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Multipartite entanglement classes of a multiport beam-splitter

F. E. S. Steinhoff

TL;DR

The paper analyzes how a multiport beam-splitter (MBS) generates multipartite entanglement across spatial modes and classifies the resulting states under SLOCC. It treats three input paradigms—finite superpositions of number states, finite superpositions of coherent states, and hybrids—deriving representative entangled states and establishing inequivalent SLOCC classes for each scenario: a photon-number-based hierarchy, a nonclassicality-rank hierarchy, and a combined hybrid structure. Although all three families feature genuine multipartite entanglement, their local-rank patterns and adjoint-density-matrix-range criteria place them in distinct SLOCC classes, with explicit representatives such as $| Psi_N\rangle$, $|GHZ_{(r)}\rangle$, and hybrid-state constructions. Practically, the results suggest feasible routes to engineer target entanglement in optical networks using truncated or cat-state inputs and point to future work on mixed-state classifications and SLOCC witnesses.

Abstract

The states generated by a multiport beam-splitter usually display genuine multipartite entanglement between the many spatial modes. Here we investigate the different classes of multipartite entangled states that arise in this practical situation, working within the paradigm of Stochastic Local Operations with Classical Communication. We highlight three scenarios, one where the multipartite entanglement classes follow a total number hierarchy, another where the various classes follow a nonclassicality degree hierarchy and a third one that is a combination of the previous two. Moreover, the multipartite entanglement of higher-dimensional versions of Dicke states relate naturally to our results.

Multipartite entanglement classes of a multiport beam-splitter

TL;DR

The paper analyzes how a multiport beam-splitter (MBS) generates multipartite entanglement across spatial modes and classifies the resulting states under SLOCC. It treats three input paradigms—finite superpositions of number states, finite superpositions of coherent states, and hybrids—deriving representative entangled states and establishing inequivalent SLOCC classes for each scenario: a photon-number-based hierarchy, a nonclassicality-rank hierarchy, and a combined hybrid structure. Although all three families feature genuine multipartite entanglement, their local-rank patterns and adjoint-density-matrix-range criteria place them in distinct SLOCC classes, with explicit representatives such as , , and hybrid-state constructions. Practically, the results suggest feasible routes to engineer target entanglement in optical networks using truncated or cat-state inputs and point to future work on mixed-state classifications and SLOCC witnesses.

Abstract

The states generated by a multiport beam-splitter usually display genuine multipartite entanglement between the many spatial modes. Here we investigate the different classes of multipartite entangled states that arise in this practical situation, working within the paradigm of Stochastic Local Operations with Classical Communication. We highlight three scenarios, one where the multipartite entanglement classes follow a total number hierarchy, another where the various classes follow a nonclassicality degree hierarchy and a third one that is a combination of the previous two. Moreover, the multipartite entanglement of higher-dimensional versions of Dicke states relate naturally to our results.
Paper Structure (14 sections, 1 theorem, 53 equations, 2 figures)

This paper contains 14 sections, 1 theorem, 53 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Multipartite entanglement classes
  4. States generated by a Multiport Beam-Splitter
  5. Results
  6. Superpositions of number states as input
  7. Superpositions of coherent states as input
  8. Hybrid superpositions
  9. Comparison between the three scenarios
  10. Conclusions and perspectives
  11. Acknowledgments
  12. Appendix
  13. Generality of a balanced multiport beam-splitter
  14. Schmidt rank of $|\Phi_N\rangle$

Key Result

Theorem 1

Two pure states of a multipartite system are equivalent under SLOCC if and only if (i) they have the same local rank of each party, and (ii) the ranges of the adjoint reduced density matrices of each party of them are related by certain ILO's.

Figures (2)

  • Figure 1: Structure of SLOCC classes for input states that are finite superpositions of number states.
  • Figure 2: Structure of SLOCC classes for input states that are finite superpositions of coherent states.

Theorems & Definitions (1)

  • Theorem 1