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Cohering Power of Unitary Operations and De-cohering of Quantum operations

Zhengjun Xi, Ming‐Liang Hu, Yongming Li, Heng Fan

Published Oct 22, 2015• 4 citations
OpenAlex

Abstract

In this paper, we consider the cohering power of unitary operation, which is defined by the maximum coherence caused by an unitary operation acting on incoherent states. We obtain that the optimizations can be reduced to a simple discrete set that depends only on the fixed reference basis. Then, we investigate that the de-cohering power of quantum channel, which is defined by the maximum decay of coherence caused by quantum channel acting on the maximally coherent states. Based on the relative entropy, we discuss the cohering power of 1-qubit and 2-qubit unitary operations and the de-cohering power of quantum channel on 1-qubit system. Finally, we study the relation between the coherence and the discord via incoherent operation.

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In this paper, we consider the cohering power of unitary operation, which is defined by the maximum coherence caused by an unitary operation acting on incoherent states. We obtain that the optimizations can be reduced to a simple discrete set that depends only on the fixed reference basis. Then, we investigate that the de-cohering power of quantum channel, which is defined by the maximum decay of coherence caused by quantum channel acting on the maximally coherent states. Based on the relative entropy, we discuss the cohering power of 1-qubit and 2-qubit unitary operations and the de-cohering power of quantum channel on 1-qubit system. Finally, we study the relation between the coherence and the discord via incoherent operation.
The coherence power of a quantum channel, that is, its maximum ability to increase the coherence of input states, is a fundamental concept within the framework of the resource theory of coherence. In this note we discuss various possible definitions of coherence power and coherence rate and their basic properties. Then we prove that the coherence power of a unitary operator acting on a qubit, computed with respect to the l1-coherence measure, can be calculated by maximizing its coherence gain over pure incoherent states. We proceed to show that this result fails in the general case, that is, the maximal coherence gain is found when acting on a state with non-vanishing coherence in the case of the l1-coherence and dimension N > 2, the relative entropy of coherence and the geometric measure of coherence.
Cohering power and de-cohering power have recently been proposed to quantify the ability of a quantum operation to produce and erase coherence respectively. In this paper, we investigate the properties of cohering power and de-cohering power. First, we prove the equivalence between two different kinds of cohering power for any quantum operation on single qubit systems, which implies that l1 norm of coherence is monotone under Maximally incoherent operation (MIO) and Dephasing-covariant operation (DIO) in 2-dimensional space. In higher dimensions, however, we show that the monotonicity under MIO or DIO does not hold. Besides, we compare the set of quantum operations with zero cohering power with Maximally incoherent operation (MIO) and Incoherent operation (IO). Moreover, two different types of de-cohering power are defined and we find that they are not equal in single qubit systems. Finally, we make a comparison between cohering power and de-cohering power for single qubit unitary operations and show that cohering power is always larger than de-cohering power.
Abstract Quantum coherence is a fundamental property of quantum systems that can be studied within the framework of a quantum resource theory. From this resource-theoretic approach, many similarities emerge between coherence and quantum entanglement, the latter being another prominent quantum resource. In this paper, we study the relationship between coherence and entanglement from the perspective of resource loss/gain by unitary dynamics. Specifically we consider how much coherence can be either generated or destroyed on a bipartite system when the action is restricted to local unitary (LU) operations. For pure states, we find that the relative entropy of entanglement and the robustness of entanglement provide tight lower bounds on the amount of coherence that can be destroyed by LUs, as measured by the relative entropy and measures of coherence, respectively. This provides new operational interpretations of the entanglement entropy and robustness of entanglement in pure states as the minimal amount of coherence that persists when the system is subjected to LU transformations. We then study the amount of bipartite pure entanglement that can be either maximized or minimized when the action is restricted to a global incoherent operation. For two-qubit pure states, maximum and minimum are shown in terms of coefficients of given state with incoherent basis. Finally we consider a generalized version of the recently introduced CCP of a quantum channel.
The coherence power of a quantum channel, that is, its ability to increase the coherence of input states, is a fundamental concept within the framework of the resource theory of coherence. In this note we discuss various possible definitions of coherence power. Then we prove that the coherence power of a unitary operator acting on a qubit, computed with respect to the l1l_1l1​-coherence measure, can be calculated by maximizing its coherence gain over pure incoherent states. We proceed to show that this result fails for dimensions N>2, that is, the maximal coherence gain is found when acting on a state with non-vanishing coherence.
Given a preferred orthonormal basis BBB in the Hilbert space of a quantum system, we define a measure of the coherence-generating power of a unitary operation with respect to BBB. This measure is the average coherence generated by the operation acting on a uniform ensemble of incoherent states. We give its explicit analytical form in any dimension and provide an operational protocol to directly detect it. We characterize the set of unitaries with maximal coherence-generating power and study the properties of our measure when the unitary is drawn at random from the Haar distribution. For a large state-space dimension a random unitary has, with overwhelming probability, nearly maximal coherence-generating power with respect to any basis. Finally, extensions to general unital quantum operations and the relation to the concept of asymmetry are discussed.
We introduce the concepts of cohering and decohering power of quantum channels. Using the axiomatic definition of the coherence measure, we show that the optimization required for calculations of these measures can be restricted to pure input states and hence greatly simplified. We then use two examples of this measure, one based on the skew information and the other based on the l1{l}_{1}l1​ norm; we find the cohering and decohering measures of a number of one-, two-, and nnn-qubit channels. Contrary to the view at first glance, it is seen that quantum channels can have cohering power. It is also shown that a specific property of a qubit unitary map is that it has equal cohering and decohering power in any basis. Finally, we derive simple relations between cohering and decohering powers of unitary qubit gates and their tensor products, results which have physically interesting implications.