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arXiv:2607.15947

Beyond the Positive Partial Transpose Squared Conjecture: The Qutrit Case

Junhyeong An, Soojoon Lee

quant-phSubmitted Jul 17, 2026• Announced Jul 20, 2026• 0 citations
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Abstract

Entanglement swapping is a fundamental operation in quantum repeaters for establishing entanglement between distant parties. The positive partial transpose (PPT) squared conjecture asks whether two PPT entangled links can generate terminal entanglement through entanglement swapping, or equivalently, whether the composition of two PPT maps is always entanglement breaking. Motivated by this conjecture, we investigate the map-composition problem beyond the PPT setting. For qutrit completely positive (CP) maps, we prove that the composition of any CP map whose Choi matrix is $1$-undistillable with any CP map whose Choi matrix has Schmidt number at most two is entanglement breaking in either order. Moreover, we show that the cone of $1$-undistillable CP maps is exactly the largest qutrit cone of CP maps whose composition with every CP map whose Choi matrix has Schmidt number at most two is entanglement breaking in both orders. Finally, although map composition captures only the standard maximally entangled outcome in entanglement swapping, we prove that any $1$-undistillable two-qutrit state and any state of Schmidt number at most two cannot generate terminal entanglement under an arbitrary selective measurement on the intermediate systems.

