Variational quantum tomography with incomplete information by means of semidefinite programs

TL;DR

This paper presents a new quantum state reconstruction method using semidefinite programming that efficiently handles incomplete and noisy data, with convergence similar to compressed sensing for low-rank states.

Contribution

It introduces a convex optimization approach for quantum tomography that is computationally efficient and effective with incomplete information.

Findings

Estimated states do not overestimate purity.

Expectation values of entanglement witnesses are accurately reconstructed.

Reconstruction efficiency is comparable to compressed sensing for low-rank states.

Abstract

We introduce a new method to reconstruct unknown quantum states out of incomplete and noisy information. The method is a linear convex optimization problem, therefore with a unique minimum, which can be efficiently solved with Semidefinite Programs. Numerical simulations indicate that the estimated state does not overestimate purity, and neither the expectation value of optimal entanglement witnesses. The convergence properties of the method are similar to compressed sensing approaches, in the sense that, in order to reconstruct low rank states, it needs just a fraction of the effort correspondig to an informationally complete measurement.