Tomographic reconstruction of quantum states in N spatial dimensions (8 pages)

Markus Grassl

Anders S. Mouritzen, Klaus Mølmer

Most quantum tomographic methods for massive particles using destructive measurement can only be used for motion in one dimension. We show how to infer the quantum state of a nonrelativistic, $N$-dimensional harmonic oscillator system by simple inverse Radon transforms. We assume a time-independent Hamiltonian for which the dynamics along each coordinate always occurs simultaneously with the others and measurements on all coordinates are performed simultaneously. This is unlike light systems where different modes can be delayed and phase shifted with respect to each other. A requirement of the procedure is that the angular frequencies of the $N$ harmonic potentials be incommensurable. We discuss the information available if the requirement of incommensurability is not fulfilled and also under what conditions the state can be reconstructed from finite-time measurements. As further examples, we consider free particles in $N$ dimensions with periodic boundary conditions and particles in an $N$-dimensional box, where we find a similar condition of incommensurability and finite recurrence time.

Augusto J. Roncaglia, Juan Pablo Paz

G. Lima, Fabian A. Torres-Ruiz, Leonardo Neves +3 more

In this paper we study the state determination for composite systems of two spatial qubits. We show, theoretically, that one can use the technique of quantum tomography to reconstruct the density matrixes of these systems. This tomographic reconstruction is based on the free evolution of the spatial qubits and a postelection process.

Tillmann Baumgratz, M. Cramer, Martin Bodo Plenio +1 more

Julio T. Barreiro, Nicholas A. Peters, Paul G. Kwiat

Simultaneous multidimensional entanglement can improve quantum communication protocols, measurements and fundamental studies. Using photons from parametric downconversion, we demonstrate a source of joint polarization and spatial-mode entanglement, and characterize the state by tomographic reconstruction.