The strong subadditivity of holographicentanglement entropy; from boundary tobulkbyAli I. RadB.Sc., Sharif University of Technology, 2015A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2017c© Ali I. RadAbstractOne decade ago, Ryu-Takayanagi explicitly introduced a formula that relates theentropy of a subregion in CFT system to a geometrical quantity which is calledminimal surface in hyperbolic space [36]. This formula extended to the idea ofconnection of gravy to quantum mechanics of gauge/gravity duality. This dualitywhich can help us to learn a more interesting feature of each side from the other.Quantum systems obey from some constraints come from the quantum informa-tion theory. I would be interesting to find out what is the dual of this constraintin the gravitation system. Dual to the specific class of quantum theories which iscalled conformal field theories. One of the most significant constraint that QFTsshould obey is the strong subadditivity of entanglement entropy. These constraintslet the theories have bound on the energy spectrum from the below; Recently therehas been the development that the combination of monotonicity of relative entropyand the strong subadditivity of entanglement entropy is equal to have a specificbound on the energy momentum tensor, called quantum null energy condition. Inthis thesis, we re-look to this argument by introducing the entanglement densityand obtain a differential operator from the strong subadditivity and exploiting fromthe Markov property of the vacuum of CFT. In the next step, by using from theRyu-Takayangi, we rewrite the strong subadditivity inequality regarding geometri-cal quantities. By using from the kinematic languages and intertwinement, we re-alize that the strong subadditivity at the boundary implies new bound on averagedenergy condition which has some common feature with the quantum null energycondition statement.iiLay AbstractStrong subadditivity of entanglement entropy, in one of the most important con-straints in quantum theories and there is enough evidence indicating there is a deepconnecting between the stability of physical theories and the strong subadditivity.In this thesis, we investigate on the aspects of strong subadditivity of entanglemententropy in the Holographic theories. At the first step, we show the combinationof Markov property of vacuum states of CFTs and strong subadditivity of entangle-ment entropy, imply a pseudo convexity of the relative entropywhich is equivalent toquantum null energy condition in quantum field theories. In the next step, we showhow does the combination of strong subadditivity for the gravitational systems atthe boundary and Ryu-Takayanagi formula, imply an averaged energy bound, whichcan be comparable to quantum null energy condition.iiiPrefaceThis thesis is discussing a problem which defines my supervisor, Prof. Mark VanRaamsdonk and guided with Dr. Felix Haehl when he was a postdoc of Stringgroup of UBC. The project started early summer of 2016, and the goal was to studythe aspect of strong subadditivity in Holography in general dimension. The ideadeveloped via discussing with "It From Qubit" during summer school event at PI atthe University of Waterloo, Simon Center for Theoretical Physics and Geometry atthe University of Stony Brook, TASI program at Univerity of Colorado, Boulder.The writer would like to thanks from all friends for their valuable comments.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Thesis Structure . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Strong Subadditivity of Quantum Entanglement Entropy . . . . . . 72.1 Von Neumann Entropy . . . . . . . . . . . . . . . . . . . . . . . 72.2 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Relations Between Inequalities . . . . . . . . . . . . . . . . . . . 102.3.1 Markov Property and Strong SuperSubAdditivity . . . . . 102.3.2 Entanglement Entropy for Lorentz Invariant Theories . . 132.4 The Stability of Theories From the Strong Subadditivity . . . . . 152.4.1 Focusing Theorem . . . . . . . . . . . . . . . . . . . . . 182.4.2 Quantum Focusing . . . . . . . . . . . . . . . . . . . . 193 Strong Subadditivity of Entanglement Entropy in CFTs . . . . . . . 213.1 Strong Subadditivity in Higher dimensions (d > 2) . . . . . . . . 263.1.1 General Deformation . . . . . . . . . . . . . . . . . . . 30vTable of Contents3.2 QNEC From Convexity of Relative Entropy . . . . . . . . . . . 344 Intertwinement and Radon Transformation . . . . . . . . . . . . . 364.1 Radon Transform and Intertwinement . . . . . . . . . . . . . . . 365 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45viAcknowledgementsI would like to thank my thesis advisor Prof.Mark Van Raamsdonk at Universityof British Columbia. The door of Mark’s office was always open whenever I raninto a trouble spot or had a question about my research or writing. He consistentlyallowed this paper to be my own work, but steered me in the right the directionwhenever he thought I needed it.I also would like to thank Dr. Felix Haehl who were involved in the projectduring the whole recent year. Without his passionate participation and input, theproblem could not have been successfully conducted.I would also like to acknowledge from Charles Rabideau, Aron Wall, JasonKoeller, Hiracio Casini, Lampros Lamprous, Amir Tajdini and Adam Levine. I’mgratefully indebted to them for their very valuable comments on this thesis. Somepart of the problem done during the ”It form Qubit” workshops at PI and SITP. I’mso grateful from Physics department of UBC that provides all things I needed tofocus on my projectFinally, I must expressmy very profound gratitude tomy parents and tomy sisterand brother for providing me with unfailing support and continuous encouragementthroughoutmy years of study and through the process of researching