Homological Tools for the Quantum Mechanic
This paper is an introduction to work motivated by the question "can multipartite entanglement be detected by homological algebra?" We introduce cochain complexes associated to multipartite density states whose cohomology detects factorizability. The $k$th cohomology components of such cochain complexes produce tuples of $(k+1)$-body operators that are non-locally correlated due to the non-factorizability of the state. Associated Poincare polynomials are invariant under local invertible linear transformations (automorphisms that decompose as tensor products). These complexes can be considered as a step toward realizing mutual information as an Euler characteristic. We motivate the definition of the "state index" associated to a multipartite state: a three-parameter function which is: invariant under local invertible transformations, well-behaved under tensor products of states, and interpolates between multipartite mutual information and the integer-valued Euler characteristics of our complexes. We compute cohomologies and state indices of multipartite W and GHZ states. The approach in this paper is directed toward practitioners of finite-dimensional quantum mechanics, although the machinery developed generalizes far beyond. Some results are applicable in infinite dimensions and should be generalizable to the context of quantum field theory. In order to compensate for the long length, a detailed summary is provided in the introduction section.