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Optimal Approximation of Nonphysical Maps via Maximum Likelihood Estimation

Vladimír Bužek, Mário Ziman, Martin Plesch

Published Mar 16, 2010• 0 citations
Publisher

Abstract

Any quantum dynamics, i.e., the process that is described by a completely positive (CP) map of a quantum-mechanical system, can be probed in two different ways. Either we use a single entangled state of a bipartite system, or we use a collection of linearly independent single-particle test states (forming a basis of the vector space of all Hermitian operators). Given the fragility of entangled states, in this chapter we will focus our attention on the process reconstruction using only single-particle states.

DOI
10.1117/3.793309.ch25

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Any quantum dynamics, i.e., the process that is described by a completely positive (CP) map of a quantum-mechanical system, can be probed in two different ways. Either we use a single entangled state of a bipartite system, or we use a collection of linearly independent single-particle test states (forming a basis of the vector space of all Hermitian operators). Given the fragility of entangled states, in this chapter we will focus our attention on the process reconstruction using only single-particle states.
We show that the method of maximum likelihood (MML) provides us with an efficient scheme for the reconstruction of quantum channels from incomplete measurement data. By construction this scheme always results in estimations of channels that are completely positive. Using this property we use the MML for a derivation of physical approximations of unphysical operations. In particular, we analyze the optimal approximation of the universal NOT gate as well as the physical approximation of a quantum nonlinear polarization rotation.
When standard methods of process (black-box) estimation are applied straightforwardly then it may happen that some sets of experimental data result in unphysical estimations of the corresponding channels (maps) describing the process. To prevent this problem, one can use the method of maximum likelihood (MML), which provides an efficient scheme for reconstruction of quantum channels. This scheme always results in estimations of channels that are fully physical, e.g. the corresponding maps are linear, positive and completely positive. To show this property, we use the MML for a derivation of physical approximations of truly unphysical operations. In particular, we analyze physical approximations of the universal-NOT gate, the quantum nonlinear polarization rotation and the map ρ → ρ2. Given the result of MML, we examine retrospectively the quality of the experiment. Depending on the resulting value of the MML functional we can determine (physical) consistency of the input data.
We present a method for the determination of the completely positive (CP) map describing a physical device based on random preparation of the input states, random measurements at the output, and the maximum-likelihood principle. In the numerical implementation the constraint of completely positivity can be imposed by exploiting the isomorphism between linear transformations from Hilbert spaces H\mathcal{H}H to K\mathcal{K}K and linear operators in $\mathcal{K}\ensuremath{\bigotimes}\mathcal{H}.$ The effectiveness of the method is shown on the basis of some examples of reconstruction of CP maps related to quantum communication channels for qubits.