Renormalized quantum tomography

Abstract

The core of quantum tomography is the possibility of writing a generally unbounded complex operator in form of an expansion over operators that are generally nonlinear functions of a generally continuous set of spectral densities--the so-called "quorum" of observables. The expansion is generally non unique, the non unicity allowing further optimization for given criteria. The mathematical problem of tomography is thus the classification of all such operator expansions for given (suitably closed) linear spaces of unbounded operators--e.g. Banach spaces of operators with an appropriate norm. Such problem is a difficult one, and remains still open, involving the theory of general basis in Banach spaces, a still unfinished chapter of analysis. In this paper we present new nontrivial operator expansions for the quorum of quadratures of the harmonic oscillator, and introduce a first very preliminary general framework to generate new expansions based on the Kolmogorov construction. The material presented in this paper is intended to be helpful for the solution of the general problem of quantum tomography in infinite dimensions, which corresponds to provide a coherent mathematical framework for operator expansions over functions of a continuous set of spectral densities.Comment: 23 pages, no figure

Quantum simulation

“…In general, for characterizing a quantum state, quantum state tomography (QST) (e.g., (D'Ariano et al, 2003)) can be used. However, QST requires resources that grow exponentially with the size of the system, making it inefficient for large quantum systems.…”

Qubit noise deconvolution

“…While this is already a known result in the literature [10,36,39,48,49], it is presented without an explicit constructive derivation, as given here.…”

Optimal Quantum Tomography

“…As we will see in the next sections, for finite dimensional systems the theory of generalized inverses is sufficient for classifying all possible expansions and consequently deriving the optimal coefficients f [X] for a fixed POVM {P l }, [7], [84]. On the other hand, the full classification of inverses Γ and consequent optimization is a still unsolved problem for infinite dimensional systems, for which alternative approaches are useful [83].…”

“…In this subsection we will review the relevant results in the theory of frames on Hilbert spaces, which is useful for dealing with POVMs on infinite dimensional systems [83] where a classification of all inverses Γ is still lacking. The method for evaluating possible inverses provided in Refs.…”