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Locality from the Spectrum

Jordan S. Cotler, Geoffrey R. Penington, Daniel H. Ranard

TL;DR

This work investigates whether a natural, basis-independent tensor product structure (TPS) of a Hilbert space can be recovered from the Hamiltonian spectrum, thereby identifying local subsystems from spectral data. It shows that generic Hamiltonians are not local in any TPS, but if a Hamiltonian is local in some TPS, that TPS is generically unique, so the spectrum often determines the local degrees of freedom. The authors prove two main theorems: (i) an infinitesimal version implying a finite number of duals, and (ii) a global, algebraic-geometry version implying a constant (often zero) number of duals across a subspace; they support these results with numerical examples in spin chains. They further generalize the TPS concept to nets of observables suitable for fermions and gauge theories and discuss implications for quantum simulation, geometry on TPS, and potential links to quantum gravity and the SYK model. Overall, the paper argues that locality, as encoded by the spectrum, can be a robust and informative guide to the natural subdivision of a quantum system into interacting subsystems.

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.

Locality from the Spectrum

TL;DR

This work investigates whether a natural, basis-independent tensor product structure (TPS) of a Hilbert space can be recovered from the Hamiltonian spectrum, thereby identifying local subsystems from spectral data. It shows that generic Hamiltonians are not local in any TPS, but if a Hamiltonian is local in some TPS, that TPS is generically unique, so the spectrum often determines the local degrees of freedom. The authors prove two main theorems: (i) an infinitesimal version implying a finite number of duals, and (ii) a global, algebraic-geometry version implying a constant (often zero) number of duals across a subspace; they support these results with numerical examples in spin chains. They further generalize the TPS concept to nets of observables suitable for fermions and gauge theories and discuss implications for quantum simulation, geometry on TPS, and potential links to quantum gravity and the SYK model. Overall, the paper argues that locality, as encoded by the spectrum, can be a robust and informative guide to the natural subdivision of a quantum system into interacting subsystems.

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.

Paper Structure

This paper contains 23 sections, 2 theorems, 44 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Defining Tensor Product Structures
  3. Defining Locality
  4. Existence and Uniqueness of Local TPS
  5. Existence of local TPS
  6. Uniqueness of local TPS
  7. Finite number of duals
  8. Constant number of duals
  9. Numerics
  10. Example showing finite duals
  11. Example showing no duals
  12. Discovering dualities
  13. Generalizations of Tensor Product Structures
  14. Discussion
  15. Summary of results
  16. ...and 8 more sections

Key Result

Theorem 1

Suppose that we have a subspace $S$ of Hermitian matrices $\textrm{Herm}(N)$ together with some subgroup $G \subset U(N)$ that preserves $S$ when acting by conjugation. Moreover suppose that there exists a matrix $H_0 \in S$ whose Jordan form is the generic Jordan formBy generic Jordan form on $S$, for some Hermitian matrix $V$ then either $[V,H_0] = 0$ or $V \in \mathfrak{g}$, where $\mathfrak{g

Figures (2)

  • Figure 1: We depict the spaces $S$ and $\textrm{Orb}_{U(N)}(H)$ intersecting in the ambient space $\textrm{Herm}(\mathcal{H})$. The Hamiltonian $H$ is depicted to have no duals. The orbit intersects $S$ in multiple disconnected components, appearing as circles in the diagram. These disconnected components together make up $\textrm{Orb}_{G}(H) =S \cap \textrm{Orb}_{U(N)}(H).$ Each disconnected component of $S \cap \textrm{Orb}_{U(N)}(H)$ contains Hamiltonians related by local unitaries (which are continuous transformations), and the sets are related to one another by permutations of qubits and transposition, which are discrete transformations. Alternatively, if the intersection contained points not related to $H$ by local unitaries, permutations, or transposition, then $H$ would have duals. The figure is only intended as a schematic representation of the spaces involved.
  • Figure 2: We visualize what cannot happen, namely the intersection of $\textrm{Orb}_{G}(H)$ and $S$ contains both isolated points and open sets. The directions of $S$ which correspond to local unitary perturbations of the Hamiltonian are suppressed.

Theorems & Definitions (2)

  • Theorem 1: Finite number of duals
  • Theorem 2: Constant number of duals