Schmidt decomposition of parity adapted coherent states for symmetric multi-quDits

Abstract

In this paper we study the entanglement in symmetric N -quDit systems. In particular we use generalizations to U (D) of spin U (2) coherent states and their projections on definite parity c ∈ Z D−1 2 (multicomponent Schrödinger cat) states and we analyse their reduced density matrices when tracing out M < N quDits. The eigenvalues (or Schmidt coefficients) of these reduced density matrices are completely characterized, allowing to proof a theorem for the decomposition of a N -quDit Schrödinger cat state with a given parity c into a sum over all possible parities of tensor products of Schrödinger cat states of N − M and M particles. Diverse asymptotic properties of the Schmidt eigenvalues are studied and, in particular, for the (rescaled) double thermodynamic limit (N, M → ∞, M/N fixed), we reproduce and generalize to quDits known results for photon loss of parity adapted coherent states of the harmonic oscillator, thus providing an unified Schmidt decomposition for both multi-quDits and (multi-mode) photons. These results allow to determine the entanglement properties of these states and also their decoherence properties under quDit loss, where we demonstrate the robustness of these states.