[Paper Review] Quantum steering in a star network

Guangming Jiang, Xiaohua Wu+1 more|arXiv (Cornell University)|Oct 4, 2022

Quantum Information and Cryptography4 citations

TL;DR

This paper introduces a framework for quantum steering in a star network where the central party is trusted and edge parties are untrusted, defining an n-local hidden state (n-LHS) model as a special case of n-local hidden variable models. It proposes three criteria—nonlinear and linear steering inequalities, and a Bell inequality—to verify network steering, demonstrating that steering can be detected even when the trusted central party performs fixed measurements, with explicit violation of inequalities for GHZ-type states by a factor of (√2)^n.

ABSTRACT

In this work, we will consider the star network scenario where the central party is trusted while all the edge parties (with a number of $n$) are untrusted. Network steering is defined with an $n$ local hidden state model which can be viewed as a special kind of $n$ local hidden variable model. Two different types of sufficient criteria, nonlinear steering inequality and linear steering inequality will be constructed to verify the quantum steering in a star network. Based on the linear steering inequality, how to detect the network steering with a fixed measurement will be discussed.

Motivation & Objective

• To define quantum steering in a star network where the central party is trusted and edge parties are untrusted.
• To introduce an n-local hidden state (n-LHS) model as a special case of n-local hidden variable models for network steering.
• To develop three sufficient criteria—nonlinear, linear, and Bell-type inequalities—for verifying quantum steering in the network.
• To demonstrate that network steering can be detected even when the trusted central party performs a fixed measurement.
• To generalize existing steering inequalities to the star network scenario, removing constraints like mutually unbiased measurements.

Proposed method

• Formalizes network steering via an n-LHS model, where the central party's state is influenced by edge parties' measurements, with the central party trusted.
• Derives a nonlinear steering inequality (Eq. 35) for two-qubit systems, generalizing prior work and relaxing the need for mutually unbiased measurements.
• Constructs a linear steering inequality (Eq. 57) that enables detection of steering even when the trusted party performs a fixed measurement.
• Develops a Bell-type inequality (Eq. 60) based on multipartite correlations, using operators defined on the central and edge parties' observables.
• Uses a GHZ-type state |Ψ⟩ = ⊗_{μ=1}^n |Φ⁺⟩^{(μ)} as a test state, with each source sharing a maximally entangled Bell pair.
• Defines measurement settings for edge parties (Eq. 37) and fixed operators for the central party (Eq. 73), enabling analytical evaluation of correlation expectations.

Experimental results

Research questions

• RQ1 Can quantum steering be verified in a star network where the central party is trusted and edge parties are untrusted?
• RQ2 Is it possible to detect network steering when the trusted central party performs a fixed measurement?
• RQ3 Can the nonlinear steering inequality be derived without requiring mutually unbiased measurements on the trusted party?
• RQ4 How do the three criteria—nonlinear, linear, and Bell inequalities—compare in detecting network steering?
• RQ5 Can the n-LHS model be viewed as a special case of the n-LHV model in network nonlocality?

Key findings

• The network steering can be detected via a linear steering inequality even when the trusted central party performs a fixed measurement, demonstrating robustness of the criterion.
• The nonlinear steering inequality (Eq. 35) generalizes prior work and removes the need for mutually unbiased measurements on the trusted party.
• For a GHZ-type state with n maximally entangled pairs, the Bell inequality (Eq. 60) is violated by a factor of (√2)^n, confirming genuine network nonlocality.
• The linear steering inequality (Eq. 57) enables detection of steering under fixed measurement settings, showing feasibility in experimental implementations.
• The n-LHS model is formally shown to be a special case of the n-LHV model, linking network steering to broader network nonlocality frameworks.
• The violation of the Bell inequality reaches 2^n × (√2)^n for n parties, demonstrating increasing nonlocality with network size.