Progress toward optimal quantum tomography with unbalanced homodyning

Balanced homodyning, heterodyning, and unbalanced homodyning are three well-known sampling techniques used in quantum optics to characterize photonic sources in the continuous-variable regime. We show that for all quantum states and all observable-parameter tomography schemes, which includes reconstructions of arbitrary operator moments and phase-space quasidistributions, localized sampling with unbalanced homodyning is always tomographically more powerful (gives more accurate estimators) than delocalized sampling with heterodyning. The latter is recently known to often give more accurate parameter reconstructions than conventional marginalized sampling with balanced homodyning. This result also holds for realistic photodetectors with subunit efficiency. With examples from first- through fourth-moment tomography, we demonstrate that unbalanced homodyning can outperform balanced homodyning when heterodyning fails to do so. This new benchmark takes us one step towards optimal continuous-variable tomography with conventional photodetectors and minimal experimental components.

Figure 1

Schema for the (a) UHOM, (b) BHOM, and (c) HET setups. The UHOM scheme consists only of one beam splitter (BS) that is almost perfectly transmissive (transmission amplitude $t\to 1$) and two photodetectors, D1 and D2, where only data corresponding to vacuum-state measurement with D1 are used to estimate the Husimi function, which is a significant experimental simplification compared to HET, which requires three balanced BSs (BBSs) and four photodetectors to randomly sample the Husimi function.

Figure 2

Wigner functions of (a, c) a squeezed Gaussian state and (b, d) a Fock state of $n=3$ reconstructed with a truncated invR for BHOM based on (a, b) perfect data and (c, d) noisy data. The wriggles of the reconstructed functions that come from truncations to a 50-dimensional Hilbert subspace, even for the case of perfect data, can lead to significant deviations from the true $\mathit{q}$.

Figure 3

Plots for the (a) first-, (b) second-, (c) third-, and (d) fourth-moment reconstructions of a Gaussian state of $1\le \mu =\lambda \le 3$, in which the $\mathrm{sCRB}$ of BHOM (squares and curve), BHOMOPT (triangles and curve), HET (circles and curve), and UHOM (diamonds and curve) are illustrated. The instability and sensitivity of the invR with the BHOM marginalized sampling strategy are clear in the plots, which behavior also depends on the truncated Hilbert space. Evidently, UHOM exhibits a more superior tomographic power than HET, BHOM, and BHOMOPT. Dashed curves represent theory derived from (3) and (5), whereas symbols are computed with Monte Carlo simulated data of the CV experiments for a 50-dimensional Hilbert space. For the purpose of illustrating the results, we take $\eta =1$ for simplicity.

Figure 4

Plots of $\mathrm{sCRB}$ for Fock states of $0\le n\le 5$. All specifications follow those in Fig. 3. For each BHOM plot, the dashed curve joins the six theoretically calculated numerical values that match the squares.