Formalize Open Quantum Problem #25: Lockable entanglement measures

### What is the conjecture

This is problem [#25](https://oqp.iqoqi.oeaw.ac.at/lockable-entanglement-measures) 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 #25: “Lockable entanglement measures”).**
> Are two-way distillible entanglement and secret key rate lockable?

A standard formalization is in terms of **single-qubit AB-locking** (loss of one local qubit by Alice or Bob):

* Fix a bipartite state `ρ_AB`, and consider an extension `ρ_Aa:B` (or `ρ_A:Bb`) where the extra subsystem `a` (or `b`) is a **qubit**, i.e. `dim(a)=2` (or `dim(b)=2`).
* For any bipartite resource measure `E`, compare the value of `E` before and after the qubit is lost:
`E(ρ_Aa:B)` versus `E(Tr_a ρ_Aa:B)`.
* `E` is **lockable** (more precisely, **AB-lockable**) iff the loss of a single local qubit can reduce `E` by an arbitrarily large amount, i.e. iff
`sup_{ρ_Aa:B, dim(a)=2} [ E(ρ_Aa:B) - E(Tr_a ρ_Aa:B) ] = +∞`,
and similarly with the extra qubit on Bob’s side.

Equivalent formulations sometimes replace “trace out the qubit” by “measure/dephase one local qubit”; the OQP wording is specifically about **loss** of a qubit.

The two operational quantities in the problem are:

* `D_↔(ρ_AB)` (also written `E_D^↔`): the **two-way distillable entanglement**, i.e. the optimal asymptotic rate at which Alice and Bob can distill maximally entangled pairs from many copies of `ρ_AB` using LOCC with **two-way classical communication**.
* `K_D(ρ_AB)`: the **distillable secret key / secret key rate**, i.e. the optimal asymptotic rate at which Alice and Bob can distill secret bits (equivalently, private states / private bits) from many copies of `ρ_AB`, secure against an adversary holding a purification.

So the open problem can be stated as two concrete questions:

* **Two-way distillable entanglement:** do there exist states `ρ_Aa:B` with `dim(a)=2` for which
`D_↔(ρ_Aa:B) - D_↔(Tr_a ρ_Aa:B)`
is arbitrarily large?
* **Secret key rate:** do there exist states `ρ_Aa:B` with `dim(a)=2` for which
`K_D(ρ_Aa:B) - K_D(Tr_a ρ_Aa:B)`
is arbitrarily large?

Equivalently, if the answer is “no”, one would like a **non-lockability theorem** giving a universal upper bound (or at least an `O(1)` bound) on the loss of `D_↔` and/or `K_D` caused by discarding one local qubit.

Known surrounding facts that make this a particularly natural formalization target:

* Many entanglement/correlation measures are known to be lockable under single-qubit loss, including entanglement of formation, entanglement cost, logarithmic negativity, **one-way** distillable entanglement, and squashed entanglement.
* By contrast, the relative entropy of entanglement is **non-lockable**: losing one qubit can reduce it by at most `2`.
* For secret key there is an important distinction between:
* **E-locking**, where extra information is given to Eve, and
* **AB-locking**, where one of the honest parties loses a local qubit.
The OQP problem is about the second notion. In fact, later work showed that distillable key is **not E-lockable**, while the **AB-lockability** question remains the relevant open one.

### Where to find the details / references

Primary sources:

* [Open Quantum Problems site (Problem #25)](https://oqp.iqoqi.oeaw.ac.at/lockable-entanglement-measures)
* [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 25):
[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:

* K. Horodecki, M. Horodecki, P. Horodecki, and J. Oppenheim, “Locking entanglement measures with a single qubit” (Phys. Rev. Lett. 94, 200501 (2005); arXiv: quant-ph/0404096)
(introduces the locking phenomenon for entanglement measures; proves lockability of `E_F`, `E_C`, logarithmic negativity, and one-way distillable entanglement; shows relative entropy of entanglement is non-lockable)
* M. Christandl and A. Winter, “Uncertainty, Monogamy, and Locking of Quantum Correlations” (IEEE Trans. Inf. Theory 51, 3159–3165 (2005); arXiv: quant-ph/0501090)
(shows squashed entanglement is lockable; also discusses the distinction between lockability questions for entanglement and key)
* C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, “Purification of Noisy Entanglement and Faithful Teleportation via Noisy Channels” (Phys. Rev. Lett. 76, 722–725 (1996); arXiv: quant-ph/9511027)
(standard starting point for entanglement distillation / two-way distillable entanglement)
* I. Devetak and A. Winter, “Distillation of secret key and entanglement from quantum states” (Proc. R. Soc. A 461, 207–235 (2005); arXiv: quant-ph/0306078)
(standard operational reference for distillable entanglement / secret key rates in the one-way setting)
* K. Horodecki, M. Horodecki, P. Horodecki, and J. Oppenheim, “General paradigm for distilling classical key from quantum states” (IEEE Trans. Inf. Theory 55, 1898–1929 (2009); arXiv: quant-ph/0506189)
(standard reference for private states and the formalism of distillable key)
* M. Christandl, A. Ekert, M. Horodecki, P. Horodecki, J. Oppenheim, and R. Renner, “Unifying classical and quantum key distillation” (TCC 2007; arXiv: quant-ph/0608199)
(proves that distillable key is not **E-lockable** and explicitly leaves open whether it is **AB-lockable**)
* G. Gour, “How many ebits can be unlocked with one classical bit?” (Phys. Rev. A 75, 054301 (2007); arXiv: quant-ph/0703246)
(related “unlocking” viewpoint; gives bounds for pure-state ensembles and discusses the connection to distillable-entanglement lockability)
* K. Horodecki, M. Studziński, R. P. Kostecki, O. Sakarya, and D. Yang, “Upper bounds on the leakage of private data and an operational approach to Markovianity” (Phys. Rev. A 104, 052422 (2021); arXiv: 2107.10737)
(partial progress on the key side: proves non-lockability results for certain private-state classes, while the general `K_D` question remains open)

### Prerequisites needed

* Finite-dimensional quantum mechanics: density matrices, purifications, tensor products, partial trace
* Entanglement theory: LOCC, maximally entangled states, entanglement distillation, one-way vs two-way classical communication
* Quantum cryptography / private states: distillable secret key, adversarial purification, public discussion / LOPC viewpoint
* Entropic quantities and continuity ideas: von Neumann entropy, mutual information, conditional mutual information
* (Helpful) Monogamy / no-cloning intuition, and basic examples of locking / data hiding

### [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)
* ams-15 (Linear and multilinear algebra; matrix 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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