On the number of spurious memories in the Hopfield model

The outer-product method for programming the Hopfield model is discussed. The method can result in many spurious stable states-exponential in the number of vectors that are to be stored-even in the case when the vectors are orthogonal

Optimal Encryption of Quantum Bits

We characterize the complete set of protocols that may be used to securely encrypt n quantum bits using secret and random classical bits. In addition to the application of such quantum encryption protocols to quantum data security, our framework allows for generalizations of many classical cryptographic protocols to quantum data. We show that the encrypted state gives no information without the secret classical data, and that 2n random classical bits are the minimum necessary for informationally secure quantum encryption. Moreover, the quantum operations are shown to have a surprising structure in a canonical inner product space. This quantum encryption protocol is a generalization of the classical one time pad concept. A connection is made between quantum encryption and quantum teleportation, and this allows for a new proof of optimality of teleportation.Comment: 12 pages, LaTeX. Replacement fixes incorrect reference [15], otherwise identica

A classification of incomparable states

Let (\{| \psi> ,| \phi>}) be an incomparable pair of states ((| \psi \nleftrightarrow | \phi>)), \emph, i.e., (| \psi>) and (| \phi>) cannot be transformed to each other with probability one by local transformations and classical communication (LOCC). We show that incomparable states can be multiple-copy transformable, \emph, i.e., there can exist a \emph{k}, such that (| \psi> ^{\otimes k+1}\to | \phi> ^{\otimes k+1}), i.e., (k+1) copies of (| \psi>) can be transformed to (k+1) copies of (| \phi>) with probability one by LOCC but (| \psi> ^{\otimes n}\nleftrightarrow | \phi> ^{\otimes n} \forall n\leq k). We call such states \emph{k}-copy LOCC incomparable. We provide a necessary condition for a given pair of states to be \emph{k}-copy LOCC incomparable for some \emph{k}. We also show that there exist states that are neither \emph{k}-copy LOCC incomparable for any \emph{k} nor catalyzable even with multiple copies. We call such states strongly incomparable. We give a sufficient condition for strong incomparability. We demonstrate that the optimal probability of a conclusive transformation involving many copies, (p_{max}(| \psi> ^{\otimes m}\to | \phi> ^{\otimes m})) can decrease exponentially with the number of source states (m), even if the source state has \emph{more} entropy of entanglement.Comment: Latex, 9 pages, 1 figur