Locality from the Spectrum (1702.06142v2)

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.

Analyzing Locality from Quantum System Spectra

The paper "Locality from the Spectrum," authored by Jordan S. Cotler, Geoffrey R. Penington, and Daniel H. Ranard, investigates the intricate relationship between the energy spectrum of quantum systems and their locality. The central argument is that, despite the apparent independence of local tensor factorization in the Hilbert space from its energy spectrum, the latter often encodes a unique description of local degrees of freedom. This essay explores the theoretical implications, methodologies, and potential consequences of these findings in quantum information and gravity.

The central theme of the paper is the distinction between Hilbert spaces with explicit locality and those whose local structure is unique once dualities are considered. A prime example is the energy spectrum's role in defining local structures, as demonstrated with the Ising model Hamiltonian. The authors argue that while dualities can allow multiple interpretations of a single Hamiltonian, like those between the Hamiltonian's $\{\sigma_i\}$ and $\{\mu_i\}$ descriptions, generic local Hamiltonians prescribe a unique local description.

The research presented in the paper employs both group theory and algebraic geometry to explore the algebraic structure underlying quantum mechanical systems. Two main results are demonstrated: Almost all Hamiltonians lack any $k$-local tensor product structure (TPS), and for those that do possess one, the local TPS is generically unique. Through Theorems 1 and 2, the authors articulate that the spectrum of a Hamiltonian is often sufficient to determine the Hamiltonian's local TPS, and more profoundly, finite numbers of duals are generically observed in local Hamiltonians.

Numerical methods complement the theoretical findings, demonstrating the finite nature of duals for 2-local Hamiltonians on qubits and spin chains, and the absence of translation-invariant duals in small systems. Such results underscore that the locality properties of Hamiltonians are derivable from their spectra, a promising direction for both quantum physics and quantum gravity.

From a theory-building standpoint, this paper contributes significantly to understanding locality's emergence and characterization in quantum systems. It also provides insight into the relevance of results in practical applications like quantum simulations, resonating with recent advances in understanding non-local systems' complex behaviors, such as in quantum gravity via AdS/CFT.

Moreover, the work could inform computational techniques in determining local TPS without a direct basis specification, an area critical for both theoretical advancements and technological implementations of quantum systems. The methodologies proposed for solutions of inverse eigenvalue problems may spur future research in finding efficient algorithms for TPS detection and quantum state complexity analysis.

Overall, "Locality from the Spectrum" advances the dialogue on quantum systems' locality, establishing foundational results that could transform our understanding of quantum dynamics, simulation, and intricate structures in theories of quantum gravity. This burgeoning frontier of exploring tensor product structures in quantum systems will likely yield further revelations with practical implications in both quantum computation and theoretical physics.