Formalize Open Quantum Problem #17: Reversibility of entanglement assisted coding

### What is the conjecture

This is problem [#17](https://oqp.iqoqi.oeaw.ac.at/reversibility-of-entanglement-assisted-coding) 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 Bennett–Devetak–Harrow–Shor–Winter; the OQP page also cites an independent concurrent proof by Berta–Christandl–Renner; see below).

> **Problem (Open Quantum Problem #17: “Reversibility of entanglement assisted coding”).**
> For quantum channels `S` and `T`, let `C_E(T,S)` denote the optimal asymptotic rate `r` such that `n` uses of `S` can simulate `rn` uses of `T`, with vanishing error, using arbitrary coding/decoding operations together with unlimited shared entanglement between sender and receiver.
> Show that `C_E(T,S) = C_E(S,T)^(-1)`.

Concretely, for a finite-dimensional quantum channel `N : B(H_A) → B(H_B)`, write `C_EA(N)` for its usual entanglement-assisted classical capacity:
`C_EA(N) = max_{ρ_A} I(R:B)_{(id_R ⊗ N)(ψ_RA)}`,
where `ψ_RA` is a purification of `ρ_A`, and `I(R:B) = H(R) + H(B) - H(RB)` is quantum mutual information (with base-2 logs).

The reversibility question asks whether every channel becomes, under free shared entanglement, asymptotically equivalent to a noiseless classical channel with exchange rate determined by `C_EA`. In the nondegenerate case `C_EA(S), C_EA(T) > 0`, this would amount to
`C_E(T,S) = C_EA(S) / C_EA(T)`,
and hence
`C_E(T,S) C_E(S,T) = 1`.

The standard motivating example is the comparison between the ideal classical 1-bit channel `S_1` and the ideal qubit channel `id_2`:
- without entanglement assistance, `C(S_1,id_2)=1` but `C(id_2,S_1)=0`;
- with entanglement assistance, superdense coding gives `C_E(S_1,id_2)=2`, while teleportation gives `C_E(id_2,S_1)=1/2`.

**Solved statement to formalize (BSST + Quantum Reverse Shannon Theorem):**
For finite-dimensional quantum channels, the standard entanglement-assisted classical capacity is given by the single-letter BSST formula
`C_EA(N) = max_{ρ_A} I(R:B)_{(id_R ⊗ N)(ψ_RA)}`.
Moreover, the quantum reverse Shannon theorem shows that `N` can be asymptotically simulated using noiseless classical communication at rate `C_EA(N)` bits per channel use, together with unlimited shared entanglement. Therefore, for any finite-dimensional channels `S,T` with `C_EA(S), C_EA(T) > 0`,
`C_E(T,S) = C_EA(S) / C_EA(T)`,
and hence
`C_E(T,S) = C_E(S,T)^(-1)`.

**Important nuance (from the solution literature):**
For tensor-power / i.i.d. source models, standard shared ebits suffice for the simulation. For fully general source-channel simulation settings, the proofs cited on the OQP page use more flexible auxiliary entanglement such as embezzling states (or equivalent auxiliary resources such as backward communication).

### Where to find the details / references

Primary sources:
- [Open Quantum Problems site (Problem #17)](https://oqp.iqoqi.oeaw.ac.at/reversibility-of-entanglement-assisted-coding)
- [Solved Quantum Problems list (shows #17 solved)](https://oqp.iqoqi.oeaw.ac.at/solved-quantum-problems)
- [Werner / Krüger–Werner arXiv list](https://arxiv.org/abs/quant-ph/0504166) (see “Problem 17”)

Key solution references (as cited on the OQP page):
- C. H. Bennett, I. Devetak, A. W. Harrow, P. W. Shor, and A. Winter, **“The Quantum Reverse Shannon Theorem and Resource Tradeoffs for Simulating Quantum Channels”**, *IEEE Trans. Inf. Theory* 60(5), 2926–2959 (2014); arXiv: [0912.5537](https://arxiv.org/abs/0912.5537)
- M. Berta, M. Christandl, and R. Renner, **“The quantum reverse Shannon theorem based on one-shot information theory”**, *Commun. Math. Phys.* 306(3), 579–615 (2011); arXiv: [0912.3805](https://arxiv.org/abs/0912.3805)

Key background / precursor references mentioned on the OQP page:
- C. H. Bennett, P. W. Shor, J. A. Smolin, and A. V. Thapliyal, **“Entanglement-assisted classical capacity of noisy quantum channels”**, *Phys. Rev. Lett.* 83, 3081 (1999); arXiv: [quant-ph/9904023](https://arxiv.org/abs/quant-ph/9904023)
- C. H. Bennett, P. W. Shor, J. A. Smolin, and A. V. Thapliyal, **“Entanglement-assisted capacity of a quantum channel and the reverse Shannon theorem”**, *IEEE Trans. Inf. Theory* 48(10), 2637–2655 (2002); arXiv: [quant-ph/0106052](https://arxiv.org/abs/quant-ph/0106052)
- W. van Dam and P. Hayden, **“Universal entanglement transformations without communication”**, *Phys. Rev. A* 67, 060302 (2003); arXiv: [quant-ph/0201041](https://arxiv.org/abs/quant-ph/0201041)

### Prerequisites needed

- Quantum channels: completely positive trace-preserving maps; Stinespring dilations / isometric extensions; tensor powers
- Shared entanglement as a resource; teleportation and superdense coding
- Entropic quantities: von Neumann entropy, purification, quantum mutual information
- Entanglement-assisted classical capacity theorem (BSST) and its single-letter formula
- Asymptotic channel simulation / resource inequalities; error notions for channel simulation (e.g. worst-case / diamond-norm style formulations)
- If formalizing the solution in full generality: one-shot information theory (state splitting / merging, post-selection) or the representation-theoretic approach of Bennett–Devetak–Harrow–Shor–Winter; embezzling states and related auxiliary resources

### [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
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Begin by reading the BSST and Quantum Reverse Shannon references linked in the issue, along with the stated prerequisites and resource assumptions. The work is done when the solved reversibility statement, including its finite-dimensional and positive-capacity conditions, is formalized in the repository with the relevant supporting results.

Written by the indexing model from the issue text.