Tensor network theory

Abstract: Brain theory is quickly developing into a central focus among disciplines as varied as Neuroscience (Reichardt and Poggio 1981), Machine Intelligence and Robotics (Marr 1982, Palm 1982, Loeb 1983), Computer Science and Mathematics (Grimson 1981), Health Care and Rehabilitation (Mann 1981), and Philosophy and Humanities (Churchland 1985). The unifying power of brain theory, however, is not a recent discovery: the assumption that all brain functions can be more coherently represented once the functioning of the brain itself is basically understood, is as old as brain theory. The desire for a unified view goes back to Plato’s idea that the external world is reflected by forming a single image in the darkness of a cave.

Abstract: Invariant theory is concerned with functions that do not change under the action of a given group. Here we communicate an approach based on tensor networks to represent polynomial local unitary invariants of quantum states. This graphical approach provides an alternative to the polynomial equations that describe invariants, which often contain a large number of terms with coefficients raised to high powers. This approach also enables one to use known methods from tensor network theory (such as the matrix product state (MPS) factorization) when studying polynomial invariants. As our main example, we consider invariants of MPSs. We generate a family of tensor contractions resulting in a complete set of local unitary invariants that can be used to express the R?nyi entropies. We find that the graphical approach to representing invariants can provide structural insight into the invariants being contracted, as well as an alternative, and sometimes much simpler, means to study polynomial invariants of quantum states. In addition, many tensor network methods, such as MPSs, contain excellent tools that can be applied in the study of invariants.