Formalize Open Quantum Problem #22: Asymptotic cloning is state estimation

### What is the conjecture

This is problem [#22](https://oqp.iqoqi.oeaw.ac.at/asymptotic-cloning-is-state-estimation) 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).

**Status:** this problem is listed as **solved** on the [Solved Quantum Problems](https://oqp.iqoqi.oeaw.ac.at/solved-quantum-problems) page (solved by J. Bae and A. Acín, 2006; see below).

> **Problem (Open Quantum Problem #22: “Asymptotic cloning is state estimation”).**
> Fix an arbitrary probability measure on the pure states of a `d`-dimensional quantum system. Let `F(N,M)` be the optimal **single-copy fidelity** for `N`-to-`M` cloning transformations, averaged with respect to the given probability measure and over all `M` output clones.
> On the other hand, let `F(N,∞)` be the best mean fidelity achievable by measuring the `N` input copies of the state and re-preparing a state according to the measured data.
> Decide whether one always has
> `lim_{M→∞} F(N,M) = F(N,∞)`.

Concretely, let `H ≅ C^d`, let `μ` be a probability measure on the pure states of `H`, and write `ψ = |ψ><ψ|`. For a quantum channel `T : B(H^{⊗ N}) → B(H^{⊗ M})`, let
`ρ^(k)_ψ = Tr_{≠k} T(ψ^{⊗ N})`
denote the `k`-th output marginal. Then the optimal average single-copy cloning fidelity is
`F(N,M) = sup_T ∫ μ(dψ) (1/M) Σ_{k=1}^M <ψ|ρ^(k)_ψ|ψ>.`

The corresponding estimation / measure-and-prepare benchmark is
`F(N,∞) = sup_{ {E_j}, {σ_j} } ∫ μ(dψ) Σ_j Tr[E_j ψ^{⊗ N}] <ψ|σ_j|ψ>,`
where `{E_j}` is a POVM on `H^{⊗ N}` and `σ_j` are output states on `H` (equivalently one may take `σ_j = |φ_j><φ_j|` pure).

It is immediate that `F(N,M)` is non-increasing in `M`, so the limit `lim_{M→∞} F(N,M)` exists. Also `F(N,M) ≥ F(N,∞)` for every `M`, since a measure-and-prepare protocol is a particular cloning strategy. The question is whether asymptotic cloning can ever do **strictly better** than optimal state estimation. The OQP page also notes a weaker variant asking only whether the equality at least holds in the limit `N→∞`.

**Solved statement to formalize (Bae–Acín, 2006):**
For every finite-dimensional pure-state ensemble / prior `μ` and every fixed number `N` of input copies,
`lim_{M→∞} F(N,M) = F(N,∞).`

A convenient structural formulation of the proof is the following. One may restrict to **symmetric** optimal cloners. If `T_M^c` denotes the **single-clone marginal channel** of an optimal symmetric `N`-to-`M` cloner `T_M`, then the Choi state `J(T_M^c)` is `M`-extendible / `M`-shareable. By compactness in finite dimensions, any subsequential limit of `(J(T_M^c))_M` is infinitely extendible; by Werner’s 1989 theorem, every infinitely shareable bipartite state is **separable**. Equivalently, every asymptotic single-clone channel is **entanglement-breaking**, hence of **measure-and-prepare** form. Therefore asymptotic optimal cloning reduces exactly to optimal state estimation, and the two fidelities coincide.

**Useful strengthening (also mentioned on the OQP page):**
Shortly after, Chiribella and D’Ariano showed a finite-`M` estimate of the form
`|F(N,M) - F(N,∞)| ≤ c/M`,
for a positive constant `c` depending on the single-particle Hilbert-space dimension. Chiribella later proved an `O(1/M)` **diamond-norm** approximation of the one-clone marginal by a suitable measure-and-prepare channel.

**Important scope note:** this problem is about **average single-copy fidelity** (equivalently, the one-particle marginal of the cloner), not the stronger question of globally approximating the full `M`-partite output state.

### Where to find the details / references

Primary sources:
- [Open Quantum Problems site (Problem #22)](https://oqp.iqoqi.oeaw.ac.at/asymptotic-cloning-is-state-estimation)
- [Solved Quantum Problems list (shows #22 solved)](https://oqp.iqoqi.oeaw.ac.at/solved-quantum-problems)
- [Werner / Krüger–Werner arXiv list](https://arxiv.org/abs/quant-ph/0504166) (see “Problem 22”)

Key solution reference (as cited on the OQP page):
- J. Bae and A. Acín, **“Asymptotic quantum cloning is state estimation”**, *Phys. Rev. Lett.* 97, 030402 (2006); arXiv: [quant-ph/0603078](https://arxiv.org/abs/quant-ph/0603078)

Related / stronger results mentioned on the OQP page (useful background for a formalization roadmap):
- M. Keyl and R. F. Werner, *J. Math. Phys.* 40, 3283 (1999) — optimal cloning with the single-clone figure of merit
- D. Bruß, M. Cinchetti, G. M. D’Ariano, and C. Macchiavello, **“Phase-covariant quantum cloning”**, *Phys. Rev. A* 62, 012302 (2000); arXiv: [quant-ph/9909046](https://arxiv.org/abs/quant-ph/9909046)
- R. F. Werner, **“An application of Bell’s inequalities to a quantum state extension problem”**, *Lett. Math. Phys.* 17, 359–363 (1989)
- G. Chiribella and G. M. D’Ariano, **“Quantum information becomes classical when distributed to many users”**, *Phys. Rev. Lett.* 97, 250503 (2006); arXiv: [quant-ph/0608007](https://arxiv.org/abs/quant-ph/0608007)
- G. Chiribella, **“On quantum estimation, quantum cloning and finite quantum de Finetti theorems”**, in *Theory of Quantum Computation, Communication, and Cryptography*, LNCS 6519, 9–25 (2011); arXiv: [1010.1875](https://arxiv.org/abs/1010.1875)

### Prerequisites needed

- Quantum cloning channels `B(H^{⊗ N}) → B(H^{⊗ M})`, symmetry / symmetrization of cloners, and reduced one-clone marginals
- State estimation from `N` copies; POVMs; measure-and-prepare channels; fidelity as the figure of merit
- Choi–Jamiołkowski representation of channels
- `k`-extendibility / shareability of bipartite states; Werner’s theorem that infinite shareability implies separability
- Entanglement-breaking channels and their equivalence with measure-and-prepare maps
- Basic finite-dimensional compactness / subsequence arguments for states or channels

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

- ams-81 (Quantum theory)
- ams-94 (Information and communication theory)
- 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

The payload names no repository files or tests. Start with the Bae–Acín reference and the listed prerequisites, then inspect existing formalized conjectures in formal-conjectures for the appropriate Lean entry point. Done means adding the stated asymptotic cloning equality as a verified formal statement.

Written by the indexing model from the issue text.