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Communication scenario enables robust self-testing of n-party Greenberger-Horne-Zeilinger basis measurements

Barnik Bhaumik, Sagnik Ray, Debashis Saha

TL;DR

This paper introduces a semi-device-independent framework to self-test entangled basis measurements, notably the $n$-party GHZ basis, using a multi-sender single-receiver communication task with a bounded system dimension. It proves that the maximum quantum value of a carefully constructed success metric self-tests the GHZ measurement and provides robust fidelity bounds via a sum-of-squares decomposition; it also shows that extremal correlations do not universally guarantee self-testing by presenting a two-sender example. Extending the approach, the authors adapt the protocol to robustly certify a three-outcome partial Bell basis measurement, demonstrating feasibility for linear-optical implementations and deriving practical error-tolerance bounds. Collectively, the results enable device-independent-like certification of complex entangled measurements in networks without requiring shared entanglement, with implications for scalable quantum networks and optical quantum information tasks.

Abstract

Entangled basis measurements play a crucial role in distributing quantum entanglement between parties across a quantum network. In this work, we adopt a semi-device-independent approach that enables the self-testing of n-qubit Greenberger-Horne-Zeilinger (GHZ) basis measurements without requiring shared entanglement between distant parties. Our method relies solely on input-output statistics from a communication scenario involving n spatially separated senders, each receiving two bits of input, and a single receiver with no input. We analyze the robustness of the proposed self-testing protocol. Additionally, we introduce a protocol for robust self-testing of the three-outcome partial Bell basis measurement that is easily implementable in an optical setup.

Communication scenario enables robust self-testing of n-party Greenberger-Horne-Zeilinger basis measurements

TL;DR

This paper introduces a semi-device-independent framework to self-test entangled basis measurements, notably the -party GHZ basis, using a multi-sender single-receiver communication task with a bounded system dimension. It proves that the maximum quantum value of a carefully constructed success metric self-tests the GHZ measurement and provides robust fidelity bounds via a sum-of-squares decomposition; it also shows that extremal correlations do not universally guarantee self-testing by presenting a two-sender example. Extending the approach, the authors adapt the protocol to robustly certify a three-outcome partial Bell basis measurement, demonstrating feasibility for linear-optical implementations and deriving practical error-tolerance bounds. Collectively, the results enable device-independent-like certification of complex entangled measurements in networks without requiring shared entanglement, with implications for scalable quantum networks and optical quantum information tasks.

Abstract

Entangled basis measurements play a crucial role in distributing quantum entanglement between parties across a quantum network. In this work, we adopt a semi-device-independent approach that enables the self-testing of n-qubit Greenberger-Horne-Zeilinger (GHZ) basis measurements without requiring shared entanglement between distant parties. Our method relies solely on input-output statistics from a communication scenario involving n spatially separated senders, each receiving two bits of input, and a single receiver with no input. We analyze the robustness of the proposed self-testing protocol. Additionally, we introduce a protocol for robust self-testing of the three-outcome partial Bell basis measurement that is easily implementable in an optical setup.

Paper Structure

This paper contains 12 sections, 6 theorems, 117 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Self-testing in a Communication Scenario
  3. Optimal quantum advantage does not always imply self-testing
  4. Self-Testing of GHZ measurements
  5. Communication task
  6. Ideal self-testing of GHZ measurements
  7. Robust Self-testing of the measurements
  8. Robust Self-testing of partial Bell basis measurements
  9. Conclusion
  10. Explicit matrix representation of states and measurements to demonstrate that optimal quantum advantage does not always imply self-testing
  11. Sum-Of-Squares decomposition of $\mathcal{W}_s$ operators
  12. Bloch vector antipodality in $n$-Sender, single-receiver configurations

Key Result

Theorem 1

The optimal quantum value of $\mathcal{S}$ in SGen is $1$. If this value is achieved from an unknown set of qubit states $\{\rho^{(j)}_{a_j|x_j}\}_{a_j,x_j}$ and an unknown measurement $\{\mathcal{M}_s\}_s$ on the $n$-qubit system, then there exists a set of qubit unitaries $\{U_j\}_j$ such that for $j\neq 1$ and where and $|\xi_s\rangle=\frac{1}{\sqrt{2}}(|0 \textrm{ } s_2 \textrm{ }\cdots \te

Figures (3)

  • Figure 1: Schematic diagram of a multi-party communication scenario in which multiple senders $\{A^{(j)}\}_j$, each transmits a message $m_d(x_j,a_j)$, with $d=2$ in our case, determined by the respective inputs $(x_j,a_j)$ where $j\in\{1,2,\cdots,n\}$. These messages are sent to the receiver, who produces an $n$-bit binary output.
  • Figure 2: A schematic diagram of a two-sender one-receiver communication scenario in which $A^{(1)}$ and $A^{(2)}$, each transmits a message $m_d(x_j,a_j)$, with $d=2$ in our case, determined by the respective inputs $(x_j,a_j)$ where $j\in\{1,2\}$. These messages are sent to a receiver, who accepts an input $k\in\{1,2,3\}$ and tries to guess the first or the second bit of $A^{(1)}$ for $k=1,2$ respectively, and produces three outputs for $k=3$.
  • Figure 3: Plot illustrating the relationship between the deviation $\epsilon$ and achievable fidelity $\mathcal{F}(\epsilon)$ as a function of the deviation $\epsilon$ for two different scenarios. The red line indicates the robust self-testing of Bell basis measurement in a two sender-one receiver scenario where $\mathcal{S}\geq 1-\epsilon$. The blue line indicates the scenario when partial Bell basis measurement is robustly self-tested and $\mathcal{S}^{Comm}\geq 1-\epsilon$. Also, note that in this scenario, the graph shows the variation when $\mathcal{S}^{RAC}$ achieves its maximum value.

Theorems & Definitions (13)

  • Definition 1
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • ...and 3 more