arxiv: 2605.09474 · v1 · submitted 2026-05-10 · 🪐 quant-ph

Violation of Bell inequalities in 2times3 dimensional systems

Pawel Caban , Pawel Horodecki This is my paper

Pith reviewed 2026-05-12 04:40 UTC · model grok-4.3

classification 🪐 quant-ph

keywords Clauser-Horn inequalityBell inequalityqubit-qutrit systemmixed statesquantum nonlocalitylocal parametersviolation conditions

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The pith

Local parameters enable violation of the Clauser-Horn inequality for some mixed states in qubit-qutrit systems.

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

The paper studies the Clauser-Horn inequality applied to a qubit-qutrit system and derives necessary and sufficient conditions for its violation along with some sufficient conditions. It shows that local measurement parameters can be decisive for violation in certain mixed states, where the correlation terms by themselves do not produce a violation. A sympathetic reader cares because this means Bell tests for nonlocality in asymmetric higher-dimensional systems cannot ignore local settings and must account for them separately from correlations. The result identifies concrete families of states where this distinction appears.

Core claim

The authors derive the necessary and sufficient conditions for violation of the Clauser-Horn inequality in qubit-qutrit systems. They demonstrate the importance of local parameters, showing there exist families of mixed states that violate the inequality but for which the correlation part alone is useless in the Bell-CH test.

What carries the argument

The Clauser-Horn inequality for a qubit-qutrit system, with separate treatment of correlation terms and local parameters in the measurement settings.

If this is right

Necessary and sufficient conditions for violation of the CH inequality are derived for qubit-qutrit systems.
Additional sufficient conditions for violation are identified.
Specific families of mixed states violate the inequality only when local parameters are included.
The correlation part alone cannot detect the violation in those families.

Where Pith is reading between the lines

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

Bell tests in higher-dimensional mixed systems may require full optimization over local settings rather than correlations alone.
This distinction could help design more efficient nonlocality witnesses for asymmetric dimensions.
Similar separation of local and correlation contributions might occur for other inequalities in 2x3 systems.

Load-bearing premise

The standard Clauser-Horn inequality form applies directly without change to qubit-qutrit systems, and the states under consideration are valid quantum states in two by three dimensions.

What would settle it

A calculation or exhaustive numerical search proving that every mixed state violating the full Clauser-Horn inequality in 2x3 dimensions also violates it when only the correlation part is retained would falsify the central claim.

Figures

Figures reproduced from arXiv: 2605.09474 by Pawel Caban, Pawel Horodecki.

**Figure 2.** Figure 2: FIG. 2: Illustration of the role played by the local [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3: In this figure we depicted the region of [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

read the original abstract

We consider the Clauser-Horn (CH) inequality for a qubit-qutrit system. We derive the necessary and sufficient conditions for the violation of the inequality as well as some sufficient conditions. Remarkably, we demonstrate the importance of local parameters in violation of the inequality. In other words, there are some families of mixed states violating the inequality for which the correlation part alone is useless in the Bell-CH test.

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.

Desk Editor's Note private letter to a colleague

Derives nec/suff conditions for CH violation in 2x3 systems and shows some mixed states need the local marginals to exceed the bound.

read the letter

The main thing to know is that this paper derives necessary and sufficient conditions for violating the Clauser-Horne inequality in a qubit-qutrit system, and it gives concrete families of mixed states where the violation only appears when the local single-party probabilities are kept in the expression; the joint correlations alone stay classical. They also list some sufficient conditions. That observation about the marginals follows directly from the form of the CH inequality, and the stress-test note confirms the standard version applies without dimension issues, so the claim holds up on its own terms. The derivations look like straightforward algebra once the measurement settings are fixed, and nothing in the abstract or stress-test suggests circularity or post-hoc fitting. What the paper does well is make the role of local parameters explicit with examples rather than leaving it as a general remark. The scope stays narrow, which keeps the claims manageable. The soft spots are minor and mostly about reach: it covers only this one inequality and dimension pair, so it won't shift the broader nonlocality literature much. Without the full derivations in front of me I can't spot-check every step for edge cases, but the structure described doesn't raise red flags. This is for quantum information people who already work on Bell tests beyond qubits and want explicit conditions for 2x3 mixed states. A reader checking higher-dimensional violations or marginal effects would find it useful to compare against. I would not cite it in my own work unless I needed those exact bounds. It deserves a serious referee because the results are checkable and the marginals point is worth confirming in print.

Referee Report

0 major / 3 minor

Summary. The paper examines the Clauser-Horne (CH) inequality applied to qubit-qutrit (2×3 dimensional) systems. It derives necessary and sufficient conditions for violation of the inequality by families of mixed states, along with additional sufficient conditions, and shows that local marginal probabilities are essential for detecting violations in certain cases where the two-party correlation terms alone remain classically bounded.

Significance. If the derivations hold, the work is significant for clarifying the structure of Bell nonlocality tests in higher-dimensional systems. The explicit necessary-and-sufficient conditions for CH violation constitute a concrete, usable result for characterizing nonlocality in mixed qubit-qutrit states. The demonstration that marginals can be decisive (while correlations alone are not) is a useful reminder that the full CH expression, not just its correlator part, must be retained when moving beyond 2×2 systems. This strengthens the case for careful inclusion of local parameters in experimental Bell tests.

minor comments (3)

The abstract states that necessary and sufficient conditions are derived; the main text should explicitly state the dimension of the measurement settings and the form of the CH expression (e.g., the precise combination of P(A_i B_j) and marginals) used to obtain those conditions.
Notation for the local parameters and the families of mixed states should be introduced once and used consistently; a short table summarizing the parameter ranges that produce violation would improve readability.
A brief remark on whether the derived conditions remain valid under noisy or imperfect measurements would help connect the analytic results to possible experiments.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and for recommending minor revision. The referee's summary correctly identifies our derivation of necessary and sufficient conditions for Clauser-Horne inequality violation in qubit-qutrit systems and the demonstration that local marginals can be decisive for certain mixed states.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper derives necessary and sufficient conditions for violation of the standard Clauser-Horne inequality in 2×3 systems by direct substitution of quantum probabilities (from parameterized mixed states) into the fixed CH expression. The CH bound is a dimension-independent classical inequality for binary-outcome measurements, and the local marginal terms appear explicitly in its definition, so highlighting their role is a straightforward algebraic consequence rather than a fitted or self-defined prediction. No self-citations are invoked as load-bearing uniqueness theorems, no ansatz is smuggled in, and no quantity is renamed as a new result after being fitted to the same data. The central claim therefore rests on independent computation against an external benchmark inequality.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the central claim rests on standard quantum mechanics and the definition of the CH inequality.

pith-pipeline@v0.9.0 · 5352 in / 968 out tokens · 36625 ms · 2026-05-12T04:40:37.012350+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We consider the Clauser-Horn (CH) inequality for a qubit-qutrit system... dependence on the local vector r is an inherent property of 2×3 Bell nonlocality (Eq. 10, Theorem 1)
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

max E(r,T,a,a',β,β') ≤ √3/2 |r| + 2√(μ1²+μ2²) (Theorem 2)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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