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arXiv:1703.02737 · Scientific Reports

The classical correlation limits the ability of the measurement-induced average coherence

Jun Zhang, Si‐ren Yang, Yang Zhang, Chang‐shui Yu

Published Apr 4, 2017• 21 citations
arXivHTMLPDF

Abstract

Coherence is the most fundamental quantum feature in quantum mechanics. For a bipartite quantum state, if a measurement is performed on one party, the other party, based on the measurement outcomes, will collapse to a corresponding state with some probability and hence gain the average coherence. It is shown that the average coherence is not less than the coherence of its reduced density matrix. In particular, it is very surprising that the extra average coherence (and the maximal extra average coherence with all the possible measurements taken into account) is upper bounded by the classical correlation of the bipartite state instead of the quantum correlation. We also find the sufficient and necessary condition for the null maximal extra average coherence. Some examples demonstrate the relation and, moreover, show that quantum correlation is neither sufficient nor necessary for the nonzero extra average coherence within a given measurement. In addition, the similar conclusions are drawn for both the basis-dependent and the basis-free coherence measure.

DOI
10.1038/srep45598

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Coherence is the most fundamental quantum feature in quantum mechanics. For a bipartite quantum state, if a measurement is performed on one party, the other party, based on the measurement outcomes, will collapse to a corresponding state with some probability and hence gain the average coherence. It is shown that the average coherence is not less than the coherence of its reduced density matrix. In particular, it is very surprising that the extra average coherence (and the maximal extra average coherence with all the possible measurements taken into account) is upper bounded by the classical correlation of the bipartite state instead of the quantum correlation. We also find the sufficient and necessary condition for the null maximal extra average coherence. Some examples demonstrate the relation and, moreover, show that quantum correlation is neither sufficient nor necessary for the nonzero extra average coherence within a given measurement. In addition, the similar conclusions are drawn for both the basis-dependent and the basis-free coherence measure.
We establish an operational connection between coherence and quantum correlation by showing that, Alice can create quantum coherence on Bob's side using local selective measurement as long as the previously shared bipartite state has non-vanishing quantum correlation on BBB, and the maximum average coherence she can extract can not surpass the BBB-side quantum correlation in the initial bipartite state. Steering-induced coherence is introduced to characterize Alice's ability to extract quantum coherence on Bob's side. For pure states and the maximally correlated states, the steering-induced relative entropy of coherence is proved to reach the initially shared quantum correlation which is measured by the relative entropy of BBB-side measurement-induced disturbance. The condition to reach the upper bound of quantum correlation varies for different measures of coherence. While the steering-induced l1l_1l1​-norm of coherence is shown reach the initially shared quantum correlation (measured by trace-norm of BBB-side measurement-induced disturbance) for any two-qubit states, an example is found whose steering-induced relative entropy of coherence is strictly less than the relative entropy of BBB-side measurement-induced disturbance.
Based on a proposed coherence measure, we show that the local coherence of a bipartite quantum pure state (coherence of its reduced density matrix) is exactly the same as the minimal average coherence with all potential pure-state realizations under consideration. In particular, it is shown that bipartite concurrence of pure states just captures the maximal difference between local coherence and the average coherence of one subsystem induced by local operations on the other subsystem with the assistance of classical communications, which provides an alternative operational meaning for bipartite concurrence of pure states. The relation between concurrence and the proposed coherence measure can also be extended to bipartite mixed states.
Quantum coherence is a fundamental aspect of quantum physics and plays a central role in quantum information science. This essential property of the quantum states could be fragile under the influence of the quantum operations. The extent to which quantum coherence is diminished depends both on the channel and the incoherent basis. Motivated by this, we propose a measure of nonclassicality of a quantum channel as the average quantum coherence of the state space after the channel acts on, minimized over all orthonormal basis sets of the state space. Utilizing the squared l1{l}_{1}l1​-norm of coherence for the qubit channels, the minimization can be treated analytically and the proposed measure takes a closed form of expression. If we allow the channels to act locally on a maximally entangled state, the quantum correlation is diminished making the states more classical. We show that the extent to which quantum correlation is preserved under local action of the channel cannot exceed the quantumness of the underlying channel. We further apply our measure to the quantum teleportation protocol and show that a nonzero quantumness for the underlying channel provides a necessary condition to overcome the best classical protocols.
Abstract Quantum coherence characterizes the non-classical feature of a single party system with respect to a local basis. Based on a recently introduced resource framework, coherence can be regarded as a resource and be systematically manipulated and quantified. Operationally, coherence quantifies the intrinsic randomness of the outcome of the projective measurement in the system’s computational basis. However, such a relation is only proven when randomness is characterized by the von Neumann entropy. In this work, we consider several recently proposed coherence measures and relate them to the general uncertainties of the projective measurement outcome conditioned on all the other systems. Our work thus provides a unified framework for redefining several coherence measures via general conditional entropies. Based on the relation, we numerically calculate the coherence measures via semi-definite programming. Furthermore, we discuss the operational meaning of the unified definition. Our result highlights the close relation between single partite coherence and bipartite quantum correlation.
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.
This paper studies the quantification and structural properties of quantum average correlation based on average coherence. Motivated by two mathematically equivalent approaches to define average coherence: one by averaging over complete sets of mutually unbiased bases, and the other by integrating over all orthogonal bases under the Haar measure, we define an average correlation for bipartite systems as the difference between global and local skew information. This correlation measure is shown to satisfy essential properties including non negativity, contractivity under local quantum channels, and local unitary invariance. We further prove the equivalence between the average correlation defined via mutually unbiased bases and that defined via unitary groups. Finally, we derive a complementarity relation that connects wave-particle duality with the average correlation between a system and its environment.