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Optical and Quantum Electronics

Relationship between quantum coherence and uncertainty bound in an arbitrary two-qubit X-state

Saeed Haddadi, Mohammad Reza Pourkarimi, Soroush Haseli

Published Aug 17, 2021• 28 citations
Publisher
DOI
10.1007/s11082-021-03176-4

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Quantum coherence is important in quantum mechanics, and its essence is from superposition principle. We study the coherence of any two pure states and that of their arbitrary superposition, and obtain the relationship between them. In the case that the two states have support on orthogonal subspaces, the relationship is simple, that is, the difference between the coherence of their superposition state and the average coherence of them is smaller than 1. In other cases, we obtain different and a little more complicated relationships. Furthermore, we also obtain the lower bound of coherence of superpositions.
Abstract Recently, Ming et al (Quantum Inf Process 17 267, 2018) have investigated the entropic uncertainty relation in the presence of quantum memory (EUR-QM) and quantum coherence in a thermal regime for a two-spin chain under the effect of Dzyaloshinskii–Moriya (DM) interaction. Here, we study thermal quantum coherence and EUR-QM for a three-spin chain in Ising model under the effect of different directions of DM interaction. Our results confirm the results for the two-qubit case. Increasing temperature can reduce the quantum coherence but oppositely the uncertainty will grow. We show that the coupling parameter and DM interaction have constructive effects on quantum coherence and they reduce uncertainty in measurement results of incompatible observables, and especially the y -direction of DM interaction has a better influence on decreasing the uncertainty than the x and z directions. Unfortunately, the effect of the third spin acts destructively on the uncertainty and its bound, as well as the quantum coherence between the first two spins. Notably, an explicit relationship between quantum coherence and the uncertainty lower bound is found.
Abstract Quantum coherence is an important quantum resource in the field of quantum information. We establish the uncertainty relations for the unified ( α , β )-relative entropy of coherence (UREOC) and the sandwiched Rényi α -relative entropy of coherence (SRREOC) under two and three orthonormal bases in qubit systems. We then prove that UREOC can be upper bounded by the purity in arbitrary finite dimensional spaces, from which a complementarity relation between UREOC and the mixedness is obtained. Finally, we show that the minimum error probability of quantum state discrimination and asymptotic exact coherence distillation rate can be upper bounded by UREOC, and can thus be easily estimated with the purity/mixedness of the state.