Certifying quantumness: Benchmarks for the optimal processing of generalized coherent and squeezed states

Quantum technology promises revolutionary advantages in information processing and transmission compared to classical technology; however, determining which specific resources are needed to surpass the capabilities of classical machines often remains a nontrivial problem. To address such a problem, one first needs to establish the best classical solutions, which set benchmarks that must be beaten by any implementation claiming to harness quantum features for an enhanced performance. Here we introduce and develop a self-contained formalism to obtain the ultimate, generally probabilistic benchmarks for quantum information protocols including teleportation and approximate cloning, with arbitrary ensembles of input states generated by a group action, so-called Gilmore-Perelomov coherent states. This allows us to construct explicit fidelity thresholds for the transmission of multimode Gaussian and non-Gaussian states of continuous-variable systems, as well as qubit and qudit pure states drawn according to nonuniform distributions on the Bloch hypersphere, which accurately model the current laboratory facilities. The performance of deterministic classical procedures such as square-root measurement strategies is further compared with the optimal probabilistic benchmarks, and the state-of-the-art performance of experimental quantum implementations against our newly derived thresholds is discussed. This work provides a comprehensive collection of directly useful criteria for the reliable certification of quantum communication technologies.

Figure 1

Scheme of a probabilistic measure-and-prepare (MP) state transformation game. The figure of merit $F$ is typically the fidelity between the output prepared by Bob and the target state set by the verifier Victor for a given input state supplied to Alice, averaged over the input set X according to a specified prior probability distribution. The maximum value $F_{\mathrm{c}}$ of $F$ over all MP protocols implementable by Alice and Bob sets a quantum benchmark for the transformation of the input ensemble of states. Any demonstration of state transformation of the input set which achieves, on average, a figure of merit exceeding $F_{\mathrm{c}}$, implies necessarily that a quantum channel has been used as a resource by Alice and Bob.

Figure 2

(a) Marginal prior probability distribution $p_{\beta }\left(z\right)$ for the $z$ component of the Bloch vector of single-qubit pure states, plotted for different values of the inverse-width parameter $\beta$ $\left(\beta =0,1,2,5\right)$. (b) Overlay of the corresponding prior probability distribution $p_{\beta }(\theta ,\phi )$ for single-qubit states $|{\psi }_{\theta ,\phi }\rangle$ on the Bloch sphere (where $\theta ,\phi$ denote the polar and azimuthal angles), for $\beta =5$.

Figure 3

Fidelity benchmark for the $N\to M$ transformation of arbitrary pure qudit states with (a)–(b) $d=2$, (c)–(d) $d=3$, and (e)–(f) $d=4$. The input states are distributed according to a prior distribution $p_{\beta }$ dependent on an inverse-width parameter $\beta$ as explained in the main text. In the first column [panels (a), (c), (e)], we set $\beta =0$, corresponding to a uniform distribution. In the second column [panels (b), (d), (f)], we set $\beta =5$, which gives a peaked distribution depicted in Fig. 2 for the qubit case $\left(d=2\right)$. The color legend for the CFT $F_{\mathrm{c}}^{\left(d\right)}$ in the bar charts is 0 (green) to 1 (red).

Figure 4

Prior probability distribution over the input ensemble of pure single-mode Gaussian states, Eq. (38), setting the inverse-width parameters to $\lambda =2$ and $\beta =6$. Panel (a) depicts the marginal prior distribution for displacement $\alpha$ and squeezing degree $s$ after integrating Eq. (38) over the squeezing phase $\theta$, yielding $p_{\lambda ,\beta }(\alpha ,s)={\pi }^{-1}\lambda \beta e^{-{\lambda \left|\alpha \right|}^2}sinhs{(coshs)}^{-\beta -2}{I}_0{\left[\lambda \right|\alpha |}^{2}tanhs]$ [11], where $I_0$ is a modified Bessel function. Panel (b) depicts cross sections of the phase-space Wigner functions for a small sample of pure single-mode Gaussian states with parameters $\alpha ,s$, and $\theta$ randomly sampled according to the considered prior distribution. The correspondence between the parameters of a Gaussian state and the form factor of the corresponding cross section (which is an ellipse obtained by cutting the two-dimensional Gaussian Wigner function) is as follows: the center of the ellipse has phase-space coordinates $(q,p)\equiv (\sqrt{2}\mathrm{Re}\alpha ,\sqrt{2}\mathrm{Im}\alpha )$, the ellipse is rotated by $\theta /2$ with respect to the horizontal axis, and the ratio between the lengths of the semiaxes is $e^{2s}$; see, e.g., [46] for more detail.

Figure 5

Fidelity benchmark for the $N\to M$ transformation of arbitrary pure single-mode Gaussian states, distributed according to the prior $p_{\lambda ,\beta }$ of Eq. (38), with (a) $\lambda =\beta =0$ (input ensemble of infinite energy), and (b) $\lambda =2,\beta =6$ (peaked distribution as in Fig. 4). The color legend for the CFT $F_{\mathrm{c}}^{\left(1cs\right)}$ in the bar charts is 0 (green) to 1 (red).

Figure 6

Fidelity benchmark for the $N\to M$ transformation of Perelomov-$j$ squeezed states with (a)–(b) $j=1$, corresponding to (Gaussian) two-mode squeezed vacuum states, (c)–(d) $j=\frac{3}{2}$, corresponding to (non-Gaussian) squeezed single-photon states, and (e)–(f) $j=3$, corresponding to (non-Gaussian) states obtained by applying a two-mode squeezing operation to the Fock states $|2\rangle |0\rangle$. The input states are distributed according to a prior distribution $p_{\beta }$ dependent on an inverse-width parameter $\beta$ as explained in the main text. In the first column [panels (a), (c), (e)], we set $\beta =0$, corresponding to a uniform distribution. In the second column [panels (b), (d), (f)], we set $\beta =4$. The color legend for the CFT $F_{\mathrm{c}}^{\left(\mathrm{P}j\right)}$ in the bar charts is 0 (green) to 1 (red).

Figure 7

The fidelity $F_{{\eta }_{\mathrm{opt}}}\left(\beta \right):={max}_{\eta }F(\beta ,\eta )$ for the optimal deterministic square-root measurement (red dashed line) and the ultimate probabilistic benchmark $F_{\mathrm{c}}^{\left(2\right)\mathrm{tele}}\left(\beta \right)$ for one-qubit teleportation (blue solid line) as functions of the inverse width $\beta$ of the prior distribution. The (green) filling between the two lines represents the gap between the benchmark and the performance of the optimal square-root measurement.