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Communications in Theoretical Physics

Uncertainty Relations for Coherence*

Shunlong Luo, Yuan Sun

Published Dec 1, 2019• 12 citations
Publisher

Abstract

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 .

DOI
10.1088/0253-6102/71/12/1443

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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.
Abstract Uncertainty principle plays a vital role in quantum physics. The Wigner–Yanase skew information characterizes the uncertainty of an observable with respect to the measured state. We generalize the uncertainty relations for two quantum channels to arbitrary N quantum channels based on Wigner–Yanase skew information. We illustrate that these uncertainty inequalities are tighter than the existing ones by detailed examples. Especially, we also discuss the uncertainty relations for N unitary channels, which could be regarded as variance-based sum uncertainty relations with respect to any pure state.
Abstract In quantum information science, systems with incomplete information are typically dealt with. To characterize quantumness from different viewpoints, several kinds of non-classical correlations should be described quantitatively. The concept of coherence within purely quantum framework is currently the subject of active research. A certain attention is paid to coherence quantifiers averaged with respect to a set of quantum ensembles or special measurements. Mutually unbiased bases and symmetric informationally complete measurements are important examples. We present uncertainty relations for quantum-coherence quantifiers based on the Tsallis relative 1/2-entropies. Together with mutually unbiased bases, the paper also deals with a measurement built of the states of an equiangular tight frame. The derived inequalities are exemplified with mutually unbiased bases and symmetric informationally complete measurement in two dimensions.
Abstract Quantum coherence stands as a core property of quantum mechanics, its origin rooted in the superposition principle that governs quantum states. Leveraging mutually unbiased bases (MUBs), this paper delves into an in-depth exploration of quantum coherence characteristics through the lens of the l p -norm and s k e w Information. To begin with, we establish rigorous equality relations for the l p -norm coherence of qubit states under Pauli MUBs. Following this, we derive the complementary relations (upper bounds) and uncertainty relations (lower bounds) associated with l p -norm coherence under MUBs within arbitrary d-dimensional systems. Then, by virtue of deducing the connection between the classical purity and quantum purity of the square root of the quantum state, i.e. ρ , we further elaborate on the strict equality relations of skew-information coherence as well as several corresponding entropy-based coherences under MUBs in d-dimensional systems. Additionally, we provide illustrative examples to instantiate each of these relations, enhancing their clarity and practical relevance.