ABSTRACT We investigate the relationships among the wave property, particle property and mixedness of quantum states based on the ‐norm coherence. By conforming that the ‐norm coherence is an appropriate measure of wave property and introducing a measure of particle property based on the differences between the maximal ‐norm coherence and the general ‐norm coherence, we present tradeoff relationships among the wave, particle and mixedness of quantum states. For , we establish two kinds of tradeoffs of the wave, particle and mixedness with respect to the upper and lower bounds of the ‐norm coherence given by the ‐norm coherence. These trade relations give rise to compressive understanding of the intrinsic connections among the wave, particle and mixedness of quantum states, and cover some existing results as particular ones.
ABSTRACT We investigate the relationships among the wave property, particle property and mixedness of quantum states based on the ‐norm coherence. By conforming that the ‐norm coherence is an appropriate measure of wave property and introducing a measure of particle property based on the differences between the maximal ‐norm coherence and the general ‐norm coherence, we present tradeoff relationships among the wave, particle and mixedness of quantum states. For , we establish two kinds of tradeoffs of the wave, particle and mixedness with respect to the upper and lower bounds of the ‐norm coherence given by the ‐norm coherence. These trade relations give rise to compressive understanding of the intrinsic connections among the wave, particle and mixedness of quantum states, and cover some existing results as particular ones.
We investigate the relationships among the wave property, particle property and mixedness of quantum states based on the lp-norm coherence. By conforming that the lp-norm coherence is an appropriate measure of wave property and introducing a measure of particle property based on the differences between the maximal l2-norm coherence and the general l2-norm coherence, we present tradeoff relationships among the wave, particle and mixedness of quantum states. For 1<= p <2, we establish two kinds of tradeoffs of the wave, particle and mixedness with respect to the upper and lower bounds of the lp-norm coherence given by the l2-norm coherence. These trade relations give rise to compressive understanding of the intrinsic connections among the wave, particle and mixedness of quantum states, and cover some existing results as particular ones.
Abstract As an important resource, quantum coherence is one of the fundamental features of the quantum world. It is essential to understand the relationship between the resource and the mixedness of an open quantum system. We present complementary relations between the l p ( 1 ⩽ p < 2 ) norm of coherence and the normalized linear entropy mixedness. We then study the geometry of the complementary relations for Bell-diagonal states, as well as the evolutions of Werner and isotropic states under bit flip, phase flip, bit-phase flip, depolarizing, generalized amplitude damping and amplitude damping channels. Our results include some existing conclusions as special cases.
Abstract Wave-particle duality as one of the expression of Bohr complementarity is a significant concept in the field of quantum mechanics. Quantitative analysis of wave-particle duality aims to establish a complementary relation between the particle and wave properties. Beyond the conventional quantitative analysis depending on special choice of quantum information measures, we are aimed to provide a measure-independent complementary relation via coherence. By employing maximally coherent states in the set of all states with fixed diagonal elements, we give a measure-independent description of wave-particle-mixedness triality in d -path interferometers. Our complementary relations reveal the relationship between wave-particle duality and quantum coherence, and also give a justification to coherence as it truly brings out the wave nature of quantum systems at its heart.
Abstract The wave-particle duality, as a manifestation of Bohr’s complementarity, is usually quantified in terms of path predictability and interference visibility. Various characterizations of the wave-particle duality have been proposed from an operational perspective, most of them are in forms of inequalities, and some of them are expressed in forms of equalities by incorporating entanglement or coherence. In this work, we shed different insights into the nature of the wave-particle duality by casting it into a form of information conservation in a multi-path interferometer, with uncertainty as a unified theme. More specifically, by employing the simple yet fundamental concept of variance, we establish a resolution of unity, which can be interpreted as a complementarity relation among wave feature, particle feature, and mixedness of a quantum state. This refines or reinterprets some conventional approaches to wave-particle duality, and highlights informational aspects of the issue. The key idea of our approach lies in that a quantum state, as a Hermitian operator, can also be naturally regarded as an observable, with measurement uncertainty (in a state) and state uncertainty (in a measurement) being exploited to quantify particle feature and wave feature of a quantum state, respectively. These two kinds of uncertainties, although both are defined via variance, have fundamentally different properties and capture different features of a state. Together with the mixedness, which is a kind of uncertainty intrinsic to a quantum state, they add up to unity, and thus lead to a characterization of the wave-particle-mixedness complementarity. This triality relation is further illustrated by examples and compared with some popular wave-particle duality or triality relations.
Abstract Quantum coherence is a relevant resource for various quantum information processing tasks, but it is fragile since it is generally affected by environmental noise. This is reflected in the loss of purity of the system, which in turn limits the amount of quantum coherence of it. As a consequence, a complementarity relation between coherence and mixedness arises. Previous works characterize this complementarity through inequalities between the ℓ 1-norm of coherence and linear entropy, and between the relative entropy of coherence and von Neumann entropy. However, coherence–mixedness complementarity is expected to be a general feature of quantum systems, regardless of the measures used. Here, an alternative approach to coherence–mixedness complementarity, based on majorization theory, is proposed. Vectorial quantifiers of coherence and mixedness, namely the coherence vector and the spectrum, respectively, are used, instead of scalar measures. A majorization relation for the tensor product of both vectorial quantifiers is obtained, capturing general aspects of the trade-off between coherence and mixedness. The optimal bound for qubit systems and numerical bounds for qutrit systems are analyzed. Finally, coherence–mixedness complementarity relations are derived for a family of symmetric, concave and additive functions. These results provide a deeper insight into the relation between quantum coherence and mixedness.
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-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-norm measure this is related to a natural ``coherence radius'' for the state and leads to a conjecture for an 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.