Quantitative Tomography for Continuous Variable Quantum Systems

We present a continuous variable tomography scheme that reconstructs the Husimi $Q$ function (Wigner function) by Lagrange interpolation, using measurements of the $Q$ function (Wigner function) at the Padua points, conjectured to be optimal sampling points for two dimensional reconstruction. Our approach drastically reduces the number of measurements required compared to using equidistant points on a regular grid, although reanalysis of such experiments is possible. The reconstruction algorithm produces a reconstructed function with exponentially decreasing error and quasilinear runtime in the number of Padua points. Moreover, using the interpolating polynomial of the $Q$ function, we present a technique to directly estimate the density matrix elements of the continuous variable state, with only a linear propagation of input measurement error. Furthermore, we derive a state-independent analytical bound on this error, such that our estimate of the density matrix is accompanied by a measure of its uncertainty.

Figure 1

(a) Exact $Q$ function of the ideal test state [cf. Eq. (3)]. (b) Raw data of $Q$-function measurements taken on a $16\times 16$ equidistant grid. (c) A Lagrange interpolation from measurements made at 231 Padua points (white dots). (d) Interpolation on an square grid (white dots) with nonpolynomial preprocessing of the data. Both (c) and (d) accurately reconstruct the ideal $Q$ function of (a). (e) and (f) show the result of thresholding the interpolations to use only the 65 largest magnitude measurements, in which case the Padua interpolation in (e) outperforms the equidistant grid in (f).

Figure 2

Relative reconstruction error [cf. Eq. (10)] as a function of the number of Padua points $N$, for the density matrix elements of the test state of Eq. (3). The error decreases exponentially for all density matrix elements as $N$ increases.