Optimal entanglement-assisted discrimination of quantum measurements

We investigate optimal discrimination between two projective single-qubit measurements in a scenario where the measurement can be performed only once. We consider general setting involving a tunable fraction of inconclusive outcomes and we prove that the optimal discrimination strategy requires an entangled probe state for any nonzero rate of inconclusive outcomes. We experimentally implement this optimal discrimination strategy for projective measurements on polarization states of single photons. Our setup involves a real-time electrooptical feed-forward loop which allows us to fully harness the benefits of entanglement in discrimination of quantum measurements. The experimental data clearly demonstrate the advantage of entanglement-based discrimination strategy as compared to unentangled single-qubit probes.

Figure 1

(a) Single-qubit measurements $\mathcal{M}$ and $\mathcal{N}$ on a Bloch sphere. (b) General measurement discrimination scheme involving entangled probe state. (c) Simple discrimination scheme with single-qubit probe.

Figure 2

Blue crosses show the dependence of $P_S$ on $P_I$ as specified by Eqs. (13) and (17); $c=0.9$. The red circles indicate the convex hull, points $A$ and $U$ correspond to minimum error and unambiguous discrimination with single-qubit probes, respectively, and point $T$ is specified by Eq. (18).

Figure 3

The scheme of the experimental setup. Bulk beam splitter (BS) 50:50; fiber beam splitter (FBS) 50:50; polarizing beam splitter (PBS); half-wave plate (HWP); collimating lens (C), phase modulator (PM), and single-photon detector (D).

Figure 4

Dependence of relative success probability ${\tilde{P}}_S$ on probability of inconclusive results $P_I$ is plotted for seven values of ${\theta }_j=j\pi /30,j=1,2,3,4,5,6,7$. The value of $j$ increases from bottom to top. Shown are the experimental data (circles) as well as the maximum ${\tilde{P}}_S$ achievable by the optimal scheme using entangled state (solid lines), and using single-qubit probes only (dashed lines).

Figure 5

Unambiguous discrimination of quantum measurements. The probabilities $P_S$ (blue circles), $P_I$ (red squares), and $P_E$ (black crosses) are plotted as functions of the VRC splitting ratio $T$. The lines represent theoretical predictions.

Figure 6

The second derivative $d^2{P}_S/d{P}_I^2$ given by Eq. (A4) is plotted as a function of $P_I$ and $c$.