Mixed-state sensitivity of several quantum-information benchmarks

We investigate an imbalance between the sensitivity of the common state measures—fidelity, trace distance, concurrence, tangle, von Neumann entropy, and linear entropy—when acted on by a depolarizing channel. Further, in this context we explore two classes of two-qubit entangled mixed states. Specifically, we illustrate a sensitivity imbalance between three of these measures for depolarized (i.e., Werner-state-like) nonmaximally entangled and maximally entangled mixed states, noting that the size of the imbalance depends on the state’s tangle and linear entropy.

Figure 1

(Color online) Constant fidelity curves for the maximally entangled state $(\mid 00⟩+\mid 11⟩)∕\sqrt{2}$ (star, upper left corner). Also shown are the Werner state curve (dotted line) and, bounding the gray region of nonphysical entropy-tangle combinations, the MEMS curve, which is solid for ${\rho }_{\mathit{MEMS}\mathrm{I}}$ and dashed for ${\rho }_{\mathit{MEMS}\mathrm{II}}$. The (horizontal) constant fidelity curves below the Werner state curve are swept out by comparing the starting state with states of the form ${\rho }_1(ϵ,\theta )$, Eq. (20), while the (nearly vertical) curves above the Werner state line are generated by varying the parameters of ${\rho }_2(ϵ,r)$ given by Eq. (22). For comparison, the pure product state $\mid 00⟩$ (lower left corner) has fidelity of 0.5 with this target.

Figure 2

(Color online) Constant 0.99-fidelity curves for several starting MEMS, plotted as stars. The constant fidelity curves are calculated by comparing the target state with ${\rho }_2(ϵ,r)$ where $ϵ$ and $r$ are varied to give different tangles and linear entropies. The dark gray regions are 20 000 points per initial MEMS, corresponding to numerically generated density matrices that have fidelity of 0.9900 or higher with the target.

Figure 3

(Color online) Constant fidelity curves for a nonmaximally entangled mixed state $\left[{\rho }_2(ϵ=0.225,\theta =11.57°)\right]$ compared with states calculated by varying $ϵ$ and $\theta$. In the case of this mixed entangled state, the constant fidelity regions are surprisingly large: for example, a 0.9 fidelity with this starting state could arise from a nearly pure entangled state or from an unentangled near fully mixed state. The 0.9 and higher fidelity region covers $\sim 60\%$ of the physically allowed region of the linear-entropy–tangle plane.