Generalized distillability conjecture and generalizations of Cauchy-Bunyakovsky-Schwarz inequality and Lagrange identity

TL;DR

This paper introduces a new conjecture about the 1-indistillability of tensor products of critical Werner states, relates it to hypermatrix inequalities generalizing classical identities, and proves some cases.

Contribution

It proposes the Generalized Distillability Conjecture (GDC), links it to hypermatrix inequalities, and extends classical identities to a hypermatrix and integral framework.

Findings

GDC holds when at most one dimension d_k is greater than 2.

Reformulates GDC as a hypermatrix inequality generalizing Cauchy-Bunyakovsky-Schwarz.

Derives an integral version of the hypermatrix Lagrange identity.

Abstract

Let rho_k, k=1,2,...,m, be the critical Werner state in a bipartite d_k by d_k quantum system, i.e., the one that separates the 1-distillable Werner states from those that are 1-indistillable. We propose a new conjecture (GDC) asserting that the tensor product of rho_k is 1-indistillable. This is much stronger than the familiar conjecture saying that a single critical Werner state is indistillable. We prove that GDC is true for arbitrary m provided that d_k is bigger than 2 for at most one index k. We reformulate GDC as an intriguing inequality for four arbitrary complex hypermatrices of type d_1 x ... x d_m. This hypermatrix inequality is just the special case n=2 of a more general conjecture (CBS conjecture) for 2n arbitrary complex hypermatrices of the same type. Surprisingly, the case n=1 turns out to be quite interesting as it provides hypermatrix generalization of the classical Lagrange identity. We also formulate the integral version of the CBS conjecture and derive the integral version of the hypermatrix Lagrange identity.