Entanglement Hamiltonians: From Field Theory to Lattice Models and Experiments

Abstract

Results about entanglement (or modular) Hamiltonians of quantum many-body systems in field theory and statistical mechanics models, and recent applications in the context of quantum information and quantum simulation, are reviewed. In the first part of the review, what is known about entanglement Hamiltonians of ground states (vacua) in quantum field theory is summarized, based on the Bisognano-Wichmann theorem and its extension to conformal field theory. This is complemented with a more rigorous mathematical discussion of the Bisognano-Wichmann theorem, within the framework of Tomita-Takesaki theorem of modular groups. The second part of the review is devoted to lattice models. There, exactly soluble cases are first considered and then the discussion is extended to non-integrable models, whose entanglement Hamiltonian is often well captured by the lattice version of the Bisognano-Wichmann theorem. In the last part of the review, recently developed applications in quantum information processing that rely upon the specific properties of entanglement Hamiltonians in many-body systems are summarized. These include protocols to measure entanglement spectra, and schemes to perform state tomography.

Universality in the tripartite information after global quenches: spin flip and semilocal charges

“…temperatures approximated well by the Hamiltonian of the subsystem [105]. Similarly, we expect that the properties of the entanglement Hamiltonian in the GGE can be captured well by the restriction of Q to the subsystem.…”

Entanglement and negativity Hamiltonians for the massless Dirac field on the half line

“…The relation (25) between the reduced free fermion correlator C A and the entanglement Hamiltonian kernel H A provides the lattice entanglement Hamiltonian. However, recovering the continuum limit of the entanglement Hamiltonian from a discrete system turns out to be a non-trivial problem, as lengthy discussed in [80][81][82][83][84][85][86][87]. Indeed, the local inverse temperature has been obtained in some cases from the lattice by taking into account hoppings to distant neighbours and not only the nearest ones.…”

Entanglement Hamiltonian in the non-Hermitian SSH model

“…(2024) 063102 entanglement spectrum, have been investigated. Arguably, the most complete understanding stems from the entanglement (or modular) Hamiltonian (EH), which is the logarithm of the (normalised) reduced density matrix [12][13][14]…”

“…Although calculating the EH is considerably more challenging than determining the entanglement entropies, a rich theoretical framework has been developed during the last decades. The most remarkable result is the Bisognano-Wichmann theorem [12][13][14][15][16]. Considering the vacuum state of a unitary relativistic quantum field theory (QFT) on R d+1 and taking as subsystem the half-space x 1 > 0, the Bisognano-Wichmann theorem asserts that the EH in the half-space is the generator of Lorentz boosts…”