andwriting thisthesis. This accomplishment would not have been possible without them. Thankyou.AliviiChapter 1IntroductionIn the recent decades, the concept of entanglement entropy has been made lots ofattraction owing to its attractive indication about the nature of space time and gravityand its deep connection to the quantum world. The relationship of entropy withgravity became more apparent when people realized the black hole has an entropy.According to on the Bekenstien-Hawking formula [5, 6] the black hole entropy isgiven by the horizon area in terms of Planck scale:Sbh =A4× (kc3G~) (1.1)where k is the Boltzmann constant, c is the light speed andG and ~ are gravity andPlanck constants. The existence of ~ andG indicates that the formula is describingthe quantum-gravity quantity. One can define aGeneralized Entropy [10, 11] basedon the above notion:Sgen =A4G~+ Sout (1.2)where Sout defined as entropy of matters are located beyond the shell with area ofA. According to the generalized second law of entanglement [4] we expect thatdSgen ≥ 0, (1.3)which is comparable to the second law of thermodynamic. This connection be-comes much more interesting when we realize that based on the Holographic prin-ciple [37], there is an upper bound on the entropy of the subsystem in the presenceof the gravity∆S ≤ ∆A4G~. (1.4).1Chapter 1. IntroductionThe fact that the entropy of a system in the presence of gravity has a boundand it’s proportional to area, not the volume, provides a new door to the quantum-gravity realm. As we know, in the general quantum field theories the entropy ofthe system is proportional to the volume. Let’s imagine a system which is made oflattice and each lattice we have a Q degree of freedom. For example, for magneticmodels, Q is 2. If we call the size of lattice site Λ, then the volume of phase spaceis proportional to Ω ∼ QV/Λ3 thusSQFT ∼ V × ( logQΛ3). (1.5)As we said before, we expect that the entropy is proportional to volume, butsurprisingly it seems in the gravitational systems, the number of degrees of freedomis much more than degrees of freedom needed to be consistent with gravitationaltheory. The clearest example is the entropy of the horizon by considering it as aquantum system. To see how holographic principle puts a bound on a number ofdegrees of freedom for gravitational systems, let’s consider the system that includesgravity and it is inside the spherical region (for simplicity) with volume V and areaA. This system has an entropy say S˜. Obviously, The total mass of the system isless that the mass of black hole with horizon area A. Now let’s add matter to thesystem until the mass of the system becomes to the mass of black hole associatedwith area A. The final entropy in this state will be now A/(4G~). Thus we expectthatS˜ ≤ A4G~. (1.6)This maximum bound on entropy for the spherical region numerically is aroundSMax ∼ [ R1cm×1033]2 , for example, this formula states that themaximal informationthat can be stored in our Brian is around SMax Brian ∼ 1068 we can compare it withtotal storage capacity of Internet which is around 1024 bits.In the same faith, It has now been a decade since Ryu and Takayanagi discoveredan elegant geometric prescription to evaluate entanglement entropy in gauge/gravityduality [34, 35]:SEE (D) =extd∼DA(d)4kc3G~. (1.7)2Chapter 1. Introductionwhere d is the bulk surface, homologous to boundary region D. The above simpleformula provides a unique frame work to re-establish the gravity from the begin-ning. The idea that quantum world and gravity are two sides of one coin let us thisopportunity to learn new thing from both sides. It will be much more valuable whenwe realize that this connection let us understand better gravity side, which we haveless information about it, from the quantum mechanics world.The first observation was that the notion of space-time, the building blocks ofrelativity are highly related to Entanglement in quantum mechanics. The thermalfield double thought experiment, shown that the entanglement is like a glue makesa course of space-time possible [39, 40].The next breakthrough observation was that quantum inequalities in the quan-tummechanical systems, the systems that live in the boundary, give us crucial infor-mation about the physics of inside the bulk which is (quantum-)gravitation systems.For example, we already know the positivity of relative entropy implies that the Ein-stein equation, at least up to first order, should be valid in bulk [24, 29]. Indeed,according to the first law of entanglement, the small variation of entangled entropyfor nearby states is equal to variation of modular Hamiltonian, associated with thosestatesδSEE = δ〈H〉. (1.8)Considering the above equality for all ball shape region with all arbitrary ori-entation provides a local constraint that for the states near to vacuum, the linearizedEinstien equation should be valid [23, 38]. In the recent years, there have beensome attempts to provide the relativistic bound on the stress tensor based on theprior knowledge of quantum mechanics or other evidence. However, the generalstatements have remained as a conjecture. One of the recent proposals of energycondition is the Quantum Null Energy Condition (QNEC) [13], which is nominatedas an improved bound on the stress tensor. This new proposal can be seen as a pureQFT bound, and it’s not necessarily related to gravity theories. One can ask howthe CFT constraint on the energy can be related to the bulk property. Recent worksindicated that positivity of relative entropy could lead to gravitational positive en-3Chapter 1. Introductionergy theorem and similarly, the strong subadditivity for two-dimensional CFT leadsto inviolable energy conditions in bulk [30, 31].The notion of energy in physics play a fundamental rule on our understandingof the universe. The theories