Device-independent certification of tripartite quantum networks with bilocal Bell inequalities

Technical Summary: Device-independent certification of tripartite quantum networks with bilocal Bell inequalities

Problem Statement
Quantum networks, which extend standard Bell scenarios to include multiple independent sources, offer a richer correlation structure but present significant challenges in the construction of nonlocality witnesses. While standard Bell scenarios rely on linear inequalities, the assumption of source independence in networks (specifically the bilocal scenario with two sources and three parties) renders the set of local correlations non-convex. Consequently, optimal witnesses for network nonlocality must be nonlinear.

Existing literature has largely focused on specific, restricted cases:

1. Limited Measurement Settings: Most constructions are restricted to scenarios where external parties perform only two measurements.
2. Limited Dimensions: Many proposals are tailored specifically to qubit systems.
3. Lack of Self-Testing: There is a notable absence of Bell inequalities capable of self-testing (device-independently certifying) the underlying quantum states and measurements in network scenarios, even in the simplest bilocal setup.

The primary challenge addressed in this work is the lack of a general framework for constructing nonlinear Bell inequalities for tripartite quantum networks that accommodate arbitrary numbers of binary measurements and higher local dimensions, while simultaneously possessing self-testing properties and analytically determinable quantum bounds.

Methodology
The authors propose a general framework for constructing nonlinear Bell inequalities tailored to the bilocal scenario (two independent sources, $S_1$ and $S_2$, distributing states to Alice, Bob, and Charlie). The construction relies on a specific class of matrices known as Real Row-Orthogonal and Column-Normalized (ROCN) matrices.

1. Inequality Construction:
   For a given ROCN matrix $h$ of size $m \times n$ (where $m$ is the number of measurements for Alice/Charlie and $n$ for Bob), the authors define linear combinations of correlators $I_y$:
   $$I_y := \sum_{x=1}^m \sum_{z=1}^m h_{xy} h_{zy} \langle A_x B_y C_z \rangle$$
   These are combined into a nonlinear expression:
   $$S_h := \sum_{y=1}^n \sqrt{|I_y|} \leq \beta_{BL}$$
   where $\beta_{BL}$ is the bilocal bound.

2. Analytical Derivation of Quantum Bounds:
   The algebraic properties of ROCN matrices allow for the analytical determination of the maximal quantum value ($\beta_Q$). The authors utilize a sum-of-squares (SOS) decomposition to prove that for any ROCN matrix, the maximal quantum value is exactly equal to the number of Bob's measurements ($n$).

3. Reference Strategy:
   To achieve this bound, the authors define a "reference strategy" involving:

   • Two maximally entangled states of local dimension $d=2^r$ (where $r = \lfloor m/2 \rfloor$) emitted by the sources.
   • Alice and Charlie measuring sets of pairwise anticommuting Clifford observables (realized via the Jordan-Wigner representation).
   • Bob performing measurements defined as bilinear combinations of the transposed operators of Alice and Charlie.

4. Self-Testing Framework:
   The paper investigates whether the maximal violation of these inequalities uniquely certifies the reference strategy. This is achieved by combining the nonlinear inequality $S_h$ with a corresponding linear ROCN Bell expression $L_h = \sum I_y$. The authors establish that simultaneous maximal violation of both inequalities leads to self-testing, provided the coefficient matrix $h$ satisfies a specific algebraic condition regarding the rank of an auxiliary matrix $M$ constructed from pairwise products of $h$'s entries.

Key Contributions and Results

• General Construction: The paper presents a general method to construct nonlinear Bell inequalities for bilocal networks with arbitrary numbers of binary measurements ($m, n$) and arbitrary local dimensions. This recovers the pioneering 22-type Branciard–Rosset–Gisin–Pironio (BRGP) inequality as a specific case.
• Analytical Quantum Bounds: A significant result is the proof that the maximal quantum value for any such inequality is exactly $n$ (the number of Bob's measurements). This analytical determinability is crucial for device-independent applications where numerical optimization is often intractable.
• Self-Testing Theorem: The authors prove that the simultaneous maximal violation of the nonlinear inequality $S_h$ and its linear counterpart $L_h$ enables the device-independent certification (self-testing) of the entire network. This includes:
  • The quantum states produced by the sources (maximally entangled states of suitable dimensions).
  • The observables measured by all parties (specifically, the anticommuting Clifford observables for external parties and the specific bilinear combinations for the central party).
• Rank Condition for Certification: The paper establishes a necessary and sufficient condition for this self-testing property: the auxiliary matrix $M$, formed by products $h_{xy}h_{zy}$, must have full column rank.
• Examples and Quantum-Classic Gaps:
  • The framework recovers the BRGP inequality ($m=n=2$) and the Gisin elegant Bell inequality ($m=3, n=4$).
  • The authors demonstrate a family of inequalities derived from the elegant Bell inequality that, for certain parameters, yield a strictly larger gap between the bilocal bound and the quantum bound compared to the standard elegant inequality, while maintaining self-testing capabilities.

Significance and Claims
The authors claim that this work provides the first general self-testing result for quantum networks that relies solely on the maximal violation of a single nonlinear Bell inequality (combined with its linear counterpart).

• Conceptual Clarity: Unlike previous protocols that may require multiple witnesses or complex assumptions, this approach relies on a single experimentally accessible witness structure.
• Device-Independence: The ability to analytically determine the quantum bound and certify the network components without trusting the devices makes these inequalities highly promising for device-independent applications.
• Foundational and Practical Relevance: The observables involved (pairwise anticommuting Clifford operators) are relevant not only for foundational physics but also for topological quantum computation and uncertainty relations.

The paper concludes by noting that while explicit constructions satisfying the rank condition exist, the scaling of the required number of columns (Bob's measurements) relative to the number of Alice/Charlie's measurements is not yet optimal, suggesting a direction for future improvement in practical feasibility.