2018/09/10 by Raimond Abt, Johanna Erdmenger · 7 citations
Computer Science · Physics and Astronomy · #Entropy (arrow of time) #Fisher information #Ground state #Hamiltonian (control theory) #Joint quantum entropy #Logarithm #Modular design #Quantum Information and Cryptography #Quantum and electron transport phenomena #Quantum entanglement #Quantum many-body systems #hep-th
paper · pdf · doi:10.1007/jhep11(2018)002
published in Journal of High Energy Physics 2018(11) (Springer Nature)
arxiv created 2018/09/10 · openalex created_date 2018/09/27 · openalex publication_date 2018/11/01 · arxiv updated 2018/11/14 · openalex updated_date 2026/08/05
A bstract The modular Hamiltonian of reduced states, given essentially by the logarithm of the reduced density matrix, plays an important role within the AdS/CFT correspondence in view of its relation to quantum information. In particular, it is an essential ingredient for quantum information measures of distances between states, such as the relative entropy and the Fisher information metric. However, the modular Hamiltonian is known explicitly only for a few examples. For a family of states ρ λ that is parametrized by a scalar λ , the first order contribution in \overset∼ λ = λ -λ0 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>λ</mml:mi> <mml:mo>∼</mml:mo> </mml:mover> <mml:mo>=</mml:mo> <mml:mspace /> <mml:mi>λ</mml:mi> <mml:mo>−</mml:mo> <mml:msub> <mml:mi>λ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> of the modular Hamiltonian to the relative entropy between ρλ and a reference state ρλ 0 is completely determined by the entanglement entropy, via the first law of entanglement. For several examples, e.g. for ball-shaped regions in the ground state of CFTs, higher order contributions are known to vanish. In these cases the modular Hamiltonian contributes to the Fisher information metric in a trivial way. We investigate under which conditions the modular Hamiltonian provides a non-trivial contribution to the Fisher information metric, i.e. when the contribution of the modular Hamiltonian to the relative entropy is of higher order in \overset∼ λ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>λ</mml:mi> <mml:mo>∼</mml:mo> </mml:mover> </mml:math> . We consider one-parameter families of reduced states on two entangling regions that form an entanglement plateau, i.e. the entanglement entropies of the two regions saturate the Araki-Lieb inequality. We show that in general, at least one of the relative entropies of the two entangling regions is expected to involve \overset∼ λ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>λ</mml:mi> <mml:mo>∼</mml:mo> </mml:mover> </mml:math> contributions of higher order from the modular Hamiltonian. Furthermore, we consider the implications of this observation for prominent AdS/CFT examples that form entanglement plateaux in the large N limit.