Quantum-State Reconstruction by Maximizing Likelihood and Entropy

Quantum-state reconstruction on a finite number of copies of a quantum system with informationally incomplete measurements, as a rule, does not yield a unique result. We derive a reconstruction scheme where both the likelihood and the von Neumann entropy functionals are maximized in order to systematically select the most-likely estimator with the largest entropy, that is, the least-bias estimator, consistent with a given set of measurement data. This is equivalent to the joint consideration of our partial knowledge and ignorance about the ensemble to reconstruct its identity. An interesting structure of such estimators will also be explored.

Figure 1

A simulation on quantum tomography on a randomly generated mixed state of light in the five-dimensional Fock space. In this plot, the number of copies of quantum systems measured is fixed at $N={10}^4$. A choice of 20 quadrature eigenstates made up of four different $\vartheta$ settings, with five $x$ values corresponding to each setting, which are projected onto this space was used, and state estimators are constructed for different values of $\lambda$. As $\lambda$ decreases, both the entropy and likelihood functionals approach their respective optimal values obtained from MLME (i.e., when $\lambda \to 0$). When $\lambda$ is zero, there is a convex set of estimators giving the optimal likelihood value. For very large $\lambda$ values, the estimators approach the maximally mixed state, and hence $S(\rho )$ approaches the maximal value $\log 5$.

Figure 2

A simulation on quantum tomography on a randomly generated mixed state ${\rho }_{\mathrm{true}}$ of light in the 20-dimensional Fock space with a slightly positive $W_{00}=0.141$. ${\overline{D}}_{\mathrm{tr}}$ and ${\overline{W}}_{00}$, respectively, denote the trace-class distance between the reconstructed estimator and the true state and the Wigner function at the phase space origin, both averaged over 50 experiments with $N={10}^4$. The same set of 20 quadrature eigenstates as in Fig. 1, projected onto this space, was used, and this set of measurements is informationally complete in the two-, three-, and four-dimensional Fock subspaces (shaded region). The values ${\overline{W}}_{00}$ and ${\overline{D}}_{\mathrm{tr}}$ were obtained by ML [9] in subspaces of dimensions two to four and by the MLME scheme in dimensions greater than four. The plot shows a strong dependence of ${\overline{W}}_{00}$ and ${\overline{D}}_{\mathrm{tr}}$ on the subspace dimension. In this case, it is obvious that the negativity of ${\overline{W}}_{00}$ inferred by a reconstruction in a subspace too small is just an artifact of the truncations. Also, ${\overline{D}}_{\mathrm{tr}}$ decreases as the reconstruction subspace increases in dimension. This demonstrates the advantages of the MLME scheme over the ML method.