Quantifying coherence relative to channels via metric-adjusted skew information

In terms of the metric-adjusted skew information (an important and versatile class of quantum Fisher information), which generalizes the seminal Wigner-Yanase skew information arising naturally from the study of quantum measurement, we propose a family of coherence measures of states relative to quantum channels, and reveal their basic properties such as unitary covariance, convexity, and monotonicity. Furthermore, we evaluate these coherence measures of states relative to several prototypical quantum channels, and make a comparative study for this family of coherence measures with relative entropy of coherence. This provides a general approach to coherence of states relative to quantum channels, which also captures decoherence on the states caused by quantum channels and asymmetry of states relative to quantum channels.

Figure 1

Comparison between the two coherence measures $F_f\left(\right|{\xi }_{\varphi }\rangle ,{\mathrm{\Phi }}_{\mathrm{BF}})$ and $C_{\mathrm{rel}}\left(\right|{\xi }_{\varphi }\rangle ,{\mathrm{\Phi }}_{\mathrm{BF}})$ as functions of $\varphi \in [0,2\pi )$ for the bit flip channel ${\mathrm{\Phi }}_{\mathrm{BF}}$ with $p=1/2$ (example 0). Here $|{\xi }_{\varphi }\rangle =\left(\right|0\rangle +e^{i\varphi }|1\rangle )/\sqrt{2}$. We see that they display quite different behaviors in this special case: $F_f\left(\right|{\xi }_{\varphi }\rangle ,{\mathrm{\Phi }}_{\mathrm{BF}})$ is oscillating in the parameter $\varphi$, while $C_{\mathrm{rel}}\left(\right|{\xi }_{\varphi }\rangle ,{\mathrm{\Phi }}_{\mathrm{BF}})=ln2$ is a constant independent of $\varphi$.

Figure 2

Comparison between the two coherence measures $F_f\left(\right|{\psi }_{\theta }\rangle ,{\mathrm{\Phi }}_{\mathrm{AD}})$ and $C_{\mathrm{rel}}\left(\right|{\psi }_{\theta }\rangle ,{\mathrm{\Phi }}_{\mathrm{AD}})$ as functions of $\theta \in [0,\pi ]$ for the amplitude damping channel ${\mathrm{\Phi }}_{\mathrm{AD}}$ with $p=1/2$ (example 1). Here $|{\psi }_{\theta }\rangle =\cos \frac{\theta }{2}|0\rangle +\sin \frac{\theta }{2}|1\rangle$. We see that they display radically different behaviors in this special case: $F_f\left(\right|{\psi }_{\theta }\rangle ,{\mathrm{\Phi }}_{\mathrm{AD}})$ is decreasing in the parameter $\theta \in [0,\pi ]$, while $C_{\mathrm{rel}}\left(\right|{\psi }_{\theta }\rangle ,{\mathrm{\Phi }}_{\mathrm{AD}})$ is increasing in $\theta \in [0,\pi ]$. Consequently, they yield different orderings of coherence of states relative to the same amplitude damping channel.

Figure 3

Comparison between the coherence measures $F_{f_i}({\rho }_{\lambda },{\mathrm{\Phi }}_{\mathrm{AD}})$ and $C_{\mathrm{rel}}({\rho }_{\lambda },{\mathrm{\Phi }}_{\mathrm{AD}})$, as functions of the mixing parameter $\lambda \in (0,1)$ for the amplitude damping channel ${\mathrm{\Phi }}_{\mathrm{AD}}$ with $p=1/2$ (example 1). Here ${\rho }_{\lambda }=\lambda |0\rangle \langle 0|+(1-\lambda )|1\rangle \langle 1|$. We see that $F_{f_i}({\rho }_{\lambda },{\mathrm{\Phi }}_{\mathrm{AD}})$ share similar properties for $i=1,2,3$ (all are special instances of coherence measures defined via metric-adjusted skew information). In sharp contrast, $F_{f_i}({\rho }_{\lambda },{\mathrm{\Phi }}_{\mathrm{AD}})$ and $C_{\mathrm{rel}}({\rho }_{\lambda },{\mathrm{\Phi }}_{\mathrm{AD}})$ are radically different in the sense that they display different monotonicity: $F_{f_i}({\rho }_{\lambda },{\mathrm{\Phi }}_{\mathrm{AD}})$ is decreasing in $\lambda \in (0,1/2]$ and increasing in $\lambda \in [1/2,1)$, but $C_{\mathrm{rel}}({\rho }_{\lambda },{\mathrm{\Phi }}_{\mathrm{AD}})$ is monotonically decreasing in $\lambda \in (0,1)$. $F_{f_i}({\rho }_{\lambda },{\mathrm{\Phi }}_{\mathrm{AD}})$ and $C_{\mathrm{rel}}({\rho }_{\lambda },{\mathrm{\Phi }}_{\mathrm{AD}})$ yield different orderings of coherence.