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

TL;DR

This paper provides a detailed Schmidt decomposition for symmetric multi-quDit Schr"odinger cat states with parity, characterizing their entanglement and robustness under quDit loss, generalizing known photon loss results.

Contribution

It introduces a Schmidt decomposition for parity-adapted coherent states in symmetric multi-quDit systems, extending photon loss results to quDits and analyzing their entanglement properties.

Findings

Eigenvalues of reduced density matrices are fully characterized.

A theorem for decomposing multi-quDit Schr"odinger cat states by parity is proven.

States show robustness under quDit loss, with asymptotic properties analyzed.

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 $\mathbb{C}\in\mathbb{Z}_2^{D-1}$ (multicomponent Schr\"odinger 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 $\mathbb{C}$ 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.