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Characterizing high-dimensional multipartite entanglement beyond Greenberger-Horne-Zeilinger fidelities

Shuheng Liu, Qiongyi He, Marcus Huber, Giuseppe Vitagliano

TL;DR

The paper tackles certifying high-dimensional multipartite entanglement by introducing a covariance-matrix–based witness that bounds the entanglement-dimensionality vector $\boldsymbol{\mathcal{SN}}^\downarrow$ across all bipartitions. The method constructs cross-covariances $X_\varrho^{(\alpha)}$ and a bound $f_\alpha(\varrho)$ for each bipartition, and then solves a linear program to determine a feasible set of SN-vectors, yielding a bound on the largest component $\mathcal{SN}_1^{\downarrow}(\varrho)$. This framework generalizes and strengthens GHZ-fidelity witnesses, with a derived special case that recovers $F_{\max}(\boldsymbol{v},\Psi^d_{GHZ})\leq v_{\mathcal{N}}/d$, corresponding to 1-uniform targets. Empirically, the method outperforms existing criteria on random $3\times3\times3$ states and GHZ-noise mixtures, and combining it with GHZ-based bounds yields even higher detection rates. The results offer a practical, scalable route to certify genuine high-dimensional multipartite entanglement in networks and experiments, with potential extensions to incorporate additional witnesses and nonlinear criteria.

Abstract

Characterizing entanglement of systems composed of multiple particles is a very complex problem that is attracting increasing attention across different disciplines related to quantum physics. The task becomes even more complex when the particles have many accessible levels, i.e., they are of high dimension, which leads to a potentially high-dimensional multipartite entangled state. These are important resources for an ever-increasing number of tasks, especially when a network of parties needs to share highly entangled states, e.g., for communicating more efficiently and securely. For these applications, as well as for purely theoretical arguments, it is important to be able to certify both the high-dimensional and the genuine multipartite nature of entangled states, possibly based on simple measurements. Here we derive a novel method that achieves this and improves over typical entanglement witnesses like the fidelity with respect to states of a Greenberger-Horne-Zeilinger (GHZ) form, without needing more complex measurements. We test our condition on paradigmatic classes of high-dimensional multipartite entangled states like imperfect GHZ states with random noise, as well as on purely randomly chosen ones and find that, in comparison with other available criteria our method provides a significant advantage and is often also simpler to evaluate.

Characterizing high-dimensional multipartite entanglement beyond Greenberger-Horne-Zeilinger fidelities

TL;DR

The paper tackles certifying high-dimensional multipartite entanglement by introducing a covariance-matrix–based witness that bounds the entanglement-dimensionality vector across all bipartitions. The method constructs cross-covariances and a bound for each bipartition, and then solves a linear program to determine a feasible set of SN-vectors, yielding a bound on the largest component . This framework generalizes and strengthens GHZ-fidelity witnesses, with a derived special case that recovers , corresponding to 1-uniform targets. Empirically, the method outperforms existing criteria on random states and GHZ-noise mixtures, and combining it with GHZ-based bounds yields even higher detection rates. The results offer a practical, scalable route to certify genuine high-dimensional multipartite entanglement in networks and experiments, with potential extensions to incorporate additional witnesses and nonlinear criteria.

Abstract

Characterizing entanglement of systems composed of multiple particles is a very complex problem that is attracting increasing attention across different disciplines related to quantum physics. The task becomes even more complex when the particles have many accessible levels, i.e., they are of high dimension, which leads to a potentially high-dimensional multipartite entangled state. These are important resources for an ever-increasing number of tasks, especially when a network of parties needs to share highly entangled states, e.g., for communicating more efficiently and securely. For these applications, as well as for purely theoretical arguments, it is important to be able to certify both the high-dimensional and the genuine multipartite nature of entangled states, possibly based on simple measurements. Here we derive a novel method that achieves this and improves over typical entanglement witnesses like the fidelity with respect to states of a Greenberger-Horne-Zeilinger (GHZ) form, without needing more complex measurements. We test our condition on paradigmatic classes of high-dimensional multipartite entangled states like imperfect GHZ states with random noise, as well as on purely randomly chosen ones and find that, in comparison with other available criteria our method provides a significant advantage and is often also simpler to evaluate.
Paper Structure (19 sections, 97 equations, 2 figures, 2 tables)

This paper contains 19 sections, 97 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Methods
  3. Definition of Schmidt-number vector
  4. Known witnesses of entanglement-dimensionality vector
  5. Results
  6. New method to characterize the Schmidt-number vector
  7. Recovering GHZ-like fidelity witnesses
  8. Comparison with the other methods
  9. Conclusions and Outlook
  10. Acknowledgments
  11. Appendix
  12. Further details on known entanglement-dimensionality-vector witnesses
  13. Fidelity-based witnesses
  14. Entanglement-dimensionality criterion from the linear entropy vector
  15. Detection of the single-particle rank vector from the correlation tensor norm
  16. ...and 4 more sections

Figures (2)

  • Figure 1: (a) The structure of all possible Schmidt number vectors in a $4\times 3\times 2$ state space. Ellipses are only for illustration as the sets are not convex. Moreover, the extremal states have a complicated characterization and the structure is only partially nested. This is demonstrated more clearly in (b). (b) Taking vector $(322)$ as an example, it lies in the region covered by both $\boldsymbol{v}\overset{{\rm el}}{\le} (422)$ and $\boldsymbol{v}\overset{{\rm el}}{\le} (332)$, but outside $\boldsymbol{v}\overset{{\rm el}}{\le} (222)$. The regions (I) and (II) require careful distinction. Region (I) represents $(322)$ states that can be formed by mixtures of pure states with $\boldsymbol{v} \overset{{\rm el}}{\le} (322)$, while region (II) represents $(322)$ mixed states that cannot. For example, there can be states that only have decompositions with pure states with $\boldsymbol{v}= (422)$ or $\boldsymbol{v}= (332)$.
  • Figure 2: Comparison of fidelity-witness based criterion and \ref{['eq:obs2sys']} over random states of the form \ref{['eq:varrho432']}, visualized with a 2-dimensional clustering made using t-SNE. (a) Blue points represent states detected with higher white-noise $(1-p)$ by our criterion \ref{['eq:obs2sys']}, while red points correspond to states that are detected with a higher tolerance by the fidelity witness with respect to $\ket{\psi_{432}}(\boldsymbol{c})$. (b) Blue points represent states for which $C_{\mathrm{GM}}>0.8$, while orange points correspond to those with $C_{\mathrm{GM}}\leq 0.8$.