Lockability of two-way distillable entanglement
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- Topics
Problem
Can discarding one local qubit reduce two-way distillable entanglement by an arbitrarily large amount? Let \(D_{\leftrightarrow}(A:B)_\rho\) denote the asymptotic singlet-distillation rate of \(\rho_{AB}\) under local operations and unrestricted two-way classical communication. The question is whether there are finite-dimensional states \(\rho^{(r)}_{A_ra:B_r}\), with \(\dim a=2\), such that
Equation (1) requires an unbounded loss while the discarded subsystem has fixed dimension two.
Source
Horodecki et al. prove lockability of one-way distillable entanglement and explicitly leave the two-way quantity unresolved [HHHO05].
Progress
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Horodecki, Horodecki, Horodecki, and Oppenheim constructed families in which discarding one qubit causes an unbounded loss of entanglement of formation, entanglement cost, logarithmic negativity, or one-way distillable entanglement. These results do not establish Eq. (1), which allows unrestricted two-way communication [HHHO05].
The same work proved that discarding one qubit decreases the relative entropy of entanglement by at most two ebits, but explicitly left lockability of unrestricted distillable entanglement open. No corresponding universal bound is known for \(D_{\leftrightarrow}\) [HHHO05].
Comment
The remaining gap is to construct a family satisfying Eq. (1) or prove a dimension-independent upper bound on the loss of \(D_{\leftrightarrow}\). The one-way analogue is already known to be lockable.