of physics are formulated to guaranty the energy ofthe system is always bounded from the below. However, it is not clear what kindof energy condition holds exactly in each theory. There is some evidence that theenergy conditions in quantum field theory are entirely related to the informationtheoric quantities. There is an observation that the stability of the QFT is related tothe strong subadditivity of entanglement entropy which implies the more strongly aquantum system A is entangled with a system B, the less strongly it can be entangledwith another system C. According to [41], the strong subadditivity of entanglemententropy in 1 + 1 QFT is enough to put a bound on energy-momentum tensor∫ ∞u0〈T 〉du ≥ − ~2piS′(ρu>u0), (1.9)where here u is the spatial coordinate of the system. The similar bound has beenproofed recently [2] for QFTswithMinkowski space time based on themonotonicityof the relative entropy:∫ ∂B∂Adλ−〈T−−〉 ≥ ~2piδS(B)δλ−(y)− ~2piδS(A)δλ−(y)(1.10)where λ parametrize the null line pathing from entangling surface ∂A to ∂B andy is a coordinates along the entangling surface. This inequality can be seen as asemi-local inequality by taking the derivative[12, 13]〈T 〉 ≥ ~2piS′′. (1.11)In addtion for CFT1+1, we can find more stronger bound on the energy momentumtensor [13]〈T 〉 ≥ ~2pi(S′′ +6cS′2), (1.12)where c is the central charge of theory. Despite these attempts in the QFT side, up tonow, there is no proof for bounding the condition in the Einstein gravity, and indeedthere is no covariant positive energy momentum tensor for this theory. According41.1. Thesis Structureto Einstein general relativity, the geometry of space-time is entirely related to theenergy distribution of matter inside the space. By looking at the Einstein equation,we do not see a prior restriction on geometry property, namely the metric. One mayask this simple question that is there an intrinsic restriction on geometry or energy-momentum tensor and is there forbidden sectors in the phase space of geometry orequivalently energy distribution?We can think of some exotic cases, which are validated answer of Einstein equa-tion, but they require some special energy distribution. Wormholes, warp drives andtime machines are few examples of this exotic sector of the possible solution of Ein-stein equation. However, most of this strange cases need energy violation, whichimplies we should have negative energy density within a considerable volume ofspace-time.The there is no principle of the general relativity to narrow down the phasespace of stress tensors, but there is some famous energy bound in the literature thatmakes sense for the ordinary matter we deal in the universe or the abstract classicallimit. In this thesis, based on the concept of Holography we want to investigatemore on this energy conditions.In this project, we show that Strong Subadditivity(SSA) in the bulk implies the energy condition similar to QNEC in the boundary:∫B˜〈T 〉 ≥ −(d− 2) ~2pi`S ′(`) (1.13)for the states near to vacuum where ` is the covariant size of the subregion, andd is the dimension of CFTd. Also, we show the combination of Markov propertyof vacuum states and strong subadditivity implies convexity of the relative entropy,S(ρ||ρ0)′′ ≥ 0 which is equivalent to quantum null energy condition in quantumfield theories [26, 27].1.1 Thesis StructureIn this thesis, at first we review on the concept of entanglement entropy and it’sproperties such as strong subadditivity. Then we study the aspect of strong subad-ditivity in two dimensional QFTs. We will extend our study to general dimensions51.1. Thesis Structurein the next chapter. Then we see the connection of SSA with quantum null energycondition. Finally, we found a geometrical constraint for SSA in the bulk at lastchapter.6Chapter 2Strong Subadditivity of QuantumEntanglement Entropy2.1 Von Neumann EntropyFor a given quantum system, we can define a quantum density matrix, ρ, and basedon density matrix, it’s useful to introduce an entropy of that states, defined as VonNeumann’s entropy [7]S(ρ) = −Tr(ρ log ρ). (2.1)Alternatively if we denote the eigenvalue of the density matrix with λi, then theabove formula get a simpler form:S(ρ) = −∑iλi log λi, (2.2)where this form of the entropy reminds the Shannon entropy in classical informa-tion theory and we can have much interpretation for the meaning of entropy forgiven state. One can argue similarly to classical entropy; the Von Neumann entropyreflects how much states are more complex than the Identity case or how much in-formation is stored in our quantum system. It’s also useful to define and introducean operator that quantify the distinguishability of the tow given states to see howmuch they are different from each other. The relative entropy is useful definition toprovide this desire. For a given ρ and σ density matrix one can defines a relativeentropy as [33]S(ρ||σ) = Tr(ρ log ρ)− Tr(ρ log σ). (2.3)72.2. InequalitiesIt’s easy to show that based on the Klien’s inequality [? ], the relative entropy isalways positive:S(ρ2||ρ1) ≥ 0. (2.4)monotonicity is one of the most important property that relative entropy has. Aswewill see later, it’s has a great impact on the context of Holography. To understandbetter the reason of this property, let’s consider the quantum system B that has thesubset A, where A ⊂ B, the monotonicity of relative implies thatS(ρA||σB) ≤ S(ρB||σB). (2.5)To see the reason of above relation, it’s good to notice that there is a unitarytransformation in B such that [33]ρA ⊗ Id=∑jpjUjρBU †j , (2.6)and from the convexity of the relative entropy we expectS(ρA ⊗ Id||σA ⊗ Id) ≤∑jpjS(UjρBU †j ||Uj .