Formalize Open Quantum Problem #29: Entanglement of formation for Gaussian states

### What is the conjecture

This is problem [#29](https://oqp.iqoqi.oeaw.ac.at/entanglement-of-formation-for-gaussian-states) 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 #29: “Entanglement of formation for Gaussian states”).**
> Entanglement of formation is defined as a minimum over all convex decompositions of a bipartite state into pure states (see Problem 7). It has been shown that for certain two-mode Gaussian states this minimum can be taken over decompositions of the given state into pure states, all of which are translates of the same squeezed Gaussian state with Gaussian weights. Show (or disprove) that this is true for all Gaussian states.

A standard formalization is in terms of **bipartite bosonic Gaussian states** and the **convex-roof definition** of entanglement of formation:

* Fix a bipartite continuous-variable system with `n_A` bosonic modes for Alice and `n_B` bosonic modes for Bob.
* A state `ρ_AB` is **Gaussian** iff its Wigner function / characteristic function is Gaussian; equivalently, it is specified by first moments together with a covariance matrix `V` satisfying the canonical commutation relation uncertainty constraints.
* For a generic bipartite state, the **entanglement of formation** is
`E_F(ρ_AB) := inf { ∑_i p_i S(Tr_B |ψ_i⟩⟨ψ_i|) : ρ_AB = ∑_i p_i |ψ_i⟩⟨ψ_i| }`,
where `S` is the von Neumann entropy.
* In the continuous-variable / infinite-dimensional setting one should in principle allow more general integral decompositions (probability measures over pure states), but for Gaussian states this gives the same quantity.
* The **Gaussian entanglement of formation** `E_F^G(ρ_AB)` is the same infimum, but restricted to decompositions into **pure Gaussian states**.

The OQP page highlights that for Gaussian states, such Gaussian-restricted decompositions can be taken in the special form

* `ρ_AB = ∫ D(ξ) |ψ_G⟩⟨ψ_G| D(ξ)^† dμ_G(ξ)`

where `|ψ_G⟩` is a fixed pure Gaussian state, `D(ξ)` denotes phase-space displacement (“translation”) operators, and `μ_G` is a Gaussian probability measure on phase space.

With this, the central question can be phrased as:

* **Main question:** decide whether
`E_F(ρ_AB) = E_F^G(ρ_AB)`
for **every** bipartite Gaussian state `ρ_AB`.

Equivalently:

* Does every bipartite Gaussian state admit an **optimal** entanglement-of-formation decomposition consisting only of displaced copies of a single pure Gaussian state with Gaussian weights?
* In the covariance-matrix formulation, `E_F^G` is obtained by optimizing over pure Gaussian covariance matrices `V_p` with `V_p ≤ V`; the question is whether this Gaussian-restricted optimization always already equals the true `E_F`.

The OQP page also records important partial progress:

* For **symmetric two-mode Gaussian states**, the equality `E_F = E_F^G` is known.
* More strongly, the OQP page states that the conjecture has now been proved for **all two-mode Gaussian states**.
* Thus the genuinely remaining open direction is the **general multimode bipartite case**.

This is also the natural version to formalize: the Gaussian-restricted quantity `E_F^G` has a concrete covariance-matrix optimization formulation, while the unrestricted `E_F` is a harder infinite-dimensional convex-roof problem.

### Where to find the details / references

Primary sources:

* [Open Quantum Problems site (Problem #29)](https://oqp.iqoqi.oeaw.ac.at/entanglement-of-formation-for-gaussian-states)
* [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 29):
[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 (as listed on the OQP page):

* A. Uhlmann, “Entropy and optimal decompositions of states relative to a maximal commutative subalgebra” (Open Syst. Inf. Dyn. 5, 209–228 (1998); arXiv: quant-ph/9704017)
(background on optimal decompositions / convex roofs)
* M. E. Shirokov, “On properties of quantum channels related to their classical capacity” (Theory Probab. Its Appl. 52, 250–276 (2008))
(background on continuous ensembles / integral decompositions in infinite dimensions)
* M. M. Wolf, G. Giedke, O. Krüger, R. F. Werner, and J. I. Cirac, “Gaussian entanglement of formation” (Phys. Rev. A 69, 052320 (2004); arXiv: quant-ph/0306177)
(introduces Gaussian entanglement of formation and the covariance-matrix optimization viewpoint)
* G. Giedke, M. M. Wolf, O. Krüger, R. F. Werner, and J. I. Cirac, “Entanglement of formation for symmetric Gaussian states” (Phys. Rev. Lett. 91, 107901 (2003); arXiv: quant-ph/0304042)
(proves `E_F = E_F^G` for symmetric two-mode Gaussian states)
* J. Solomon Ivan and R. Simon, “Entanglement of Formation for Gaussian States” (arXiv: 0808.1658)
(computes the arbitrary two-mode case under an additional conjecture)
* P. Marian and T. A. Marian, “Entanglement of formation for an arbitrary two-mode Gaussian state” (Phys. Rev. Lett. 101, 220403 (2008); arXiv: 0809.0321)
(claimed a full two-mode solution; the OQP page notes that this proof was later judged incomplete)
* Y. Akbari-Kourbolagh and H. Alijanzadeh-Boura, “On the entanglement of formation of two-mode Gaussian states: a compact form” (Quantum Inf. Process. 14, 4179–4199 (2015); arXiv: 1402.3321)
(per the OQP page, removes the symmetry assumption and proves the conjecture for all two-mode Gaussian states)

(If you want to include a brief “recent progress” pointer section, there are also newer papers on explicit formulas / bounds / squeezing-based descriptions for two-mode Gaussian entanglement, but the OQP page above plus the Gaussian-EoF papers and the two-mode solution references are the core references for writing the issue.)

### Prerequisites needed

* Continuous-variable / bosonic quantum mechanics: modes, quadrature operators, canonical commutation relations
* Gaussian states and Gaussian unitaries; covariance matrices, symplectic linear algebra / Williamson normal forms
* Bipartite entanglement measures: entropy of entanglement for pure states, convex roofs, entanglement of formation
* Basic functional analysis / operator theory for infinite-dimensional Hilbert spaces; trace-class states and probability-measure decompositions
* (Optional) Optimization viewpoint on Gaussian states via covariance-matrix inequalities

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

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

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