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 $\mathbbm{c}\in\mathbb{Z}_2^{D-1}$ (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.

Localization measures of parity adapted U(D) -spin coherent states applied to the phase space analysis of the D -level Lipkin-Meshkov-Glick model

“…(C16). We shall also highlight that 2 z 0 + c L 0 is also the rank of the M-particle reduced density matrix of a c-DCAT, as it is shown in [55]. The connection of the two concepts is subject to further investigation.…”

“…The photon-added CSs has also been extended to SU(2) [53] and SU(1,1) [54]. As these states have only been studied for the Heisenberg-Weyl group, and for SU(2) [53] and SU(1,1) [54], the generalization to SU(D) presents a novel research topic [55].…”

“…The z i → 0 limits in the general c and D cases of a c-DCAT are not straightforward to compute analytically (see [55]), thus, Appendix A is devoted to show in detail these calculations. However, it is necessary to introduce the following limit and notation to progress in our discussion.…”

Study of entanglement in symmetric multi-quDits systems through information diagrams

“…This work is limited in scope but in [6] the ideas seen here are further developed, generalizing the decomposition of the RDM of DCATs into their eigenvectors using Group theoretical methods and more sophisticated interpretations.…”

“…Using the explicit expression of DCATs in terms of the Fock basis [1], one can easily compute the RDM taking partial trace. This RDM can be diagonalized and then its entropy computed [6].…”