Multinegativity and single-letter formulas for asymptotic entanglement

Abstract

The study of entanglement inevitably leads to the concept of regularization, where measures are evaluated on infinitely many copies of a state. Single-letter formulas try to make these asymptotic entanglement quantities accessible through a calculation on one copy of a state, but are rarely available. We introduce a decreasing hierarchy of computable upper bounds on the asymptotic relative entropy of entanglement with respect to positive-partial-transpose (PPT) states. The regularization of every fixed hierarchy level equals this asymptotic quantity. Additivity at any level therefore yields a single-letter formula, even when the usual one-copy relative entropy is nonadditive. We establish such additivity at the first nontrivial level for two broad multiparameter families, both containing all Werner states. The upper-bound construction also extends to sandwiched Renyi divergences. The hierarchy motivates $k$-multinegative states, a generalization of binegative states and the associated $k$-multinegativity, which gives explicit upper bounds on both the asymptotic relative entropy and exact PPT entanglement cost. We construct states which are $k$-multinegative at arbitrarily large depths $k$ that separate consecutive levels of the previously introduced entanglement-cost hierarchy, disproving its conjectured finite collapse. These results provide state-dependent single-letter formulas while identifying a limitation of universal finite-level characterizations.