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International Journal of Quantum Information

QUANTUM OPERATIONS: A FANO-REPRESENTATION APPROACH

Giuliano Benenti, G. Strini

Published Jan 1, 2011• 0 citations
Publisher

Abstract

The Fano representation is a particularly simple and physically appealing description of quantum operations, in that it directly provides the evolution of the expectation value of polarization measurements. The usefulness of this approach is illustrated in quantum process tomography and in computing the distance between two quantum channels.

DOI
10.1142/s0219749911007083

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The Fano representation is a particularly simple and physically appealing description of quantum operations, in that it directly provides the evolution of the expectation value of polarization measurements. The usefulness of this approach is illustrated in quantum process tomography and in computing the distance between two quantum channels.
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.
The diamond norm measures the distance between two quantum channels. From an operational vewpoint, this norm measures how well we can distinguish between two channels by applying them to input states of arbitrarily large dimensions. In this paper, we show that the diamond norm can be conveniently and in a physically transparent way computed by means of a Monte-Carlo algorithm based on the Fano representation of quantum states and quantum operations. The effectiveness of this algorithm is illustrated for several single-qubit quantum channels.
As optical systems are one of the candidates for implementation of a scalable quantum computer, it is important to develop an adequate method of description of both quantum states of light and operations performed by optical elements. Using the concept of chi-matrix representation of quantum operations and Choi-Jamiolkowski isomorphism we expand Jones calculus to allow description of evolution of mixed polarization states in linear optical systems. The developed method is then used to give a full description of polarization echo effect, which was described in 1 based on an analogy between the effects of polarization optics and spin dynamics. Theoretical predictions are confirmed by reconstructing operations performed by a series of quartz waveplates using quantum process tomography protocols.
We introduce a mathematical framework for the description of measurements of quantum processes. Using this framework, process estimation problems can be treated in a similar way as state estimation problems, only replacing the concept of positive-operator-valued measure (POVM) by the concept of process POVM (PPOVM). In particular, we will show that any measurement of qudit channels can be described by a collection of effects (positive operators) defined on a two-qudit system. However, the effects forming a PPOVM are not normalized in the usual sense. We will demonstrate the use of this formalism in discrimination problems by showing that perfect channel discrimination is equivalent to a specific unambiguous state discrimination.
In quantum-process tomography it is possible to express the experimenter's prior information as a sequence of quantum operations, i.e., trace-preserving completely positive maps. In analogy to de Finetti's concept of exchangeability for probability distributions, we give a definition of exchangeability for sequences of quantum operations. We then state and prove a representation theorem for such exchangeable sequences. The theorem leads to a simple characterization of admissible priors for quantum process tomography and solves to a Bayesian's satisfaction the problem of an unknown quantum operation.
The weak value concept has enabled fundamental studies of quantum measurement and, recently, found potential applications in quantum and classical metrology. However, most weak value experiments reported to date do not require quantum mechanical descriptions, as they only exploit the classical wave nature of the physical systems. In this work, we demonstrate measurement of the sequential weak value of two incompatible observables by making use of two-photon quantum interference so that the results can only be explained quantum physically. We then demonstrate that the sequential weak value measurement can be used to perform direct quantum process tomography of a qubit channel. Our work not only demonstrates the quantum nature of weak values but also presents potential new applications of weak values in analyzing quantum channels and operations.