bibkey: chiribelladavidsonpaulsenrahaman2023real authors: Giulio Chiribella; Kenneth R. Davidson; Vern I. Paulsen; Mizanur Rahaman year: 2023 title: “Positive maps and entanglement in real Hilbert spaces” doi: 10.1007/s00023-023-01325-x url: https://arxiv.org/abs/2207.02510v2 claim: “Section 9.2 asks whether the complex PPT-squared and real and complex IPT-squared conjectures are equivalent.” strata_touched:
- D5/S3/Quantum/QuantumChannels/IptPptSquaredEquivalence license: citation-only triage: anchor
Verified locator
DOI: https://doi.org/10.1007/s00023-023-01325-x
Source: https://arxiv.org/abs/2207.02510v2
Source statements
Section 9.2, Conjecture 9.5, PDF p. 23:
For every PPT map Φ : Mₙ(ℂ) → Mₙ(ℂ), Φ ◦ Φ is ℂ-entanglement breaking.
Section 9.2, Conjecture 9.7, PDF p. 24:
For 𝕂 ∈ {ℝ, ℂ}, every IPT map Φ : Mₙ(𝕂) → Mₙ(𝕂) satisfies Φ ◦ Φ is 𝕂-entanglement breaking.
Section 9.2, after Proposition 9.9, PDF p. 25:
We do not know if these conjectures are equivalent.
Section 9.1, PDF p. 21:
A completely positive map Φ : Mₙ(𝕂) → Mₘ(𝕂) is called PPT (IPT) if its Choi matrix is PPT (IPT).
The preceding definitions say that a positive matrix P is PPT when (τₙ ⊗ idₘ)(P) ≥ 0, and IPT when (τₙ ⊗ idₘ)(P) = P. Section 8, PDF p. 18, defines 𝕂-entanglement p-breaking by (Φ ⊗ idᵣ)(P) ∈ SEPₚ(𝕂ᵐ ⊗ 𝕂ʳ) for every positive P and r ≥ 1, and states:
For p = 1, we just call the map Φ 𝕂-entanglement breaking.
The separable cone is generated by rank-one products xx* ⊗ yy*; coefficients are absorbed into their vectors. Theorem 8.1 gives the Choi and measure-and-prepare characterizations over both fields. Proposition 9.2 identifies CSEP ∩ IPT with real separability. Proposition 9.9 gives PPT²(ℂ) ⇒ IPT²(ℂ) ⇒ IPT²(ℝ).
The Lean encoding uses Fin n for each matrix index, RCLike scalars, input-first Choi coordinates, and ptFirst for τₙ ⊗ idₘ. Universal quantification includes n = 0 and r = 0; those cases are vacuous and do not strengthen the positive-dimensional conjectures.
Applicability
The equivalence question concerns all dimensions, without a trace-preserving or unital hypothesis. It does not assert that any of the three conjectures is true. Chen, Yang and Tang, arXiv:1807.03636v2, Theorem 3 (PDF p. 2), prove the complex PPT-squared conjecture for n = 3. Their Section II also recalls the n = 2 case from the Peres–Horodecki criterion. Proposition 9.9 of the primary source transfers the dimension-three conclusion to IPT² over both fields. The equivalence proof preserves the original implications in each fixed dimension and sends IPT² in dimension 2n to complex PPT² in dimension n in the converse direction. The dimension-doubling converse does not transfer a dimension-three IPT² result to dimension three PPT².