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On genuine multipartite entanglement signals

Abhijit Gadde

Abstract

We give a general construction of genuinely multipartite entanglement signals from families of lower-partite symmetric local-unitary invariants satisfying a natural compatibility condition. Möbius inversion on the partition lattice plays a key role in this construction. We show that many examples of multipartite entanglement signals considered in the literature fit naturally into this framework. We also explain how the genuinely multipartite signal can be extracted from a general, not necessarily symmetric, multi-invariant.

On genuine multipartite entanglement signals

Abstract

We give a general construction of genuinely multipartite entanglement signals from families of lower-partite symmetric local-unitary invariants satisfying a natural compatibility condition. Möbius inversion on the partition lattice plays a key role in this construction. We show that many examples of multipartite entanglement signals considered in the literature fit naturally into this framework. We also explain how the genuinely multipartite signal can be extracted from a general, not necessarily symmetric, multi-invariant.
Paper Structure (14 sections, 9 theorems, 101 equations)

This paper contains 14 sections, 9 theorems, 101 equations.

Table of Contents

  1. States and separability
  2. General construction
  3. Symmetric signal and pre-signal
  4. Examples
  5. Rényi entropy
  6. Sum of Rényi entropy and reflected entropy
  7. Rényi multi-entropy
  8. Multipartite entanglement of purification
  9. Non-symmetric Signals from multi-invariants
  10. Conclusion and outlook
  11. Proof of Theorem 1
  12. Proof of Lemma 2
  13. Useful lemmas
  14. Vanishing signals

Key Result

Lemma 1

If an LU-invariant function is additive and vanishes for $\pi$-separable state $|\psi\rangle$ of type $(1,q-1)$ then it is a signal.

Theorems & Definitions (28)

  • Definition 1
  • Definition 2: Signal
  • Definition 3: Pre-signal
  • Remark 1
  • Lemma 1
  • Remark 2
  • Theorem 1
  • Definition 4
  • Definition 5
  • Definition 6
  • ...and 18 more