Geometric characterization of true quantum decoherence

Surprisingly often decoherence is due to classical fluctuations of ambient fields and may thus be described in terms of random unitary (RU) dynamics. However, there are decoherence channels where such a representation cannot exist. Based on a simple and intuitive geometric measure for the distance of an extremal channel to the convex set of RU channels we are able to characterize the set of true quantum phase-damping channels. Remarkably, using the Caley-Menger determinant, our measure may be assessed directly from the matrix representation of the channel. We find that the channel of maximum quantumness is closely related to a symmetric, informationally complete positive operator-valued measure on the environment. Our findings are in line with numerical results based on the entanglement of assistance.

Figure 1

Schematic plot of the Bloch vectors (black, dashed arrows) corresponding to the “maximum quantum” channel (big) and to a selection of MCMQ (small) spanning tetrahedra (red, solid lines) of decreasing volume.

Figure 2

Quantumness $Q_A$ vs Bloch volume $V_B$ for a set of random phase-damping channels of rank 2. The color coding indicates the purity of the channel in the following way: blue (the rightmost black circles) ($0.5\le P<0.6$), red (dark gray circles) ($0.6\le P<0.7$), yellow (light gray circles) ($0.7\le P<0.8$), green (gray circles) ($0.8\le P<0.9$), and purple (the leftmost black circles) ($0.9\le P<1$). The dashed line corresponds to the one-parameter set of MCMQ, the channel with maximum Bloch volume $D_{\mathrm{\Delta }}$ is indicated with a black dot, as are the MCMQ channels with $P=0.6$, 0.7, 0.8, and 0.9 (from right to left).

Figure 3

Quantumness in terms of purity for a set of random phase-damping channels. The color coding indicates the rank of the respective channels: rank 2 (red) (dark gray), rank 3 (yellow) (gray), and rank 4 black (black). Upper bound is given by MCMQ channels (solid line). Quantumness of the convex channel $D_{\lambda }$ is plotted with a dashed line with few values of the parameter $\lambda$ indicated with black triangles. For further details see text.