Distance between quantum states in the presence of initial qubit-environment correlations: A comparative study

The time evolution of the trace distance between two states of an open quantum system may increase due to initial system-environment correlations, thus exhibiting a breakdown of distance contractivity of the reduced dynamics. We analyze how the time evolution of the distance depends on the chosen distance measure. Here we elucidate the behavior of the trace distance, the Hilbert-Schmidt distance, the Bures distance, the Hellinger distance, and the quantum Jensen-Shannon divergence for two system-environment setups, namely a qubit bilinearly coupled to an infinite and a finite-size environment with the latter composed of harmonic oscillators.

Figure 1

Time evolution of distances between two qubit states in the case of infinite environment. (a) The trace $D_T=D_{\mathrm{HS}}/\sqrt{2}$, (b) Bures $D_B$, (c) Hellinger $D_H$, and (d) quantum Jensen-Shannon $D_{\mathrm{JS}}$ distances, respectively. The distances $D\left(t\right)=D[{\rho }_0\left(t\right),{\rho }_{\lambda }\left(t\right)]$ are between the initially noncorrelated and correlated states for selected values of the correlation parameter $\lambda$. Time is in units of ${\omega }_c$, the dimensionless coupling $\alpha {\omega }_c^{\mu }=0.01$ and $\gamma {\omega }_c^{\nu }=0.05$. The remaining parameters are $\varepsilon =1,\mu =0.01$, $\nu =0.2$, and $|b_{+}^{\left(1\right)}{|}^2={\left|b_{+}^{\left(2\right)}\right|}^2=1/2$.

Figure 2

Time evolution of the (a) trace $D_T=D_{\mathrm{HS}}/\sqrt{2}$, (b) Bures $D_B$, (c) Hellinger $D_H$, and (d) Jensen-Shannon $D_{\mathrm{JS}}$ distances of qubit states for a finite environment. Initially the boson is in the mixture of the ground $|0\rangle$ and coherent states $|z\rangle$ with $z=\left|z\right|e^{i\phi }$. The impact of initial correlations quantified by the parameter $\lambda$ is depicted. As in Fig. 1, the distances $D\left(t\right)=D[{\rho }_0\left(t\right),{\rho }_{\lambda }\left(t\right)]$ are between the initially noncorrelated and correlated states. Time is in units of $\omega$. The chosen system parameters are $\varepsilon =1,g=0.1,\left|z\right|=1,\phi =0$, and $|b_{+}^{\left(1\right)}{|}^2={\left|b_{+}^{\left(2\right)}\right|}^2=1/2$.

Figure 3

Illustration of the role of the amplitude $\left|z\right|$ [(a) and (b)] and phase $\phi$ [(c) and (d)] of the environment coherent state $|z\rangle$ ($z=\left|z\right|e^{i\phi }$) on trace and Jensen-Shannon distances of qubit states. Qualitatively, the Bures and Hellinger distances behave like the Jensen-Shannon distance. max$\left[D\right(t)-D(0\left)\right]$ is the amplitude of distance time oscillations shown in Fig. 2. In (a) and (b) $\phi =0$. In (c) and (d) $\left|z\right|=1$. The remaining parameters are the same as in Fig. 2.

Figure 4

Illustration of the role of initial qubit states on trace and Jensen-Shannon distances. Qualitatively, the Bures and Hellinger distances behave like the Jensen-Shannon distance. max$\left[D\right(t)-D(0\left)\right]$ is the amplitude of distance time oscillations shown in Fig. 2. The parameters characterizing two initial states (20) are ${\lambda }_1={\lambda }_2=1$, for the first state and $b_{+}=b_{-}=1/\sqrt{2}$, for the second state: $b_{+}=\cos (\theta /2),b_{-}=\exp \left(i\zeta \right)\sin (\theta /2)$ with angle parametrization $\theta$ and $\zeta$ on the Bloch sphere. The remaining parameters are $g=0.1,z=1$.

Figure 5

The trace and Jensen-Shannon distances of qubit states for the finite environment: boson in the mixture of the ground $|0\rangle$ and number $|N\rangle$ states. The Bures and Hellinger distances display similar time dependence as the Jensen-Shanon distance with the exception that they lie below zero. The parameters are ${\lambda }_1=0,{\lambda }_2=1,\epsilon =1,g=0.1$, and $|b_{+}^{\left(1\right)}{|}^2={\left|b_{+}^{\left(2\right)}\right|}^2=1/2$.