Interpolating between R\'enyi entanglement entropies for arbitrary bipartitions via operator geometric means

TL;DR

This paper introduces a novel method to interpolate between Rényi entanglement entropies for arbitrary bipartitions using operator geometric means, providing new monotones with potential applications in quantum entanglement theory.

Contribution

It develops a new construction of entanglement monotones via regularized Rényi divergence and operator geometric means, extending previous entanglement measures.

Findings

Introduces new subadditive and submultiplicative monotones

Provides explicit operator families for functional interpolation

Establishes bounds and relationships between monotones

Abstract

The asymptotic restriction problem for tensors can be reduced to finding all parameters that are normalized, monotone under restrictions, additive under direct sums and multiplicative under tensor products, the simplest of which are the flattening ranks. Over the complex numbers, a refinement of this problem, originating in the theory of quantum entanglement, is to find the optimal rate of entanglement transformations as a function of the error exponent. This trade-off can also be characterized in terms of the set of normalized, additive, multiplicative functionals that are monotone in a suitable sense, which includes the restriction-monotones as well. For example, the flattening ranks generalize to the (exponentiated) R\'enyi entanglement entropies of order $\alpha\in[0,1]$. More complicated parameters of this type are known, which interpolate between the flattening ranks or R\'enyi entropies for special bipartitions, with one of the parts being a single tensor factor. We introduce a new construction of subadditive and submultiplicative monotones in terms of a regularized R\'enyi divergence between many copies of the pure state represented by the tensor and a suitable sequence of positive operators. We give explicit families of operators that correspond to the flattening-based functionals, and show that they can be combined in a nontrivial way using weighted operator geometric means. This leads to a new characterization of the previously known additive and multiplicative monotones, and gives new submultiplicative and subadditive monotones that interpolate between the R\'enyi entropies for all bipartitions. We show that for each such monotone there exist pointwise smaller multiplicative and additive ones as well. In addition, we find lower bounds on the new functionals that are superadditive and supermultiplicative.