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arxiv: 2412.10505 · v3 · submitted 2024-12-13 · 🪐 quant-ph

Nonlocality of Quantum States can be Transitive

Kai-Siang Chen , Gelo Noel M. Tabia , Chung-Yun Hsieh , Yu-Chun Yin , Yeong-Cherng Liang This is my paper

Pith reviewed 2026-05-23 06:42 UTC · model grok-4.3

classification 🪐 quant-ph
keywords nonlocality transitivityquantum statesbipartite marginalstripartite systemsBell inequalityW statequantum steeringHaar-random states
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0 comments X

The pith

Nonlocality of quantum states can be transitive, with two nonlocal bipartite marginals forcing nonlocality in the third.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that nonlocality transitivity exists for quantum states by giving the first explicit construction of a pair of nonlocal bipartite states. When these states appear together as marginals of one tripartite quantum state, the third bipartite subsystem must also be nonlocal. This settles an open question from 2011 on whether the effect known in abstract non-signaling theories appears inside quantum mechanics itself. The same transitivity shows up in random three-qutrit states and for steering with the W state, and the authors supply a method to build states that are nonlocal across all bipartite marginals.

Core claim

By leveraging Bell-inequality violation by tensoring, we analytically construct a pair of nonlocal bipartite states such that simultaneously realizing them in a tripartite system induces nonlocality in the remaining bipartite subsystem. We also prove that multiple copies of the W-state marginals uniquely determine the global compatible state. The nonlocality transitivity of quantum states occurs among the reduced states of Haar-random three-qutrit pure states, and the transitivity of quantum steering can already be demonstrated with the marginals of a three-qubit W state.

What carries the argument

The analytically constructed pair of nonlocal bipartite states that can be realized as marginals of a tripartite quantum state to induce nonlocality in the third pair.

If this is right

  • Nonlocality transitivity occurs among the reduced states of Haar-random three-qutrit pure states.
  • The transitivity of quantum steering can be demonstrated with the marginals of a three-qubit W state.
  • A simple method constructs quantum states and correlations that are nonlocal in all their non-unipartite marginals.
  • Multiple copies of the W-state marginals uniquely determine the global compatible state.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The uniqueness result for W-state marginals may extend to other families of entangled states and limit how much freedom global states have.
  • The construction technique could be adapted to produce examples of transitivity for other multipartite correlation types.
  • Experimental tests with few-qubit systems would directly check whether the predicted induced nonlocality appears in the laboratory.
  • This form of transitivity might supply a new diagnostic for detecting multipartite entanglement from pairwise measurements alone.

Load-bearing premise

The chosen nonlocal bipartite states can be realized simultaneously as marginals of a single tripartite quantum state while the induced nonlocality on the third pair is preserved.

What would settle it

A proof that no tripartite quantum state exists whose marginals include the constructed pair and still make the third marginal nonlocal.

Figures

Figures reproduced from arXiv: 2412.10505 by Chung-Yun Hsieh, Gelo Noel M. Tabia, Kai-Siang Chen, Yeong-Cherng Liang, Yu-Chun Yin.

Figure 1
Figure 1. Figure 1: FIG. 1. Spacetime diagram illustrating a hypothetical trip [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Summary of the relationships between the various Defi [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. For the problem of nonlocality transitivity in a trip [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Schematic showing an intuitive but unsuccessful att [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. In our analytic construction of the nonlocality tran [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Number line for [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Summary of the relationships between the various Defi [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Schematic representation of the [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
read the original abstract

As a striking manifestation of quantum entanglement, nonlocality has long played a pivotal role in shaping our understanding of the quantum world. When considering a Bell test involving three parties, we may even find a remarkable situation where the nonlocality in two bipartite subsystems {\em forces} the remaining bipartite subsystem to exhibit nonlocality. This intriguing effect, dubbed nonlocality transitivity, was first identified in the non-quantum non-signaling world in 2011. However, whether such transitivity could manifest within quantum theory has remained unresolved -- until now. Here, we provide the first affirmative answer to this open problem at the level of quantum states, thereby showing that there exists a quantum-realizable notion of nonlocality transitivity. Specifically, by leveraging the possibility of Bell-inequality violation by tensoring, we analytically construct a pair of nonlocal bipartite states such that simultaneously realizing them in a tripartite system induces nonlocality in the remaining bipartite subsystem. En route to showing this, we also prove that multiple copies of the $W$-state marginals uniquely determine the global compatible state, thus establishing another instance when the parts determine the whole. Surprisingly, the nonlocality transitivity of quantum states also occurs among the reduced states of Haar-random three-qutrit pure states. We further show that the transitivity of quantum steering can already be demonstrated with the marginals of a three-qubit $W$ state, showing again another noteworthy difference between the two forms of quantum correlations. Finally, we present a simple method to construct quantum states and correlations that are nonlocal in all their non-unipartite marginals, which may be of independent interest.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript claims to resolve an open question by providing the first explicit analytical construction of quantum states exhibiting nonlocality transitivity: two nonlocal bipartite states ρ_AB and ρ_BC (built via Bell-inequality violation under tensoring) are realized as marginals of a single tripartite state ρ_ABC, inducing nonlocality in the remaining marginal ρ_AC. Supporting results include a uniqueness theorem showing that multiple copies of W-state marginals determine the global tripartite state, occurrence of the transitivity effect in Haar-random three-qutrit states, transitivity of steering already for the three-qubit W state, and a general method for constructing states nonlocal in all non-unipartite marginals.

