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arXiv:1607.05522 · Mathematics

Uncertainty Relations for Quantum Coherence

Uttam Singh, Arun Kumar Pati, Manabendra Nath Bera

Published Jul 16, 2016• 50 citations
arXivHTMLPDF

Abstract

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.

DOI
10.3390/math4030047

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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.
The concept of quantum coherence, including various ways to quantify the degree of coherence with respect to the prescribed basis, is currently the subject of active research. The complementarity of quantum coherence in different bases was studied by deriving upper bounds on the sum of the corresponding measures. To obtain a two-sided estimate, lower bounds on the coherence quantifiers are also of interest. Such bounds are naturally referred to as uncertainty relations for quantum coherence. We obtain new uncertainty relations for coherence quantifiers averaged with respect to a set of mutually unbiased bases (MUBs). To quantify the degree of coherence, the relative entropy of coherence and the geometric coherence are used. Further, we also derive novel state-independent uncertainty relations for a set of MUBs in terms of the min-entropy.
As one of the most essential properties of quantum mechanics, quantum coherence is also considered a key physical resource which can be used to accomplish certain tasks of quantum information processing. Significant efforts are involved in the quantification of quantum coherence using measures such as the l1{l}_{1}l1​-norm and the relative entropy of coherence. The quantum coherence as a physical resource is usually defined with respect to a given basis, which also arouses an issue of interest concerning the variation of quantum coherence for a quantum state in different reference bases. In this paper, using an all optical setup, we experimentally investigate a trade-off relation between the coherence measures quantified by relative entropy of coherence in two noncommuting reference bases. Our result shows that the sum of quantum coherence under these bases is bounded in a region defined by its lower bound and upper bound.
Various measures have been suggested recently for quantifying the coherence of a quantum state with respect to a given basis. We first use two of these, the l1{l}_{1}l1​-norm and relative entropy measures, to investigate tradeoffs between the coherences of mutually unbiased bases. Results include relations between coherence, uncertainty, and purity; tight general bounds restricting the coherences of mutually unbiased bases; and an exact complementarity relation for qubit coherences. We further define the average coherence of a quantum state. For the l1{l}_{1}l1​-norm measure this is related to a natural ``coherence radius'' for the state and leads to a conjecture for an l2{l}_{2}l2​-norm measure of coherence. For relative entropy the average coherence is determined by the difference between the von Neumann entropy and the quantum subentropy of the state and leads to upper bounds for the latter quantity. Finally, we point out that the relative entropy of coherence is a special case of G-asymmetry, which immediately yields several operational interpretations in contexts as diverse as frame alignment, quantum communication, and metrology, and suggests generalizing the property of quantum coherence to arbitrary groups of physical transformations.
Abstract Quantum mechanical uncertainty relations are fundamental consequences of the incompatible nature of noncommuting observables. In terms of the coherence measure based on the Wigner-Yanase skew information, we establish several uncertainty relations for coherence with respect to von Neumann measurements, mutually unbiased bases (MUBs), and general symmetric informationally complete positive operator valued measurements (SIC-POVMs), respectively. Since coherence is intimately connected with quantum uncertainties, the obtained uncertainty relations are of intrinsically quantum nature, in contrast to the conventional uncertainty relations expressed in terms of variance, which are of hybrid nature (mixing both classical and quantum uncertainties). From a dual viewpoint, we also derive some uncertainty relations for coherence of quantum states with respect to a fixed measurement. In particular, it is shown that if the density operators representing the quantum states do not commute, then there is no measurement (reference basis) such that the coherence of these states can be simultaneously small .
Measurement outcomes of a quantum state can be genuinely random (unpredictable) according to the basic laws of quantum mechanics. The Heisenberg-Robertson uncertainty relation puts constraints on the accuracy of two noncommuting observables. The existing uncertainty relations adopt variance or entropic measures, which are functions of observed outcome distributions, to quantify the uncertainty. According to recent studies of quantum coherence, such uncertainty measures contain both classical (predictable) and quantum (unpredictable) components. In order to extract out the quantum effects, we define quantum uncertainty to be the coherence of the state on the measurement basis. We discover a quantum uncertainty relation using coherence between two measurement noncommuting bases. Furthermore, we analytically derive the quantum uncertainty relation for the qubit case with three widely adopted coherence measures, the relative entropy of coherence, the coherence of formation, and the l1{l}_{1}l1​ norm of coherence.
Abstract The quantum uncertainty relation characterizes the restriction of the quantum coherence on different measurement bases. The skew information of coherence is a coherence measure both operationally and informatically. We study the quantum uncertainty relation for the skew information of coherence under any two orthonormal bases. In qubit and qutrit systems, we derive the quantum uncertainty relations with the lower bounds in the factorized forms by the purity of quantum states and the incompatibility between two orthonormal bases. Especially the lower bound in qubit systems is strictly positive for non-maximally mixed states and incompatible orthonormal bases. These quantum uncertainty relations reveal the intrinsic relation among the coherence, the purity and the incompatibility between two orthonormal bases quantitatively. The advantages of these quantum uncertainty relations are illustrated by some specific examples. This method we developed here is beneficial to some other quantum uncertainty relations.