Two convergent NPA-like hierarchies for the quantum bilocal scenario

TL;DR

The paper addresses the problem of characterising quantum network correlations in bilocal networks by recasting it as noncommutative polynomial optimization. It introduces two convergent outer-approximation hierarchies—the factorisation bilocal NPA hierarchy and a convergent bilocal scalar-extension hierarchy—and proves that they converge to the same relaxation, the set of Projector Bilocal Quantum Distributions, thereby providing SDP-friendly tests for bilocal quantum correlations. It further connects these hierarchies to the inflation-NPA framework and discusses generalisations to broader networks, including stopping criteria for finite-dimensional representations. The results clarify the relationships between different bilocal network formalisms, resolve convergence concerns from earlier work, and offer practical tools for device-independent certification in networked quantum information regimes.

Abstract

Characterising the correlations that arise from locally measuring a single part of a joint quantum system is one of the main problems of quantum information theory. The seminal work [M. Navascués et al., New J. Phys. 10, 073013 (2008)], known as the Navascués-Pironio-Acín (NPA) hierarchy, reformulated this question as a polynomial optimisation problem over noncommutative variables and proposed a convergent hierarchy of necessary conditions, each testable using semidefinite programming. More recently, the problem of characterising the quantum network correlations, which arise when locally measuring several independent quantum systems distributed in a network, has received considerable interest. Several generalisations of the NPA hierarchy, such as the scalar extension [A. Pozas-Kerstjens et al., Phys. Rev. Lett. 123, 140503 (2019)], were introduced while their converging sets remain unknown. In this work, we introduce a new bilocal factorisation NPA hierarchy, prove its equivalence to a modified bilocal scalar extension NPA hierarchy, and characterise its convergence in the case of the simplest network, the bilocal scenario. We further explore its relations with the other known generalisations.

Figures (3)

• Figure 1: (a) Standard three-party Bell scenario. A three-particles quantum state, mathematically represented by a projector over a pure state $\tau\in\mathcal{B}(\mathcal{H}_A\otimes\mathcal{H}_B\otimes\mathcal{H}_C)$ (a positive operator such that $\mathop{}\!\mathrm{Tr}{\left(\tau\right)}=1$ and $\tau^2=\tau$), is created, each particle is sent to one of three separated parties $A, B, C$. $A$ measures the received particle according to some input $x$, obtaining an output $a$, mathematically represented by PVMs $A_{a|x}\in\mathcal{B}(\mathcal{H}_A)$ (a set of positive operators such that $\sum_a A_{a|x}=\mathbbm{1}$), and $B,C$ do the same. The behavior of the experiment is described by a probability distribution $\vec{P}=\{p(abc|xyz)\}$ with $p(abc|xyz)=\mathop{}\!\mathrm{Tr}_{\tau}{\left(A_{a|x}\otimes B_{b|y}\otimes C_{c|z}\right)}$. (b) Bilocal scenario. Two two-particles quantum state (mathematically represented by two projectors over pure states $\rho\in\mathcal{B}(\mathcal{H}_A\otimes\mathcal{H}_{B_L}), \sigma\in\mathcal{B}(\mathcal{H}_{B_R}\otimes\mathcal{H}_{C})$ such that $\mathop{}\!\mathrm{Tr}{\left(\rho\right)}=\mathop{}\!\mathrm{Tr}{\left(\sigma\right)}=1$ and $\rho^2=\rho, \sigma^2=\sigma$) are created, $A$ (resp. $C$) receiving one particle from $\rho$ (resp. $\sigma$) and $B$ one of each state, as depicted. $A, B, C$ measurement operators are mathematically represented by positive operators $A_{a|x}\in\mathcal{B}(\mathcal{H}_A), B_{b|y}\in\mathcal{B}(\mathcal{H}_{B_L}\otimes\mathcal{H}_{B_R}), C_{c|z}\in\mathcal{B}(\mathcal{H}_C)$ such that $\sum_a A_{a|x}=\sum_b B_{b|y}=\sum_c C_{c|z}=\mathbbm{1}$). The behavior of the experiment is described by a probability distribution $\vec{P}=\{p(abc|xyz)\}$ with $p(abc|xyz)=\mathop{}\!\mathrm{Tr}_{\rho\otimes\sigma}{\left(A_{a|x}\otimes B_{b|y}\otimes C_{c|z}\right)}.$
• Figure 2: Physical scenario corresponding to the inflation technique (to simplify, inputs $x,y,z$ and outputs $a,b,c$ are omitted). In case a distribution $\vec{P}$ is obtain from the bilocal scenario, one can consider the thought experiment in which the sources are duplicated, and each party is given additional inputs $i$; $j,k$; and $l$. The parties apply the measurement specified in the original scenario to the copies of the states given by these extra inputs. Here, we represented the case where $A$ performs here measurement $A_{a|x}$ over her share of state $\rho^{1}$, corresponding to the new operator $A^1_{a|x}$. Similarly, here Bob measures operator $B^{12}_{b|y}$ over his shares of states $\rho^{1},\sigma^2$ and Charlie measures $C^{1}_{c|z}$ over his share of state $\sigma^2$. Note that in that case Charlie measures a state not considered by the other parties: his behavior factorises from the rest, which is imposed by the condition (iv) of the main text. Moreover, as the states and PVMs are copies of each other, the behavior should be invariant under $S_2\times S_2$ permutation group, as specified by condition (iii) of the main text.
• Figure 3: Some generalised network scenario: the triangle scenario (a) and the four party star network (b).

Theorems & Definitions (57)

• Theorem : Convergence of the factorisation bilocal and scalar extension bilocal hierarchies
• Definition : Projector Bilocal Quantum Distributions
• Definition 2.1: Tensor Tripartite Quantum Distributions
• Theorem 2.1: Convergence of the standard NPA hierarchy
• Definition 2.2: Commutator Tripartite Quantum Distributions
• Definition 2.3: Tensor Bilocal Quantum Distributions
• Definition 2.4: Projector Bilocal Quantum Distributions
• Remark 2.2
• Definition 2.5: Factorisation Bilocal Moment Matrix
• Definition 2.6: Factorisation bilocal NPA hierarchy