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arxiv: 2312.02588 · v2 · submitted 2023-12-05 · 🪐 quant-ph

Lower-bounding entanglement with nonlocality in a general Bell's scenario

Liang-Liang Sun , Xiang Zhou , Zhen-Peng Xu , Sixia Yu This is my paper

Pith reviewed 2026-05-24 04:47 UTC · model grok-4.3

classification 🪐 quant-ph
keywords entanglementBell nonlocalityminimal distancelower boundsquantum correlationsseparable stateslocal correlationsBell scenarios
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The pith

The minimal distance from observed correlations to local ones lower-bounds the minimal distance from the underlying state to separable states.

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

Two minimal distances sit at the center of quantifying entanglement and Bell nonlocality. The first is the shortest distance from a given quantum state to the set of separable states. The second is the shortest distance from the observed correlation to the set of local correlations. The paper shows that the second distance always lower-bounds the first, for any Bell scenario and any nonlocal correlation. This single inequality immediately supplies lower bounds on many standard entanglement measures once a nonlocal correlation is given. The bound becomes tighter in the (n,2,2) scenario by using the known geometry of the two sets.

Core claim

In a general Bell scenario the minimal correlation distance to local correlations is always at least as large as the minimal state distance to separable states. The inequality holds without restrictions on dimension or the form of the measurements. It therefore converts any observed nonlocal correlation into a concrete lower bound on entanglement quantifiers such as entanglement of formation, concurrence, or negativity. The estimate can be sharpened substantially once the structural properties of entanglement and nonlocality in the (n,2,2) scenario are inserted.

What carries the argument

The direct inequality between the minimal correlation distance (to local correlations) and the minimal state distance (to separable states), which converts any nonlocal correlation into a lower bound on entanglement.

If this is right

  • Any observed nonlocal correlation yields a nontrivial lower bound on multiple entanglement measures without state tomography.
  • The bound applies in completely general Bell scenarios with arbitrary numbers of parties, settings, and outcomes.
  • In the (n,2,2) scenario the bound improves once the known convex geometry of the local and separable sets is used.
  • The method supplies quantitative, device-independent estimates of entanglement from Bell-test data alone.

Where Pith is reading between the lines

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

  • The same distance relation might be adapted to bound other resources such as quantum steering or discord when only correlations are available.
  • Experimental Bell tests could be re-analyzed to extract not only the presence but also a numerical lower limit on entanglement.
  • Choosing different distance measures (trace distance, Hilbert-Schmidt, etc.) could produce families of bounds with different tightness and computational cost.
  • The approach suggests a systematic way to turn any known facet of the local-correlation polytope into an entanglement witness.

Load-bearing premise

A universal inequality always relates the shortest distance from a correlation to local correlations with the shortest distance from the underlying state to separable states, without extra conditions on dimension or operators.

What would settle it

A concrete quantum state, its generated correlation, and a local correlation for which the correlation distance exceeds the state distance would falsify the claimed inequality.

Figures

Figures reproduced from arXiv: 2312.02588 by Liang-Liang Sun, Sixia Yu, Xiang Zhou, Zhen-Peng Xu.

Figure 1
Figure 1. Figure 1: FIG. 1. Comparison between the quantifications of entanglement and nonlocality: Entanglement (E) essentially captures the [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
read the original abstract

Understanding the quantitative relation between entanglement and Bell nonlocality is a long-standing open problem of fundamental and practical interest. Here, we tackle this problem in a general Bell scenario. {We observe that lying in the center of quantifying these properties are two minimal distances: one from a state to separable states (entanglement), and the other from a correlation to local correlations (nonlocality).} We find that these two distances can be related to each other -- the minimal correlation distance provides a lower bound for the minimal state distance, which allows us to derive nontrivial bounds on many entanglement measures with an arbitrary nonlocal correlation. Moreover, with the on-hand structural knowledge of entanglement and nonlocality in the $(n, 2, 2)$ Bell scenario, we refine our estimate significantly.

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

0 major / 4 minor

Summary. The manuscript establishes that, in a general Bell scenario, the minimal distance from an observed correlation to the set of local correlations lower-bounds the minimal distance from the underlying quantum state to the set of separable states. This relation is used to derive lower bounds on standard entanglement measures (e.g., entanglement of formation, concurrence) directly from arbitrary nonlocal correlations. The authors further tighten the estimate in the (n,2,2) scenario by exploiting known structural properties of entanglement and nonlocality.

Significance. If the central inequality holds, the work supplies a dimension-independent, measurement-independent route to quantitative lower bounds on entanglement from Bell nonlocality. This is a concrete advance on a long-standing quantitative question and supplies falsifiable predictions that can be checked with existing Bell-test data. The geometric formulation via minimal distances is clean and leverages only the contractivity of the trace distance under POVM maps.

minor comments (4)
  1. §2, definition of the correlation distance: the precise metric (total variation, Hellinger, or other) on the probability vectors should be stated explicitly, together with the normalization factor that makes the inequality min D_corr ≤ min D_state hold with constant 1.
  2. §3, proof of the main inequality: while the contractivity argument is standard, the manuscript should include a one-line derivation showing that the total-variation distance between the two probability distributions is at most the trace distance of the states, so that the infima satisfy the claimed relation for arbitrary POVMs.
  3. Table 1 and Fig. 2: the numerical values for the refined (n,2,2) bounds should be accompanied by a short statement of the optimization method and the dimension cutoff used, to allow independent verification.
  4. §4, discussion of tightness: the manuscript notes that equality can hold for certain states, but does not exhibit an explicit example; adding one (e.g., a two-qubit Werner state) would strengthen the presentation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the accurate summary of its contributions, and the recommendation of minor revision. We are pleased that the geometric approach and its implications for quantitative entanglement-nonlocality relations are viewed as a concrete advance.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The central claim rests on the inequality min correlation distance ≤ min state distance, which follows directly from the contractivity of the quantum-to-classical map (probability differences bounded by trace distance between states). This holds by definition for arbitrary POVMs and Bell scenarios without requiring fitted parameters, self-referential definitions, or load-bearing self-citations. The derivation is self-contained against external benchmarks and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed from abstract only; no explicit free parameters, invented entities, or non-standard axioms are stated in the provided text.

axioms (1)
  • domain assumption Standard quantum-mechanical description of states, measurements, and correlations in a Bell scenario
    Implicit background framework required for any discussion of entanglement and nonlocality distances.

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