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arXiv:2103.09409 · Physical Review Research

Quantifying the entanglement of quantum channel

Huaqi Zhou, Ting Gao, Fengli Yan

Published Mar 14, 2022• 9 citations
arXivHTMLPDF

Abstract

To describe the nonclassicality of quantum channels, it is necessary to quantify their properties. In the framework of resource theory of quantum channel, we present three new entanglement measures of channels for a quantum system of arbitrary finite dimension, based on the Choi relative entropy of channels, concurrence, and $k$-ME concurrence, respectively. Rigorous proofs show that these proposed measures fulfill all the requirements dictated by the resource theory of quantum channels, including non-negativity, weak monotonicity, strong monotone, convexity, additivity, and subadditivity. By using the measure, we describe the probability that the initial hypothesis is true in the hypothesis testing of channels. Compared with other known measures of the quantum channels, these measures have an obvious advantage in computability. According to the power of generating entanglement, we give a classification of quantum channels. Moreover, we provide several examples that characterize the properties of quantum channels and the conversions between quantum coherence states and entangled states and between fully separable states and multipartite entangled states.

DOI
10.1103/physrevresearch.4.013200

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To describe the nonclassicality of quantum channels, it is necessary to quantify their properties. In the framework of resource theory of quantum channel, we present three new entanglement measures of channels for a quantum system of arbitrary finite dimension, based on the Choi relative entropy of channels, concurrence, and kkk-ME concurrence, respectively. Rigorous proofs show that these proposed measures fulfill all the requirements dictated by the resource theory of quantum channels, including non-negativity, weak monotonicity, strong monotone, convexity, additivity, and subadditivity. By using the measure, we describe the probability that the initial hypothesis is true in the hypothesis testing of channels. Compared with other known measures of the quantum channels, these measures have an obvious advantage in computability. According to the power of generating entanglement, we give a classification of quantum channels. Moreover, we provide several examples that characterize the properties of quantum channels and the conversions between quantum coherence states and entangled states and between fully separable states and multipartite entangled states.
In this paper, we investigate the characterization of multipartite entanglement based on the unified (q,s)(q,s)(q,s)-entropy framework. For bipartite quantum states, we propose an entanglement measure Eq,sA∣B(ρ)E_{q,s}^{A|B}(ρ)Eq,sA∣B​(ρ) based on unified entropy and derive several analytical lower bounds for it using local orthonormal observables. We demonstrate with concrete examples that our lower bounds provide tighter estimates of quantum entanglement than several existing analytical bounds. For multipartite quantum states, based on unified entropy, we propose two measures Aq,sk(ρ)A^k_{q,s}(ρ)Aq,sk​(ρ) and Gq,sk(ρ)G^k_{q,s}(ρ)Gq,sk​(ρ) for quantifying kkk-nonseparability, and prove that these measures satisfy several desirable properties, such as faithfulness, local unitary invariance, and convexity. Moreover, we rigorously prove that when the parameters qqq and sss are in certain ranges, Aq,sk(ρ)A^k_{q,s}(ρ)Aq,sk​(ρ) and Gq,sk(ρ)G^k_{q,s}(ρ)Gq,sk​(ρ) also satisfy monotonicity and strong monotonicity. Furthermore, we establish the order relations of the measures Aq,sk(ρ)A^k_{q,s}(ρ)Aq,sk​(ρ) and Gq,sk(ρ)G^k_{q,s}(ρ)Gq,sk​(ρ) with respect to the parameters qqq and sss for fixed quantum states. In addition, for fixed qqq and sss, we also give their order relations for any two pure states.
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 nongenerating transformations. Conversely, a quantum entanglement monotone of quantum measurements induces a quantum coherence monotone of quantum measurements. Our results confirm that the understanding on the link between quantum coherence and quantum entanglement is valid even for quantum measurements which do not generate any quantum resource.
We study two distinct diagnostics of bipartite quantum channels: the average fidelity between a desired state and noisy output across different classes of input states, and generation of entanglement from pure product input states, quantified by well-defined entangling power. These diagnostics are generally independent, but together they provide a more complete description of bipartite channels. We show that the fidelity averaged over any local-unitary orbit with fixed Schmidt coefficients for equal local dimensions is completely determined by the average input-output fidelity and its restriction to product inputs. We also introduce concurrence- and negativity-based entangling power for two-qubit channels, prove their convexity and monotonicity under local postprocessing, and show that, unlike the previously proposed linear-entropy quantity, they vanish for all separable channels. Representative separable and nonseparable channels are investigated. Finally, we generalize our definitions of entangling power to nonproduct inputs, and provide an analytical lower bound for the concurrence-based entanglement variation, showing its effectiveness with specific examples.
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.
Quantum coherence and nonclassical correlation are key features of a quantum world. Quantifying coherence and nonclassical correlation are two key tasks in quantum information theory. First, we present a bona fide measure of quantum coherence by utilizing the Hellinger distance. This coherence measure is proven to fulfill all the criteria of a well-defined coherence measure, including the strong monotonicity in the resource theories of quantum coherence. In terms of this coherence measure, the distribution of quantum coherence in multipartite systems is studied and a corresponding polygamy relation is proposed. Its operational meanings and the relations between the generation of quantum correlations and the coherence are also investigated. Moreover, we present the Hellinger distance-based measure of nonclassical correlation, which not only inherits the nice properties of the Hellinger distance including contractivity, but also shows a powerful analytic computability for a large class of quantum states. We show that there is an explicit tradeoff relation satisfied by the quantum coherence and this nonclassical correlation.
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.