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arXiv:2410.13499v2

Bell nonlocality

from compatibility of entanglement-breaking channels

Local descriptions can look classical while every compatible global broadcast realization is forced to be nonclassical.

Gelo Noel Tabia portrait
01
Gelo Noel Tabia
Hon Hai Research Institute
Chung-Yun Hsieh portrait from slide 2
02
Chung-Yun Hsieh
University of Bristol
Min-Hsiu Hsieh portrait
03
Min-Hsiu Hsieh
Hon Hai Research Institute
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Why expect local → global?

Many theories infer global structure from local information.

Classical probability
Compatible marginals
Joint distribution
Local tomography
Local measurement data
Global density operator
Differential equations
Local equations of motion
Global dynamics
Error-correcting codes
Local parity constraints
Valid global codeword
Tree-structured CSPs
Local constraints
Global assignment
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When does it break?

Known failures usually enlarge the scenario.

Additivity violation
Multiple channel uses
[Hastings 2009]
Superactivation
Channel composition
[Smith-Yard 2008]
Hidden nonlocality
Filtering on copies
[Palazuelos 2012]
Contextuality
Peres–Mermin magic square
[Peres 1990; Mermin 1990]
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Definition

Entanglement-breaking (EB) channels

Operational form and Choi characterization.

Operational form

Measure, then prepare

\( \mathcal{E}(\rho)=\sum_i \operatorname{Tr}(\rho M_i)\sigma_i \)
Choi characterization

Separable iff EB

The Choi state is separable exactly when the channel is entanglement-breaking.

[Horodecki-Shor-Ruskai 2003]
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Another possibility

Can compatibility alone enforce nonclassicality?

Classical marginals + no extra operations
Output A B C Entanglement-breaking channels BC state separable?
\( \mathcal{E}_{A\to B} \)
\( \mathcal{E}_{A\to C} \)
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Definition

Broadcast compatibility as marginals.

\( \operatorname{Tr}_C \circ \mathcal{G}_{A\to BC} = \mathcal{E}_{A\to B} \)
\( \operatorname{Tr}_B \circ \mathcal{G}_{A\to BC} = \mathcal{E}_{A\to C} \)

Trace out the unused output system to recover each marginal channel.

Broadcast channel A B C
\( \mathcal{G}_{A\to BC} \)
\( \mathcal{E}_{A\to B} \)
\( \mathcal{E}_{A\to C} \)
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Main result

Compatible EB channels can force every joint broadcast channel to output CHSH-nonlocal states.

Consequence
  • EB + EB + compatibility \( \to \) Bell nonlocality
  • Every compatible realization is nonlocal.
Broadcast compatibility
Broadcast channel Output A B C BC state nonlocal
\( \mathcal{G}_{A\to BC} \)
\( \mathcal{E}_{A\to B} \)
\( \mathcal{E}_{A\to C} \)
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Why is this unusual?

No extra activation machinery is used.

Multiple copies
No
Filtering
No
Post-selection
No
Special network
No
Compatibility constraints onlyYes.
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Nonlocality meta-transitivity

This result is stranger than ordinary nonlocality transitivity.

Traditional transitivity

Nonlocal + nonlocal → forced nonlocal

  • Bell-nonlocal in AB
  • Bell-nonlocal in AC
  • Every compatible BC is Bell-nonlocal
[Chen-Tabia-Hsieh-Yin-Liang 2026]
This work

Local + local → forced nonlocal

  • Bell-local in AB
  • Bell-local in AC
  • Every compatible BC is Bell-nonlocal
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Proof details

Certifying all compatible extensions

Translate channels to Choi states, rule out separable BC by SDP, then exhibit explicit states and the broader NPT family.

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Formulation in terms of states

The channel problem becomes a tripartite marginal problem.

Do separable \( \rho_{AB} \) and \( \rho_{AC} \) with \( \rho_A = I_2/2 \) have a joint \( \rho_{ABC} \) with separable \( \rho_{BC} \)?

If the answer is no, EB marginals can be compatible while every global realization is nonclassical.

