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arXiv:quant-ph/0009104 · Physical Review A

Maximum-likelihood reconstruction of completely positive maps

Massimiliano Federico Sacchi

Published Apr 13, 2001• 64 citations
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Abstract

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 $\mathcal{H}$ to $\mathcal{K}$ 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.

DOI
10.1103/physreva.63.054104

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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 K\ensuremath⨂H.\mathcal{K}\ensuremath{\bigotimes}\mathcal{H}.K\ensuremath⨂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.
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 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.
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.
For a given set of input-output pairs of quantum states or observables, we ask the question whether there exists a physically implementable transformation that maps each of the inputs to the corresponding output. The physical maps on quantum states are trace-preserving completely positive maps, but we also consider variants of these requirements. We generalize the definition of complete positivity to linear maps defined on arbitrary subspaces, then formulate this notion as a semidefinite program, and relate it by duality to approximative extensions of this map. This gives a characterization of the maps which can be approximated arbitrarily well as the restriction of a map that is completely positive on the whole algebra, also yielding the familiar extension theorems on operator spaces. For quantum channel extensions and extensions by probabilistic operations we obtain semidefinite characterizations, and we also elucidate the special case of Abelian inputs or outputs. Finally, revisiting a theorem by Alberti and Uhlmann, we provide simpler and more widely applicable conditions for certain extension problems on qubits, and by using a semidefinite programming formulation we exhibit counterexamples to seemingly reasonable but false generalizations of the Alberti-Uhlmann theorem.
We investigate the evolution of open quantum systems in the presence of initial correlations with an environment. Here the standard formalism of describing evolution by completely positive trace preserving (CPTP) quantum operations can fail and non-completely positive (non-CP) maps may be observed. A new classification of correlations between a system and environment using quantum discord is explored. However, we find quantum discord is not a symmetric quantity between exchange of systems and this leads to ambiguity in classifications - states which are both quantum and classically correlated depending on the order of the two systems. State preparation in quantum process tomography is investigated with regard to non-CP maps. In SQPT the preparation procedure can influence the complete-positivity of the reconstructed quantum operation if our system is initially correlated with an environment. We examine a recently proposed preparation procedures using projective measurements, and propose our own protocol that uses a single measurement followed by unitary rotations. The former can give rise to non-CP evolution while the later will always give rise to a CP map. State preparation in AAPT was found always to give rise to CP evolution. We examine the effect of statistical noise in process tomography and find it can result in the identification of a non-CP when the evolution should be CP. The variance of the distribution for reconstructed processes is found to be inversely proportional to the number of copies of a state used to perform tomography. Finally, we detail an experiment using currently available linear optics QC devices to demonstrate non-CP maps arising in SQPT.