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arXiv:2110.10891 · Journal of Physics A Mathematical and Theoretical

l 1 norm of coherence is not equal to its convex roof quantifier

Jianwei Xu

Published Feb 15, 2022• 4 citations
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Abstract

Abstract Since a rigorous framework for quantifying quantum coherence was established by Baumgratz et al (2014 Phys. Rev. Lett. 113 140401), many coherence measures have been found. For a given coherence measure C , extending the values of C on pure states to mixed states by the convex roof construction, we will get a valid coherence measure C ¯ , we call C ¯ the corresponding convex roof quantifier of C . Whether C = C ¯ for a given coherence measure is an important question. In this work, we show that for the widely used coherence measure, l 1 norm of coherence C l 1 , it holds that C l 1 ≠ C l 1 ¯ .

DOI
10.1088/1751-8121/ac552a

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Abstract Since a rigorous framework for quantifying quantum coherence was established by Baumgratz et al (2014 Phys. Rev. Lett. 113 140401), many coherence measures have been found. For a given coherence measure C , extending the values of C on pure states to mixed states by the convex roof construction, we will get a valid coherence measure C ¯ , we call C ¯ the corresponding convex roof quantifier of C . Whether C = C ¯ for a given coherence measure is an important question. In this work, we show that for the widely used coherence measure, l 1 norm of coherence C l 1 , it holds that C l 1 ≠ C l 1 ¯ .
While various coherence measures based on the framework for quantifying coherence [T. Baumgratz, M. Cramer, and M. B. Plenio, Phys. Rev. Lett. 113, 140401 (2014)] have been proposed, it is often asked whether a coherence measure fulfills the maximal value condition that only maximally coherent states should achieve the maximal value of a coherence measure [Y. Peng, Y. Jiang, and H. Fan, Phys. Rev. A 93, 032326 (2016)]. Although it may be easy to rule out the measures that violate the condition by giving a counterexample, it is usually complicated to confirm the validity of the condition for a general coherence measure due to the deal with all mixed states. In this paper, we prove that the maximal value condition of coherence measures holds for mixed states if and only if it holds for a special subset of pure states. Our finding can greatly reduce the examination of whether a coherence measure fulfills the maximal value condition since one only needs to consider a subset of pure states, avoiding the work of considering mixed states.
Abstract We study the closed expressions of the convex roof coherence measures for one-qubit states in this paper. We present the analytical expressions for the convex roof coherence measures, C f ( ρ ) , of one-qubit states with C f ( φ ) := f ( c 0 2 , c 1 2 ) (where | φ ⟩ = c 0 | 0 ⟩ + c 1 | 1 ⟩ ) being convex with respect to the l 1 norm of coherence of ϕ (i.e. C l 1 ( φ ) ), such coherence measures including the coherence of formation, the geometric measure of coherence, the coherence concurrence, and the coherence rank. We further present the operational interpretations of these measures. Finally, we present the usefulness of the convex roof coherence measures C f ( φ ) being non-convex with respect to C l 1 ( φ ) by giving the necessary and sufficient conditions for the transformations p φ 1 ⊕ ( 1 − p ) φ 2 → q ϕ 1 ⊕ ( 1 − q ) ϕ 2 via incoherent operations, where ϕ i , φ j ( i , j = 1 , 2 ) are one-qubit pure states and 0 ⩽ p , q ⩽ 1 .
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.
Quantification is one of the primary goals of quantum coherence resource theory. Here, we put forward two coherence measures, analytically prove their validity in the pure-state regime following the axiomatic definition of a legitimate pure-state coherence measure (PSCM), and provide their general normalized forms. We further discuss their extensions in the domain of mixed states with the assistance of the convex roof of coherence, and top coherence (quantification in terms of pure state coherence) and define their classes (coherence \emph{monotone} or \emph{measure}); in this regard, we thoroughly investigate the top coherence class. For the quantitative demonstration, we pick up different laser pulse-two-qubit interaction scenarios and compare the evolutions of these two coherence measures along with l1l_{1}l1​-norm of coherence and relative entropy of coherence; this study also signifies the importance of coherence in probing an atomic or molecular system.