Regular language quantum states

TL;DR

Regular language states (RLS) fuse formal language theory with quantum many-body physics by associating regular languages to quantum state superpositions; they are amenable to an MPS description with constant bond dimension via unambiguous automata, and a canonical DFA-based embedding yields a unique representation. A fundamental LU-equivalence theorem for sparse RLS reduces equivalence to permutation relabelings and local unitaries, while efficient criteria for translational and shift-invariance arise from the MPS/algebraic framework. The work also establishes canonical decompositions for RLs, discusses the potential for richer coefficient structures, and sketches a path to 2D generalizations, signaling a fruitful cross-fertilization between tensor networks and automata theory with potential broader impact in quantum information and many-body physics. The results open new avenues for classifying and manipulating structured quantum states using language-theoretic tools and tensor-network techniques, and they suggest practical routes to analyze symmetry, interconvertibility, and higher-dimensional generalizations.

Abstract

We introduce regular language states, a family of quantum many-body states. They are built from a special class of formal languages, called regular, which has been thoroughly studied in the field of computer science. They can be understood as the superposition of all the words in a regular language and encompass physically relevant states such as the GHZ-, W- or Dicke-states. By leveraging the theory of regular languages, we develop a theoretical framework to describe them. First, we express them in terms of matrix product states, providing efficient criteria to recognize them. We then develop a canonical form which allows us to formulate a fundamental theorem for the equivalence of regular language states, including under local unitary operations. We also exploit the theory of tensor networks to find an efficient criterion to determine when regular languages are shift-invariant.

Figures (1)

• Figure 1: Example of how the partition in Eq. \ref{['eq:sets_Li_intersection']} looks like in the proof of Theorem \ref{['prop:canonical_decomposition_RL']}, given three initial sets $L_{x_1 \dots x_m}, L_{y_1 \dots y_m}, L_{z_1 \dots z_m}$.

Theorems & Definitions (22)

• Definition 1
• Theorem 1
• Lemma 1
• Lemma 1
• Corollary 1
• Theorem 2: Canonical decomposition of a RL
• Corollary 3
• Definition 2: Canonical form of RLS
• Theorem 4: Fundamental theorem of sparse RLS
• Theorem 4