracking of quantum wavefunctions in real-time with minimal disruption of the dynamics. Here we demonstrate that high fidelity state monitoring, and hence quantum control, is possible even in the presence of classical dephasing and amplitude noise, by simulating such measurements on a two-level system undergoing Rabi oscillations. Finite estimation fidelity is found to persist indefinitely long after the decoherence times set by the noise fields in the absence of measurement.
Coherence, treated as a resource in quantum information theory, is a basis-dependent quantity. Looking for states that have constant coherence under canonical changes of basis yields highly symmetric structures in state space. For the case of a qubit, we find an easy construction of qubit SICs (Symmetric Informationally Complete POVMs). SICs in dimension 3 and 8 are also shown to be equicoherent.
In this work we investigate how to quantify the coherence of quantum measurements. First, we establish a resource theoretical framework to address the coherence of measurement and show that any statistical distance can be adopted to define a coherence monotone of measurement. For instance, the relative entropy fulfills all the required properties as a proper monotone. We specifically introduce a coherence monotone of measurement in terms of off-diagonal elements of Positive-Operator-Valued Measure (POVM) components. This quantification provides a lower bound on the robustness of measurement-co…
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…
Non-local Advantage of Quantum Coherence(NAQC) or steerability of local quantum coherence is a strong non-local resource based on coherence complementarity relations. In this work, we provide an upper bound on the number of observers who can independently steer the coherence of the observer in the other wing in a scenario where half of an entangled pair of spin-$\frac{1}{2}$ particles is shared between a single observer (Bob) in one wing and several observers (Alices) on the other, who can act sequentially and independently of each other. We consider one-parameter dichotomic POVMs for the Alic…
Quantum coherence represented by a superposition of energy eigenstates is, together with energy, an important resource for quantum technology and thermodynamics. Energy and quantum coherence however, can be complementary. The increase of energy can reduce quantum coherence and vice versa. Recently, it was realized that steady-state quantum coherence could be autonomously harnessed from a cold environment. We propose a conditional synthesis of $N$ independent two-level systems (TLS) with partial quantum coherence obtained from an environment to one coherent system using a measurement able to in…
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 subjective Bayesian interpretation of probability asserts that the rules of the probability calculus follow from the normative principle of Dutch-book coherence: A decision-making agent should not assign probabilities such that a series of monetary transactions based on those probabilities would lead them to expect a sure loss. Similarly, the subjective Bayesian interpretation of quantum mechanics (QBism) asserts that the Born rule is a normative rule in analogy to Dutch-book coherence, but with the addition of one or more empirically based assumptions -- i.e., the "only a little more" tha…
Wave-particle duality, an important and fundamental concept established upon pure quantum systems, is central to the complementarity principle in quantum mechanics. However, due to the environment effects or even the entanglement between the quanton and the which-way detector (WWD), the quanton should be described by a mixed quantum state but not a pure quantum state. Although there are some attempts to clarify the complementarity principle for mixed quantum systems, it is still unclear how the mixedness affects the complementary relation. Here, we give a ternary complementary relation (TCR) a…
School of Mathematics and Statistics, Hainan University（海南大学数学与统计学院）；Hainan International Exchange Center for Theoretical Physics（海南理论物理交流中心）；School of Physics, Huazhong University of Science and Technology（华中科技大学物理学院）；Center for Gravitational Physics and Quantum Information, Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所引力物理与量子信息中心）