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Our construction

Three-qubit state with separable AB, AC and nonlocal BC

State form
\( \rho_{ABC}=\lambda_1|\psi_1\rangle\langle\psi_1|+\lambda_2|\psi_2\rangle\langle\psi_2| \)
Eigenvalues
\( \lambda_1=0.4733175264 \)
\( \lambda_2=0.5266824836 \)
\( |\psi_1\rangle_{ABC}=\left(\begin{array}{r} -0.209999952108617\\ 0.174094868773554\\ 0.117224725538884\\ 0.182754562207009\\ 0.338480049085443\\ -0.850589433171125\\ 0.194658379550856\\ 0.049871016786613 \end{array}\right) \)
\( |\psi_2\rangle_{ABC}=\left(\begin{array}{r} 0.413378163566997\\ -0.732628740143910\\ 0.202664162888498\\ 0.302333182263915\\ -0.047229605421459\\ -0.088639894678373\\ 0.345304731167627\\ 0.174849887218713 \end{array}\right) \)
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A compatible extension

One nonlocal extension is not enough.

\( \rho_{AB} \) and \( \rho_{AC} \) are separable.
\( \forall\,\tau_{ABC} \) satisfying \( \operatorname{Tr}_C(\tau_{ABC})=\rho_{AB} \) and \( \operatorname{Tr}_B(\tau_{ABC})=\rho_{AC} \)
\( \tau_{BC} \) is CHSH-nonlocal.
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Certifying unavoidable nonlocality

A fixed CHSH witness stays violated over the entire compatibility set.

The SDP minimizes the same Bell operator over all compatible extensions.

Source: SDP certificate
\( \displaystyle \gamma_*=\min_{\tau_{ABC}}\operatorname{Tr}(\tau_{BC}\beta_{\mathrm{CHSH}}) \)
\( \operatorname{Tr}_C(\tau_{ABC})=\rho_{AB} \)
\( \operatorname{Tr}_B(\tau_{ABC})=\rho_{AC} \)
\( \tau_{ABC}\ge 0 \)
\( \gamma_* \approx 2.0485 > 2 \)
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Interpretation

Compatibility constraints exclude all possible CHSH-local extensions.

Bell-local set CHSH local boundary crossed by the certificate Compatible extensions
all feasible extensions satisfy \( \gamma_* > 2 \)
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Channel interpretation

Choi extensions are broadcast channels.

Thus every joint broadcast channel outputs a CHSH-nonlocal BC state on a maximally mixed input.

whole Choi state
\( \tau_{ABC} \)
 A
\( \mathcal{G}_{A\to BC} \)
 broadcast map BC marginal B C
\( \rho_{BC} \) CHSH-nonlocal
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PPT obstruction

Joint broadcast channels are not entanglement-breaking.

If the Choi state were separable across \( A|BC \), it would be PPT across the same bipartition.

\( \mathrm{EB\ jointly} \Rightarrow \tau_{ABC} \) separable across \( A|BC \Rightarrow \tau_{ABC}^{\Gamma_A}\ge 0 \)
PPT SDP across A|BC
\( \displaystyle \mu_* := \max_{\tau_{ABC}}\lambda_{\min}\!\left(\tau_{ABC}^{\Gamma_A}\right) \)
\( \operatorname{Tr}_C(\tau_{ABC})=\rho_{AB} \)
\( \operatorname{Tr}_B(\tau_{ABC})=\rho_{AC} \)
\( \tau_{ABC}\ge 0 \)
\( \mu_* < 0 \Rightarrow \) every compatible \( \tau_{ABC} \) is NPT across \( A|BC \).
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Generic inputs

Generic inputs: CHSH versus negativity.

CHSH locality region
CHSH locality regions over theta and phi input angles
Negativity over inputs
Negativity contour plot over theta and phi input angles
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Take home messages

Compatibility constraints are not compositional.

Local classical realizability does not imply global classical realizability.

Compatibility can enforce Bell nonlocality even when marginals are classically simulable.
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Open questions

Open questions for compatibility-induced nonlocality

Question 01

Characterize compatibility-induced nonlocality

Which marginal structures force global nonclassicality?

Question 02

Compatibility as a structural resource

When can consistency constraints enforce quantum resources?

Question 03

Beyond the CHSH example

Can analytic families with separable \( AB, AC \) force NPT or other resources in every compatible \( BC \)?

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Foxconn Research X U. Bristol

Thank you!

Source: arXiv:2410.13499v2.
Questions?
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