Quantum benchmarking with realistic states of light

The goal of quantum benchmarking is to certify that imperfect quantum communication devices (e.g., quantum channels, quantum memories, quantum key distribution systems) can still be used for meaningful quantum communication. However, the test states used in quantum benchmarking experiments may be imperfect as well. Many quantum benchmarks are only valid for states which match some ideal form, such as pure states or Gaussian states. We outline how to perform quantum benchmarking using arbitrary states of light. We demonstrate these results using real data taken from a continuous-variable quantum memory.

Figure 1

Upper and lower bounds for the maximum purity $\mathcal{P}$ of experimentally compatible Gram matrices, corresponding to rotationally symmetric test ensembles of different cardinality. We have removed the factor $\frac{1}{M^2}$, which is common to both the upper and the lower bounds and independent of the purifications. The difference between the two bounds is on the order of ${10}^{-3}$ or less (inset), meaning that the obtained Gram matrix has purity which is near-optimal (represented by the lower curve). The estimated numerical precision of the optimization is $\approx$${10}^{-3}$.

Figure 2

Benchmarking results based on rotationally symmetric test ensembles using tomographically reconstructed density matrices. The top curve is the negativity of the input state ${\rho }_{A{A}^{\prime }}^{\mathrm{in}}$ found by optimizing the heuristic function $h$. The other curves are lower bounds to the negativity of ${\rho }_{AB}^{\mathrm{out}}$ based on (in descending order) (i) tomographic reconstruction and (ii) quadrature values obtained from the tomographic reconstruction. A benchmarking procedure is successful when the output state has negativity larger than zero.

Figure 3

Benchmarking results based on rotationally symmetric test ensembles using direct quadrature measurement data. The top curve is the same as in Fig. 2. The other curves are lower bounds to the negativity of ${\rho }_{AB}^{\mathrm{out}}$ based on (in descending order) (iii) quadrature values obtained directly from measurement data and (iv) quadrature data with error bars included in the optimization ($1\sigma$, $2\sigma$, and $3\sigma$ levels). For more than two test states, there is nonzero entanglement remaining even when including error estimates.