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arXiv:1508.02538

Hessian geometry and entanglement thermodynamics

Hiroaki Matsueda

Published Aug 11, 2015• 2 citations
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Abstract

We reconstruct entanglement thermodynamics by means of Hessian geometry, since this method exactly generalizes thermodynamics into much wider exponential family cases including quantum entanglement. Starting with the correct first law of entanglement thermodynamics, we derive that a proper choice of the Hessian potential leads to both of the entanglement entropy scaling for quantum critical systems and hyperbolic metric (or AdS space with imaginary time). We also derive geometric representation of the entanglement entropy in which the entropy is described as integration of local conserved current of information flowing across an entangling surface. We find that the entangling surface is equivalent to the domain boundary of the Hessian potential. This feature originates in a special property of critical systems in which we can identify the entanglement entropy with the Hessian potential after the second derivative by the canonical parameters, and this identification guarantees violation of extensive nature of the entropy.

DOI
10.48550/arxiv.1508.02538

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We reconstruct entanglement thermodynamics by means of Hessian geometry, since this method exactly generalizes thermodynamics into much wider exponential family cases including quantum entanglement. Starting with the correct first law of entanglement thermodynamics, we derive that a proper choice of the Hessian potential leads to both of the entanglement entropy scaling for quantum critical systems and hyperbolic metric (or AdS space with imaginary time). We also derive geometric representation of the entanglement entropy in which the entropy is described as integration of local conserved current of information flowing across an entangling surface. We find that the entangling surface is equivalent to the domain boundary of the Hessian potential. This feature originates in a special property of critical systems in which we can identify the entanglement entropy with the Hessian potential after the second derivative by the canonical parameters, and this identification guarantees violation of extensive nature of the entropy.
We examine geometry and dynamics of classical spacetime derived from entanglement spectrum. The spacetime is a kind of canonical parameter space defined by the Fisher information metric. As a concrete example, we focus on the spectrum for free fermions in spatially one dimension. The spectrum has exponential family form like thermal probability distribution owing to mixed-state feature emerging from truncation of environmental degrees of freedom. In this case, the Fisher metric is given by the second derivative of the Hessian potential that can be identified with the entanglement entropy. We emphasize that the canonical parameters are nontrivial functions of partial system size by the truncation, filling fraction of fermions, and time. Then, the precise determination of this nontrivial mapping is necessary to derive the functional form of the Hessian potential that leads to correct entanglement entropy scaling. By this potential, we find that the emergent geometry becomes anti-de Sitter spacetime with imaginary time, and a radial axis as well as spacetime coordinates appears spontaneously. We also find that the information of the UV limit of the original free fermions lives in the boundary of the anti-de Sitter spacetime. These findings strongly suggest that the Hessian potential for free fermions has enough geometrical meaning associated with gauge-gravity correspondence. Furthermore, some deformation of the spectrum near the conformal fixed point is mapped onto spacetime dynamics. The fluctuation of the entanglement entropy embedded into the spacetime behaves like free scaler field, and the dynamics is described by the Einstein equation with a negative cosmological constant. Therefore, the Einstein equation can be regarded as the equation of original quantum state.
Entanglement entropy is a proposal to quantify quantum entanglement of two disjoint regions in a pure system. It is a relatively new topic and is developing rapidly. The current main motives to study it are its applications to studying quantum gravity, thermalization and phase transitions in condensed matter systems. Mutual information is a related quantity and it can be used to measure the amount of information two disjoint regions share. The purpose of this thesis is to give an introduction to the general results of the field without considering specific systems. The two prominent approaches used are two-dimensional conformal field theory and the holographic entanglement entropy conjecture. The first approach is to calculate entanglement entropy using conformal field theory, the results of which are known to be exact although technically more difficult to derive and only available for 1+1 dimensional systems. Both static and dynamic systems will be discussed. The results are reproduced and generalized to higher dimensions using holography. As a more recent topic, the holographic approach is used to rederive the entanglement entropy of systems with a Fermi surface using AdS/Vaidya metric with Lifshitz scaling and hyperscaling violation. The final chapter of the thesis discusses mutual information in various static and dynamic cases considered in the previous chapters. For the first time, the evolution of mutual information is calculated in AdS/Vaidya metric with Lifshitz scaling and hyperscaling violation in the critical theta=d-1 case.
We explore the connection between the area law for entanglement and geometry by representing the entanglement entropies corresponding to all 2N{2}^{N}2N bipartitions of an NNN-party pure quantum system by means of a (generalized) adjacency matrix. In the cases where the representation is exact, the elements of that matrix coincide with the mutual information between pairs of sites. In others, it provides a very good approximation, and in all the cases it yields a natural entanglement contour which is similar to previous proposals. Moreover, for one-dimensional conformal invariant systems, the generalized adjacency matrix is given by the two-point correlator of an entanglement current operator. We conjecture how this entanglement current may give rise to a metric entirely built from entanglement.
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A measure of how sensitive the entanglement entropy is in a quantum system, has been proposed and its information geometric origin is discussed. It has been demonstrated for two exactly solvable spin systems, that thermodynamic criticality is directly \textit{indicated} by finite size scaling of the global maxima and turning points of the susceptibility of entanglement entropy through numerical analysis - obtaining power laws. Analytically we have proved those power laws for ∣ λc(N)−λc∞∣| \ λ_c(N)-λ_c^{\infty}|∣ λc​(N)−λc∞​∣ as N→∞N\to \inftyN→∞ in the cases of finite 1D transverse field ising model (TFIM) (λ=hλ=hλ=h) and XY chain (λ=γλ=γλ=γ). The integer power law appearing for XY model has been verified using perturbation theory in O(1N)\mathcal{O}(\frac{1}{N})O(N1​) and the fractional power law appearing in the case of TFIM, is verified by an exact approach involving Chebyshev polynomials, hypergeometric functions and complete elliptic integrals. Furthermore a set of potential applications of this quantity under quantum dynamics and also for non-integrable systems, are briefly discussed. The simplicity of this setup for understanding quantum criticality is emphasized as it takes in only the reduced density matrix of appropriate rank.
Generalizing the first law of thermodynamics, the increase in entropy density $\ensuremath{\delta}S(x)$ of a conformal field theory (CFT) is proportional to the increase in energy density, $\ensuremath{\delta}E(x)$, of a subsystem divided by a spatially dependent entanglement temperature, TE(x){T}_{E}(x)TE​(x), a fixed parameter determined by the geometry of the subsystem, crossing over to thermodynamic temperature at high temperatures. In this paper we derive a generalization of the thermodynamic Clausius relation, showing that deformations of the CFT by marginal operators are associated with spatial temperature variations, $\ensuremath{\delta}{T}_{E}(x)$, and spatial energy correlations play the role of specific heat. Using AdS/CFT duality we develop a relationship between a perturbation in the local entanglement temperature of the CFT and the perturbation of the bulk AdS metric. In two dimensions, we demonstrate a method through which direct diagonalizations of the boundary quantum theory may be used to construct geometric perturbations of AdS3{\mathrm{AdS}}_{3}AdS3​.