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arXiv:1906.09789 · Physical Review A

Quantum coherence fraction

Yao Yao, Dong Li, C. P. Sun

Published Sep 18, 2019• 16 citations
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Abstract

As an analogy of the fully entangled fraction in the framework of entanglement theory, we have introduced the notion of quantum coherence fraction ${C}_{\mathcal{F}}$, which quantifies the closeness between a given state and the set of maximally coherent states. By providing an alternative formulation of the robustness of coherence ${C}_{\mathcal{R}}$, we have elucidated the relationship between the quantum coherence fraction and the normalized version of ${C}_{\mathcal{R}}$, i.e., ${\overline{C}}_{\mathcal{R}}$, where the role of genuinely incoherent operations is highlighted. Numerical simulation shows that, though as expected ${C}_{\mathcal{F}}$ is upper bounded by ${\overline{C}}_{\mathcal{R}}$, ${C}_{\mathcal{F}}$ constitutes a good approximation to ${\overline{C}}_{\mathcal{R}}$ especially in low-dimensional Hilbert spaces. Even more intriguingly, we can analytically prove that ${C}_{\mathcal{F}}$ is exactly equivalent to ${\overline{C}}_{\mathcal{R}}$ for qubit and qutrit states. Moreover, some intuitive properties and implications of ${C}_{\mathcal{F}}$ are also indicated.

