Quantifying correlations relative to channels via metric-adjusted skew information

Abstract

Quantum coherence and quantum correlations are important components of quantum information theory, which play important roles in quantum information processing. In this paper, we quantify correlations from coherence. The correlations relative to channels via metric-adjusted skew information are put forward. The corresponding results are suitable for special channels, such as positive operator-valued measurements (POVMs), projection measurements and von Neumann measurements. In addition, we discuss the dynamics of quantum correlations in the Bell-diagonal state relative to some classical channels via metric-adjusted skew information. The typical two are the Werner state and the isotropic state. We find that some separable states in entanglement resources theory possess correlations. It also shows quantum channels will disturb quantum states and have a great influence on the correlations. The correlations relative to different channels via different versions of metric-adjusted skew information have their advantages. This has inspired one to study the problem of the existence and action of various quantum correlations in quantum theory to be easily applied to experiments.

Appendix: The proof of contractivity (x)

(x)   (Contractivity)

$$\begin{aligned} I_f\left( \left( \mathcal {I}^a\otimes \Phi ^b\right) \left( \rho ^{ab}\right) , \Phi ^a\otimes \mathcal {I}^b\right) \le I_f\left( \rho ^{ab},\Phi ^a\otimes \mathcal {I}^b\right) , \end{aligned}$$

where $\mathcal {I}^a$ and $\Phi ^a$ are channels on party a, and $\mathcal {I}^b$ and $\Phi ^b$ are channels on part b.

Proof

Similarly to Ref. [68], we prove this property in two ways.

Method 1: Obviously, the channel $\mathcal {I}^a\otimes \Phi ^b$ is not disturb the channel $\Phi ^a\otimes \mathcal {I}^b$. Therefore, according to the item (vii), we have item (x).

Method 2: For any channel $\Phi ^b$, there is an auxiliary system c with a state $\rho ^c$ and a unitary operator $U^{bc}$ on the combined system bc, such that

$$\begin{aligned}{} & {} I_f\left( \left( \mathcal {I}^a\otimes \Phi ^b\right) \left( \rho ^{ab}\right) , \Phi ^a\otimes \mathcal {I}^b\right) \\{} & {} \quad =I_f\left( \textrm{Tr}_c\left( \mathcal {I}^a\otimes U^{bc}\right) \left( \rho ^{ab}\otimes \rho ^c\right) \left( \mathcal {I}^a\otimes U^{bc}\right) ^\dagger , \Phi ^a\otimes \mathcal {I}^b\right) \\{} & {} \quad \le I_f\left( \left( \mathcal {I}^a\otimes U^{bc}\right) \left( \rho ^{ab}\otimes \rho ^c\right) \left( \mathcal {I}^a\otimes U^{bc}\right) ^\dagger , \Phi ^a\otimes \mathcal {I}^b\otimes \mathcal {I}^c\right) \\{} & {} \quad = I_f\left( \rho ^{ab}\otimes \rho ^c, \left( \mathcal {I}^a\otimes U^{bc}\right) ^\dagger \left( \Phi ^a\otimes \mathcal {I}^b\otimes \mathcal {I}^c\right) \left( \mathcal {I}^a\otimes U^{bc}\right) \right) \\{} & {} \quad = I_f\left( \rho ^{ab}\otimes \rho ^c, \Phi ^a\otimes \mathcal {I}^b\otimes \mathcal {I}^c\right) \\{} & {} \quad = I_f\left( \rho ^{ab},\Phi ^a\otimes \mathcal {I}^b\right) , \end{aligned}$$

where the inequality is given by item (vi), the second equality is given by item (ii), and the last equality is given by item (v). $\square$