Entanglement purification of unknown quantum states - art. no. 042309

A concern has been expressed that "the Jaynes principle can produce fake en tanglement" [R. Horodecki et al., Phys. Rev. A 59, 1799 (1999)]. In this pa per we discuss the general problem of distilling maximally entangled states from N copies of a bipartite quantum system about which only partial infor mation is known, for instance, in the form of a given expectation value. We point out that there is indeed a problem with applying the Jaynes principl e of maximum entropy to more than one copy of a system, but the nature of t his problem is classical and was discussed extensively by Jaynes. Under the additional assumption that the state rho ((N)) of the N copies of the quan tum system is exchangeable, one can write down a simple general expression for rho ((N)). By measuring one or more of the subsystems, one can gain inf ormation and update the state estimate for the remaining subsystems with th e quantum version of the Bayes rule. Using this rule. we show how to modify two standard entanglement purification protocols, one-way hashing and recu rrence, so that they can be applied to exchangeable states. We thus give an explicit algorithm for distilling entanglement from an unknown or partiall y known quantum state.