Finite-Function-Encoding Quantum States

Abstract

We investigate the encoding of higher-dimensional logic into quantum states. To that end we introduce finite-function-encoding (FFE) states which encode arbitrary d-valued logic functions and investigate their structure as an algebra over the ring of integers modulo d. We point out that the polynomiality of the function is the deciding property for associating hypergraphs to states. Given a polynomial, we map it to a tensor-edge hypergraph, where each edge of the hypergraph is associated with a tensor. We observe how these states generalize the previously defined qudit hypergraph states, especially through the study of a group of finite-function-encoding Pauli stabilizers. Finally, we investigate the structure of FFE states under local unitary operations, with a focus on the bipartite scenario and its connections to the theory of complex Hadamard matrices.