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PPT-square conjecture for block Schur products

Let MAM_A be a matrix algebra, and let ϱ1,ϱ2∈MA⊗MA\varrho_1,\varrho_2\in M_A\otimes M_A be states. Let ϱ1□ϱ2∈MA⊗(MA⊗MA)\varrho_1\square\varrho_2\in M_A\otimes(M_A\otimes M_A) denote their block Schur product, followed by block-wise summation. A state is PPT if it has positive partial transpose, and is separable if it belongs to the separable cone. PPT-square conjecture. For all states ϱ1,ϱ2∈MA⊗MA\varrho_1,\varrho_2\in M_A\otimes M_A, if both ϱ1\varrho_1 and ϱ2\varrho_2 are PPT, then the block-wise summation of the block Schur product ϱ1□ϱ2\varrho_1\square\varrho_2 is separable. This is a state-level reformulation of the PPT-square conjecture and concerns whether the PPT condition is sufficient for separability after this block-wise product operation.

Sources & referencesView supporting material

Primary source

Mark Girard, Seung-Hyeok Kye and Erling Størmer, “Convex cones in mapping spaces between matrix algebras”, arXiv:2002.09614 (2020).

positive-mapsquantum-informationseparability

Progress summary

Refreshed
Open

The conjecture remains unproved in general: only special dimensions and restricted families are known.

Proposed by Christandl, the conjecture says that combining two states satisfying the positive-partial-transpose condition by the stated block operation always produces a separable state. Equivalent formulations concern whether composing two PPT\mathrm{PPT} maps is always entanglement breaking.

Known results

  • The n=3n=3 case is proved, using that 3×33\times3 states with positive partial transpose have Schmidt number at most two (Johnston, 2019).
  • The conjecture holds for certain graph-based Schur-product maps (2017).
  • For unital PPT\mathrm{PPT} maps, some finite iterate is entanglement breaking; related iterates approach the entanglement-breaking cone for unital or trace-preserving maps.
  • A generic-result paper establishes the conjecture for some induced-state regimes, but not in general.

Current status (as of August 2026): The n=3n=3 case and several restricted or generic families are settled, but the general block-Schur-product PPT\mathrm{PPT}-square conjecture has no recorded proof or counterexample.

Sources

Solutions 0

No solutions have been posted yet.