Adaptive procedures for discriminating between arbitrary tensor-product quantum states

Discriminating between quantum states is a fundamental task in quantum information theory. Given two quantum states ${\rho }_{+}$ and ${\rho }_{-}$, the Helstrom measurement distinguishes between them with minimal probability of error. However, finding and experimentally implementing the Helstrom measurement can be challenging for quantum states on many qubits. Due to this difficulty, there is great interest in identifying local measurement schemes which are close to optimal. In the first part of this work, we generalize previous work by Acin et al. [Phys. Rev. A 71, 032338 (2005)] and show that a locally greedy scheme using Bayesian updating can optimally distinguish between any two states that can be written as a tensor product of arbitrary pure states. We then show that the same algorithm cannot distinguish tensor products of mixed states with vanishing error probability (even in a large subsystem limit), and introduce a modified locally greedy scheme with strictly better performance. In the second part of this work, we compare these simple local schemes with a general dynamic programming approach which finds both the optimal series of local measurements as well as the optimal order in which subsystems are measured.

Figure 1

Comparison of the probability of success as a function of the number of available systems for the case of identical copies. As the depolarizing parameter $\gamma$ increases, the LG probability of success levels off for large $N$ while the MLG probability of success converges to 1.

Figure 2

Left-hand side depicts the average probability of success for the best and worst ordering using both ternary and binary projective measurements for qutrit product states when $N=3$. Results are averaged over 1000 trials. Right-hand side depicts difference in average success probability for the various methods, namely, $P_{\mathrm{diff},\mathrm{order}}(\gamma ,\mathcal{A})$ as a function of $\gamma$ when $N=3$. Results are averaged over 1000 trials.

Figure 3

Left-hand side depicts probabilities $P_{\mathrm{s},\mathrm{best}}(N=3,\gamma )$ and $P_{\mathrm{s},\mathrm{worst}}(N=3,\gamma )$ as a function of the depolarising parameter $\gamma$ over 1000 trials. Although $P_{\mathrm{s},\mathrm{best}}(N=3,\gamma )\ne P_{\mathrm{s},\mathrm{worst}}(N=3,\gamma )$, the relative difference is small. Right-hand side depicts difference in maximum and minimum probability of success, $P_{\mathrm{succ},\mathrm{diff}}(N,\gamma )$, as a function of the depolarizing parameter $\gamma$ over 1000 trials for $N=3,4,5,6,7$.

Figure 4

Left-hand side depicts probability of success for varying $\gamma$ in the distinct subsystems scenario as a function of the number of available systems, $N$. Results are average over 1000 trials. Right-hand side compares probability of success as a function of the number of available systems, $N$, for depolarizing parameter $\gamma =0.3$. Results are averaged over 1000 trials.