A Classification of Incomparable States

Let {|ψ〉 , |φ〉} be an incomparable pair of states (|ψ〉 = |φ〉), 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, i.e., there can exist a k, such that |ψ〉 → |φ〉, i.e., k+1 copies of |ψ〉 can be transformed to k + 1 copies of |φ〉 with probability one by LOCC but |ψ〉 = |φ〉 ∀n ≤ k. We call such states k -copy LOCC incomparable. We provide a necessary condition for a given pair of states to be k -copy LOCC incomparable for some k. We also show that there exist states that are neither k -copy LOCC incomparable for any 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, pmax ( |ψ〉 → |φ〉 ) can decrease exponentially with the number of source states m, even if the source state has more entropy of entanglement. We also show that the probability of a conclusive conversion might not be a monotonic function of the number of copies. Fascinating developments in quantum information theory [1] and quantum computing [2] during the past decade has led us to view entanglement as a valued physical resource. Consequently, recent studies have largely been devoted towards its quantification in appropriate limits (finite or asymptotic), optimal manipulation, and transformation properties under local operations and classical communication (LOCC) [3, 4, 5, 6, 7, 8]. Since the specific tasks that can be accomplished with entanglement as a resource is closely related to its transformation properties, it is of importance to know what transformations are allowed under LOCC. Suppose Alice and Bob share a pure state |ψ〉 (source state), which they wish to convert to another entangled state |φ〉 (target state) under LOCC. A necessary and sufficient condition for this transformation to be possible with certainty (denoted by |ψ〉 → |φ〉) has been obtained by Nielsen [3]. If such a deterministic transformation is not possible but |ψ〉 has at least as many Schmidt coefficients as |φ〉, then one 1 [email protected], [email protected] 2 [email protected] 3 [email protected]