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UC Berkeley UC Berkeley Electronic Theses and Dissertations UC Berkeley UC Berkeley Electronic Theses and Dissertations Title Entropy Bounds and Entanglement Permalink https://escholarship.org/uc/item/8c90p060 Author Fisher, Zachary Kenneth Publication Date 2017 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California Entropy Bounds and Entanglement by Zachary Fisher A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Physics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Raphael Bousso, Chair Professor Yasunori Nomura Professor Nicolai Reshetikhin Spring 2017 Entropy Bounds and Entanglement Copyright 2017 by Zachary Fisher 1 Abstract Entropy Bounds and Entanglement by Zachary Fisher Doctor of Philosophy in Physics University of California, Berkeley Professor Raphael Bousso, Chair The generalized covariant entropy bound, or Bousso bound, is a holographic bound on the entropy of a region of space in a gravitational theory. It bounds the entropy passing through certain null surfaces. The bound remains nontrivial in the weak-gravity limit, and provides non-trivial constraints on the entropy of ordinary quantum states even in a regime where gravity is negligible. In the first half of this thesis, we present a proof of the Bousso bound in the weak-gravity regime within the framework of quantum field theory. The bound uses techniques from quantum information theory which relate the energy and entropy of quantum states. We present two proofs of the bound in free and interacting field theory. In the second half, we present a generalization of the Bousso bound called the quantum focussing conjecture. Our conjecture is a bound on the rate of entropy generation in a quan- tum field theory coupled semiclassically to gravity. The conjecture unifies and generalizes several ideas in holography. In particular, the quantum focussing conjecture implies a bound on entropies which is similar to, but subtly different from, the Bousso bound proven in the first half. The quantum focussing conjecture implies a novel non-gravitational energy condition, the quantum null energy condition, which gives a point-wise lower bound on the null-null component of the stress tensor of quantum matter. We give a proof of this bound in the context of free and superrenormalizable bosonic quantum field theory. i For Melanie, Dennis, Jeremy and Laura. ii Contents Contents ii List of Figures iv 1 Introduction1 1.1 The Holographic Principle............................ 1 1.2 The Bousso Bound................................ 3 1.3 Holography and Quantum Field Theory..................... 5 1.4 Entropy, Energy and Geometry ......................... 7 2 The Bousso Bound in Free Quantum Field Theory9 2.1 Regulated Entropy ∆S .............................. 10 2.2 Proof that ∆S ∆ K ............................. 13 ≤ h i 2.3 Proof that ∆ K ∆A=4GN ~ ......................... 13 2.4 Discussion.....................................h i ≤ 16 2.A Monotonicity of ∆A(c;b) ∆S .......................... 21 4GN ~ − 3 The Bousso Bound in Interacting Quantum Field Theory 22 3.1 Entropies for Null Intervals in Interacting Theories .............. 25 3.2 Bousso Bound Proof ............................... 30 3.3 Holographic Computation of ∆S for Light-Sheets ............... 33 3.4 Why is ∆S = ∆ K on Null Surfaces?..................... 37 3.5 Discussion.....................................h i 40 3.A Extremal Surfaces and Phase Transitions on a Black Brane Background . 43 3.B Toy Model with ∆ K = ∆S = 0........................ 49 h i 6 4 The Quantum Focussing Conjecture 53 4.1 Classical Focussing and Bousso Bound ..................... 56 4.2 Quantum Expansion and Focussing Conjecture................. 58 4.3 Quantum Bousso Bound............................. 62 4.4 Quantum Null Energy Condition ........................ 66 4.5 Relationship to Other Works........................... 70 iii 4.A Renormalization of the Entropy......................... 75 5 Proof of the Quantum Null Energy Condition 82 5.1 Statement of the Quantum Null Energy Condition............... 86 5.2 Reduction to a 1+1 CFT and Auxiliary System................ 87 5.3 Calculation of the Entropy............................ 92 5.4 Extension to D = 2, Higher Spin, and Interactions . 103 5.A Correlation Functions............................... 105 Bibliography 107 iv List of Figures 2.1 (a) The light-sheet L is a subset of the light-front x− = 0, consisting of points + with b(x?) x c(x?). (b) The light-sheet can be viewed as the disjoint union of small transverse≤ ≤ neighborhoods of its null generators with infinitesimal areas Ai ........................................... 11 2.2f Operatorg algebras associated to various regions. (a) Operator algebra associated to the domain of dependence (yellow) of a space-like interval. (b) The domain of dependence of a boosted interval. (c) In the null limit, the domain of dependence degenerates to the interval itself. .......................... 