Useful States and Entanglement Distillation

Felix Leditzky, Nilanjana Datta, Graeme Smith

Research output: Contribution to journalArticlepeer-review


We derive general upper bounds on the distillable entanglement of a mixed state under one-way and two-way local operations and classical communication (LOCC). In both cases, the upper bound is based on a convex decomposition of the state into 'useful' and 'useless' quantum states. By 'useful,' we mean a state whose distillable entanglement is non-negative and equal to its coherent information (and thus given by a single-letter, tractable formula). On the other hand, 'useless' states are undistillable, i.e., their distillable entanglement is zero. We prove that in both settings, the distillable entanglement is convex on such decompositions. Hence, an upper bound on the distillable entanglement is obtained from the contributions of the useful states alone, being equal to the convex combination of their coherent informations. Optimizing over all such decompositions of the input state yields our upper bound. The useful and useless states are given by degradable and antidegradable states in the one-way LOCC setting, and by maximally correlated and positive partial transpose (PPT) states in the two-way LOCC setting, respectively. We also illustrate how our method can be extended to quantum channels. Interpreting our upper bound as a convex roof extension, we show that it reduces to a particularly simple, non-convex optimization problem for the classes of isotropic states and Werner states. In the one-way LOCC setting, this non-convex optimization yields an upper bound on the quantum capacity of the qubit depolarizing channel that is strictly tighter than previously known bounds for large values of the depolarizing parameter. In the two-way LOCC setting, the non-convex optimization achieves the PPT-relative entropy of entanglement for both isotropic and Werner states.

Original languageEnglish (US)
Pages (from-to)4689-4708
Number of pages20
JournalIEEE Transactions on Information Theory
Issue number7
StatePublished - Jul 2018
Externally publishedYes


  • Quantum information
  • entanglement distillation
  • quantum entanglement

ASJC Scopus subject areas

  • Information Systems
  • Computer Science Applications
  • Library and Information Sciences


Dive into the research topics of 'Useful States and Entanglement Distillation'. Together they form a unique fingerprint.

Cite this