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International Journal of Quantum Information

Cohering and decohering power of Gaussian channels

Huazhen Yao, Jianwen Zhang, Yangyang Wang

Published Jun 1, 2020• 0 citations
Publisher

Abstract

In this paper, we generalized the concepts of the cohering power and decohering power to Gaussian channels for continuous-variable systems with respect to coherence measures based on the relative entropy and obtained some basic properties. In addition, the cohering power and decohering power of classical noise channels, thermal noise channels and amplifier channels are calculated.

DOI
10.1142/s0219749920500173

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•Jan 1, 2014•16 citations
In this paper, we generalized the concepts of the cohering power and decohering power to Gaussian channels for continuous-variable systems with respect to coherence measures based on the relative entropy and obtained some basic properties. In addition, the cohering power and decohering power of classical noise channels, thermal noise channels and amplifier channels are calculated.
Having a suitable measure to quantify the coherence of quantum states, a natural task is to evaluate the power of quantum channels for creating or destroying the coherence of input quantum states. In the present paper, by introducing the maximal coherent Gaussian states based on the relative entropy measure of coherence, we propose the (generalized) cohering power and the (generalized) decohering power of Gaussian unitary operations for continuous-variable systems. Some basic properties are obtained and the cohering power and decohering power of two special kinds of Gaussian unitary operations are calculated.
Abstract In this paper, we investigate the cohering and decohering power of the one-qubit Markovian channels with respect to coherence measures based on the l 1 - norm, the Rényi α - relative entropy and the Tsallis α - relative entropy of coherence, respectively. The amplitude damping channel, phase damping channel, depolarizing channel, and flip channels are analytically calculated. It shows that the decohering power of the amplitude damping channel on the x , y , and z basis is equal to each other. The same phenomenon can be seen for the phase damping channel and the flip channels. The cohering power for the phase damping channel and the flip channels on the x , y basis also equals to that on the z basis. However, the cohering and decohering power of the depolarizing channel is independent to the reference basises. And the cohering power of the amplitude damping channel on the x , y basis is different to that on the z basis.
Abstract Quantum coherence stems from the superposition principle of states. It shows an essential feature of quantum systems which is different from the classical phenomena. We investigate the concept of cohering and decohering power of single-mode quantum Gaussian noises. For this purpose, we use the Hellinger distance-based and Bures distance-based measure to quantify the coherence of single-mode quantum Gaussian states. We will show that both Hellinger and Bures distance-based measures are the same when quantifying the quantum coherence of pure single-mode quantum Gaussian states. In this work we will study the cohering and decohering power of single-mode attenuation and amplification noise. We will investigate the effects of the amplification parameter k and attenuation parameter on the cohering and decohering power of corresponding channels. We find that for single-mode amplification noise the cohering power is increased by increasing k , while the decohering power is decreased by increasing k . Conversely, we will see that for a single-mode attenuation channel the cohering power is decreased by increasing the attenuation parameter , while in general the decohering power is increased by increasing . We also investigate the effect of the mean photon number of the environment N on the cohering and decohering power of single-mode attenuation and amplification noises. For both of the channels the cohering power is increased, while the decohering power is decreased, by increasing N .
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.
We prove that whenever the coherent information of a one-mode Gaussian (OMG) channel is non-zero its supremum is achieved for the infinite input power. This is a well established fact for zero added classical noise, whereas the non-zero case has not been studied in detail. The presented analysis fills the gap for three canonical classes of OMG channels: the lossy, amplifying and additive classical noise channel class. For the remaining OMG channel classes the coherent information is known to vanish.
We propose a generalization of the quantum entropy power inequality involving condi-tional entropies. For the special case of Gaussian states, we give a proof based on perturba-tion theory for symplectic spectra. We discuss some implications for entanglement-assisted classical communication over additive bosonic noise channels. 1 Classical and quantum entropy-power inequalities The entropy power inequality, proposed by Shannon [27] and later established with increasing rigor by Stam [29] and Blachman [5], has become a fundamental tool in classical informa-tion theory. Shannon’s original application of the entropy power inequality is a lower bound on the capacity of an additive (but potentially non-Gaussian) noise channel [27, Theorem 18]. However, the usefulness of the entropy power inequality is especially evident in multi-terminal information theory. Among the most well-known applications are the characterization of the Gaussian broadcast channel by Bergman [4], the Gaussian two-description problem by Ozarow [25] and the quadratic Gaussian CEO problem by Oohama [24]. In these multi-user settings, Fano’s inequality by itself is insufficient to characterize different tradeoffs. A more recent application proposed by Liu and Viswanath [23] uses entropy power inequalities to solve certain optimization problems. The entropy power inequality lower bounds the differential entropy of the convolution of two independent random variables X, Y taking values in Rn. Its covariance-preserving version states that H( λX + 1 − λY) ≥ λH(X) + (1 − λ)H(Y) for all 0 ≤ λ ≤ 1. (1) Eq. (1) can be shown [22, 32] to be equivalent to the more commonly used statement e2H(X+Y)/n ≥ e2H(X)/n + e2H(Y)/n. The latter explains the terminology as e2H(X)/n is the power, i.e., variance of a Gaussian ran-dom variable with identical entropy as H(X). Inequalities such as (1) are closely related to Log-Sobolev inequalities (see e.g., [31]) as well as Brunn-Minkowski-type inequalities [6]. Gen-eralizations to free probability [30] and quantum states have been considered. 1 ar