DOI
10.1103/physreva.100.032324

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As an analogy of the fully entangled fraction in the framework of entanglement theory, we have introduced the notion of quantum coherence fraction CF{C}_{\mathcal{F}}CF​, which quantifies the closeness between a given state and the set of maximally coherent states. By providing an alternative formulation of the robustness of coherence CR{C}_{\mathcal{R}}CR​, we have elucidated the relationship between the quantum coherence fraction and the normalized version of CR{C}_{\mathcal{R}}CR​, i.e., C‾R{\overline{C}}_{\mathcal{R}}CR​, where the role of genuinely incoherent operations is highlighted. Numerical simulation shows that, though as expected CF{C}_{\mathcal{F}}CF​ is upper bounded by C‾R{\overline{C}}_{\mathcal{R}}CR​, CF{C}_{\mathcal{F}}CF​ constitutes a good approximation to C‾R{\overline{C}}_{\mathcal{R}}CR​ especially in low-dimensional Hilbert spaces. Even more intriguingly, we can analytically prove that CF{C}_{\mathcal{F}}CF​ is exactly equivalent to C‾R{\overline{C}}_{\mathcal{R}}CR​ for qubit and qutrit states. Moreover, some intuitive properties and implications of CF{C}_{\mathcal{F}}CF​ are also indicated.
Quantum coherence, present whenever a quantum system exists in a superposition of multiple classically distinct states, marks one of the fundamental departures from classical physics.Quantum coherence has recently been investigated rigorously within a resource-theoretic formalism.However, the finer-grained notion of multilevel coherence, which explicitly takes into account the number of superposed classical states, has remained relatively unexplored.A comprehensive analysis of multilevel coherence, which acts as the singleparty analogue to multipartite entanglement, is essential for understanding natural quantum processes as well as for gauging the performance of quantum technologies.Here, we develop the theoretical and experimental groundwork for characterizing and quantifying multilevel coherence.We prove that nontrivial levels of purity are required for multilevel coherence, as there is a ball of states around the maximally mixed state that do not exhibit multilevel coherence in any basis.We provide a simple, necessary, and sufficient analytical criterion to verify the presence of multilevel coherence, which leads to a complete classification of multilevel coherence for three-level systems.We present the robustness of multilevel coherence, a bona fide quantifier, which we show to be numerically computable via semidefinite programming and experimentally accessible via multilevel coherence witnesses, which we introduce and characterize.We further verify and lower bound the robustness of multilevel coherence by performing a semi-device-independent phase discrimination task, which is implemented experimentally with four-level quantum probes in a photonic setup.Our results contribute to understanding the operational relevance of genuine multilevel coherence, also by demonstrating the key role it plays in enhanced phase discrimination-a primitive for quantum communication and metrology-and suggest new ways to reliably and effectively test the quantum behavior of physical systems.
We generalize the recently proposed resource theory of coherence (or superposition) [T. Baumgratz et al., Phys. Rev. Lett. 113, 140401 (2014); A. Winter and D. Yang, Phys. Rev. Lett. 116, 120404 (2016)] to the setting where not just the free (``incoherent'') resources, but also the manipulated objects, are quantum operations rather than states. In particular, we discuss an information theoretic notion of the 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 unitaries and general channels acting on a ddd-dimensional system; we show that a maximally coherent state of rank ddd is always sufficient as a resource if incoherent operations are allowed, and one of rank d2{d}^{2}d2 for ``strictly incoherent'' operations. We also show lower bounds on the simulation cost of channels that allow us to conclude that there exists bound coherence in operations, i.e., maps with nonzero cost of implementing them but zero coherence capacity; this is in contrast to states, which do not exhibit bound coherence.
As an analogy of best separable approximation (BSA) in the framework of entanglement theory, here we concentrate on the notion of best incoherent approximation, with application to characterizing and quantifying quantum coherence. From both analytical and numerical perspectives, we have demonstrated that the weight-based coherence measure displays some unusual properties, in sharp contrast to other popular coherence quantifiers. First, by deriving a closed formula for single-qubit states, we have shown the exact value of weight-based coherence measure does not depend solely on the off-diagonal element but relies on the geometry of a given qubit state, thus exhibiting a rich structure even in this simplest case. Second, albeit the fact that almost all popular coherence measures only assign a maximal value to pure maximally coherent states, we have demonstrated the existence of mixed maximally coherent states (MMCS) with respect to this coherence measure and discussed the characteristic feature of MMCS in high-dimensional Hilbert spaces. Especially, we present several important families of MMCS by gaining insights from the numerical simulations. Moreover, it is pointed out that some considerations in this work can be generalized to general convex resource theories and a numerical method of improving the computational efficiency for finding the BSA is also discussed.
The notion of measurement is of both foundational and instrumental significance in quantum mechanics, and coherence destroyed by measurements (decoherence) lies at the very heart of quantum to classical transition. Qualitative aspects of this spirit have been widely recognized and analyzed ever since the inception of quantum theory. However, axiomatic and quantitative investigations of coherence are attracting great interest only recently with several figures of merit for coherence introduced [Baumgratz, Cramer, and Plenio, Phys. Rev. Lett. 113, 140401 (2014)]. While these resource theoretic approaches have many appealing and intuitive features, they rely crucially on various notions of incoherent operations which are sophisticated, subtle, and not uniquely defined, as have been critically assessed [Chitambar and Gour, Phys. Rev. Lett. 117, 030401 (2016)]. In this paper, we elaborate on the idea that coherence and quantum uncertainty are dual viewpoints of the same quantum substrate, and address coherence quantification by identifying coherence of a state (with respect to a measurement) with quantum uncertainty of a measurement (with respect to a state). Consequently, coherence measures may be set into correspondence with measures of quantum uncertainty. In particular, we take average quantum Fisher information as a measure of quantum uncertainty, and introduce the corresponding measure of coherence, which is demonstrated to exhibit desirable properties. Implications for interpreting quantum purity as maximal coherence, and quantum discord as minimal coherence, are illustrated.
Recently, there has been a renewed interest in the quantification of coherence or other coherencelike 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 two sides of the same coin. Therefore, a natural question is raised: How can the conventional decoherence processes, such as the von Neumann--L\"uders (projective) measurement postulation or partially dephasing channels, fit into the bigger picture of the recently established theoretical framework? Here we show that the state collapse rules of the von Neumann or L\"uders-type measurements, as special cases of genuinely incoherent operations (GIOs), are consistent with the resource theories of quantum coherence. New hierarchical measures of coherence are proposed for the L\"uders-type measurement and their relationship with measurement-dependent discord is addressed. Moreover, utilizing the fixed-point theory for C∗{C}^{*}C∗ algebra, we prove that GIOs 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 dilation theorem, the physical realizations of incoherent operations are investigated in detail and we find that GIOs in fact constitute the core of strictly incoherent operations and generally incoherent operations and the unspeakable notion of coherence induced by GIOs can be transferred to the theories of speakable coherence by the corresponding permutation or relabeling operators.
In recent years, several measures have been proposed for characterizing the coherence of a given quantum state. We derive several results that illuminate how these measures behave when restricted to pure states. Notably, we present an explicit characterization of the closest incoherent state to a given pure state under the trace distance measure of coherence. We then use this result to show that the states maximizing the trace distance of coherence are exactly the maximally coherent states. We define the trace distance of entanglement and show that it coincides with the trace distance of coherence for pure states. Finally, we give an alternate proof to a recent result that the ${\ensuremath{\ell}}_{1}$ measure of coherence of a pure state is never smaller than its relative entropy of coherence.