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arxiv: 2606.19765 · v1 · pith:NGKDVUMJnew · submitted 2026-06-18 · 🪐 quant-ph

Sparse positive maps on qutrits with exact nondecomposability thresholds and PPT-entanglement transitions

Davide Poderini , Angela Rosy Morgillo , Fabio Benatti , Fabio Anselmi , Chiara Macchiavello , Massimiliano F. Sacchi This is my paper

Pith reviewed 2026-06-26 17:38 UTC · model grok-4.3

classification 🪐 quant-ph
keywords positive mapsqutritsChoi matricesPPT entanglementdecomposabilitybound entanglementsparse mapsquantum maps
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The pith

Sparse positive maps on qutrits admit exact positivity boundaries and PPT-entanglement thresholds via block-structured Choi matrices.

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

The paper examines families of sparse positive maps on three-level quantum systems, showing that the structure of their Choi matrices simplifies positivity checks to a solvable Hermitian biquadratic form. This simplification produces exact boundaries for when the maps remain positive for three specific parametric families. The same approach pins down the precise point where the maps switch from decomposable to non-decomposable and identifies exact thresholds separating separable states from bound-entangled PPT states. In the trace-preserving case, it also reveals the explicit difference between positivity and a known bound for 2-positive maps.

Core claim

For three representative parametric families of sparse positive maps on qutrits, the block structure of the associated Choi matrices reduces the positivity condition to the analysis of a Hermitian biquadratic form. This reduction yields exact positivity boundaries, the exact transition point between decomposable and non-decomposable maps, and exact separability-to-bound-entanglement thresholds for two classes of associated PPT states. In the trace-preserving subclass the gap between positivity and 2-positivity is made fully explicit through comparison with an eigenvalue bound.

What carries the argument

The block structure of the Choi matrices, which reduces positivity verification to the positivity of a Hermitian biquadratic form.

Load-bearing premise

The block structure of the associated Choi matrices reduces positivity to a Hermitian biquadratic form that permits explicit analysis for the chosen parametric families.

What would settle it

A numerical check that finds a map in one of the three families violating positivity inside the analytically claimed positive region, or a PPT state that is separable below the claimed threshold.

Figures

Figures reproduced from arXiv: 2606.19765 by Angela Rosy Morgillo, Chiara Macchiavello, Davide Poderini, Fabio Anselmi, Fabio Benatti, Massimiliano F. Sacchi.

Figure 1
Figure 1. Figure 1: FIG. 1. Boundaries for different properties of the map in [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Boundaries for different properties of the map in [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Positivity region for the map [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Boundaries for different properties of the map in [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Values of the matrix elements of the rank-four bound-entangled state in [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
read the original abstract

We study a family of sparse positive maps on qutrits for which positivity, decomposability, and PPT entanglement can all be analysed explicitly. The block structure of the associated Choi matrices reduces positivity to a Hermitian biquadratic form and leads to exact positivity boundaries for three representative parametric families. For the same families we determine the exact transition between decomposable and non-decomposable maps and construct associated PPT states of two classes. The first consists of witness-adapted deformations naturally tied to the non-decomposability analysis. The second consists of analytically tractable families whose full PPT-entangled branch is detected by fixed positive maps, yielding exact thresholds between separability and bound entanglement. For the trace-preserving subclass, we further compare positivity with a recent eigenvalue bound for 2-positive maps, thereby making the gap between positivity and higher-order positivity fully explicit within this family.

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 / 2 minor

Summary. The manuscript analyzes sparse positive maps on qutrits for which the block structure of the associated Choi matrices reduces positivity to a Hermitian biquadratic form. For three representative parametric families this reduction yields exact positivity boundaries, exact decomposable-to-non-decomposable transitions, and two classes of associated PPT states whose separability-to-bound-entanglement thresholds are also obtained exactly. For the trace-preserving subclass the gap between positivity and an eigenvalue bound for 2-positive maps is made explicit.

Significance. The explicit algebraic treatment of positivity, decomposability, and PPT entanglement for deliberately chosen sparse families supplies concrete, fully analytical examples in a setting where such thresholds are typically intractable. The construction of witness-adapted PPT states and the fixed-map detection of entire PPT-entangled branches illustrate the utility of sparsity for obtaining falsifiable, parameter-free boundaries. The direct comparison with the 2-positivity eigenvalue bound further clarifies the distinction between positivity and higher-order positivity within the family.

minor comments (2)
  1. [Abstract] Abstract, final sentence: the phrase 'making the gap between positivity and higher-order positivity fully explicit' would benefit from a parenthetical reference to the specific eigenvalue bound employed, to orient readers unfamiliar with the cited result.
  2. [Introduction] The three parametric families are introduced as 'representative'; a brief paragraph early in the manuscript explaining the selection criteria (e.g., sparsity pattern, number of free parameters, or coverage of distinct block structures) would strengthen the claim of representativeness without altering the central results.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the recommendation of minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper selects three specific parametric families of sparse positive maps on qutrits such that the block structure of their Choi matrices reduces positivity to a Hermitian biquadratic form, then derives exact boundaries and thresholds by direct algebraic solution of the resulting equations. This reduction is a deliberate feature of the chosen families rather than a general assertion, and the explicit results follow from standard Choi correspondence and polynomial analysis without any parameter fitting, self-definitional loops, or load-bearing self-citations. The derivation chain is therefore self-contained within the stated scope.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The analysis rests on standard quantum-information properties of Choi matrices and positivity; the parametric families introduce variables whose thresholds are derived rather than fitted constants. No new entities are postulated.

axioms (2)
  • domain assumption Positivity of a linear map on quantum states is equivalent to positivity of its associated Choi matrix.
    Standard equivalence used throughout quantum information to translate map properties into matrix conditions.
  • domain assumption The block structure of the sparse maps reduces the positivity condition to a Hermitian biquadratic form amenable to explicit solution.
    Invoked as the enabling step for obtaining exact boundaries in the three families.

