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Laser Physics Letters

Maximum coherence under noisy channels

Zhiyong Ding, Huan Yang, Cheng-Cheng Liu, Wen‐Yang Sun, Juan He, Liu Ye

Published Aug 28, 2018• 0 citations
Publisher

Abstract

Abstract For the well-known relative entropy of coherence (REC), a quantum state attains maximum coherence only in optimal bases, which are mutually unbiased bases to the eigenvectors of the given state. In this paper, we explore the conditions of obtaining the maximum of REC under noisy channels and explain the geometric significance of optimal basis form the viewpoint of the Bloch sphere. We also show that the eigenvectors of the state will remain constant throughout the whole process of evolution, as long as the effects of channels induce a synchronous change for the components of the initial state Bloch vector. Furthermore, as a special example, we investigate the dynamics of the maximum REC for any finite dimensional single quantum system under a depolarizing (DP) channel, and obtain an analytical form of maximum coherence for a d -dimensional maximally coherent state. Meanwhile, a concise analytical expression of maximum coherence for two classes of an N -qubit system under the DP channel is also provided.

DOI
10.1088/1612-202x/aad98e

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Abstract For the well-known relative entropy of coherence (REC), a quantum state attains maximum coherence only in optimal bases, which are mutually unbiased bases to the eigenvectors of the given state. In this paper, we explore the conditions of obtaining the maximum of REC under noisy channels and explain the geometric significance of optimal basis form the viewpoint of the Bloch sphere. We also show that the eigenvectors of the state will remain constant throughout the whole process of evolution, as long as the effects of channels induce a synchronous change for the components of the initial state Bloch vector. Furthermore, as a special example, we investigate the dynamics of the maximum REC for any finite dimensional single quantum system under a depolarizing (DP) channel, and obtain an analytical form of maximum coherence for a d -dimensional maximally coherent state. Meanwhile, a concise analytical expression of maximum coherence for two classes of an N -qubit system under the DP channel is also provided.
The resource theoretic measure of quantum coherence is basis dependent, and the amount of coherence contained in a state is different in different bases. We obtained analytical solutions for the maximum coherence by optimizing the reference basis and highlighted the essential role of the mutually unbiased bases (MUBs) on attaining the maximum coherence. Apart from the relative entropy of coherence, we showed that the MUBs are optimal for the robustness of coherence, the coherence weight, and the modified skew information measure of coherence for any state. Moreover, the MUBs are optimal for all the faithful coherence measures if the state is pure or is of the single qubit. We also highlighted an upper bound for the l1{l}_{1}l1​ norm of coherence and compared it with the other bounds as well as the maximum one attainable by optimizing the reference basis.
In order to study the decoherence process of quantum system and find the ways to delay it, in this paper we investigate the decoherence process under correlated depolarizing channels by calculating the evolution of norm coherence, relative entropy coherence and basis-independent coherence of the three initial states: special initial state, maximum coherent state and isotropic state. From the analytic results, numerical results and evolution images, it can be found that 1) completely correlated channels can suppress decoherence to a greatest extent,. 2) the time point at which maximum decoherence occurs is irrelevant to the degree of correlation, but system’s decoherence evolution behavior depends on it. 3)the correlation of depolarizing channels can reinforce the collective quantum coherence between subsystems, and an inequality relation of basis-independent coherence is proved in this system.
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.
Is it always possible to explain random stochastic transitions between states of a finite-dimensional system as arising from the deterministic quantum evolution of the system? If not, then what is the minimal amount of randomness required by quantum theory to explain a given stochastic process? Here, we address this problem by studying possible coherifications of a quantum channel Φ, i.e., we look for channels Φ C that induce the same classical transitions T , but are ‘more coherent’. To quantify the coherence of a channel Φ we measure the coherence of the corresponding Jamiołkowski state J Φ . We show that the classical transition matrix T can be coherified to reversible unitary dynamics if and only if T is unistochastic. Otherwise the Jamiołkowski state J Φ C of the optimally coherified channel is mixed, and the dynamics must necessarily be irreversible. To assess the extent to which an optimal process Φ C is indeterministic we find explicit bounds on the entropy and purity of J Φ C , and relate the latter to the unitarity of Φ C . We also find optimal coherifications for several classes of channels, including all one-qubit channels. Finally, we provide a non-optimal coherification procedure that works for an arbitrary channel Φ and reduces its rank (the minimal number of required Kraus operators) from d 2 to d .
We explore methods to generate quantum coherence through unitary evolutions, by introducing and studying the coherence generating capacity of Hamiltonians. This quantity is defined as the maximum derivative of coherence that can be achieved by a Hamiltonian. By adopting the relative entropy of coherence as our figure of merit, we evaluate the maximal coherence generating capacity with the constraint of a bounded Hilbert-Schmidt norm for the Hamiltonian. Our investigation yields closed-form expressions for both Hamiltonians and quantum states that induce the maximal derivative of coherence under these conditions. Specifically, for qubit systems, we solve this problem comprehensively for any given Hamiltonian, identifying the quantum states that lead to the largest coherence derivative induced by the Hamiltonian. Our investigation enables a precise identification of conditions under which quantum coherence is optimally enhanced, offering valuable insights for the manipulation and control of quantum coherence in quantum systems.