Multi-directional unitarity and maximal entanglement in spatially symmetric quantum states
We consider dual unitary operators and their multi-leg generalizations
that have appeared at various places in the literature. These objects
can be related to multi-party quantum states with special entanglement
patterns: the sites are arranged in a spatially symmetric pattern and
the states have maximal entanglement for all bipartitions that follow
from the reflection symmetries of the given geometry. We consider
those cases where the state itself is invariant with respect to the
geometrical symmetry group. The simplest examples are those dual
unitary operators which are also self dual and reflection invariant,
but we also consider the generalizations in the hexagonal, cubic, and
octahedral geometries. We provide a number of constructions and
concrete examples for these objects for various local dimensions. All
of our examples can be used to build quantum cellular automata in 1+1
or 2+1 dimensions, with multiple equivalent choices for the
"direction of time".