Quantum tomography meets dynamical systems and bifurcations theory

Abstract

A powerful tool for studying geometrical problems in Hilbert space is developed. In particular, we study the quantum pure state tomography problem in finite dimensions from the point of view of dynamical systems and bifurcations theory. First, we introduce a generalization of the Hellinger metric for probability distributions which allows us to find a geometrical interpretation of the quantum state tomography problem. Thereafter, we prove that every solution to the state tomography problem is an attractive fixed point of the so-called physical imposition operator. Additionally, we demonstrate that multiple states corresponding to the same experimental data are associated to bifurcations of this operator. Such a kind of bifurcations only occurs when informationally incomplete set of observables are considered. Finally, we prove that the physical imposition operator has a non-contractive Lipschitz constant 2 for the Bures metric. This value of the Lipschitz constant manifests the existence of the quantum tomography problem for pure states.Comment: 16 pages, 2 figures. Submitted to Journal of Mathematical Physic

Mutually unbiased triplets from non-affine families of complex Hadamard matrices in dimension 6

“…, d−1. Interestingly, when the eigenvectors bases are MU, every basin of attraction is found to be of the same size, verified numerically in every prime dimension 2 ≤ d ≤ 37 [27], as well as in every simulation reported below for d = 6. This property indicates that the efficiency of the algorithm is maximal when the eigenvector bases of the observables are MU, because the number of randomly chosen seed states needed to find all solutions is minimized.…”

“…Moreover, both triplets occur with nearly equal frequency: we found {I, D(0), H 1 } 48 times while {I, D(0), H 2 } occurred 52 times, an observation which can be explained if one assumes that the basin of attraction of every MU vector has the same size. This apparent symmetry has been noticed so far in each imposition-operator search for MU bases, whatever the dimension d [26,27].…”

“…The iterative method used here allows us to efficiently generate highly accurate approximations to the solutions of the defining set of equations. The desired states are attractive fixed points of the physical imposition operator [26] which has been used previously to find those quantum states known as Pauli partners [27]. The problem has a unique answer if the given probability distributions are informationally complete; otherwise a finite or infinitely many number of solutions may exist.…”

“…It has been proven [27] that every solution of the system of coupled equations (7,8) The problem of constructing MU vectors is a particular case of the quantum state reconstruction problem just described. Let A and B be two observables with a pair of MU eigenbases B A and B B .…”

“…In the present context, we need to consider more than one observable and the corresponding probability distributions. Therefore, we introduce the Hellinger metric for m observables, the so-called distributional metric [27],…”