A classification of incomparable states

Let (\{| ψ> ,| ϕ>}) be an incomparable pair of states ((| ψ\nleftrightarrow | ϕ>)), \emph, i.e., (| ψ>) and (| ϕ>) 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 (| ψ> ^{\otimes k+1}\to | ϕ> ^{\otimes k+1}), i.e., (k+1) copies of (| ψ>) can be transformed to (k+1) copies of (| ϕ>) with probability one by LOCC but (| ψ> ^{\otimes n}\nleftrightarrow | ϕ> ^{\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}(| ψ> ^{\otimes m}\to | ϕ> ^{\otimes m})) can decrease exponentially with the number of source states (m), even if the source state has \emph{more} entropy of entanglement.

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