Formalize Open Quantum Problem #25: Lockable entanglement measures

[MarioKrenn6240]

What is the conjecture

This is problem #25 in Reinhard F. Werner’s collection, later collected by a community of quantum researchers.

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)
• Open Quantum Problems master list (for numbering/metadata)
• O. Krüger & R. F. Werner, “Some Open Problems in Quantum Information Theory” (Problem 25):
  https://arxiv.org/abs/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

• 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
• This issue is up for grabs: I would like to see this conjecture added by somebody else

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