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Communications in Theoretical Physics

Coherence of Superposition States*

Shanshan Yuwen, Lian-He Shao, Zhengjun Xi

Published Sep 1, 2019• 9 citations
Publisher

Abstract

Abstract We find tight upper bound on the coherence of a superposition of two states in terms of the coherence of the two states constituting the superposition with l 1 -norm of coherence. Our upper bound is tighter than the one presented by Liu, et al. [Quantum Inf. Process. 15 (2016) 4209.] We also generalize the results to the case that the superposition is constituted with more than two states in high dimension, and we give the corresponding upper bounds.

DOI
10.1088/0253-6102/71/9/1084

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Abstract We find tight upper bound on the coherence of a superposition of two states in terms of the coherence of the two states constituting the superposition with l 1 -norm of coherence. Our upper bound is tighter than the one presented by Liu, et al. [Quantum Inf. Process. 15 (2016) 4209.] We also generalize the results to the case that the superposition is constituted with more than two states in high dimension, and we give the corresponding upper bounds.
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We find tight lower and upper bounds on the entanglement of a superposition of two bipartite states in terms of the entanglement of the two states constituting the superposition. Our upper bound is dramatically tighter than the one presented by Linden et al. [Phys. Rev. Lett. 97, 100502 (2006)] and our lower bound can be used to provide lower bounds on different measures of entanglement such as the entanglement of formation and the entanglement of subspaces. We also find that in the case in which the two states are one-sided orthogonal, the entanglement of the superposition state can be expressed explicitly in terms of the entanglement of the two states in the superposition.
Abstract Quantum coherence plays a major role in the promotion for quantum information processing and designing quantum technology. Since coherence is rooted in superposition principle, it is vital to understand the coherence change with respect to superpositions. Here we study the bounds for coherence of quantum superpositions in high dimension. We consider three most frequently used measures of coherence, i.e. the relative entropy of coherence,l1norm of coherence and robustness of coherence. For a quantum state (an arbitrary dimension) and its arbitrary decomposition, we give the upper and lower bounds for coherence of the superposition state in terms of the coherence of the states being superposed.
The entanglement content of superpositions of quantum states is investigated based on a measure called concurrence. Given a bipartite pure state in arbitrary dimension written as the quantum superposition of two other such states, we find simple inequalities relating the concurrence of the state to that of its components. We derive an exact expression for the concurrence when the component states are biorthogonal and provide elegant upper and lower bounds in all other cases. For quantum bits, our upper bound is tighter than the previously derived bound [N. Linden et al., Phys. Rev. Lett. 97, 100502 (2006)].
In this paper, we use the concurrence vector as a measure of entanglement, and investigate lower and upper bounds on the concurrence of a superposition of bipartite states as a function of the concurrence of the superposed states. We show that the amount of entanglement quantified by the concurrence vector is exactly the same as that quantified by III concurrence, so that our results can be compared to those given in Phys. Rev. A 76, 042328 (2007). We obtain a tighter lower bound in the case in which the two superposed states are orthogonal. We also show that when the two superposed states are not necessarily orthogonal, both lower and upper bounds are, in general, tighter than the bounds given in terms of the III concurrence. An extension of the results to the case with more than two states in the superpositions is also given.
Quantum coherence is a fundamental feature of quantum physics and plays a significant role in quantum information processing. By generalizing the resource theory of coherence from von Neumann measurements to positive operator-valued measures (POVMs), POVM-based coherence measures have been proposed with respect to the relative entropy of coherence, the l 1 norm of coherence, the robustness of coherence and the Tsallis relative entropy of coherence. We derive analytically the lower and upper bounds on these POVM-based coherence of an arbitrary given superposed pure state in terms of the POVM-based coherence of the states in superposition. Our results can be used to estimate range of quantum coherence of superposed states. Detailed examples are presented to verify our analytical bounds.