Homological Tools for the Quantum Mechanic

Understand

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.