Formalize Open Quantum Problem #38: The PPT-squared conjecture

[MarioKrenn6240]

What is the conjecture

This is problem #38 in the community-maintained Open Quantum Problems collection.

Problem (Open Quantum Problem #38: “The PPT-squared conjecture”).
Let ρ_AB and ρ_CD be quantum states with positive partial transpose (i.e. they are PPT states), and let M be a positive operator describing a yes/no measurement on the BC-system. Then consider the state on AD, conditional on the result of M being “yes”,
σ_AD = λ Tr_{BC}((ρ_AB ⊗ ρ_CD)(1_A ⊗ M ⊗ 1_D)),
where λ is a normalization factor.
The conjecture states that all such σ_AD are separable.

A standard formalization is as follows:

• Fix finite-dimensional Hilbert spaces H_A, H_B, H_C, H_D.
• ρ_AB and ρ_CD are density operators on H_A ⊗ H_B and H_C ⊗ H_D, respectively.
• A bipartite state ω_XY is PPT iff its partial transpose is positive semidefinite:
  (id_X ⊗ T_Y)(ω_XY) ≥ 0
  (with respect to a fixed local basis on Y).
• A bipartite state ω_XY is separable iff it can be written as a convex combination of product states:
  ω_XY = ∑_i p_i (α_i ⊗ β_i),
  where p_i ≥ 0, ∑_i p_i = 1, and α_i, β_i are local density operators.
• Let M ≥ 0 be a positive operator / postselected effect on H_B ⊗ H_C with nonzero success probability
  p = Tr[(ρ_AB ⊗ ρ_CD)(1_A ⊗ M ⊗ 1_D)] > 0.
  Define the postselected entanglement-swapping output
  σ_AD = p^{-1} Tr_{BC}[(ρ_AB ⊗ ρ_CD)(1_A ⊗ M ⊗ 1_D)].

Then the conjecture is:

• Main conjecture: whenever ρ_AB and ρ_CD are PPT, the swapped state σ_AD is always separable across the bipartition A:D, for every positive operator M with nonzero success probability.

A standard equivalent reformulation is in terms of PPT maps and entanglement-breaking maps (via the Choi–Jamiołkowski correspondence):

• A completely positive map Φ is PPT iff T ∘ Φ (equivalently Φ ∘ T) is completely positive, i.e. its Choi matrix is PPT.
• A completely positive map Φ is entanglement breaking iff its Choi matrix is separable
  (equivalently, in the trace-preserving case, iff (id ⊗ Φ)(τ) is separable for every bipartite state τ).

In that language, the conjecture can be written as:

• Equivalent map formulation: if Φ and Ψ are PPT maps, then Ψ ∘ Φ is entanglement breaking.

The name “PPT-squared” comes from the especially natural special case Ψ = Φ, but the formulation commonly studied in the literature allows two a priori different PPT maps.

Known partial results / current open regime:

• The conjecture is known in several special cases (e.g. for maps on M_2, for the n = 3 / two-qutrit case, for Gaussian channels, for all Choi-type maps, and in certain random/asymptotic regimes).
• The general finite-dimensional case remains open.

Where to find the details / references

Primary sources:

• Open Quantum Problems site (Problem #38)
• Open Quantum Problems master list (for numbering/metadata)
• M. B. Ruskai, M. Junge, D. Kribs, P. Hayden, and A. Winter (eds.), “Operator structures in quantum information theory” (BIRS workshop report 12w5084, 2012):
  https://www.birs.ca/workshops/2012/12w5084/report12w5084.pdf

Key related references (from the OQP page and later core literature):

• K. Horodecki, M. Horodecki, P. Horodecki, and J. Oppenheim, “Secure key from bound entanglement” (Phys. Rev. Lett. 94, 160502 (2005); arXiv: quant-ph/0309110)
  (motivation: PPT states can still carry secret key)
• S. Bäuml, M. Christandl, K. Horodecki, and A. Winter, “Limitations on quantum key repeaters” (Nature Communications 6, 6908 (2015); arXiv: 1402.5927)
  (explains the repeater motivation behind the conjecture)
• M. Christandl and R. Ferrara, “Private states, quantum data hiding and the swapping of perfect secrecy” (Phys. Rev. Lett. 119, 220506 (2017); arXiv: 1609.04696)
  (directly related entanglement-swapping / private-state background; cited on the OQP page)
• M. Kennedy, N. A. Manor, and V. I. Paulsen, “Compositions of PPT Maps” (Quantum Inf. Comput. 18, 472–480 (2018); arXiv: 1710.08475)
  (records the state-vs-map formulation, proves asymptotic results, and discusses the 2×2 case)
• B. Collins, Z. Yin, and P. Zhong, “The PPT square conjecture holds generically for some classes of independent states” (J. Phys. A 51, 425301 (2018); arXiv: 1803.00143)
  (random / asymptotic evidence)
• M. Christandl, A. Müller-Hermes, and M. M. Wolf, “When Do Composed Maps Become Entanglement Breaking?” (Ann. Henri Poincaré 20, 2295–2322 (2019); arXiv: 1807.01266)
  (systematic treatment; proves important special cases including dimension 3 and Gaussian channels; gives equivalent conjectures)
• L. Chen, Y. Yang, and W.-S. Tang, “Positive-partial-transpose square conjecture for n = 3” (Phys. Rev. A 99, 012337 (2019); arXiv: 1807.03636)
  (independent proof of the n = 3 / two-qutrit case; states that the case n ≥ 4 remains open)
• S. Singh and I. Nechita, “The PPT^2 conjecture holds for all Choi-type maps” (Ann. Henri Poincaré 23, 3311–3329 (2022); arXiv: 2011.03809)
  (proves the conjecture for a large structured family of maps including Choi-type maps)

(If you want to include a brief “recent progress” pointer section, see also Sang-Jun Park, “k-Positivity and high-dimensional bound entanglement under symplectic group symmetry” (arXiv:2602.09860, 2026), which proves the conjecture for symplectic-covariant / conjugate-symplectic-covariant PPT maps. But the references above are already enough for writing the issue.)

Prerequisites needed

• Finite-dimensional quantum mechanics: density matrices, tensor products, partial trace, postselected measurements / POVM effects
• Separability vs entanglement; convex decompositions into product states
• Partial transpose and the PPT criterion
• Quantum channels / completely positive maps and the Choi–Jamiołkowski correspondence
• Linear algebra / matrix analysis: positive semidefinite operators, transpose, tensor products, traces
• (Optional) Positive maps, decomposable maps, entanglement-breaking maps, and Schmidt number methods

AMS categories

• ams-81 (Quantum theory)
• ams-15 (Linear and multilinear algebra; matrix theory)
• ams-47 (Operator theory)
• ams-94 (Information and communication theory)

Choose either option

• I plan on adding this conjecture to the repository
• This issue is up for grabs: I would like to see this conjecture added by somebody else

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