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Laser Physics

Complementary relations between l p norm coherence and mixedness of quantum states

Liu Sun, Yuan-Hong Tao, Shao-Ming Fei

Published Dec 13, 2022• 4 citations
Publisher

Abstract

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.

DOI
10.1088/1555-6611/aca4cc

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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.
Generally quantum processes tend to mix the pure input states and degrade the original coherence. Such processes display exactly the trade-offs between the coherence and the mixedness. In this paper, we present trade-offs between the coherence measures (l2 norm of coherence, coherence via skew information, geometric measure of coherence) and the mixedness (normalized linear entropy mixedness, geometric measure of mixedness, mixedness via skew information). The evolution of these trade-offs are also investigated under four quantum noise channels: the bit flip, phase flip, depolarizing and amplitude damping channels.
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.
Abstract We study the curves of coherence for the Bell-diagonal states including l 1 -norm of coherence and relative entropy of coherence under the Markovian channels in the first subsystem once. For a special Bell-diagonal state under bit-phase ip channel, we find frozen coherence under l 1 norm occurs, but relative entropy of coherence decrease. It illustrates that the occurrence of frozen coherence depends on the type of the measure of coherence. Also, we study the coherence evolution of Bell-diagonal states under Markovian channels in the first subsystem n times and find that coherence under depolarizing channel decreases initially then increases for small n and tends to zero for large n. The dynamics of coherence of the Bell-diagonal state under two independent same type local Markovian channels is discussed.
Abstract We study the curves of coherence for the Bell-diagonal states including l 1 -norm of coherence and relative entropy of coherence under the Markovian channels in the first subsystem once. For a special Bell-diagonal state under bit-phase ip channel, we find frozen coherence under l 1 norm occurs, but relative entropy of coherence decrease. It illustrates that the occurrence of frozen coherence depends on the type of the measure of coherence. Also, we study the coherence evolution of Bell-diagonal states under Markovian channels in the first subsystem n times and find that coherence under depolarizing channel decreases initially then increases for small n and tends to zero for large n. The dynamics of coherence of the Bell-diagonal state under two independent same type local Markovian channels is discussed.
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.
We focus our attention on tripartite mixed states as initial states, and apply coherence concurrence to investigate quantum coherence properties in the background of a Schwarzschild black hole under phase damping, phase flip and bit flip channels, respectively. Several analytic complementary relationships based on coherence concurrence for tripartite subsystems are proposed. In the case of the bit flip channel, the behavior of the coherence concurrence is similar to the one of the phase damping channel, the accessible coherence concurrence always degrades as the Hawking acceleration rising, but sudden death never occurs, while the inaccessible coherence increases from zero monotonically. Interestingly, the coherence concurrence is decreasing at first and then increasing as the decay probability rising under phase flip channel. Unlike the case of tripartite pure states, the coherence concurrence of mixed state with X shape is equal to l1l_1l1​ -norm of coherence.