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Genuine multipartite Rains entanglement

Hailey S. Murray, Sagnik Bhattacharya, M. Cerezo, Liuke Lyu, Mark M. Wilde

TL;DR

The paper defines the genuine multipartite Rains entanglement (GMRE), a computable measure of genuine multipartite entanglement that extends the bipartite Rains relative entropy and is monotone under selective PPT operations. It expresses GMRE as a convex optimization with a semidefinite-programming (SDP) formulation, enabling efficient numerical computation, and proves that GMRE upper-bounds one-shot GHZ-distillable entanglement and probabilistic approximate distillable entanglement (GHZ-PADME). A general D-Rains framework is introduced, including a sandwiched Rényi version $\widetilde{R}_\alpha$, with asymptotic bounds connecting GMRE to GHZ-distillable entanglement in the regularized limit. Additional results establish nonnegativity, zero value for biseparable states, and lower bounds based on GHZ fidelity, along with an explicit SDP constraint that facilitates practical computation. The work lays a foundation for applying GMRE to condensed matter and cryptographic contexts and suggests avenues for quantum-computer implementations and symmetry-driven simplifications.

Abstract

We introduce the genuine multipartite Rains entanglement (GMRE) as a measure of genuine multipartite entanglement that can be computed using semi-definite programming. Similar to the Rains relative entropy (its bipartite counterpart), the GMRE is monotone under selective quantum operations that completely preserve the positivity of the partial transpose, implying that it is a multipartite entanglement monotone. As a consequence, we show that the GMRE bounds both the one-shot standard and probabilistic approximate GHZ-distillable entanglement from above. We also develop a generalization of this quantity that incorporates other entropies, including quantum Renyi relative entropies.

Genuine multipartite Rains entanglement

TL;DR

The paper defines the genuine multipartite Rains entanglement (GMRE), a computable measure of genuine multipartite entanglement that extends the bipartite Rains relative entropy and is monotone under selective PPT operations. It expresses GMRE as a convex optimization with a semidefinite-programming (SDP) formulation, enabling efficient numerical computation, and proves that GMRE upper-bounds one-shot GHZ-distillable entanglement and probabilistic approximate distillable entanglement (GHZ-PADME). A general D-Rains framework is introduced, including a sandwiched Rényi version , with asymptotic bounds connecting GMRE to GHZ-distillable entanglement in the regularized limit. Additional results establish nonnegativity, zero value for biseparable states, and lower bounds based on GHZ fidelity, along with an explicit SDP constraint that facilitates practical computation. The work lays a foundation for applying GMRE to condensed matter and cryptographic contexts and suggests avenues for quantum-computer implementations and symmetry-driven simplifications.

Abstract

We introduce the genuine multipartite Rains entanglement (GMRE) as a measure of genuine multipartite entanglement that can be computed using semi-definite programming. Similar to the Rains relative entropy (its bipartite counterpart), the GMRE is monotone under selective quantum operations that completely preserve the positivity of the partial transpose, implying that it is a multipartite entanglement monotone. As a consequence, we show that the GMRE bounds both the one-shot standard and probabilistic approximate GHZ-distillable entanglement from above. We also develop a generalization of this quantity that incorporates other entropies, including quantum Renyi relative entropies.
Paper Structure (20 sections, 13 theorems, 83 equations, 1 figure)

This paper contains 20 sections, 13 theorems, 83 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. GMRE definition
  4. Selective PPT monotonicity of GMRE
  5. Other Properties of GMRE
  6. Lower bound on the GMRE and sandwiched Rényi--Rains entanglement
  7. Upper bound on GHZ-distillable entanglement
  8. Semi-definite constraint for GMRE
  9. Conclusion
  10. Proof of Theorem \ref{['thm:general_convex_opt_theorem']}
  11. Comparison of GMRE and genuine multipartite log-negativity in the transverse field Ising model
  12. GMRE as the limit of the genuine multipartite sandwiched Rényi--Rains entanglement
  13. Proof of Proposition \ref{['prop:multi_rains_nonnegative']}
  14. Monotonicity of binary relative entropy
  15. Proof of Lemma \ref{['lem:multi_rains_ghz']}
  16. ...and 5 more sections

Key Result

Theorem 1

The following equality holds for every multipartite $\boldsymbol{D}$-Rains entanglement measure and $\rho\in\mathcal{S}(\mathcal{H}_{k})$:

Figures (1)

  • Figure 1: The GMRE and genuine multipartite log-negativity for tripartite reduced density matrices in the 1D transverse field Ising model are plotted versus the magnetic field $h$. The reduced density matrices are obtained exactly via the Jordan-Wigner transformation Fagotti2013RDM, and the code for calculating the entanglement uses cvxcvxquadqetlab. We used the relationship $E_N=\log_2(2N+1)$ to calculate the log-negativity $E_N$, where hofmann2014 was used for calculating the genuine multipartite negativity $N$.

Theorems & Definitions (36)

  • Theorem 1
  • proof
  • Remark 2
  • Definition 3
  • Theorem 4
  • proof
  • Remark 5
  • Proposition 6
  • proof
  • Lemma 7
  • ...and 26 more