σBU †j ). (2.7)In the last step, we can use from the invariant property of the relative entropyunder unitary transformation to getS(ρA||σA) = S(ρA ⊗ Id||σA ⊗ Id) ≤∑jpjS(ρB||σB) = S(ρB||σB) (2.8)Intuitively, the monotonicity implies that if we ignore the part of our quantumsystem, it becomes harder to distinguish the given states. We can also extend theresult from to any completely positive trace preserving (CPTP) operations, N :S(N (ρ)||N (σ)) ≤ S(ρ||σ) (2.9)2.2 InequalitiesEntanglement entropy has some fundamental properties that can be described insome distinguishable inequalities. The number of these inequalities can be infinite82.2. Inequalitiesin general. The Subadditivity is one of them, which express for two subregion Aand B, the following inequality always holds|S(A)− S(B)| ≤ S(A ∪B) ≤ S(A) + S(B) (2.10)The Strong Subadditivity is another profound inequality which plays the mainrole in our thesis. For given three subregions this inequality implies [33]S(A ∪B ∪ C) + S(B) ≤ S(A ∪B) + S(B ∪ C) (2.11)To see the reason of above relation, let’s start with Golden-Thompson-Lieb in-equality [33] where stares that for a given A,B and C Hermitian matrix we have:TreA+B+C ≤∫ ∞0Tr(t+ e−A)−1eB(t+ e−A)−1eCdt. (2.12)By using this inequality we observe thatTrelog ρAB−log ρb+log ρBC ≤∫ ∞0TrρAB(tI + ρB)−1ρBC(tI + ρB)−1dt=∫ ∞0TrρB(tI + ρB)−1ρB(tI + ρB)−1dt = 1(2.13)therefor ther is a λ ≤ 1 and a state Ω such thatelog ρAB−log ρb+log ρBC = λΩ (2.14)Now, by using from Von Neumann definition of entropy we realize thatS(ρAB) + S(ρBC)− S(ρABC)− S(ρB)= TrρABC(log ρABC − (log ρAB − log ρB + log ρBC))= S(ρABC ||λΩ) = S(ρABC ||Ω)− log λ.(2.15)By Assuming that λ ≤ 1 and positivity of relative entropy, the last term in theabove relation will be always positive. Thus the first term is greater than zero whichis exactly the strong subadditivity. Finally, it’s good to notice that Carlen and Liebshowed [33] that we can improve the strong subadditivity by the following relationS(ρAB)+S(ρBC)−S(ρABC)−S(ρB) ≥ 2Max[S(ρA)−S(ρAB), S(ρB)−S(ρAB), 0],(2.16)92.3. Relations Between Inequalitiesin quantum information, it’s completely likely that the right hand side be positiveand then the improved inequality has an extra information for us.2.3 Relations Between InequalitiesIt would be interesting to find out is there any connection between the Strong Sub-additivity (SSA) and monotonicity of the relative entropy (MON) and monotonicityof relative entropy under partial trace (MPT) or not. It turns out one can show themonotonicity of the relative entropy leads to monotonicity of relative entropy underpartial trace and vice versa. In addition, Strong Subadditivity is equivalent to mono-tonicity of relative entropy under partial trace. Thus it seems SSA and MON arenot the independent inequalities. Let’s just proof of one side, to show how mono-tonicity of relative entropy under partial trace leads to SSA inequality; the otherdirection of proof is similar. By considering the special CPTP operator, which isthe partial traceT : HAB → HAB, T = IHA ⊗ TrHB (2.17)and assuming ρ = ρ123, σ = ρ1⊗ρ23, T = IH∞∈⊗H3 and based on theS(Tρ||Tσ) ≤S(ρ||σ) inequality, we getS(ρ12||ρ1 ⊗ ρ2) ≤ S(ρ123||ρ1 ⊗ ρ23) (2.18)which is equal to say [33]S(ρ123||ρ1 ⊗ ρ23)− S(ρ12||ρ1 ⊗ ρ2) = S(ρ12) + S(ρ23)− S(ρ123)− S(ρ2) ≥ 0.(2.19)The proof of other side is similar as well.2.3.1 Markov Property and Strong SuperSubAdditivityIn this section, we want to discuss about the specific property entanglement entropyof vacuum states which will plays a key role in our later discussions. Let’s recall at102.3. Relations Between Inequalitiesfirst that we can define the condition mutual information:I(A1 : A3|A2) = S(ρ12) + S(ρ23)− S(ρ2)− S(ρ123) ≥ 0. (2.20)Now, let’s consider two subregion AA and AB , where AB ⊆ AA. In general,the associated density matrix of these two regions have different information aboutthe of underlying physics in the regions and they are not related to each other in anobvious way. But there is a special case of states that ρB can be recovered for thedensity matrix of the smaller subregion, ρA. This statement implies that the physicsof the system doesn’t depend on the scales. The states with this property are calledMarkov states. It will be interesting to check that vacuum states of the conformalfield theory have the Markov property or not because the theory is scale invariantand the vacuum state is the special case that information are non-local. It’s usefulto represent the density matrix with path integral. we can divide the space in xidirection to discrete sub-regions, Ai = A(i ≤ {x} ≤ (i + 1)). Then in thisrepresentation we get [28]:ρ =∫ N∏i=1[Dφi]ρi(φi, φi+1) =∫ N∏i=1[Dφi]〈φi|ρi|φi+1〉 (2.21)If we try to define new subregions, by deforming the main subregion,A, in twodifferent directions, at two different points on the border of subregionwewill be ableto compute the relative entropy, mutual information and other QI tools to measurethe share information between this regions and gives us more straightforward pathto check the Markov property. let’s deform the sub region at points a and b, thenthe new subregion will given by:A′ = A+ δaA+ δbA, (2.22)Following by [25] notation, the density matrix of new given density matrix can befound by unitary transformation for the original one:ρA(η) = U †ρA(g)U. (2.23)112.3. Relations Between InequalitiesWe can restore the subregionA by applying the diffeomorphism and change the flatbackground metric from η to g wheregµν = ∂µξv + ∂νξµ + ∂µξα∂νξα, (2.24)here if we just take the deformation in the null direction, the result become verysimple and favorable. If Fa be the translation function that deform the u to u +λf(xa), then for subregion define by x = xa the metric component change toλ∂xf(xa) . As a result we need to recalculate the path integral for matrix elementassociated to x = xa[28]:〈φ(∂[i]−)|ρ′(φa, φa+1)|φ(∂[i]+)〉 =∫ φ(xa+1)=φa+1φ(xa)=φa[Dφ]e−S[φ,g] (2.25)Here the action is a function of a newmetric g. We are able to expand the actionin term of a flat