We study the geometric measure of quantum coherence recently proposed in [Phys. Rev. Lett. 115 (2015) 020403]. Both lower and upper bounds of this measure are provided. These bounds are shown to be tight for a class of important coherent states — maximally coherent mixed states. The trade-off relation between quantum coherence and mixedness for this measure is also discussed.
We study the geometric measure of quantum coherence recently proposed in [Phys. Rev. Lett. 115 (2015) 020403]. Both lower and upper bounds of this measure are provided. These bounds are shown to be tight for a class of important coherent states — maximally coherent mixed states. The trade-off relation between quantum coherence and mixedness for this measure is also discussed.
Abstract Quantum coherence is an important quantum resource and it is intimately related to various research fields. The geometric coherence is a coherence measure both operationally and geometrically. We study the trade-off relation of geometric coherence in qubit systems. We first derive an upper bound for the geometric coherence by the purity of quantum states. Based on this, a complementarity relation between the quantum coherence and the mixedness is established. We then derive the quantum uncertainty relations of the geometric coherence on two and three general measurement bases in terms of the incompatibility respectively, which turn out to be state-independent for pure states. These trade-off relations provide the limit to the amount of quantum coherence. As a byproduct, the complementarity relation between the minimum error probability for discriminating a pure-states ensemble and the mixedness of quantum states is established.
Quantum coherence is a key element in topical research on quantum resource theories and a primary facilitator for design and implementation of quantum technologies. However, the resourcefulness of quantum coherence is severely restricted by environmental noise, which is indicated by the loss of information in a quantum system, measured in terms of its purity. In this work, we derive the limits imposed by the mixedness of a quantum system on the amount of quantum coherence that it can possess. We obtain an analytical trade-off between the two quantities that upperbound the maximum quantum coherence for fixed mixedness in a system. This gives rise to a class of quantum states, ``maximally coherent mixed states,'' whose coherence cannot be increased further under any purity-preserving operation. For the above class of states, quantum coherence and mixedness satisfy a complementarity relation, which is crucial to understand the interplay between a resource and noise in open quantum systems.
We study the average quantum coherence over the pure state decompositions of a mixed quantum state. An upper bound of the average quantum coherence is provided, and sufficient conditions for the saturation of the upper bound are shown. These sufficient conditions always hold for two- and three-dimensional systems. This provides a tool to estimate the average coherence experimentally by measuring only the diagonal elements, which remarkably requires less measurements compared with state tomography. We then describe the pure state decompositions of the qubit state in the Bloch sphere geometrically. For any given qubit state, the optimal pure state decomposition achieving the maximal average quantum coherence as well as three other pure state decompositions are shown in the Bloch sphere. The order relations among their average quantum coherence are invariant for any coherence measure. The results presented in this paper are universal and suitable for all coherence measures.
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.
Detecting multipartite quantum coherence usually requires quantum state reconstruction, which is quite inefficient for large-scale quantum systems. Along this line of research, several efficient procedures have been proposed to detect multipartite quantum coherence without quantum state reconstruction, among which the spectrum-estimation-based method is suitable for various coherence measures. Here, we first generalize the spectrum-estimation-based method for the geometric measure of coherence. Then, we investigate the tightness of the estimated lower bound of various coherence measures, including the geometric measure of coherence, the l1-norm of coherence, the robustness of coherence, and some convex roof quantifiers of coherence multiqubit GHZ states and linear cluster states. Finally, we demonstrate the spectrum-estimation-based method as well as the other two efficient methods. We observe that the spectrum-estimation-based method outperforms other methods in various coherence measures, which significantly enhances the accuracy of estimation.
Recently, a proper genuine multipartite entanglement measure has been found for three-qubit pure states [see Xie and Eberly, Phys. Rev. Lett. 127, 040403 (2021)PRLTAO0031-900710.1103/PhysRevLett.127.040403], but capturing useful entanglement measures for mixed states has remained an open challenge. So far, it requires not only a full tomography in experiments, but also huge calculational labor. A leading proposal was made by Gühne, Reimpell, and Werner [Phys. Rev. Lett. 98, 110502 (2007)PRLTAO0031-900710.1103/PhysRevLett.98.110502], who used expectation values of entanglement witnesses to describe a lower bound estimation of entanglement. We provide here an extension that also gives genuine upper bounds of entanglement. This advance requires only the expectation value of any Hermitian operator. Moreover, we identify a class of operators A_{1} that not only give good estimates, but also require a remarkably small number of experimental measurements. In this Letter, we define our approach and illustrate it by estimating entanglement measures for a number of pure and mixed states prepared in our recent experiments.