Witnessing nonlocality in quantum network of continuous-variable systems by generalized quasiprobability functions

1. Problem Statement

The paper addresses a critical gap in the study of quantum network nonlocality within Continuous-Variable (CV) systems.

• The Limitation of Gaussian Measurements: While Gaussian states are the standard resource in CV quantum information due to their ease of generation and manipulation, they possess a positive-definite Wigner function. Consequently, they cannot exhibit Bell nonlocality when measured using standard Gaussian measurements.
• The Challenge: Detecting network nonlocality in CV systems requires non-Gaussian measurements. However, existing literature on network nonlocality (e.g., chain, star, cyclic networks) has largely focused on finite-dimensional (Discrete Variable) systems or specific two-input/two-output scenarios. There is a lack of general, nonlinear Bell-type inequalities applicable to infinite-dimensional CV networks with arbitrary measurement outcomes.
• The Goal: To develop a general framework and specific inequalities to witness network nonlocality in CV networks where source states are arbitrary multipartite multimode Gaussian states, utilizing feasible non-Gaussian measurement schemes.

2. Methodology

The authors propose a comprehensive approach combining theoretical inequality derivation with a specific measurement strategy based on generalized quasiprobability functions.

A. General Nonlinear Bell-Type Inequality (Theorem 2.1)

The authors first derive a new nonlinear Bell-type inequality applicable to any finite-size quantum network (both finite and infinite dimensional).

• Setup: A network with $y$ parties and $z$ independent sources.
• Constraints: All parties have two inputs (dichotomic), but the number of measurement outputs is unrestricted (potentially infinite).
• Independence: The inequality depends on $k$, the number of independent parties (nodes that do not share a common source).
• The Inequality: For a network-local scenario, the following must hold:
  $$B = |I|^{1/k} + |J|^{1/k} \leq 1$$
  Where $I$ and $J$ are correlation terms constructed from the expectation values of measurement operators. Violation of this inequality ($B > 1$) witnesses network nonlocality.

B. The Supremum Strategy (Theorem 3.1)

To apply this to CV systems with Gaussian sources, the authors introduce the Supremum Strategy:

1. Measurement Scheme: They utilize non-Gaussian measurements based on generalized quasiprobability functions ($Q(\alpha; s)$). These functions are parameterized by a real number $s \leq 0$.
   • The measurement operators are constructed from the generalized quasiprobability operators $\hat{\Pi}(\alpha; s)$.
   • Crucially, these measurements are experimentally feasible, requiring only beam splitters and photon detectors (specifically, distinguishing vacuum from non-vacuum states when $s=-1$).
2. Transformation: The nonlinear inequality terms ($I$ and $J$) are transformed into expressions involving the generalized quasiprobability functions of the Gaussian source states.
3. Optimization: Since the inequality must hold for any local hidden variable model, the authors define a supremum function:
   $$B_{\Xi}(s, \rho) = \sup_{\alpha_0, \alpha_1} B_{\Xi}^{\rho}(\alpha_0, \alpha_1; s)$$
   If $B_{\Xi}(s, \rho) > 1$ for any $s \leq 0$, the network is proven to be nonlocal.

C. Application to Specific Topologies

The authors apply this framework to four specific network configurations with source states being $(1+1)$-mode Gaussian states (EPR states or Squeezed Thermal States):

1. Chain Networks ($Cha(y)$)
2. Star Networks ($S(y)$)
3. Tree-shaped Networks ($T(m, f)$)
4. Cyclic Networks ($Cyc(y)$)

For each topology, they derive specific nonlinear Bell-type inequalities (Theorems 4.1, 5.1, 6.1, 7.1, 7.2) tailored to the network's independence structure.

3. Key Contributions

1. General Inequality for CV Systems: The derivation of a nonlinear Bell-type inequality valid for infinite-dimensional systems with arbitrary output cardinality, overcoming the limitations of previous DV-centric approaches.
2. The Supremum Strategy: A novel, general method to detect network nonlocality in Gaussian networks by optimizing over the parameters of generalized quasiprobability functions.
3. Experimental Feasibility: The proposal relies on measurements that can be implemented in current optical laboratories (beam splitters + photon detectors), specifically highlighting the case where the parameter $s = -1$ (corresponding to vacuum/non-vacuum discrimination).
4. Comprehensive Topological Analysis: The paper provides explicit inequalities and numerical verification for chain, star, tree, and cyclic networks, filling a void in the literature regarding CV network nonlocality.

4. Key Results

The authors validated their method through numerical simulations on specific examples:

• Entanglement Swapping (Chain $y=3$):

  • Pure States: If sources emit identical entangled EPR states, the network is nonlocal for all squeezing parameters $r > 0$ when $s = -1$.
  • Mixed States (Symmetric STS): For mixed Squeezed Thermal States, nonlocality is detected for a wide range of parameters, provided $s = -1$.
  • Sensitivity: The parameter $s = -1$ was found to be the most effective for witnessing nonlocality across various scenarios.

• Chain Networks ($y > 3$):

  • For chains with identical entangled Gaussian pure states, the inequality is violated for any $r > 0$ using appropriate $s$ (specifically $s=-1$ for large $r$, and $s \in (-1, 0]$ for small $r$).
  • For mixed states, nonlocality is witnessed for sufficiently large squeezing parameters.

• Star and Tree Networks:

  • Similar to chains, networks with identical entangled pure Gaussian sources exhibit nonlocality for all $r > 0$ when $s = -1$.
  • For mixed sources, nonlocality is detected above specific thresholds of the squeezing parameter.

• Cyclic Networks:

  • The method successfully detects nonlocality in cyclic topologies (e.g., $y=5$) for both pure and mixed Gaussian sources, demonstrating the robustness of the approach even in closed-loop network structures.

General Conclusion from Results: If all sources in a CV network emit identical entangled Gaussian pure states, the network is inherently nonlocal, and this can be witnessed using the proposed generalized quasiprobability measurements.

5. Significance

• Bridging Theory and Experiment: The work provides a "recipe" for experimentalists to verify network nonlocality in CV systems, which are the backbone of future quantum communication networks (e.g., quantum internet).
• Overcoming Gaussian Limitations: It definitively shows that while Gaussian states are "local" under Gaussian measurements, they possess intrinsic network nonlocality that can be unlocked via specific non-Gaussian measurements.
• Scalability: The framework is not limited to simple two-party scenarios but scales to complex, multi-node networks (chains, trees, cycles), which is essential for distributed quantum computing and sensing.
• Hybrid Network Potential: The authors note that their scheme is adaptable to "hybrid" networks combining Discrete Variable (DV) and Continuous Variable (CV) systems, suggesting broad applicability in next-generation quantum architectures.

In summary, this paper establishes a fundamental connection between Bell nonlocality and CV quantum networks, offering a practical, mathematically rigorous, and experimentally viable method to detect nonlocality in Gaussian-based quantum networks.