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Reference graph

Works this paper leans on

50 extracted references · 2 linked inside Pith

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    Moreover, Tr[C(3) a,wρ2] = 1 5(2a+ 1−2 √ 2w).(75) 14 The solution ofTr[C(3) a,wρ2] = 0is w= 2a+ 1 2 √ 2

    ={3/10,3/10,1/5,1/10,1/10,0,0,0,0}. Moreover, Tr[C(3) a,wρ2] = 1 5(2a+ 1−2 √ 2w).(75) 14 The solution ofTr[C(3) a,wρ2] = 0is w= 2a+ 1 2 √ 2 . This line is tangent to the ellipse in Eq. (73) at the point a= 1 4 , w= 3 √ 2 8 . The second state is ρ3 = 1 10   2· · · · · · · − √ 2 ·2· − √ 2· · · · · · ·1 · · · · · · · − √ 2· 1· · · · · · · · ·...

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    the fixed witnessC(1) 1/2,1/2 detects entanglement whenever 1 + √ 2 2 < k≤ √ 2

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    the stateρ1(1)is separable and admits the explicit product decomposition given in Eq.(88)

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    according to the level-2DPS hierarchy with PPT constraints,ρ1(k)is entangled for everyk >1. Proof.Consider the family ρ1(k) = √ 2−1 4   2· · · · · · · −k · √ 2· −k· · · · · · · √ 2 · · · · · · · −k· √ 2· · · · · · · · ·1· · · · · · · · · · · · · · · · · · · √ 2· · · · · · · · · · · −k· · · · · · ·1   ,(85) which is PPT for0...

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    according to the level-2DPS hierarchy with PPT constraints, both families are entangled for everyk >1. Proof.Consider ρ2(k) = 1 10   2· · · · · · · −k ·1· −k· · · · · · ·2 · · · · · · · −k· 2· · · · · · · · ·1· · · · · · · · · · · · · · · · · · · 1· · · · · · · · · · · −k· · · · · · ·1   , ρ 3(k) = 1 10   2·...

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    according to the level-2DPS hierarchy with PPT constraints, the state is bound entangled forv < k≤ √ 2v. Proof.Consider the family ξ(a, k) =   q· · · · · · · −k ·r· −k· · · · · · ·s · · · · · · · −k· s· · · · · · · · ·v· · · · · · · · · · · · · · · · · · · r· · · · · · · · · · · −k· · · · · · ·v   ,(94) with0≤k≤g. Hereq, r,...

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    Henceσ 1(k)is PPT entangled and detected by the fixed witnessC(1) 1/2,1/2 for every1< k≤ √ 2

    =ρ 1, the stateσ1(k)is separable for0≤k≤1and PPT for0≤k≤ √ 2, and Tr C(1) 1/2,1/2 σ1(k) = 1−k 2(1 +k) . Henceσ 1(k)is PPT entangled and detected by the fixed witnessC(1) 1/2,1/2 for every1< k≤ √ 2. Proof.The identityσ 1( √

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    Moreover, Tr C(1) a,b σ1(k) = a+kb−k 1 +k ,(97) so the choicea=b= 1/2gives the stated expectation value and proves entanglement fork >1

    =ρ 1 is immediate from the definitions. Moreover, Tr C(1) a,b σ1(k) = a+kb−k 1 +k ,(97) so the choicea=b= 1/2gives the stated expectation value and proves entanglement fork >1. The stated PPT interval0≤k≤ √ 2follows directly from the2×2principal minors ofσ1(k)and its partial transpose. For0≤k≤1, write σ1(k) = 1 4(1 +k) M1(k) +N 1(k) ,(98) with M1(k) =  ...

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    both families are separable for0≤k≤1

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    Tr C(3) 1/3,2/3 σ2(k) = Tr C(3) 1/3,2/3 σ3(k) = √ 2−3 15 (k−1), so both are PPT entangled and detected byC(3) 1/3,2/3 for every1< k≤ √ 2

    both families are PPT for0≤k≤ √ 2; 4. Tr C(3) 1/3,2/3 σ2(k) = Tr C(3) 1/3,2/3 σ3(k) = √ 2−3 15 (k−1), so both are PPT entangled and detected byC(3) 1/3,2/3 for every1< k≤ √ 2. Proof.For the lower branch0≤k≤1, the two families coincide and admit the block decomposition σ2(k) =σ 3(k) = 1 8 M0(k) +N 0(k) ,(102) with M0(k) =   1 0 0 0 0 1−k0 0−k1 0 0 0 0...

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    To verify PPT for1≤k≤ √ 2, it is enough to check the two non-trivial determinants of the partial transpose

    =ρ 3. To verify PPT for1≤k≤ √ 2, it is enough to check the two non-trivial determinants of the partial transpose. For both families one finds rj(k)sj(k)−g j(k)2 = (k−1)( √ 2−k) 80 ≥0, j= 2,3,(106) and qj(k)vj(k)−g j(k)2 = 14 + 4 √ 2−(8 + 6 √ 2)k+ (2 √ 2−1)k 2 320 ≥0, j= 2,3,(107) throughout the interval1≤k≤ √

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