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International Journal of Theoretical Physics

Is the Relative Entropy of Coherence Always the Minimal-Coherence-Value Measure?

Dong-Mei Gao, Ting-Hua Lv

Published Feb 20, 2019• 2 citations
Publisher
DOI
10.1007/s10773-019-04011-z

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In quantum mechanics, quantum coherence of a state relative to a quantum measurement can be identified with the quantumness that has to be destroyed by the measurement. In particular, quantum coherence of a bipartite state relative to a local quantum measurement encodes quantum correlations in the state. If one takes minimization with respect to the local measurements, then one is led to quantifiers which capture quantum correlations from the perspective of coherence. In this vein, quantum discord, which quantifies the minimal correlations that have to be destroyed by quantum measurements, can be identified as the minimal coherence, with the coherence measured by the relative entropy of coherence. To advocate and formulate this idea in a general context, we first review coherence relative to Lüders measurements which extends the notion of coherence relative to von Neumann measurements (or equivalently, orthonomal bases), and highlight the observation that quantum discord arises as minimal coherence through two prototypical examples. Then, we introduce some novel measures of quantum correlations in terms of coherence, illustrate them through examples, investigate their fundamental properties and implications, and indicate their applications to quantum metrology.
We investigate the generic aspects of quantum coherence guided by the concentration of measure phenomenon. We find the average relative entropy of coherence of pure quantum states sampled randomly from the uniform Haar measure and show that it is typical, i.e., the probability that the relative entropy of coherence of a randomly chosen pure state is not equal to the average relative entropy of coherence (within an arbitrarily small error) is exponentially small in the dimension of the Hilbert space. We find the dimension of a random subspace of the total Hilbert space such that all pure states that reside on it almost always have at least a fixed nonzero amount of the relative entropy of coherence that is arbitrarily close to the typical value of coherence. Further, we show, with high probability, every state (pure or mixed) in this subspace also has the coherence of formation at least equal to the same fixed nonzero amount of the typical value of coherence. Thus, the states from these random subspaces can be useful in the relevant coherence consuming tasks like catalysis in the coherence resource theory. Moreover, we calculate the expected trace distance between the diagonal part of a random pure quantum state and the maximally mixed state and find that it does not approach to zero in the asymptotic limit. This establishes that randomly chosen pure states are not typically maximally coherent (within an arbitrarily small error). Additionally, we find the lower bound on the relative entropy of coherence for the set of pure states whose diagonal parts are at a fixed most probable distance from the maximally mixed state.
Coherence of a quantum state intrinsically depends on the choice of the reference basis. A natural question to ask is the following: if we use two or more incompatible reference bases, can there be some trade-off relation between the coherence measures in different reference bases? We show that the quantum coherence of a state as quantified by the relative entropy of coherence in two or more noncommuting reference bases respects uncertainty like relations for a given state of single and bipartite quantum systems. In the case of bipartite systems, we find that the presence of entanglement may tighten the above relation. Further, we find an upper bound on the sum of the relative entropies of coherence of bipartite quantum states in two noncommuting reference bases. Moreover, we provide an upper bound on the absolute value of the difference of the relative entropies of coherence calculated with respect to two incompatible bases.