Comparative investigation of the freezing phenomena for quantum correlations under nondissipative decoherence

We show that the phenomenon of frozen discord, exhibited by specific classes of two-qubit states under local nondissipative decoherent evolutions, is a common feature of all known bona fide measures of general quantum correlations. All those measures, despite inducing typically inequivalent orderings on the set of nonclassically correlated states, return a constant value in the considered settings. Every communication protocol that relies on quantum correlations as a resource will run with a performance completely unaffected by noise in the specified dynamical conditions. We provide a geometric interpretation of this phenomenon.

Figure 1

Time evolution of various measures of quantum correlations for BD states with initial condition ${\vec{c}}^{\left(k\right)}\left(0\right)$ given by Eq. (5), subject to local bit flip ($k=1$), bit-phase flip ($k=2$), or phase flip ($k=3$) channels with rate $\gamma$. In the range of parameters such that $2\gamma t<-\ln c$, depicted as a shaded (yellow) region on the horizontal planes in each panel, all the considered measures remain frozen to their respective initial values. The plotted quantities are (a) quantum discord $\mathcal{D}$, equal to one-way quantum deficit ${\Delta }^{\to }$, equal to relative entropy of discord $D_R$; (b) adjusted geometric discord ${\tilde{D}}_G$; (c) trace-distance discord $D_1$, equal to negativity of quantumness $Q_N$; (d) Bures-distance discord $D_B$; and (e) fidelity-based measure $D_F$, equal to local quantum uncertainty $\mathcal{U}$. The analytical expressions are given by combining Eq. (6) with Table .

Figure 2

Contours of constant $\mathcal{Q}$ in the plane $(c_1,c_2)$ for physical BD states with $c_3=\frac{1}{4}$, where $\mathcal{Q}$ represents (a) $\mathcal{D}\equiv {\Delta }^{\to }\equiv D_R$, (b) ${\tilde{D}}_G$, (c) $D_1\equiv Q_N$, and (d) $D_B$. The diagonals (traced from the vertices to the center) represent the evolution of BD states with the initial condition ${\vec{c}}^{\left(3\right)}\left(0\right)$ [Eq. (5)] under local phase flip channels. For $t<t^{☆}$ (continuous traits) all the measures $\mathcal{Q}$ are constant; they then decay to zero for later times (dashed traits). The color legend for the $\mathcal{Q}$'s in the contour plots is 0 1.