Comments
9 pages, 3 figures

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Entanglement swapping is a fundamental operation in quantum repeaters for establishing entanglement between distant parties. The positive partial transpose (PPT) squared conjecture asks whether two PPT entangled links can generate terminal entanglement through entanglement swapping, or equivalently, whether the composition of two PPT maps is always entanglement breaking. Motivated by this conjecture, we investigate the map-composition problem beyond the PPT setting. For qutrit completely positive (CP) maps, we prove that the composition of any CP map whose Choi matrix is 111-undistillable with any CP map whose Choi matrix has Schmidt number at most two is entanglement breaking in either order. Moreover, we show that the cone of 111-undistillable CP maps is exactly the largest qutrit cone of CP maps whose composition with every CP map whose Choi matrix has Schmidt number at most two is entanglement breaking in both orders. Finally, although map composition captures only the standard maximally entangled outcome in entanglement swapping, we prove that any 111-undistillable two-qutrit state and any state of Schmidt number at most two cannot generate terminal entanglement under an arbitrary selective measurement on the intermediate systems.
We present the positive-partial-transpose (PPT) square conjecture introduced by M. Christandl Banff International Research Station Workshop: Operator Structures in Quantum Information Theory (Banff International Research Station, Alberta, 2012). We prove the conjecture in the case n=3n=3n=3 as a consequence of the fact that two-qutrit PPT states have Schmidt number of at most 2. The PPT square conjecture in the case of $n\ensuremath{\ge}4$ is still open. We present an example to support the conjecture for n=4n=4n=4.
The positive-partial-transpose-squared (PPT-squared) conjecture asks whether the composition of two positive-partial-transpose (PPT) quantum channels must be entanglement breaking (EB). The problem remains open in general and, within the diagonal orthogonal covariant (DOC) class, the first unresolved deterministic dimension is four. A sufficient condition for PPT-squared composition is derived for DOC maps in arbitrary finite dimension. The proof isolates the quantum coherence data into two correlation matrices, mixes each with the identity until it lies in the convex hull of rank-one correlation matrices, and converts the resulting objects into an explicit triplewise completely positive (TCP) core. The remaining contribution is purely classical and is TCP whenever a simple entrywise mixing inequality is satisfied. For PPT DOC maps ΦA,B,CΦ_{A,B,C}ΦA,B,C​ and ΦD,E,FΦ_{D,E,F}ΦD,E,F​ on MdM_dMd​, the entrywise bound (AD)ij≥2⌊d⌋AiiAjjDiiDjj(AD)_{ij}\ge 2\lfloor\sqrt{d}\rfloor\sqrt{A_{ii}A_{jj}D_{ii}D_{jj}}(AD)ij​≥2⌊d​⌋Aii​Ajj​Dii​Djj​​ for all i,ji,ji,j guarantees that ΦA,B,C∘ΦD,E,FΦ_{A,B,C}\circΦ_{D,E,F}ΦA,B,C​∘ΦD,E,F​ is EB. In dimension four the universal coefficient becomes four. A continuous family of bistochastic PPT channels is then constructed; every member with a≠1a\ne1a=1 is not EB, while every pairwise composition within the family is proved to be EB. A rank-sensitive refinement shows explicitly how low coherence rank can improve the universal coefficient. The resulting certificate is sufficient rather than necessary and is complementary to factor-width and semidefinite-hierarchy approaches.
We prove that the composition of two PPT maps through an intermediate qubit system is entanglement breaking. Specifically, for arbitrary finite dimensions m and n, if Φ: Mₘ → M₂ and Ψ: M₂ → Mₙ are completely positive and completely copositive, then Ψ ∘ Φ is entanglement breaking. The proof uses the special 2 × 2 block structure imposed by the qubit intermediate system, Schur-complement inequalities, an operator interpolation argument, and an explicit separability decomposition. As a consequence, no counterexample to the PPT-squared conjecture can occur through the 4 → 2 → 4 composition pattern.
Positive but not completely positive maps provide one of the most direct ways to detect entanglement beyond the positive-partial-transpose (PPT) criterion. We introduce and analyze an exactly solvable two-parameter family of sparse bistochastic positive maps on qutrits, in which two coherence channels are independently tuned by parameters www and zzz. The sparse structure makes the full phase diagram analytic: positivity holds exactly on the square 0≤w,z≤2/30\le w,z\le2/30≤w,z≤2/3, complete positivity on the smaller square 0≤w,z≤1/30\le w,z\le1/30≤w,z≤1/3, and decomposability is lost precisely outside a quarter circle in the corner w,z≥1/3w,z\ge1/3w,z≥1/3. The indecomposable region is certified by explicit PPT entangled state adapted to the same witness geometry. At the endpoint W∗=W(2/3,2/3)W_*=W(2/3,2/3)W∗​=W(2/3,2/3) we construct a four-parameter family of PPT edge states of rank type (5,5)(5,5)(5,5), derive their analytic detection region, and show that the corresponding rays are exposed faces of the PPT cone. Finally, although W∗W_*W∗​ is not optimal, we give an explicit optimal refinement whose detection region on this family is strictly larger. The result is an analytically tractable qutrit setting in which positivity, indecomposability, PPT entanglement, optimality, and exposed convex geometry can be studied in a single framework.
The PPT2 conjecture asserts that composites of PPT (completely positive and completely copositive) maps are entanglement breaking. For PPT maps Φ: Md1 → M3 and Ψ: M3 → Md2 let W be the Choi matrix of Ψ ∘ Φ. With the theorem that composites of PPT maps through a qubit are entanglement breaking (the qubit theorem), it is known that W + (1/5) Φ*(I)⊤ ⊗ Ψ(I) is separable, while PPT2 through a qutrit is the same statement with 1/5 replaced by 0. We ask how far 1/5 can be lowered. Certificates built from the qubit theorem independently of the pair cannot lower it, and for d1, d2 ≥ 3 with max(d1, d2) ≥ 4 neither can averages of certificates that apply maps separately to the input, middle and output spaces, even with PPT2 on M3 added. The least λ for which J(Φ) + λ Φ*(I)⊤ ⊗ I is a sum of PPT operators supported on ℂd1 ⊗ V for planes V is the one-sided constant λQD(Φ) ≤ 1/5; by the qubit theorem it is valid for every partner. It is below 1/5 when one compression of the middle qutrit to a qubit is strictly PPT of full rank, and its largest value Λ(d) over d1 = d is below 1/5 unless some extreme PPT channel is rigid, that is, has λQD = 1/5. We prove, without the qubit theorem, that no PPT channel M3 → M3 is rigid, so that some c < 1/5 makes ρ + c I ⊗ I separable for the Choi matrix ρ of every PPT channel M3 → M3. With the qubit theorem, W + c Φ*(I)⊤ ⊗ Ψ(I) is then separable for all PPT maps Φ: M3 → M3 and Ψ: M3 → Md2, and exact rational certificates beat 1/5 at explicit factors with d1 = 4. The exact computations used in the proofs are also checked in Lean 4. This paper develops Section 10, Appendix A and supplement S12-S13 of version 3 of "Diagonal-unitary twirls of composites of PPT maps through a qutrit" (doi:10.5281/zenodo.23675449) with a new introduction and preliminaries; its supplement S1-S2 is S12-S13 there. The statements and proofs are those of that version, renumbered, with references to the companion paper, version 4 of that record (doi:10.5281/zenodo.23680469). Programs, certificates and Lean checks are in the deposit, version 4 (doi:10.5281/zenodo.23680464). Dagger k of this paper is dagger k+24 in version 3 and in the deposit's folders, map files, replay record and Lean targets; the deposit's README uses this paper's numbering. Mathematics Subject Classification (2020): 81P40; 15A69, 47L07.
We analyze linear maps on matrix algebras that become entanglement breaking after composing a finite or infinite number of times with themselves. This means that the Choi matrix of the iterated linear map becomes separable in the tensor product space. If a linear map becomes entanglement breaking after finitely many iterations, we say that the map has a finite index of separability. In particular, we show that every unital positive partial transpose (PPT) channel has a finite index of separability and that the class of unital channels that have a finite index of separability is a dense subset of the unital channels. We construct concrete examples of maps which are not PPT but have a finite index of separability. We prove that there is a large class of unital channels that are asymptotically entanglement breaking. Our work is motivated by Christandl’s PPT-squared conjecture. This conjecture states that every PPT channel, when composed with itself, becomes entanglement breaking.