Optimal verification of two-qubit pure states

In a recent work [Pallister, Linden, and Montanaro, Phys. Rev. Lett. 120, 170502 (2018)], Pallister et al. proposed an optimal strategy to verify nonmaximally entangled two-qubit pure states under the constraint that the accessible measurements are locally projective and nonadaptive. Their good result leads naturally to the following question: What is the optimal strategy among general local operations and classical communication (LOCC) measurements? In this paper, we answer this problem completely for two-qubit pure states. To be specific, we give the optimal strategy for each of the following available classes of measurements: (i) local operations and one-way classical communication (one-way LOCC) measurements; (ii) local operations and two-way classical communication (two-way LOCC) measurements; and (iii) separable measurements. Surprisingly, our results reveal that for the two-qubit pure state verification problem two-way LOCC measurements remarkably outperform one-way LOCC measurements and have the same power as the separable measurements.

Figure 1

The two-way measurement $\{T_1^{A\to B},\mathbb{1}-T_1^{A\to B}\}$. Alice first performs measurement $\left\{\delta \right|0\rangle \langle 0|,(1-\delta \left)\right|0\rangle \langle 0|+|1\rangle \langle 1\left|\right\}$ and sends the outcome to Bob. Conditioned on the outcome, Bob adopts different measurements on his postmeasurement state and sends the outcome to Alice if necessary. Alice then performs the corresponding two-outcome measurement $\{{\tau }_j,\mathbb{1}-{\tau }_j\}$ to detect the final state she holds.

Figure 2

The number of measurements required to verify the two-qubit pure state $|\mathrm{\Psi }\rangle$ using various strategies – ${\mathrm{\Omega }}_{PLM}$, ${\mathrm{\Omega }}_{\to }$, ${\widehat{\mathrm{\Omega }}}_{\leftrightarrow }$, and ${\mathrm{\Omega }}_{\leftrightarrow }$ – as a function of $\lambda$. Here, $\epsilon =0.01$ and $\delta =0.1$. These parameters are chosen in accordance with Fig. 1 of [32] for better comparison.