[Paper Review] Entanglement of free-fermion systems, signal processing and algebraic combinatorics

Pierre-Antoine Bernard, Nicolas Crampé+3 more|arXiv (Cornell University)|Jan 13, 2024

Molecular spectroscopy and chirality4 citations

TL;DR

This paper develops a unified framework for analyzing entanglement in free-fermion systems on graphs by leveraging signal processing and algebraic combinatorics. It introduces a commuting tridiagonal Heun operator and uses the Terwilliger algebra of P-polynomial association schemes to simplify diagonalization of the chopped correlation matrix, enabling exact computation of entanglement entropy in systems like the hypercube and Hamming graphs.

ABSTRACT

This paper offers a review of recent studies on the entanglement of free-fermion systems on graphs that take advantage of methods pertaining to signal processing and algebraic combinatorics. On the one hand, a parallel with time and band limiting problems is used to obtain a tridiagonal matrix commuting with the chopped correlation matrix in bispectral situations and on the other, the irreducible decomposition of the Terwilliger algebra arising in the context of $P$-polynomial association schemes is seen to yield a simplifying framework.

Motivation & Objective

• To establish a systematic method for computing entanglement entropy in free-fermion systems on graphs with complex symmetries.
• To bridge quantum information theory with algebraic combinatorics by linking entanglement to association schemes and Terwilliger algebras.
• To simplify the diagonalization of the chopped correlation matrix—central to entanglement entropy—by exploiting commuting operators in bispectral and symmetric settings.
• To extend known results on entanglement in spin chains and hypercubes to broader classes of graphs, including Hamming, Johnson, and folded cube graphs.
• To identify conditions under which the Terwilliger algebra enables irreducible decomposition that reduces the entanglement computation to lower-dimensional subproblems.

Proposed method

• The entanglement entropy is computed via the formula $\mathfrak{S} = -\text{Tr}[C\log C + (1-C)\log(1-C)]$, where $C = \Pi_S \Pi_E \Pi_S$ is the chopped correlation matrix restricted to subsystem 1.
• A commuting tridiagonal Heun operator $\bar{T} = \{A, A^*\} + \mu A^* + \nu A$ is constructed such that $[\bar{T}, C] = 0$, enabling simultaneous diagonalization.
• The structure of $P$-polynomial and $Q$-polynomial association schemes is used to define the adjacency matrix $A$ and dual adjacency matrix $A^*$, which generate the Terwilliger algebra $\mathfrak{T}$.
• The regular representation of $\mathfrak{T}$ on $\mathbb{C}^{|V|}$ is decomposed into irreducible components, reducing the entanglement computation to block-diagonal subspaces.
• For the binary Hamming scheme, $A$ and $A^*$ are identified with Pauli matrices $\sigma_x$ and $\sigma_z$, and the algebraic structure reduces to $\mathfrak{su}(2)$, enabling exact solutions.
• Wavefunctions between energy and position bases are shown to solve bispectral problems, linking the system to orthogonal polynomials and time-band limiting problems.

Experimental results

Research questions

• RQ1 How can the entanglement entropy of free-fermion systems on symmetric graphs be computed exactly despite the complexity of the chopped correlation matrix?
• RQ2 What role does the Terwilliger algebra play in simplifying the diagonalization of the correlation matrix in association schemes?
• RQ3 In which classes of graphs (e.g., Hamming, Johnson, folded cubes) does a commuting Heun operator exist that preserves the structure of the entanglement computation?
• RQ4 How do the symmetries of $P$- and $Q$-polynomial association schemes lead to a decomposition of the entanglement problem into irreducible subspaces?
• RQ5 Can the framework be generalized to $P$-multivariate association schemes and non-self-dual graphs?

Key findings

• A commuting Heun operator $\bar{T}$ is constructed such that $[\bar{T}, C] = 0$, allowing exact diagonalization of the chopped correlation matrix $C$ in bispectral settings.
• For $P$-polynomial association schemes, the Terwilliger algebra $\mathfrak{T}$ is generated by $A$ and $A^*$, and its irreducible decomposition simplifies the entanglement analysis into lower-dimensional blocks.
• In the binary Hamming scheme $H_d = (K_2)^{\square d}$, the adjacency and dual adjacency matrices correspond to $\sigma_x$ and $\sigma_z$, and the algebra reduces to $\mathfrak{su}(2)$, enabling exact solutions via known spin chain techniques.
• The entanglement entropy computation for the hypercube reduces to a combination of Krawtchouk chains, with exact results obtainable via orthogonal polynomial methods.
• The framework applies to graphs such as Hamming, Johnson, Hadamard, and folded cubes, with the entanglement structure determined by the algebraic properties of the underlying association scheme.
• The wavefunctions $\phi_n(k) = \langle n|\omega_k\rangle$ are shown to be solutions of bispectral problems, linking the system to time- and band-limiting problems in signal processing.