Reconstructing quantum states from local data

TL;DR

This paper investigates reconstructing global quantum states from local data using maximum entropy principles, demonstrating exact and approximate reconstructions for ground and thermal states, with implications for holography.

Contribution

It introduces a maximum entropy approach for quantum state reconstruction from local data, linking it to ground state properties and holographic duality.

Findings

Unique ground states are exactly reconstructed as maximum entropy states.

Reconstruction is approximately valid for degenerate ground states.

Thermal states of local Hamiltonians can also be reconstructed.

Abstract

We consider the problem of reconstructing global quantum states from local data. Because the reconstruction problem has many solutions in general, we consider the reconstructed state of maximum global entropy consistent with the local data. We show that unique ground states of local Hamiltonians are exactly reconstructed as the maximal entropy state. More generally, we show that if the state in question is a ground state of a local Hamiltonian with a degenerate space of locally indistinguishable ground states, then the maximal entropy state is close to the ground state projector. We also show that local reconstruction is possible for thermal states of local Hamiltonians. Finally, we discuss a procedure to certify that the reconstructed state is close to the true global state. We call the entropy of our reconstructed maximum entropy state the "reconstruction entropy", and we discuss its relation to emergent geometry in the context of holographic duality.