Formalize Open Quantum Problem #2: Undistillability implies PPT?

### What is the conjecture

This is problem [#2](https://oqp.iqoqi.oeaw.ac.at/undistillability-implies-ppt) in [Reinhard F. Werner's collection](https://arxiv.org/abs/quant-ph/0504166), later [collected by a community of quantum researchers](https://oqp.iqoqi.oeaw.ac.at/open-quantum-problems).

> **Problem (Open Quantum Problem #2: “Undistillability implies PPT?”).**
> It is well-known that states with positive partial transpose (PPT) are not distillable. The problem is to decide the converse.

Concretely, in standard bipartite entanglement theory:

* A bipartite state is represented by a **density operator** `ρ` on `H_A ⊗ H_B` (positive semidefinite, trace 1), where `H_A, H_B` are finite-dimensional complex Hilbert spaces.
* The **partial transpose** `ρ^{T_A}` (with respect to a fixed product basis) is defined by matrix elements
`⟨ij | ρ^{T_A} | kl⟩ = ⟨kj | ρ | il⟩`.
* `ρ` is **PPT** iff `ρ^{T_A} ≥ 0` (i.e. `ρ^{T_A}` is positive semidefinite / has no negative eigenvalues).
* `ρ` is **distillable** if there exists some `n ≥ 1` such that `ρ^{⊗ n}` can locally be projected onto an **entangled two-qubit state**: i.e. there exist two-dimensional projectors `P` (Alice) and `Q` (Bob) acting on `H_A^{⊗ n}` and `H_B^{⊗ n}` such that
`((P ⊗ Q) ρ^{⊗ n} (P ⊗ Q))^{T_A}` has at least one **negative eigenvalue**.
(If this holds for a given `n`, the state is called `n`-distillable.)

It is already known that **PPT ⇒ undistillable**. The open direction is the converse:

* **Conjecture / Open question:** `undistillable(ρ) ⇒ PPT(ρ)`.

Equivalently: **every NPPT/NPT state is distillable** (i.e. there are *no* “NPT bound entangled states”).

The problem page also records (optional) alternative reformulations in terms of **2-positivity** of certain maps (via the Horodecki distillability criterion).

### Where to find the details / references

Primary sources:

* [Open Quantum Problems site (Problem #2)](https://oqp.iqoqi.oeaw.ac.at/undistillability-implies-ppt)
* [Open Quantum Problems master list (for numbering/metadata)](https://oqp.iqoqi.oeaw.ac.at/open-quantum-problems)
* O. Krüger & R. F. Werner, “Some Open Problems in Quantum Information Theory” (Problem 2):
[https://arxiv.org/abs/quant-ph/0504166](https://arxiv.org/abs/quant-ph/0504166)
[https://arxiv.org/pdf/quant-ph/0504166](https://arxiv.org/pdf/quant-ph/0504166)

Key related references mentioned on the problem page (selection):

* M. Horodecki, P. Horodecki, R. Horodecki, “Mixed-State Entanglement and Distillation: Is there a ‘Bound’ Entanglement in Nature?” (1998), arXiv: quant-ph/9801069
* M. Horodecki & P. Horodecki, “Reduction criterion of separability and limits for a class of protocols of entanglement distillation” (Phys. Rev. A 59, 4206 (1999)), arXiv: quant-ph/9708015
* D. P. DiVincenzo et al., “Evidence for bound entangled states with negative partial transpose” (Phys. Rev. A 61, 062312 (2000)), arXiv: quant-ph/9910026
* W. Dür et al., “Distillability and transposition in bipartite systems” (Phys. Rev. A 61, 062313 (2000)), arXiv: quant-ph/9910022

### Prerequisites needed

* Finite-dimensional quantum mechanics: density matrices, tensor products, partial trace
* Entanglement basics: separability vs entanglement; PPT/NPPT (partial transpose criterion)
* LOCC and distillation: the operational notion of distillability; the Horodecki distillability criterion (2-qubit projections / “negative partial transpose after projection”)
* Linear algebra / matrix analysis: positive semidefinite operators, eigenvalues, projections
* (Optional, for the map-based reformulation) positive maps, complete positivity, `k`-positivity / 2-positivity

### [AMS categories](https://github.com/google-deepmind/formal-conjectures/labels?q=ams-)

* ams-81 (Quantum theory)
* ams-47 (Operator theory)
* ams-15 (Linear and multilinear algebra; matrix theory)

### Choose either option

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