Strong quantitative benchmarking of quantum optical devices

Quantum communication devices, such as quantum repeaters, quantum memories, or quantum channels, are unavoidably exposed to imperfections. However, the presence of imperfections can be tolerated, as long as we can verify that such devices retain their quantum advantages. Benchmarks based on witnessing entanglement have proven useful for verifying the true quantum nature of these devices. The next challenge is to characterize how strongly a device is within the quantum domain. We present a method, based on entanglement measures and rigorous state truncation, which allows us to characterize the degree of quantumness of optical devices. This method serves as a quantitative extension to a large class of previously known quantum benchmarks, requiring no additional information beyond what is already used for the nonquantitative benchmarks.

Figure 1

(a) Device implementing a measure and prepare strategy: A measurement is made on the input state, the result is stored or transmitted classically, and a new quantum state is prepared based on this measurement result. (b) Example distributions of test states: Entanglement-based benchmarks have been designed for ensembles consisting of two or more coherent states on a ring, squeezed and antisqueezed vacuum, as well as for generalizations to mixed states.

Figure 2

(a) Entanglement-based benchmarking: Is there any entanglement remaining after part of the state $|{\psi }^{\mathrm{ent}}\rangle$ passes through the device? How much remains? (b) Effective entanglement: Any source which produces a classical mixture of the test states $|{\psi }_k\rangle$ can be pictured as internally preparing the entangled state in Eq. (1) and performing a projective measurement on subsystem A.

Figure 3

Example benchmark results using two coherent states ($T=0.448$). The upper line was found using the EVM method; any data below this line are in the quantum domain. The lower line represents the limits of the quantitative domain using the methods of [20]. There is clearly room for improvement on the quantitative benchmarks.

Figure 4

Entanglement bounds for the two-coherent-state benchmark, using our quantification scheme (with $N=20$), assuming no loss ($T=1.0$). The dashed black line represents the boundary of the quantum domain, found using the EVM method. Variances lower than this line correspond to devices within the quantum domain, whereas variances higher than this line are known to be compatible with separable states [17] (note that the axes are inverted in order to see the results better). Direct comparison with the results of [20] (Fig. 2a) shows a dramatic improvement in the entanglement bounds at higher variances.

Figure 5

Entanglement quantification for squeezed and antisqueezed state benchmarks, using initial squeezing magnitude $r=0.35$ and cutoff $N=20$ (cf. Fig. 1 of [19]). The bottom left region is forbidden by the uncertainty relation. The quantum domain lies between the forbidden region and the upper dotted line (found with EVM method; variances above this line are compatible with separable states [19]). The quantitative domain, shown using level curves of the minimal negativity, is very faithful in this parameter regime; the zero contour and the quantum domain boundary coincide within the numerical resolution. For much larger values of ${\mathrm{Var}}_0(\mathrm{x̂})$, the two domains begin to diverge due to reduced numerical accuracy. The small “sawtooth” gap near the unphysical boundary is due to the finite resolution of sampled points.

Figure 6

Entanglement quantification for the three-coherent-state benchmark, with no loss ($T=1.0$) and cutoff $N=15$ (cf. Fig. 3 of [18]). Variance values below the dashed black line are known to be in the quantum domain (note again that the axes are flipped to aid visualization of the results). Unlike benchmarks using two test states, it is not known whether variances higher than the dashed line must come from separable or PPT entangled states. Although the quantitative domain is larger than the case with two coherent states, it does not cover the entire known quantum domain due to the smaller cutoff and higher energies involved.

Figure 7

Upper and lower bounding the minimal entanglement. The lower curve is a slice through the surface in Fig. 6 at $\left|\alpha \right|=0.2$ and the upper curve corresponds to results of the hybrid optimization (49). We see that the minimal entanglement must be quite small after Var$(x)\approx 1$. Higher values of the cutoff $N$ would be needed to have small enough error bounds to detect this entanglement.