metric, η:S[φ, g] = exp[∫a∂µξνδδgµν]S[φ, η], (2.26)which can be more simplified as a following:S[φ, g] = exp[−∫aξν∂µδδgµν+∫∂[a]dΣξνδδgµν]S[φ, η]. (2.27)Considering this fact that the diffeomorphism vanishes at the boundary termsdoesn’t contribute and the according to the fact that ∂νTµν = 0, the first term inthe above relation vanishes and we can state that:S[φ, g] = S[φ, η] (2.28)for this special deformation. Therefore, the two density matrix can be transform toeach other by unitary transformation:ρ′ = (I ⊗ U †a ⊗ U †b )ρ(I ⊗ Ua ⊗ Ub). (2.29)Let’s consider the half space region of vacuum QFT in a flat space. If we rep-resent the null surface with u = 0:HAA =∫dd−2xi∫duuTuu. (2.30)122.3. Relations Between InequalitiesWe can now see the relation between modular Hamiltonian for the null deformationstates:HA+δa = H + (U†HaUa −Ha) + (U †bHbUb −Hb) (2.31)as a result [18, 28]:HA+δaA+δbA = HA+δaA +HA+δbA −HA (2.32)Therefore the vacuum of QFT of half-space is Markov state.It can be easily shown that the same relation works for the entanglement entropyas well [17, 28]SA+δaA+δbA = SA+δaA + SA+δbA − SA (2.33)It’s well known that the by conformal transformation the half-space can be trans-formed into a ball shape region. Thus the ball shape region of vacuum states on anull cone in CFT are Markov states and they saturate the strong subadditivity. Wecall this situation the strong supersubadditivity.2.3.2 Entanglement Entropy for Lorentz Invariant TheoriesIn the quantum systems, there are sets of non-trivial inequalities that entanglemententropy should satisfy. Positivity of entanglement, subadditivity of or positivityof mutual information, Araki-Lieb and strong subadditivity are most noticeableinequalities in the quantum theories. In quantum systems that have holographicdual, it has been shown that the phase space of inequalities, which is called entropycone [3], is much more limited than general theories. For example for a numberof subregions less than five, the strong subadditivity and monotonicity of mutualinformation make a complete set for quantum theories with HRT holographic pre-scription. Therefore it seems the Strong subadditivity plays fundamental rule inconformal field theories and We hope to investigate more into the strong subaddi-tivity discover interesting property in the boundary and the bulk. Some times theentanglement entropy of the spherical region in CFT is much more favorable. The132.3. Relations Between InequalitiesFigure 2.1: For Lorentz invariant theories, the entanglement entropy is a functionof All regions with the same casual diamond structure have the same entanglemententropy, and entropy is a just function of (x, y) on that defined theory. For thisreason, all Ai have the same associated entropy as A0.spheres have more symmetries, and they are invariant under SO(2, d) conformalgroup. The causal development of the d− 2 dimensional sphere in the past and fu-ture creates a specific volume in the space-time which is known as causal diamond.Each causal diamond has the past and future tip, xµ, and yµ. The 2d set points(x, y) live in the moduli space which is called the kinematic space. Indeed for eachspherical subregion, we have a unique casual diamond, and the metric space thatdescribes the relationship between these sets of causal diamonds is kinematic space.It is essential to mention that the entanglement entropy of two subregions with thesame causal diamond is the same. To show this, we need to notice the density ma-trix of the deformed subregion, see Fig.2.1, with the same causal diamond is relatedto the original one with unitary relationU †ρ0U = ρ˜, (2.34)thus the Von-Neuman entropy of both density matrix is the sameS(ρ) = S(ρ˜). (2.35)142.4. The Stability of Theories From the Strong Subadditivity2.4 The Stability of Theories From the StrongSubadditivityThe concept of energy in physics plays a fundamental role in our notions of the uni-verse.However, the meaning of energy, it’s definition is still ambiguous. Sometimesdefining the covariant energy is difficult or impossible in the proposed theory, butwe are sure about one property of the associated energy in our theories, it shouldhave a bound from the below. Otherwise, the theory is unstable and it will be ill-defined. Having this faith that the theory should have abound from below, motivateus to quantify the bound more precisely. Unfortunately up to know, there is noproof for bounding condition in the classical gravitation theories (Einstein Gravity)and there is no covariant positive energy momentum tensor for this theory. Indeedthere are some proposal for energy conditions that can be valid in some regime butthe physical motivation for considering the bound on the energy momentum is stillunclear. The similar situation happens in the quantum field theories coupled or un-coupled to the gravity. The recent proposal of Quantum Null Energy Condition,QNEC and quantum focusing conjecture, QFC [13], is an attempt to answer to thisopen puzzle in physics. It seems this bound saturates in the vacuum states whichmake it a suitable strong bound on the energy density of the quantum systems. Ac-cording to the QNEC the energy momentum tensor is bounded from below as afollowing [13]Tkk ≥ ~2pi√AS′′ (2.36)whereA is an area element limiting to zero andS′′ is second order derivative respectto λ, and Affine parameter deformation which is in null direction . Based on thisproposal one can conjecture there might be general bound in theories as a following:〈Tab〉lanb ≥ [ ~2picadb∂c∂dS]lanb (2.37)To support this conjecture, [41] have provided a neat proof which we would like topresent the essence of this work here. At first, let’s consider the classical theory (incontrast with quantum theories), the theory can be described by some fields, {φi}152.4. The Stability of Theories From the Strong Subadditivityand some