Finite-Function-Encoding Quantum States

TL;DR

This paper introduces finite-function-encoding states that encode multivariate functions over finite rings, explores their structural properties, and classifies their entanglement under local operations, revealing connections to hypergraph states and Hadamard matrices.

Contribution

It develops a new framework for encoding quantum states with finite functions, analyzes their stabilizers, and classifies bipartite states under local unitaries and Pauli operations.

Findings

FFE states encode arbitrary finite-valued functions.

Classification of bipartite FFE states under LU and LFP is achieved for specific dimensions.

Connections established between FFE states, hypergraph states, and complex Hadamard matrices.

Abstract

We introduce finite-function-encoding (FFE) states which encode arbitrary $d$-valued logic functions, i.e., multivariate functions over the ring of integers modulo $d$, and investigate some of their structural properties. We also point out some differences between polynomial and non-polynomial function encoding states: The former can be associated to graphical objects, that we dub tensor-edge hypergraphs (TEH), which are a generalization of hypergraphs with a tensor attached to each hyperedge encoding the coefficients of the different monomials. To complete the framework, we also introduce a notion of finite-function-encoding Pauli (FP) operators, which correspond to elements of what is known as the generalized symmetric group in mathematics. First, using this machinery, we study the stabilizer group associated to FFE states and observe how qudit hypergraph states introduced in arXiv:1612.06418v2 admit stabilizers of a particularly simpler form. Afterwards, we investigate the classification of FFE states under local unitaries (LU), and, after showing the complexity of this problem, we focus on the case of bipartite states and especially on the classification under local FP operations (LFP). We find all LU and LFP classes for two qutrits and two ququarts and study several other special classes, pointing out the relation between maximally entangled FFE states and complex Butson-type Hadamard matrices. Our investigation showcases also the relation between the properties of FFE states, especially their LU classification, and the theory of finite rings over the integers.