Formalize Open Quantum Problem #36: Composition of decoherence functionals

### What is the conjecture

This is problem [#36](https://oqp.iqoqi.oeaw.ac.at/composition-of-decoherence-functionals) in the [Open Quantum Problems collection](https://oqp.iqoqi.oeaw.ac.at/open-quantum-problems).

> **Problem (Open Quantum Problem #36: “Composition of decoherence functionals”).**
> In a histories-based / quantum-measure-theory framework, identify a set of decoherence functionals that is closed under tensor products and that contains elements which are not strongly positive.
> Equivalently, find a decoherence functional `D` with `D \not\ge 0` such that `D^{\otimes n}` satisfies weak positivity for every `n ≥ 1`.

A standard finite-dimensional formalization is as follows:

* Fix a finite **joint measurement framework** with experiments `x = 1, ..., m`, outcome sets `A_x`, and **history space**
`Ω = A_1 × ... × A_m`.
* Let `H_Ω = ℂ^Ω` with orthonormal basis `{|a⟩ : a ∈ Ω}`.
* For any event `A ⊆ Ω`, define the event vector
`|A⟩ = ∑_{a∈A} |a⟩`.
* A **decoherence functional** can be represented (in this finite setting) by a Hermitian operator `D` on `H_Ω`, equivalently by the sesquilinear form `D(A,B) = ⟨A|D|B⟩`.

The axioms are:

* **Hermiticity:** `D = D^†`.
* **Weak positivity:** `⟨A|D|A⟩ ≥ 0` for every `A ⊆ Ω`.
* **Normalization:** `⟨Ω|D|Ω⟩ = 1`.
* **Strong decoherence on measurable partitions:** if `{A_k}_k` is a partition of `Ω` corresponding to a feasible experiment, then
`⟨A_k|D|A_j⟩ = P(A_k) δ_{kj}`.

The key distinction is between the following positivity notions:

* **Weak positivity** tests `D` only on event vectors `|A⟩` (equivalently, on `0-1` indicator vectors).
* **Strong positivity** requires `D ≥ 0` as a positive semidefinite matrix on all vectors of `H_Ω`.

For independent systems with history spaces `Ω_1, Ω_2, ...` and decoherence functionals `D_1, D_2, ...`, the composite system is modeled on
`Ω = Ω_1 × Ω_2 × ...`
with decoherence functional
`D = D_1 ⊗ D_2 ⊗ ...`.

The difficulty is that weak positivity is **not** preserved under tensor products in general, while strong positivity **is**.

So the natural formal open problem is:

* **Main question / open existence problem:** does there exist a valid decoherence functional `D` such that
* `D` is not strongly positive (`D \not\ge 0`), but
* `D^{\otimes n}` is weakly positive for every `n ≥ 1`?

Equivalent reformulation:

* **Tensor-closed class formulation:** characterize the class of decoherence functionals whose tensor powers remain weakly positive for all `n`, and decide whether this class is strictly larger than the strongly positive class.

Important partial results already known:

* For every fixed `n`, there exists a decoherence functional `D` such that `D^{\otimes (n-1)}` is weakly positive but `D^{\otimes n}` is not.
* The set of strongly positive decoherence functionals cannot be enlarged while preserving closure under tensor products if one requires the enlarged class to contain all quantum decoherence functionals.
* The OQP page also records a sharper obstruction: there is a family `F` of quantum decoherence functionals such that any tensor-product-closed set containing `F` can contain only strongly positive elements.

### Where to find the details / references

Primary sources:

* [Open Quantum Problems site (Problem #36)](https://oqp.iqoqi.oeaw.ac.at/composition-of-decoherence-functionals)
* [Open Quantum Problems master list (for numbering/metadata)](https://oqp.iqoqi.oeaw.ac.at/open-quantum-problems)
* P. Boës and M. Navascués, “Composing decoherence functionals” (Phys. Rev. A 95, 022114 (2017); [arXiv: 1609.09723](https://arxiv.org/abs/1609.09723))

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

* F. Dowker, J. Henson, and P. Wallden, “A histories perspective on characterising quantum non-locality” (New J. Phys. 16, 033033 (2014); arXiv: 1311.6287)
* R. D. Sorkin, “Quantum Mechanics as Quantum Measure Theory” (Mod. Phys. Lett. A 9, 3119–3128 (1994); arXiv: gr-qc/9401003)
* X. Martin, D. O’Connor, and R. D. Sorkin, “The Random Walk in Generalized Quantum Theory” (Phys. Rev. D 71, 024029 (2005); arXiv: gr-qc/0403085)
* J. B. Hartle, “The Quantum Mechanics of Cosmology” (Jerusalem Winter School lectures; arXiv: 1805.12246)

(If you want to include a brief “recent progress” pointer section, there is also Fay Dowker and Henry Wilkes, “An argument for strong positivity of the decoherence functional in the path integral approach to the foundations of quantum theory” (AVS Quantum Science 4, 012601 (2022); arXiv: 2011.06120), which extends the composability/maximality discussion to infinite systems and argues further for strong positivity.)

### Prerequisites needed

* Finite-dimensional Hilbert spaces, tensor products, Hermitian operators
* Basic histories / quantum-measure-theory language: events, coarse grainings, partitions, decoherence functionals
* Linear algebra of quadratic forms and positive semidefinite matrices
* Difference between weak positivity (on event vectors) and strong positivity (PSD on all vectors)
* (Optional) quantum foundations / generalized quantum mechanics / contextuality background

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

* ams-81 (Quantum theory)
* ams-28 (Measure and integration)
* ams-15 (Linear algebra)
* ams-47 (Operator 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