sort their derivative. For now, let’s focus to 1 + 1 dimensional theories.If we define (x) as an amount of integration of energy-momentum tensor,T , alongminus infinity to position x, then we can observe that(u0) ≤∫ u0+δu0Tdu+ (u0 + δ) (2.38)This implies the local energy bound on the energy momentum tensorT ≥ ′. (2.39)In the next step, we would like to consider the quantum theories where the sys-tem is entangled and defining the local object needs more caution.Similarly, if we define the quantityM where provides a bound on the amountof energy on the u ≥ u0M(u0) = inf(∫ ∞u0〈T 〉du|ρu≤u0). (2.40)We expect thatM has some properties, for example, it should be invariant underunitary transformation.M(Uρu≤u0U†) = M(ρx≤x0). (2.41)Also we expect that M(u) that behave in a proper manner under scale invariantand Lorentz boost. In this case the most natural candidate for theM is first orderderivative of entropy, −S′. From the monotonocity of relative entropy and Unruheffect we get2pi~∫ ∞u0〈T 〉du ≥ −S′(ρu>u0) (2.42)while based on the strong subadditivitywe realize that−S′(ρu>u0) ≥ −S′(ρu<u0).In addition, for near the vacuum, the inequality comes to equality and we can matchthe coefficients. One can say clearly that in the most general formM is given byM = − ~2piS′ +G, (2.43)then the semi-local inequality will obtain by taking the first order derivatives〈T 〉 ≥ ~2piS′′ +G′ (2.44)162.4. The Stability of Theories From the Strong Subadditivityfind the G needs more information and the feeling is that for the relativistic theoryis goes to zero. Despite the 1 + 1 case, the mentioned argument doesn’t work forhigher dimension and therefore showing energy condition like QNEC needs moreand different efforts. We continue this investigation in the gravitational systems.According to Einstein general relativity, the geometry of space-time is completelyrelated to the energy distribution of matter inside the space. By looking to the Ein-stein equation we don’t see a prior restriction on geometry property, more specifi-cally the metric. Onmay ask this simple question that is there an intrinsic restrictionon geometry or energy-momentum tensor and is there forbidden sectors in the phasespace of geometry or equivalently energy distribution?We can think of some exotic cases, which are validated answer of Einstein equa-tion, but they require some special energy distribution. wormholes, warp drives andtime machines are few examples of this exotic sector of the possible solution of Ein-stein equation. But most of this strange cases need energy violation, which implieswe should have negative energy density within a considerable volume of space-time.Basically, the there is no principle of the general relativity to narrow down thephase space of stress tensors, but there is some famous energy bound in the litera-ture that makes sense for the normal matter we deal in the universe or in the abstractclassical limit. It’s useful to mention them here for future purposes[15]. For givennull vector k and future time like vectors t and u, TheWeak energy condition statesthatTαβtαtβ ≥ 0, (2.45)where for a perfect fluid with energy momentum tensor Tαβ = ρuαuβ + phαβ , theweak energy condition implies positive energy in any frameTαβtαtβ ≥ 0→ ρ ≥ 0, ρ+ p ≥ 0. (2.46)In the quantum field theories, we can find the acceptable states that violate thepositivity of energy density, ρ. We can count the Casimir experiment, the movingmirror problem and Hawking Radiation as examples. The Dominant energy con-dition, is another assumption and suggested bound that implies the energy can not172.4. The Stability of Theories From the Strong Subadditivitygo faster that lightTαβtαuβ ≥ 0→ ρ ≥ |p|. (2.47)Another energy bound, is Null energy condition which is a strong bound andcan be violatesTαβkαkβ ≥ 0→ ρ+ p ≥ 0. (2.48)and finally the strong energy condition implies(Tαβ − 12gαβT )tαtβ ≥ 0→ ρ+ p ≥ 0, ρ+ 3p ≥ 0. (2.49)The null energy condition mostly excludes exotic phenomena in general rela-tivity. But in some cases assumption of null energy condition cause a problem, foeexample in the singularity theorem.2.4.1 Focusing TheoremIn the Einstein gravity theory, the focusing theorem states the light rays never anti-focus unless we have a negative energy density. Quantitatively, to see the argument,let’s consider two nearby light ray with accosted Affine parameters λ’s. We denotethe area and spacing between this two ray symbolically by Ξ and θ as a derivationof spanned area between two light rays:θ = limΞ→01ΞdΞdλ(2.50)it’s not hard to show that θ is equal to a trace of null extrinsic curvature as well.Indeed if we define the kµ as null tangent vector to light ray normalized to λ thenwe can say θ = ∇µkµ. According to Raychaudhuri equation, we are also able tocalculate the evolution of θ along the the light raysdθdλ= − 1d− 2θ2 − σabσab −Rabkakb (2.51)where the σ as a trace free of null extrinsic curvature.The focusing theorem states [15] if we have the null curvature condition (NCC)is the space-time then the θ only decrease along the congruenceNCC→ dθdλ≥ 0 (2.52)182.4. The Stability of Theories From the Strong Subadditivityin the Einstein gravity theory the NCC is equal to Null Energy Condition (NEC)2.48.2.4.2 Quantum FocusingIf we consider a Cauchy slice Σ, we can divide the volume of space to two part byσ imaginary line. at each point, at σ we can project the null ray towards to futureparameterized by λ. we call the null plane that will construct by N . each point onσ has an assigned coordinate say y. Then the boundary of left or right subregioncan be parameterized by some V (y) function and V (y) define a slice in N . Basedon these notations, the generalized entropy given by [13]Sgen[V (y)] =A[V (y)]4G+ SOut[V (y)], (2.53)and based the above definition, we can introduce Θ, a generalized version of θ as afollowingΘ[V (y); y1] :=4G~√g(y1)δSgenδV (y1). (2.54)Now if we define V(y) := V (y) + ∂y1V (y) we can rewrite the previous formulaasΘ[V (y); y1] = limΞ→04GΞdSgend(2.55)and then, based on the above definitions it’s easy to check thatΘ = θ +4G~ΞS′Out. (2.56)The Quantum Focusing Conjecture (QFC) states[13]θ′ +4G~Ξ(S′′ − θS′) = Θ′ ≤ 0 (2.57)or− 1D − 2θ2 − σabσab − 8pigN 〈Tkk〉+ 4G~Ξ(S′′ − θS′) ≤ 0 (2.58)We can always choose special congruence that θ and σ be zero. Thus the QFCimplies very interestingly bound on the energy momentum tensor which is calledQuantum Null Energy Condition:〈Tkk〉 ≥ ~2piΞS′′out (2.59)192.4. The Stability of Theories From the Strong SubadditivitySome time the integral form of QNEC refereed as a quantum half energy condition:∫ ∞λ〈Tkk〉dλ′ ≥ − ~2piΞS′ (2.60)which is not exactly the same statement and in order to get an integral version ofQNEC, we need to investigate more.20Chapter 3Strong Subadditivity ofEntanglement Entropy in CFTsAccording to the strong subadditivity, there is a relation between entanglement en-tropy of three different subregions. If we particularly select those three subregion,then we can extract more useful information from this property of entanglemententropy. As we will see in this chapter, the strong subadditivity of the entanglemententropy can be seen as differential operator inequality. The aspects of this inequalitywill be clarified in the next chapters.For the sake of simplicity, we start our analysis in CFT2. Let’s consider anarbitrary spacelike regionwith two endpoints, andβ. We can define new subregionsfrom this interval by moving the endpoints in any favorable directions, α+λ1δ andβ + λ2. By this notation, one can getS(α+ λ1δ, β) + S(α, β + λ2) ≥ S(α, β) + S(α+ λ1δ, β + λ2) (3.1)If we expand the the entropies interms of λ1,2 we get a local differential operatorλ1λ2DS(α, β) +O(λ21, λ22) ≥ 0 (3.2)In this chapter would like to investigate more on the structure of the above dif-ferential operator. In d = 2, If the tips of the lightcone in Cartesian coordinates are(t0, x0) and (t1, x1). By using from coordinate system (u, v) which we will oftenemploy is related to the previous one by (u, v) = (t+ x , t− x) we can denote thecoordinates of the tips of the lightcone by (u0, v0) and (u1, v1). For four differentconfiguration of λ1,2 6= 0 or λ1,2 = 0 we can get four different set of differential21Chapter 3. Strong Subadditivity of Entanglement Entropy in CFTsoperator:∂αu∂βuS(α, β) ≤ 0∂αv ∂βv S(α, β) ≤ 0∂αu∂βv S(α, β) ≤ 0∂αv ∂βuS(α, β) ≤ 0(3.3)However, the general form of differential operator when we don’t consider anyassumption of λ1,2 makes more complicated formD = [δu0δu1∂u0∂u1 + δv1δv2∂v0∂v1 − δu0δv0∂u0∂v0 − δu1δv1∂u1∂v1 ] (3.4)It is interesting to notice that, in the above equations, for some special cases, thedifferential operator look like a wave equation. For example −∂u0∂v0 ≡ 14(−∂2t0 +∂2x0) ≡ 0 and −∂u1∂v1 ≡ 14(−∂2t1 + ∂2x1) ≡ 1[8]. And also by consideringδui = δvi = δ we getDSEE (u0, v1;u1, v0)= δ2 [∂u0∂u1 + ∂v0∂v1 − ∂u0∂v0 − ∂u1∂v1 ]SEE=δ24[−(∂t1 − ∂t0)2 + (∂x1 + ∂x0)2]SEE≥ 0 .(3.5)Although the Eq.3.5 sounds complicated, indeed it has a nice interpretation.The differential operatormysteriously related to kinematic space, the space of spher-ical causal diamonds or space of OPE blocks [21].Let’s considwer 2+2 dimensionalspace, K, with coordinate system (u0, u1, v0, v1) which has the following metric:ds2K ≡ −2du0du1(u0 − u1)2 − 2dv0dv1(v0 − v1)2 (3.6)A Laplacian, associate to this space given byK(2,2) = 2(u0 − u1)2∂u0∂u1 + 2(v0 − v1)2∂v0∂v1 (3.7)The space K is called the kinematic space [1, 19, 20] It will be useful to reviewthe concept of the kinematic space quickly. To describe a sphere in CFTd, we need22Chapter 3. Strong Subadditivity of Entanglement Entropy in CFTsa one time like vector, t, and one space like vector, r, where the inner product of thistwo vector is zero. This selection brakes the symmetry of SO(2, d) to SO(1, d− 1)and remains the symmetries of SO(1, 1) for these two vectors. Therefore it’s rea-sonable to guess that the space that all ball shape regions in conformal field theorieshas the SO(2, d)/SO(1, 1) × SO(1, d − 1) symmetry. We call this space a kine-matic space. An intuitive way to obtain the form of metric of kinematic space lieson the conformal transformation properties. Let’s consider two point in the modulispace, p1 and p2. Generally, we can represent the metric in ds2 = gµνdpµ1dpν2 form.If we apply the conformal transformation to the space, we then expect the metrictransform likegµν(p′1, p′2) =∂p′1α∂p1µ∂p′2β∂p2νgαβ (3.8)We can simply guess the functionality of metic, up to scaling factor, by lookingto two point fucnction transformation in vacuum of CFT under conformal tranas-formation. The transformation given by inversion matrix and therefore we expectds2K ∝ Iµν(p1 − p2)dp1µdp2ν(p1 − p2)2 (3.9)A more concreate way to find the metric can be obtian by using the concept ofembdeig space with two extra dimension. The most general metric that obeys fromSO(1, d− 1) given byds2 = a1〈t, t〉+ a2〈r, r〉+ a3〈r, t〉 (3.10)due to SO(1, 1) symmetry, a3 needs to be zero and a1 = a2. Assymptotic behaviorat the boundary implies that 〈t, t〉 = −1, 〈r, r〉 = 1, 〈r, t〉 = 0 and lim∂〈(αr +βt, u〉 for any α and β and other basis vectors u. It turns out that the following23Chapter 3. Strong Subadditivity of Entanglement Entropy in CFTsvectors match with our requirmentst =1√−(x− y)2(1− x2)(1− y2)× ((x2 − y2),−(−1 + y2)xµ + (−1 + x2)yµ, 0)r =1√−(x− y)2(1− x2)(1− y2)× ((−1 + x2y2),−(−1 + y2)xµ − (−1 + x2)yµ,(−1 + x2)(−1 + y2))(3.11)putting this vector in Eq.3.10 and selecting a1 = a2 := 4L2 gives:ds2K =4L2(x− y)2 [−ηµν +2(xµ − yµ)(xν − yν)(x− y)2 ]dxνdyµ, (3.12)we define the space with above metric as an auxillary de Sitter space, K.Instead of using from the tips coordinate of causal diamond (xµ, yµ) in ourcalcualation, we