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.
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 $\mathbbm{c}\in\mathbb{Z}_2^{D-1}$ (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 Schr"odinger cat state with a given parity $\mathbbm{c}$ into a sum over all possible parities of tensor products of Schr"odinger 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 ($N,M\rightarrow\infty,\,M/N$ 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.
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