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.
