From particle counting to Gaussian tomography (Q2790328)

\textit{Tomography} is a word borrowed from the world of medical imaging, and usually associated to techniques that produce sliced pictures of the human body, but is also adopted to indicate similar methods in archaeology, geophysics, oceanography, material science and so on. In all these instances the aim is to use some kind of penetrating wave in order to get a reliable image of the interior of a system without really opening -- or utterly destroying -- it. \textit{Quantum state tomography}, in analogy, is the process by which a quantum state is reconstructed by using measurements on an ensemble of identical quantum states. Because indeed measurement of a quantum state (in general) changes the state being measured, getting a complete picture of that state requires measurements on many state copies. This means ultimately that the quantum tomography is applied to the source of systems, to determine what the quantum state is as the output of that source. In other words, unlike a measurement on a single system, which determines the system's current state \textit{after} the measurement (because the measurement alters the quantum state), quantum tomography works to determine the state(s) \textit{prior} to the measurements.NEWLINENEWLINENEWLINEFrom this standpoint, adding to an already rich panoply of tomographic procedures, the present paper proposes a new method which, by means of a finite number of (counting) measurements, is able to determine all the mean and covariance parameters of an \(n\)-mode Gaussian state in the Fock space \(\Gamma(\mathbb{C}^n)\). This means that we take for granted the Gaussian nature of the state from the beginning, and hence that we need just to find out its \(n(2n+3)\) parameters to completely know it. The authors are indeed able to construct ``\(n(2n + 3)\) number observables which have the discrete spectrum \(\{0,1,2, \dots\}\) and the property that all the means and covariances of the unknown Gaussian state can be easily determined from their expectation values.'' The needed observables are then produced by combining the ``observables \(a_j^\dagger a_j\), the number of particles in the \(j\)-th mode for each \(j = 1,2,\dots, n\) and also their unitary equivalents in different frames which are obtained by Weyl (displacement) operators, as well as unitary operators which implement the symplectic linear transformations in the position and momentum observables obeying the canonical commutation relations.''NEWLINENEWLINENEWLINEThe \(n\)-mode Gaussian state of interest is carefully defined in the Section \(2\) by means of the Fourier transform \(\hat{\rho}\) of its density matrix \(\rho_g(\mathbf{l},\mathbf{m},S)\): here \(\mathbf{l},\mathbf{m}\) (vectors in \(\mathbb{R}^n\)) and \(S\) (\(2n\times2n\) symmetric real matrix) -- whose independent elements amount to \(n(2n+3)\) free parameters -- respectively are the expectations of position and momentum observables and the covariance matrix in the state \(\rho_g\). In the subsequent Section \(3\) the authors introduce the usual \(j\)-th mode and total number operators NEWLINE\[NEWLINE N_j=a_j^\dagger a_j=\frac{1}{2}\,(p_j^2+q_j^2-1),\qquad1\leq j\leq n,\qquad N=\sum_{j=1}^nN_j NEWLINE\]NEWLINE and they are able to explicitly find the probability generating function \(G_N(x)\) of the total number \(N\) in the state \(\rho_g(\mathbf{l},\mathbf{m},S)\) (see Theorem 3.1), along with its distribution, expectation and variance (see Corollary 3.1): all that in terms of \(\mathbf{l}, \mathbf{m}\) and \(S\). Now these results ``indicate the possibility of estimating the mean and covariance parameters of a Gaussian state by measuring the number operator under different displacements''. Remark that, when \(\rho_g\) is a pure Gaussian state, it turns out that \(N\) has an infinitely divisible distribution. However ``whether the distribution of the total number operator \(N\) in every Gaussian state is infinitely divisible and hence of a mixed Poisson type'' is still an open question.NEWLINENEWLINENEWLINEThe subsequent Section \(4\) contains a detailed analysis of the procedures needed to express the vectors \(\mathbf{l},\mathbf{ m}\), and the covariance matrix \(S\) of a Gaussian state with \(n\) modes ``in terms of the expectation values of conjugates of the total number operator by a few elementary gates in the Hilbert space \(\Gamma(\mathbb{C}^n)\)'' (namely the suitable \(1\)- and \(2\)-modes unitary gates). Here indeed the authors put on display first the counting observables needed to find \(\mathbf{l}, \mathbf{m}\) and \(\mathrm{Tr}\, S\); then that useful to get the diagonal elements of \(S\); and finally the way to obtain also the off-diagonal elements of \(S\). In this case they use exactly \(n(2n+3)\) measurements to determine the \(n(2n+3)\) parameters of the Gaussian state. Having completed this task, in the final Section \(5\) they also briefly turn their attention to the tomography of an \(n\)-mode Gaussian channel. In this endeavor they find however that their approach ``requires \(6n^2 + 3n - 1\) measurements for getting the \(6n^2 + n\) parameters'', and they are left again with the open question whether we can determine the required parameters with less measurements.