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Certification of energy-restricted entanglement depth with simple measurements

Carles Roch I Carceller

TL;DR

The paper addresses certifying the depth of multipartite entanglement under energy constraints by framing a distributed state discrimination game among distant, uncharacterized parties. It shows that entangled state ensembles outperform all separable ones under a fixed energy bound $\omega$, while a single fixed local measurement suffices for optimal discrimination across all entanglement structures. The authors derive exact optimal success probabilities for bipartite and multipartite scenarios, establish a hierarchy of separability classes, and demonstrate exponential scaling of the entanglement advantage with the number of parties, along with robustness to white noise. This semi-device-independent approach provides a practical, scalable method for entanglement-depth certification in quantum networks and suggests pathways for experimental realization in photonic and solid-state platforms.

Abstract

The certification of entanglement in multipartite scenarios is crucial for the advancement of quantum technologies, particularly for the realization of large-scale quantum networks. Here, we introduce a method to certify entanglement in ensembles of quantum states with limited energy based on a state discrimination game played by multiple distant and uncharacterized parties. This game exhibits a hierarchy in its optimal success probability, which depends strictly on the entanglement structure of the underlying ensemble. On the other hand, the game can be optimally won using a single, fixed measurement setting shared by all parties, regardless of the specific entanglement structure. We further demonstrate that both the performance and noise robustness of our method improve in the multipartite regime, scaling exponentially with the number of parties. Consequently, our approach enables the exclusion of entire separability classes, thereby certifying the depth of multipartite entanglement.

Certification of energy-restricted entanglement depth with simple measurements

TL;DR

The paper addresses certifying the depth of multipartite entanglement under energy constraints by framing a distributed state discrimination game among distant, uncharacterized parties. It shows that entangled state ensembles outperform all separable ones under a fixed energy bound , while a single fixed local measurement suffices for optimal discrimination across all entanglement structures. The authors derive exact optimal success probabilities for bipartite and multipartite scenarios, establish a hierarchy of separability classes, and demonstrate exponential scaling of the entanglement advantage with the number of parties, along with robustness to white noise. This semi-device-independent approach provides a practical, scalable method for entanglement-depth certification in quantum networks and suggests pathways for experimental realization in photonic and solid-state platforms.

Abstract

The certification of entanglement in multipartite scenarios is crucial for the advancement of quantum technologies, particularly for the realization of large-scale quantum networks. Here, we introduce a method to certify entanglement in ensembles of quantum states with limited energy based on a state discrimination game played by multiple distant and uncharacterized parties. This game exhibits a hierarchy in its optimal success probability, which depends strictly on the entanglement structure of the underlying ensemble. On the other hand, the game can be optimally won using a single, fixed measurement setting shared by all parties, regardless of the specific entanglement structure. We further demonstrate that both the performance and noise robustness of our method improve in the multipartite regime, scaling exponentially with the number of parties. Consequently, our approach enables the exclusion of entire separability classes, thereby certifying the depth of multipartite entanglement.

Paper Structure

This paper contains 10 sections, 1 theorem, 30 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Two-distributed quantum state discrimination
  3. Dealing with shared randomness
  4. Pure state preparations without shared randomness
  5. Arbitrary state preparations with shared randomness
  6. Entanglement-based strategy
  7. Separable strategy
  8. Multipartite distributed state discrimination
  9. Certification of entanglement depth
  10. Conclusion

Key Result

Corollary 1

Let $\rho_{\vec{x}}$ be entangled across groups of the exact same number of parties, i.e. $k_i=k_j:=k$, $\forall i,j$. The maximum probability for each party to successfully discriminate their corresponding symbol $x_i\in\{0,1\}$ is that one from Eq. (eq:ps_gme) with $\tilde{\omega}_j=1-\left(1-\ome

Figures (3)

  • Figure 1: Two-distributed quantum state discrimination. One device prepares a bipartite quantum state with energy bounded by $\omega$ according to the pair of inputs $\{x_0,x_1\}$. This state is distributed among two parties, whose goal is to correctly discriminate their corresponding message $x_i$.
  • Figure 2: Entanglement advantage in distributed quantum state discrimination. Success probability for all $n$ parties to simultaneously discriminate their corresponding bits in a distributed state discrimination scenario.
  • Figure 3: Entanglement depth certification. The quantum state $\rho_{\vec{x}}$ is distributed among $4$ parties: $A$, $B$, $C$ and $D$. The success probability of each party correctly discriminating their symbol depends on the depth of the entanglement. Above the $A|BCD$ bound, the state is GME.

Theorems & Definitions (1)

  • Corollary 1