Schmidt-number benchmarks for continuous-variable quantum devices

We present quantum fidelity benchmarks for continuous-variable (CV) quantum devices to outperform quantum channels which can transmit at most $k$-dimensional coherences for positive integers $k$. We determine an upper bound of an average fidelity over Gaussian distributed coherent states for quantum channels whose Schmidt class is $k$. This settles fundamental fidelity steps where the known classical limit and quantum limit correspond to the two end points of $k=1$ and $k=\infty$, respectively. It turns out that the average fidelity is useful to verify to what extent an experimental CV gate can transmit a high-dimensional coherence. The result is further extended to be applicable to general quantum operations or stochastic quantum channels. Although the fidelity is often associated with heterodyne measurements in quantum optics, we can also obtain similar criteria based on quadrature deviations determined via homodyne measurements.

Figure 1

Our upper bound of the Schmidt-class-$k$ fidelity $F^{\left(k\right)}(\eta ,\lambda )$ [$U_k$ of Eq. (18)] for $\lambda =0.01$ and $\eta \in \{0.5,0.75,1.0,1.5,2.0\}$. If an experimental fidelity $F_{\eta ,\lambda }\left(\mathcal{E}\right)$ stays above the $k\mathrm{th}$ bound, the experimental CV gate $\mathcal{E}$ cannot be described by a $k$-PEB channel. This certifies an existence of the $(k+1)\mathrm{th}$ coherence in the CV gate $\mathcal{E}$. The classical limit fidelity of the fundamental quantum benchmark corresponds to the bound of $k=1$. In the inset, the classical limit fidelity ($k=1$) and the fidelity bound for $k=2$ due to the right-hand side of Eq. (20) are shown as a function of the Gaussian inverse width $\lambda$ for the case of unit-gain condition $\eta =1$. The dots represent the average fidelity $F_{1,\lambda }$ given in Fig. 5 of Ref. [24]. This experimental fidelity is not high enough to give evidence to outperform an arbitrary qubit gate with regard to our criterion.

Figure 2

Orders of principal submatrices.

Figure 3

To determine the order of the maximum eigenvalues of the different submatrices (a) a primitive step is to compare submatrices which can be chosen by shifts on the diagonal direction. (b) Another primitive step is to compare submatrices which have the same elements but some of the last block have higher number indices (row and columns). (c) The case with the last element is shifted. (d) The case where the last two elements are shifted. (e) When we compare the submatrices which have many blocks we consider spreading of the largest sub-block, such as $m2+m3$, first. Then, this submatrix, say $(m1;m2+m3)$, can be connected with another submatrix, say $(m1;m2;m3)$ by considering further spreading of the second largest sub-block, such as $m3$.