Eur. Phys. J. D 32, 215-222 (2005)
https://doi.org/10.1140/epjd/e2005-00020-2

Process reconstruction from incomplete and/or inconsistent data

¹ Research Center for Quantum Information, Slovak Academy of Sciences, Dúbravská cesta 9, 84511 Bratislava, Slovakia
² Faculty of Informatics, Masaryk University, Botanická 68a, 60200 Brno, Czech Republic

Corresponding author: ^a buzek@savba.sk

Received: 18 October 2004
Published online: 9 February 2005

Abstract

We analyze how an action of a qubit channel (map) can be estimated from the measured data that are incomplete or even inconsistent. That is, we consider situations when measurement statistics is insufficient to determine consistent probability distributions. As a consequence either the estimation (reconstruction) of the channel completely fails or it results in an unphysical channel (i.e., the corresponding map is not completely positive). We present a regularization procedure that allows us to derive physically reasonable estimates (approximations) of quantum channels. We illustrate our procedure on specific examples and we show that the procedure can be also used for a derivation of optimal approximations of operations that are forbidden by the laws of quantum mechanics (e.g., the universal NOT gate).