Significance. If the central construction is verified, the result is significant: it supplies the first quantum-realizable example of nonlocality transitivity, closing a question left open since the 2011 non-signaling result. The W-state uniqueness theorem is a clean, independent contribution establishing a case where parts determine the whole. The random-state observation indicates the phenomenon is not isolated, and the steering example highlights a distinction between correlation types. The constructions are analytical and leverage standard tools (tensoring of Bell violations, quantum marginals), making them potentially reproducible.

major comments (2)
  1. [main construction section] Main construction (the section presenting the analytical pair of states and the tripartite extension): the existence of a tripartite quantum state whose reductions exactly match the chosen nonlocal ρ_AB and ρ_BC while simultaneously rendering ρ_AC nonlocal is the load-bearing step for the transitivity claim. The manuscript must explicitly exhibit the tripartite state (or the extension map) and verify both marginal compatibility and the induced nonlocality on the third pair; without this, the tensoring-based construction of the bipartite marginals alone does not establish the result.
  2. [W-state uniqueness theorem] W-state uniqueness result (the theorem and proof that multiple copies of the W marginals fix the global state): while independent of the transitivity claim, the manuscript should clarify whether this uniqueness is invoked to guarantee the tripartite extension in the main construction or stands alone; if the former, the dependence must be stated explicitly.
minor comments (3)
  1. Notation for the constructed states (ρ_AB, ρ_BC, ρ_AC) should be introduced with an explicit equation number at first use and kept consistent throughout.
  2. The abstract and introduction cite the 2011 non-signaling result but should add a reference to the specific paper that posed the quantum version of the transitivity question.
  3. Figure captions for any plots of Bell violation values or random-state statistics should include the precise number of samples and the Bell inequality used.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We address each major comment below. Both points can be resolved by adding explicit details and clarifications; we will incorporate these changes in the revised version.

read point-by-point responses

  1. Referee: [main construction section] Main construction (the section presenting the analytical pair of states and the tripartite extension): the existence of a tripartite quantum state whose reductions exactly match the chosen nonlocal ρ_AB and ρ_BC while simultaneously rendering ρ_AC nonlocal is the load-bearing step for the transitivity claim. The manuscript must explicitly exhibit the tripartite state (or the extension map) and verify both marginal compatibility and the induced nonlocality on the third pair; without this, the tensoring-based construction of the bipartite marginals alone does not establish the result.

    Authors: We agree that the tripartite state must be exhibited explicitly for the claim to be fully substantiated. Our analytical construction proceeds by selecting bipartite marginals ρ_AB and ρ_BC that admit a unique tripartite extension (via the W-state uniqueness result) whose third marginal ρ_AC violates a Bell inequality. In the revised manuscript we will add an explicit presentation of the tripartite density operator, direct verification that its reductions recover the chosen ρ_AB and ρ_BC, and a calculation confirming that ρ_AC violates the relevant Bell inequality. revision: yes

  2. Referee: [W-state uniqueness theorem] W-state uniqueness result (the theorem and proof that multiple copies of the W marginals fix the global state): while independent of the transitivity claim, the manuscript should clarify whether this uniqueness is invoked to guarantee the tripartite extension in the main construction or stands alone; if the former, the dependence must be stated explicitly.

    Authors: The referee is correct that the dependence is not stated with sufficient clarity. The W-state uniqueness theorem is used to guarantee the existence and uniqueness of the tripartite extension in the main construction. We will revise the text to state this dependence explicitly, both when the theorem is introduced and when it is applied in the construction section. revision: yes

Circularity Check

0 steps flagged

No significant circularity; explicit analytical construction is self-contained

full rationale

The paper's central result is an explicit analytical construction of nonlocal bipartite marginals ρ_AB and ρ_BC (leveraging Bell-inequality violation by tensoring) such that a tripartite extension exists with the induced ρ_AC also nonlocal. The W-state uniqueness result is proven internally in this work ('we also prove that multiple copies of the W-state marginals uniquely determine the global compatible state'), not imported via self-citation. No equations reduce the target nonlocality transitivity to a fitted parameter, self-definition, or prior author result by construction. The derivation chain is independent of the inputs it claims to derive.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work relies on standard quantum mechanics and previously established facts about Bell violations and W states; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (2)
  • domain assumption Bell inequality violations can be amplified by tensoring certain quantum states
    Invoked to construct the nonlocal bipartite marginals that induce the third-party nonlocality.
  • domain assumption Multiple copies of W-state marginals uniquely determine the compatible global state
    Used as an auxiliary result en route to the main construction.

pith-pipeline@v0.9.0 · 5849 in / 1366 out tokens · 55953 ms · 2026-05-23T06:42:05.945308+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

120 extracted references · 120 canonical work pages · 1 internal anchor

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    ( 6a) to ( 6c), respectively, as ⃗PBC, ⃗PAC, and ⃗PAB

    holds, we may simi- larly define the marginal conditional probabilities arising from 3 either side of Eqs. ( 6a) to ( 6c), respectively, as ⃗PBC, ⃗PAC, and ⃗PAB. To ease notation, we again denote the set of correlations respecting the NS conditions of Eq. ( 6) as N S, even though it is clear that the NS set to which ⃗PABC belongs is different from the one ...

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    Defi- nition 1, in the spacetime configuration shown in Fig

    v-causal model generates Bell-nonlocal ⃗PAB and Bell- nonlocal ⃗PBC exhibiting nonlocal transitivity, cf. Defi- nition 1, in the spacetime configuration shown in Fig. 1. The first assumption above is the working hypothesis of v- causal models for finite v; the second assumption embodies the requirement that these influences are also hidden in the sense that th...

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