12 2.3 A possible approach to defining the entropy on a light-sheet beyond the weak- gravity limit. One divides the light-sheet into pieces which are small compared to the affine distance over which the area changes by a factor of order unity. The entropy is defined as the sum of the differential entropies on each segment. 19 3.1 The R´enyi entropies for an interval A involve the two point function of defect operators D inserted at the endpoints of the interval. An operator in the ith CFT becomes an operator in the (i + 1)th CFT when we go around the defect. 25 3.2 The functions g(v) in the expression for the modular Hamiltonian of the null slab, for conformal field theories with a bulk dual. Here d = 2; 3; 4; 8; from bottom to top. Near the boundaries (v 0, v 1), we find g 0, g0 1 1, in agreement with the modular Hamiltonian! of a Rindler! wedge. We! also note! ± that the functions are concave. In particular, we see that g0 1, in agreeement with our general argument of section 3.2..........................j j ≤ 36 3.3 Operator algebras associated to various regions. (a) Operator algebra associated to the domain of dependence (yellow) of a space-like interval. (b) The domain of dependence of a boosted interval. (c) In the null limit, the domain of dependence degenerates to the interval itself. .......................... 38 3.A.1The maximum value Emax(p) of E for getting a surface that returns to the bound- ary (solid line). For comparison, the line E = p 1 is plotted (the dashed line). The extremal surface solutions of interest appear− in the region p > 1, 0 < E < Emax(p). Here, we have taken d = 3. ................... 45 v 3.A.2Curves of constant ∆x+ (black solid curves) and ∆x− (blue dashed curves), in the logarithmic parameter space defined by (log(p 1); log(Emax(p) E)=Emax(p)). The value p = 1 maps to and p = maps− to− + on the horizontal− axis, −∞ 1 1 while E = 0 maps to 0 and E = Emax(p) maps to + on the vertical axis. The thick blue contour represent the null solutions with1 ∆x− = 0. Above this contour, the boundary interval is time-like. If ∆x+ & 15 and we follow a contour of constant ∆x+, we find two solutions with exact ∆x− = 0. For all contours of fixed ∆x+, there exists an asymptotic null solution in the limit p . 46 3.A.3The vacuum-subtracted extremal surface area versus ∆x− for fixed! ∆ 1x+ (∆x+ = 20 and ∆x+ = 10 for d = 3 is shown). This numerical simulation demonstrates that, for sufficiently large ∆x+ (in d = 3, the condition is ∆x+ & 15), there exists a phase transition at finite ∆x− to a different, perturbative class of solutions. At smaller ∆x+, there is no such phase transition.................... 49 4.1 (a) A spatial surface σ of area A splits a Cauchy surface Σ into two parts. The generalized entropy is defined by Sgen = Sout+A=4GN ~, where Sout is the von Neu- mann entropy of the quantum state on one side of σ. To define the quantum expansion Θ at σ, we erect an orthogonal null hypersurface N, and we consider the response of Sgen to deformations of σ along N. (b) More precisely, N can be divided into pencils of width around its null generators; the surface σ is deformed an affine parameter lengthA along one of the generators, shown in green. 60 4.2 (a) For an unentangled isolated matter system localized to N, the quantum Bousso bound reduces to the original bound. (b) With the opposite choice of \exterior," one can also recover the original entropy bound, by adding a distant auxiliary system that purifies the state........................ 64 4.3 (a) A portion of the null surface N, which we have chosen to coincide with Σout in the vicinity of the diagram. The horizontal line at the bottom is the surface V (y), and the orange and blue lines represent deformations at the transverse locations y1 and y2. The region above both deformations is the region outside of V1,2 (y) and is shaded beige and labeled B. The region between V (y) and V1 (y) is labeled A and shaded lighter orange. The region between V (y) and V2 (y)is labeled C and shaded lighter blue. Strong subadditivity applied to these three regions proves the off-diagonal QFC. (b) A similar construction for the diagonal part of the QFC. In this case, the sign of the second derivative with respect to the affine parameter is not related to strong subadditivity............. 67 vi 5.1 The spatial surface Σ splits a Cauchy surface, one side of which is shown in yellow. The generalized entropy Sgen is the area of Σ plus the von Neumann entropy Sout of the yellow region. The quantum expansion Θ at one point of Σ is the rate at which Sgen changes under a small variation dλ of Σ, per cross-sectional area of the variation. The quantum focussing conjecture states that the quantum expansionA cannot increase under a second variation in the same direction.
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