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arXiv:0905.0578 · Physical Review A

Simple representation of quantum process tomography

Giuliano Benenti, G. Strini

Published Aug 13, 2009• 9 citations
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Abstract

We show that the Fano representation leads to a particularly simple and appealing form of the quantum process tomography matrix ${\ensuremath{\chi}}_{F}$, in that the matrix ${\ensuremath{\chi}}_{F}$ is real, the number of matrix elements is exactly equal to the number of free parameters required for the complete characterization of a quantum operation, and these matrix elements are directly related to evolution of the expectation values of the system's polarization measurements. These facts are illustrated in the examples of one- and two-qubit quantum noise channels.

DOI
10.1103/physreva.80.022318

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We show that the Fano representation leads to a particularly simple and appealing form of the quantum process tomography matrix \ensuremathχF{\ensuremath{\chi}}_{F}\ensuremathχF​, in that the matrix \ensuremathχF{\ensuremath{\chi}}_{F}\ensuremathχF​ is real, the number of matrix elements is exactly equal to the number of free parameters required for the complete characterization of a quantum operation, and these matrix elements are directly related to evolution of the expectation values of the system's polarization measurements. These facts are illustrated in the examples of one- and two-qubit quantum noise channels.
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We present a new method for quantum process tomography. The method enables us to efficiently estimate, with fixed precision, any of the parameters characterizing a quantum channel. It is selective since one can choose to estimate the value of any specific set of matrix elements of the super-operator describing the channel. Also, we show how to efficiently estimate all the average survival probabilities associated with the channel (i.e., all the diagonal elements of its χχχ--matrix.
The characterization of the evolution of a quantum system is one of the main tasks to accomplish to achieve quantum information processing. The standard quantum process tomography (SQPT) has the unique property that it can be applied without introducing any additional quantum resources. In present work, we shall focus on the following two topics about the SQPT. At first, in the SQPT protocol for a ddd-dimensional system, one should encounter a problem in solving of a set of d4d^4d4 linear equations in order to get the matrix containing the complete information about the unknown quantum channel. Until now, the general form of the solution is unknown. And a long existed conviction is that the solutions are not unique. Here, we shall develop a self-consistent scheme, in which bounded linear operators are presented by vectors, to construct the set of linear equations. With the famous Cramer's rule for the set of linear equations, we are able to give the general form of the solution and prove that it is unique. In second, the central idea of the SQPT is to prepare a set of linearly independent inputs and measuring the outputs via the quantum state tomography (QST). Letting the inputs and the measurements be prepared by two sets of the rank-one positive-operator-valued measures [POVMs], where each POVM is supposed to be linearly independent and informationally complete (IC), we observe that SQPT now is equivalent to deciding a unknown state with a set of product IC-POVM in the d2d^2d2-dimensional Hilbert space. Following the general linear state tomography theory, we show that the product symmetric IC-POVM should minimize the mean-square Hilbert-Schmidt distance between the estimator and the true states. So, an optimal SQPT can be realized by preparing both the inputs and the measurements as the symmetric IC-POVM.
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