Maximum-likelihood estimation of the density matrix

Abstract

We present a universal technique for quantum state estimation based on the maximum-likelihood method. This approach provides a positive definite estimate for the density matrix from a sequence of measurements performed on identically prepared copies of the system. The method is versatile and can be applied to multimode radiation fields as well as to spin systems. The incorporation of physical constraints, which is natural in the maximum-likelihood strategy, leads to a substantial reduction of statistical errors. Numerical implementation of the method is based on a particular form of the Gauss decomposition for positive definite Hermitian matrices.Comment: 4 pages, 3 figures (5 eps files). Submitted to Phys. Rev. A as a Rapid Communicatio

Diluted maximum-likelihood algorithm for quantum tomography

“…This approach was used by James et al in their work on tomography of optical qubits [9]. In application to homodyne tomography, the method was elaborated by Banaszek et al [10] and used in an experiment by D'Angelo et al [11].…”

Tomographic test of Bell’s inequality for a time-delocalized single photon

“…The density matrix of the measured system has also been reconstructed by means of the maximum-likelihood method (ML) [35]: the density matrixρ that most likely represents the homodyne data is retrieved by maximizing a functional L(ρ) involving the POVM associated with two-mode homodyne measurements. The only constraint we impose is that the density matrixρ is a positivedefinite hermitian matrix with unitary trace.…”

Information Geometry of Randomized Quantum State Tomography

“…The maximum likelihood method is a canonical one to obtain a corrected estimate [2,[5][6][7][8][9][10]. From the point of view of information geometry [11][12][13], the maximum likelihood estimate (MLE) is the orthogonal projection from the temporary estimatex onto the Bloch ball B with respect to the standard Fisher metric along the ∇ (m) -geodesic [14], (cf., Appendix A).…”