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Generalized Coherence Concurrence and Coherence Number.

Seungbeom Chin

Published Feb 20, 2017• 0 citations
OpenAlex

Abstract

We propose a family of coherence monotones, named the \emph{generalized coherence concurrence} (or coherence $k$-concurrence), which has a close relation with the generalized entanglement concurrence. The coherence $k$-concurrence of a state is nonzero if and only if the coherence number (a recently introduced discrete coherence monotone) of the state is not smaller than $k$, and a state can be converted to a state with nonzero entanglement $k$-concurrence via incoherent operations if and only if the state has nonzero coherence $k$-concurrence. The generalized coherence concurrence can be considered as the entanglement-based coherence monotone.

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We propose a family of coherence monotones, named the \emph{generalized coherence concurrence} (or coherence kkk-concurrence), which has a close relation with the generalized entanglement concurrence. The coherence kkk-concurrence of a state is nonzero if and only if the coherence number (a recently introduced discrete coherence monotone) of the state is not smaller than kkk, and a state can be converted to a state with nonzero entanglement kkk-concurrence via incoherent operations if and only if the state has nonzero coherence kkk-concurrence. The generalized coherence concurrence can be considered as the entanglement-based coherence monotone.
We extend the definition of concurrence into a family of entanglement monotones, which we call concurrence monotones. We discuss their properties and advantages as computational manageable measures of entanglement, and show that for pure bipartite states all measures of entanglement can be written as functions of the concurrence monotones. We then show that the concurrence monotones provide bounds on quantum information tasks. As an example, we discuss their applications to remote entanglement distributions (RED) such as entanglement swapping and remote preparation of bipartite entangled states (RPBES). We prove a powerful theorem which states what kind of (possibly mixed) bipartite states or distributions of bipartite states cannot be remotely prepared. The theorem establishes an upper bound on the amount of GGG-concurrence (one member in the concurrence family) that can be created between two single-qudit nodes of quantum networks by means of tripartite RED. For pure bipartite states the bound on the GGG-concurrence can always be saturated by RPBES.
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.
Entanglement monotones, such as the concurrence, are useful tools to characterize quantum correlations in various physical systems. The computation of the concurrence involves, however, difficult optimizations and only for the simplest case of two qubits a closed formula was found by Wootters [Phys. Rev. Lett. 80, 2245 (1998)]. We show how this approach can be generalized, resulting in lower bounds on the concurrence for higher dimensional systems as well as for multipartite systems. We demonstrate that for certain families of states our results constitute the strongest bipartite entanglement criterion so far; moreover, they allow us to recognize novel families of multiparticle bound entangled states.
Abstract Quantum coherence is a fundamental manifestation of the quantum superposition principle. Recently, Baumgratz et al (2014 Phys. Rev. Lett . 113 140401) 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, coherence concurrence, fulfills all the requirements dictated by the resource theory of quantum coherence measures. Moreover, strong links between the resource frameworks of coherence concurrence and entanglement concurrence is derived, which shows that any degree of coherence with respect to some reference basis can be converted to entanglement via incoherent operations. Our work provides a clear quantitative and operational connection between coherence and entanglement based on two kinds of concurrence. This new coherence measure, coherence concurrence, may also be beneficial to the study of quantum coherence.
We derive a family of entanglement monotones, one member of which turns out to be the negativity. Two others are shown to be lower bounds on the I-concurrence, and on the I-tangle, respectively [P. Rungta and C. M. Caves, to appear in Phys. Rev. A]. We compare these bounds with the I-concurrence and I-tangle on the isotropic states, and on rank-two density operators resulting from a Tavis-Cummings interaction. Our results provide a global structure relating several different entanglement measures. Additionally, they possess analytic forms which are easily evaluated in the most general cases.
Although numerous measures of entanglement have been proposed so far, the calculation of a given faithful entanglement measure is a hard work since it is always involved in some optimization process. It is, therefore, important to estimate the lower bound of a given entanglement measure for an arbitrary quantum state. This results in a subject of intensive mathematical research. In particular, along this line, the lower bounds of concurrence or other measures that are induced from concurrence have been explored a lot. Here, we investigate the lower bounds of two kinds of entanglement monotones, i.e., qqq-concurrence (q>1q>1q>1) and ααα-concurrence (0<α<10<α<10<α<1), or termed the parameterized entanglement monotone together. We obtain, in the light of the informationally complete (NNN, MMM)-positive operator-valued measure [(NNN, MMM)-POVM], the lower bounds for the case of 12<α<1\frac12<α<121​<α<1, 1<q<21<q<21<q<2 for two-qudit states, and the case of 2⩽q<32\leqslant q<32⩽q<3 for two-qubit states. We list several examples which show that the lower bounds based on (NNN, MMM)-POVM outperform that of GSIC-POVM and SIC-POVM, and all these measurement based bounds are better then the ones induced by positive partial transpose (PPT) and realignment criteria in literature. In addition, we obtain an analytical formula of the parameterized entanglement monotone with 12<α<1\frac12<α<121​<α<1 and 1<q<21<q<21<q<2 for the isotropic state.