Alternative fidelity measure between quantum states

We propose an alternative fidelity measure (namely, a measure of the degree of similarity) between quantum states and benchmark it against a number of properties of the standard Uhlmann-Jozsa fidelity. This measure is a simple function of the linear entropy and the Hilbert-Schmidt inner product between the given states and is thus, in comparison, not as computationally demanding. It also features several remarkable properties such as being jointly concave and satisfying all of Jozsa’s axioms. The trade-off, however, is that it is supermultiplicative and does not behave monotonically under quantum operations. In addition, metrics for the space of density matrices are identified and the joint concavity of the Uhlmann-Jozsa fidelity for qubit states is established.

Figure 1

(Color online) Plot of the trace distance $\mathcal{D}(\mathbf{\rho },\mathbf{\sigma })$ vs $1-{\mathcal{F}}_N(\mathbf{\rho },\mathbf{\sigma })$ for $4\times {10}^6$ pairs of randomly generated $\mathbf{\rho }$ and $\mathbf{\sigma }$ with (a) $d=3$ and (b) $d=6$. The darker (blue) points are generated using pairs of mixed states whereas the lighter (green) points are generated using at least one pure state. The antidiagonal solid line is the conjectured lower bound, whereas the upper bounds given by Eq. (38) are represented by the dashed curves (cyan)—one for each integer value of $\mathfrak{r}∊[2,d]$. A gap can be clearly noticed in plot (a) between the distribution of states and the absolute upper bound—i.e., the right-hand side of inequality (38) with $\mathfrak{r}=d$. Such a gap occurs whenever $d$ is odd. However, no gap is observed in plot (b) between the bulk of randomly generated states and the absolute upper bound. In fact, this bound can be saturated by density matrices of the form given by Eq. (39) whenever $d$ is even.

Figure 2

(Color online) A semilog plot of the average computation time in Matlab and C for the fidelity measures $\mathcal{F}$ (○), ${\mathcal{F}}_N$ (×), the nonlogarithmic variety of the quantum Chernoff bound $Q$ (+), and the trace distance $\mathcal{D}$ (◻) as a function of the dimension $d$ of the state space. The smaller and larger markers correspond to timings from Matlab and C, respectively. Computations were performed on a $2.6\text{−}\mathrm{GHz}$ Intel Pentium 4 CPU.