Schmidt-number benchmarks for continuous-variable quantum devices

Abstract

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 endpoints of k = 1 and k = ∞, 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. While 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.

Modes and states in Quantum Optics

“…1/2 being the minimum value of the product ∆x∆k allowed by the Fourier-Heisenberg inequality. The complexity of a bipartite quantum state, in particular its entanglement properties, is related to the Schmidt number (Dyakonov et al, 2014;Guo and Fan, 2013;Namiki, 2016;Sharapova et al, 2015), i.e. the number of terms in the Schmidt decomposition.…”

Parametric down-conversion beyond the semiclassical approximation

“…Being based on linear algebra, the Bloch-Messiah reduction assumes a finite dimensional system. The kernels are then represented by matrices, which can be diagonalized, reminiscent of a Schmidt decomposition [31,[47][48][49]. Thus, the set of equations in Eq.…”

“…The two equations that are independent of {α, α * } give the same solution for the pump parameter function that is obtained in Eq. (49). The remaining six equations are…”

Parametric down-conversion beyond the semi-classical approximation

“…Being based on linear algebra, the Bloch-Messiah reduction assumes a finite dimensional system. The kernels are then represented by matrices, which can be diagonalized, reminiscent of a Schmidt decomposition [31,[46][47][48]. Thus, the set of equations in Eq.…”