Test One to Test Many: A Unified Approach to Quantum Benchmarks

Quantum benchmarks are routinely used to validate the experimental demonstration of quantum information protocols. Many relevant protocols, however, involve an infinite set of input states, of which only a finite subset can be used to test the quality of the implementation. This is a problem, because the benchmark for the finitely many states used in the test can be higher than the original benchmark calculated for infinitely many states. This situation arises in the teleportation and storage of coherent states, for which the benchmark of 50% fidelity is commonly used in experiments, although finite sets of coherent states normally lead to higher benchmarks. Here, we show that the average fidelity over all coherent states can be indirectly probed with a single setup, requiring only two-mode squeezing, a 50-50 beam splitter, and homodyne detection. Our setup enables a rigorous experimental validation of quantum teleportation, storage, amplification, attenuation, and purification of noisy coherent states. More generally, we prove that every quantum benchmark can be tested by preparing a single entangled state and measuring a single observable.

Figure 1

Input-output test of a quantum device. To test the device $\mathcal{C}$, the verifier prepares an input state, randomly drawn from the set $\{{\rho }_x\}$. Upon receiving the input, the device generates an output, which is then measured by the verifier with the POVM $\{P_y^{(x)}\}$. The outcome is assigned a score and the average score is used as a measure of performance.

Figure 2

Test with a single input and a single observable. A composite system $AR$ in a joint state $\sigma$. Then, system $A$ is sent to the device $\mathcal{C}$, which transforms it into the output system $A^{\prime }$. Finally, systems $A^{\prime }$ and $R$ undergo a joint measurement, described by the observable $O$. The expectation value of $O$ is then used as the figure of merit.

Figure 3

Canonical test for coherent state amplifiers. The input mode and a reference are prepared in the two-mode squeezed vacuum (TMSV). After the action of the amplifier, the output mode and the reference are sent through a two mode squeezer $S_{\theta }$, followed by a 50-50 beam splitter and two quadrature measurements on the output modes.