Impossibility of Local State Transformation via Hypercontractivity

TL;DR

This work introduces two quantum hypercontractivity ribbons as necessary conditions governing asymptotic local state transformation under local CPTP maps. By mapping bipartite states to CP super-operators via the Choi–Jamio2kowski isomorphism and employing complex interpolation, the authors derive data-processing-type properties and show that ribbons contract under local processing, thereby blocking certain transformations when the resource ribbon is larger than the target. A key result is a concrete impossibility bound: transforming $\rho_{AB}^{(\alpha)}$ to $\zeta_{UV}$ is impossible for $\alpha< (1-\log 2/\log 3)^{1/2} \approx 0.6075$, which cannot be captured by simpler entanglement measures. The paper also links ribbons to maximal correlation and log-Sobolev techniques, computes ribbons for product and pure states, and discusses implications for quantum information theory and potential extensions to quantum Rényi divergences.

Abstract

Local state transformation is the problem of transforming an arbitrary number of copies of a bipartite resource state to a bipartite target state under local operations. That is, given two bipartite states, is it possible to transform an arbitrary number of copies of one of them to one copy of the other state under local operations only? This problem is a hard one in general since we assume that the number of copies of the resource state is arbitrarily large. In this paper we prove some bounds on this problem using the hypercontractivity properties of some super-operators corresponding to bipartite states. We measure hypercontractivity in terms of both the usual super-operator norms as well as completely bounded norms.

Theorems & Definitions (30)

• Theorem 1
• Theorem 2
• Theorem 3
• Definition 4
• Remark 5
• Remark 6
• Remark 7
• Remark 8
• Lemma 9
• proof