# Locality from the Spectrum

## TL;DR

This paper demonstrates that the energy spectrum of a quantum system generally encodes a unique local structure, enabling the identification of subsystems and interactions solely from spectral data, despite some dualities.

## Contribution

It shows that the local degrees of freedom in quantum systems are almost always uniquely determined by their spectra, allowing reconstruction of local structure from spectral data.

## Key findings

- Energy spectrum typically encodes unique local degrees of freedom.
- Dual local descriptions can exist but are rare.
- Hamiltonians can often be expressed in local form from spectral information.

## Abstract

Essential to the description of a quantum system are its local degrees of freedom, which enable the interpretation of subsystems and dynamics in the Hilbert space. While a choice of local tensor factorization of the Hilbert space is often implicit in the writing of a Hamiltonian or Lagrangian, the identification of local tensor factors is not intrinsic to the Hilbert space itself. Instead, the only basis-invariant data of a Hamiltonian is its spectrum, which does not manifestly determine the local structure. This ambiguity is highlighted by the existence of dualities, in which the same energy spectrum may describe two systems with very different local degrees of freedom. We argue that in fact, the energy spectrum alone almost always encodes a unique description of local degrees of freedom when such a description exists, allowing one to explicitly identify local subsystems and how they interact. As a consequence, we can almost always write a Hamiltonian in its local presentation given only its spectrum. In special cases, multiple dual local descriptions can be extracted from a given spectrum, but generically the local description is unique.