Quantum state tomography with measurement imperfections and decoherence

Pierre Six, Philippe Campagne-Ibarcq, Igor Dotsenko, Alain Sarlette, Benjamin Huard, Pierre Rouchon

Published 2015-10-06Version 1

Tomography of a quantum-state is usually based on positive operator-valued measure (POVM) and on their experimental statistics. Among the available reconstructions, the maximum-likelihood (MaxLike) technique is an efficient one. We propose an extension of this technique when the measurement process cannot be simply described by POVM. Here, the tomography relies on a set of quantum trajectories and their measurement records. It includes the fact that, in practice, each measurement could be corrupted by imperfections and decoherence, and could also be associated with the record of continuous-time signals over a finite amount of time. The proposed extension relies on an explicit expression of the likelihood function via the effective matrices appearing in quantum smoothing and solutions of the adjoint quantum filter. It also provides, aside the MaxLike estimate of the quantum state, confidence intervals for any observable. Such confidence intervals are derived, as the MaxLike estimate, from an asymptotic expansion of multi-dimensional Laplace integrals appearing in Bayesian Mean estimation. A validation is performed on two sets of experimental data: photon(s) trapped in a microwave cavity subject to quantum non-demolition measurements relying on Rydberg atoms; heterodyne fluorescence measurements of a superconducting qubit.