Incomplete quantum state estimation: A comprehensive study

We present a detailed account of quantum state estimation by joint maximization of the likelihood and the entropy. After establishing the algorithms for both perfect and imperfect measurements, we apply the procedure to data from simulated and actual experiments. We demonstrate that the realistic situation of incomplete data from imperfect measurements can be handled successfully.

Figure 1

Schematic diagrams of $\mathcal{I}(\lambda ,\rho )$ on the space of statistical operators. The maximally mixed state resides at the center of the square base which represents the Hilbert space. At the extremal points of $\lambda$, $\mathcal{I}(\lambda =0;\rho )=\ln \left[\mathcal{L}(\left\{n_j\right\};\rho )\right]/N$, with a convex plateau at the maximal value, and $\mathcal{I}(\lambda \to \infty ;\rho )=\lambda S\left(\rho \right)$. Plot (c) shows the functional with an appropriate choice of value for $\lambda$ for MLME. An additional hill-like structure resulting from $S\left(\rho \right)$ is introduced over the plateau, so that the estimator with the largest entropy can be selected from the convex set of ML estimators within the plateau.

Figure 2

A comparison of two different schemes with ${10}^3$ random qubit true states distributed uniformly with respect to the Hilbert-Schmidt measure. Fifty experiments were simulated for every true state, with $N=5000$ for each experiment, and the respective average trace-class distances ${\mathcal{D}}_{\mathrm{tr}}^{\mathrm{avg}}$ were computed. The entire simulation was done with a set of randomly generated, informationally incomplete POM consisting of two imperfect measurement outcomes. The plot markers denoted by “$+$” represent reconstructed states using the algorithm in Eq. (13) while ignoring the imperfection of the measurements, and those denoted by “$\square$” represent the reconstructed states using the algorithm in Eq. (18) that accounts for this imperfection. The significant improvement in tomographic efficiency with the latter algorithm is a strong indication of the importance of a proper treatment of imperfect measurements.

Figure 3

A schematic diagram representing the time-multiplexed setup with $K+1$ output ports. The $T_j$ are the respective transmission probabilities for the $j$th beam splitter. The overall efficiency for, say, the $k$th port is given by ${\tilde{\eta }}_k={\eta }_k(1-T_k+T_{K+1}{\delta }_{k,K+1}){\prod }_{j=1}^{k-1}{T}_j$.

Figure 4

Density plots of the Wigner functions, in phase space, of various statistical operators for (a) the true state (twenty-dimensional stationary state of a laser, $\mu =4$) with $\tilde{\tau }\approx 0.394$, (b) the five-dimensional ML estimator with $\tilde{\tau }\approx 0.921$ and (c) the eleven-dimensional MLME estimator with $\tilde{\tau }\approx 0.489$. Here, brighter regions indicate the locations of larger Wigner function values, and vice versa. The statistical operator for (b) is obtained using ML by assuming a five-dimensional subspace in which the displaced POM outcomes are informationally complete. The statistical operator for (c) is obtained by assuming a larger subspace of dimension 11 using MLME. Numerous artificial nonclassical features of the ML estimator, a signature of its highly oscillatory Wigner function, are manifested as an abnormally large value of $\tilde{\tau }$, an inevitable byproduct of state-space truncation. One can see that with MLME, extraneous artifacts of the Wigner function resulting from such a truncation can be largely removed.

Figure 5

Density plots of the Wigner functions, in phase space, of various statistical operators for (a) the true state (${\rho }_{{\alpha }^{\prime }}$, ${\alpha }^{\prime }=5$), (b) the eight-dimensional ML estimator, (c) the ten-dimensional, and (d) fifteen-dimensional MLME estimators. In this case, the Wigner function of the ML estimator differs greatly from that of the true state, an example of misleading information obtained via state-space truncation. A transition in the structure of the Wigner function occurs at $D_{\mathrm{sub}}=10$, with the MLME estimator for $D_{\mathrm{sub}}=15$ giving a more accurate estimated picture of the Wigner function of the true state.

Figure 6

Schematic diagram of the diffraction patterns of an incoming light beam that is obtained from a SH wave front sensor. The light beam is transformed by an array of microlenses (apertures). A CCD camera is placed at the rear focal plane of the array. The measurement data consist of the measured intensities of the beam. The intensity at the $j$th pixel, located at position $x_j$, behind the $k$th microlens aperture is denoted by $I_k\left(x_j\right)$.

Figure 7

Experimental setup involving a single-mode fiber (SMF), a spatial light modulator (SLM), an aperture stop (A), and a Shack-Hartmann (SH) sensor.

Figure 8

CCD image for the state ${\rho }_{\mathrm{coh}}^{\mathrm{true}}$. The relevant part of the SH readout used for the beam reconstruction is shown. Contributions from the individual SH apertures are indicated by bright spots, with each spot made up of multiple pixels. Note that the two void regions correspond to the phase singularities of the state ${\rho }_{\mathrm{coh}}^{\sup }$. This hints that ${\rho }_{\mathrm{coh}}^{\mathrm{true}}\approx {\rho }_{\mathrm{coh}}^{\sup }$.

Figure 9

MLME state estimation from informationally incomplete data for $D_{\mathrm{sub}}=9$. The real (left) and imaginary (right) parts of the reconstructed coherence operator ${\widehat{\rho }}_{\mathrm{coh}}^{\mathrm{MLME}}$ are shown. The reconstruction subspace is spanned by the modes ${\text{LG}}_l$, with $l=0,1,...,8$. In this case, 56 out of 91 independent outcomes, required for complete characterization of ${\rho }_{\mathrm{coh}}^{\mathrm{true}}$, are not accessible, yet the MLME estimator ${\widehat{\rho }}_{\mathrm{coh}}^{\mathrm{MLME}}$ is close to ${\rho }_{\mathrm{coh}}^{\sup }$, with a fidelity of $92\%$.

Figure 10

Average fidelities, computed over 50 random choices of computational bases, of the estimators for different dimensions $D_{\mathrm{sub}}$ of the reconstruction subspace. The unfilled (filled) circular plot markers correspond to informationally complete (incomplete) tomography, respectively.