Formalize Open Quantum Problem #44: Complexity of the separability problem

### What is the conjecture

This is problem [#44](https://oqp.iqoqi.oeaw.ac.at/complexity-of-the-separability-problem) in the [Open Quantum Problems collection](https://oqp.iqoqi.oeaw.ac.at/open-quantum-problems), a later addition to Reinhard F. Werner’s original open-problems project.

This is one of the OQP items that is most naturally phrased as an **open complexity-classification problem** rather than a single yes/no conjecture.

> **Problem (Open Quantum Problem #44: “Complexity of the separability problem”).**
> Let `Sep(d,d)` denote the set of separable density matrices on `ℂ^d ⊗ ℂ^d`.
> Given a bipartite mixed state `ρ`, promised either that `ρ ∈ Sep(d,d)` or that `ρ` is at least `ε` away in trace distance from every separable state, determine the computational complexity of deciding which case holds as a function of `d` and `ε`.

A standard formalization is as an **`ε`-weak membership problem** for the convex set of separable states:

* Fix the bipartite Hilbert space `ℂ^d ⊗ ℂ^d`.
* Let `D(d,d)` be the set of density operators on `ℂ^d ⊗ ℂ^d` (positive semidefinite, trace `1`).
* Let
`Sep(d,d) = conv { α ⊗ β : α, β are density operators on ℂ^d }`.
* Define the trace-distance-to-separable-set by
`dist_tr(ρ, Sep(d,d)) := inf_{σ ∈ Sep(d,d)} (1/2) * ||ρ - σ||_1`.

Then the promise problem is:

* **Input:** a density matrix `ρ ∈ D(d,d)` (given classically by its `d^2 × d^2` matrix entries) and a parameter `ε > 0`.
* **YES case:** `ρ ∈ Sep(d,d)`.
* **NO case:** `dist_tr(ρ, Sep(d,d)) ≥ ε`.
* **Promise:** one of the two cases above holds.

The main open problem is:

* **Complexity question:** determine the computational complexity of this promise problem as a function of the local dimension `d` and the accuracy parameter `ε`.

The OQP page also highlights why this matters:

* The separability problem is tightly connected to quantum proof systems with unentangled witnesses, especially `QMA(2)`.
* If for some constant `ε > 0` there were a quasipolynomial-time algorithm (roughly `d^{O(log d)}`) for this trace-norm weak membership problem, then `QMA(2) ⊆ EXP`.
* If such an algorithm also admitted a sufficiently parallel implementation, then `QMA(2) ⊆ PSPACE`.

The OQP page also records benchmark partial results / nearby relaxations:

* `ε`-weak membership for `Sep(d,d)` is NP-hard for inverse-polynomial `ε`.
* There is a quasipolynomial-time algorithm for related weak-membership formulations where distance is measured in **LOCC norm** (and also Euclidean norm), rather than trace norm.
* A related optimization problem,
`max_{σ ∈ Sep(d,d)} Tr(Qσ)`,
admits algorithms whose runtime can depend exponentially on the Frobenius norm `||Q||_F`.

### Where to find the details / references

Primary sources:

* [Open Quantum Problems site (Problem #44)](https://oqp.iqoqi.oeaw.ac.at/complexity-of-the-separability-problem)
* [Open Quantum Problems master list (for numbering/metadata)](https://oqp.iqoqi.oeaw.ac.at/open-quantum-problems)
* For background on the original OQP project (note that the 2005 snapshot only covers Problems `1–29`):
O. Krüger & R. F. Werner, “Some Open Problems in Quantum Information Theory”:
[https://arxiv.org/abs/quant-ph/0504166](https://arxiv.org/abs/quant-ph/0504166)

Key related references (as listed on the OQP page):

* H. Blier & A. Tapp, “All languages in NP have very short quantum proofs” (2009; arXiv:0709.0738)
* S. Aaronson, S. Beigi, A. Drucker, B. Fefferman, and P. Shor, “The power of unentanglement” (Theory Comput. 5 (2009); arXiv:0804.0802)
* J. Chen & A. Drucker, “Short multi-prover quantum proofs for SAT without entangled measurements” (arXiv:1011.0716)
* A. W. Harrow & A. Montanaro, “Testing product states, quantum Merlin-Arthur games and tensor optimisation” (J. ACM 60(1), 2013; arXiv:1001.0017)
* L. Gurvits, “Classical deterministic complexity of Edmonds’ problem and quantum entanglement” (STOC 2003; arXiv:quant-ph/0303055)
* S. Gharibian, “Strong NP-hardness of the quantum separability problem” (Quantum Inf. Comput. 10, 343–360 (2010); arXiv:0810.4507)
* F. G. S. L. Brandão, M. Christandl, and J. Yard, “A quasipolynomial-time algorithm for the quantum separability problem” (STOC 2011; arXiv:1011.2751)
* Y. Shi & X. Wu, “Epsilon-net method for optimizations over separable states” (Theor. Comput. Sci. 598, 51–63 (2015); arXiv:1112.0808)

(Useful background survey, not explicitly listed on the OQP page but very helpful for formalization context:)

* L. M. Ioannou, “Computational complexity of the quantum separability problem” (Quantum Inf. Comput. 7(4), 335–370 (2007); arXiv:quant-ph/0603199)

### Prerequisites needed

* Finite-dimensional quantum mechanics: density matrices, tensor products, partial traces
* Separability vs entanglement; convex decompositions into product states
* Matrix analysis: positivity, trace norm / Schatten `1`-norm, Frobenius norm
* Convex geometry / optimization: convex hulls, distance to a convex set, weak membership vs weak optimization
* Classical complexity theory: decision and promise problems, polynomial vs quasipolynomial time, NP-hardness
* Quantum complexity theory: `QMA`, `QMA(2)`, unentangled witnesses, LOCC measurements
* (Optional) Semidefinite programming and symmetric-extension hierarchies

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

* ams-81 (Quantum theory)
* ams-68 (Computer science)
* ams-52 (Convex and discrete geometry)
* ams-94 (Information and communication 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

Start with Open Quantum Problems Problem #44 and the cited references to understand the separability promise problem and its prerequisites. Then review the repository's existing formalized conjectures; done would mean adding this conjecture to the repository, but the issue does not identify a file, entry point, or specific formalization scope.

Written by the indexing model from the issue text.