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Locality from the Spectrum

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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.

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We would like to thank Daniel Bump, Dylan Butson, Emilio Cobanera, Benjamin Lim, Edward Mazenc, Xiao-Liang Qi, Semon Rezchikov, Leonard Susskind, Arnav Tripathy, Ravi Vakil, and Michael Walter for their valuable discussions and support. We are also especially grateful to Patrick Hayden and Frances Kirwan for their valuable insights and feedback, and for reviewing this manuscript. JC is supported by the Fannie and John Hertz Foundation and the Stanford Graduate Fellowship program. DR is supported by the Stanford Graduate Fellowship program.

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Correspondence to Jordan S. Cotler.

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Communicated by M. M. Wolf

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Cotler, J.S., Penington, G.R. & Ranard, D.H. Locality from the Spectrum. Commun. Math. Phys. 368, 1267–1296 (2019).

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