define the central coordinate (cµ, `µ)defiend as a followingcµ =xµ + yµ2, `µ =xµ − yµ2. (3.13)By this transformation the metric will becomeds2K = −L2`[ηµν − 2`2`µ`ν ][dcµdcν − d`µd`ν ] (3.14)It’s fascinating to notice that in the particular case, where the spherical subre-gions lie on the fixed time slice, the associated metric to those causal diamond hasthe de Sitter space signature. More explicitly by selecting y0 = R,x0 = −R andxi = yi we get d-dimensional de Sitter spaceds2 = dsdS =L2R2[−dR2 + dx2] (3.15)This implies that dS1,d−1 is a subspace from the larger space, Kinematic space withKd,d.In the d = 2 case, the kinematic space has a simpler structure, in fact, the metricis split to two subspace, named left and right de Sitter space:ds2K = ds2dS2,R + ds2dS2,L (3.16)24Chapter 3. Strong Subadditivity of Entanglement Entropy in CFTsBy looking at the definition of Laplacian in the de Sitter space and Eqs.??-??werealize thatK(2,2)SEE = [dS2,L +dS2,R ]SEE ≥ 0dS2SEE ≥ 0(3.17)The positivity of entanglement entropy under de Sitter’s Laplacian in d = 2 canbe check also directly from the CFT calculations.If we consider the arbitrary confromal tranasformation f(z) = z′ = z + (z),then the energy momentum tensor transform T (z) trnasfrom under this conformaltransformation such that:δT (z) = − c12∂3(z)− 2∂(z)T (z)− (z)∂T (z). (3.18)This let us to conclude thatT (z′) = (dfdz)−2[T (z) +c12(f ′′′(z)f ′(z)− 32f ′′(z)2f ′(z)2)]. (3.19)the last term in above equation is called Schawartzian derivative {z′, z} = (2∂3zz′∂zz′−3(∂2zz′)2)/2(∂zz′)2. It’s good to notice that for the small perturbation z′ = z +α(z), the Schawarzian term is equal to α(z)′′′+O(α2). We are able to obtain theentanglement entropy from the replica trick. By considering the twist operators,Φ±with weight of c/24(n− 1/n) it turns oute(1−n)S(n)= 〈Φ+(z1)Φ−(z2)〉 = 1(z2 − z1) c24 (n− 1n ). (3.20)Thus the vacuum entropy given by:SVac = limn→1S(n) =c6logz2 − z1(3.21)there are some conformal transformation∈ SL(2, C) that map vacuum state to vac-uum state, but in general there are other transformation, say U that map vacuumstate,|0〉 to |f〉 = U |0〉. In this case〈f |Φ+(z1)Φ−(z2)|f〉 = (dfdz)−hnz1 (dfdz)−hnz2 (df¯dz¯)−h¯nz¯1 (df¯dz¯)−h¯nz¯2 〈0|Φ+(z1)Φ−(z)|0〉,(3.22)253.1. Strong Subadditivity in Higher dimensions (d > 2)thereforeSexc = limn→1S(n)exc =c12log |f′(z1)f ′(z2)f¯ ′(z¯1)f¯ ′(z¯2)2|. (3.23)We can rewrite the above equation in more familiar coordinate:SEE,exc(u, u¯, v, v¯) =c12log[f(u0)− f(u1)]2f ′(u0)f ′(u1)+c12logf¯(v0)− f¯(v1)f ′(v0)f¯(v1)− c6log ≡ SEE,L + SEE,R(3.24)we observe that we can also introduced a splitting between holomorphic and anti-holomorphic part. The difference split the entropy to two left and right moving part.In the special case, for the vacuum state where f(z) = z, f¯(z¯) = z¯, the entropyeasily given from the above equation as a followingSEE,vac =c6log |u1 − u0|+ c6log |v1 − v0| − c6log (3.25)It’s very useful to work on the re-normalized entropy: S ≡ SEE,exc − SEE,vac.Now, it’s interesting to investigate on the action of SSA differential operator overthe explicit formula of the entropy. The simple calculation showsdS2S =12KS = c6e−12cS = −2S + 12S2c− 48S3c2+ · · · ≥ 0 (3.26)as we can see for large c and near to vacuum, the entanglement entropy follow fromthe famous wave equation in the de Sitter space [22](dS2 + 2)δS = limc→∞O(1/c) = 0 (3.27)3.1 Strong Subadditivity in Higher dimensions (d > 2)In the cases that dimension of space time is greater than two, finding the differentialequation that entanglement entropy obeys is more subtle. For the rest of discussion,we restrict to the ball shape region to gain from its useful symmetries. Also, we263.1. Strong Subadditivity in Higher dimensions (d > 2)a. b.Figure 3.1: a: Applying strong subadditivity for N subregion for d = 3, madefrom boosting the initial subregion along null cone. The union of Intersection ofa set of subregion creates a circle with wiggle with a particular size. b: The samestrategy for d = 4, but in this case, the subregions are a sphere, and the union andintersections create a continuous singular points and crests. To find the singularpoints contribution, it’s useful to circumscribe a polygon to a systemneed to apply the strong subadditivity inequality to subregions in such way that in-tersection and unions of subsystems become the ball shape region again. We can dothis based on the method introduced in the [9, 16], but we will see that we need todeal with divergences come from corners contribution to our inequality. To elabo-rate this idea, according to Fig.3.1 we select an arbitrary space like region with tipspositions (~x, ~y). In the next step, we consider N different deformed subregions,{Ai} made by boosting the original ones along with the light cone in different di-rections uniformly similar to what illustrated in the figure. Thenwe apply the Strongsubadditivity inequality to these subsystems:∑iS(Ai)− S(∪iAi)− S(∪i,jAi ∩ Aj)− · · · − S(∩iAi) ≥ 0 (3.28)The union of all subregions gives the not perfect sphere with the highest size,matched to the original subregion, and in the same manner, the intersection of allmakes the approximated sphere with the smallest size, illustrated in the Fig.3.1. The273.1. Strong Subadditivity in Higher dimensions (d > 2)rest ofN − 2 terms in Eq.3.28 are equal to other spheres, with different sizes, cov-ering the range between two mentioned sphere in a special distribution. At the largeN limit, we expect the get sphere but we always expect to have wiggles, includingcorners and crests singularity issue on the boundary of the spheres.To elaborate this issue, it good to recall that from [14], having singular angleat the boundary of a given subsystem Rwith its complement region, R¯, leads to auniversal divergence contribution in the entanglement entropy%′(`,Ω) = T (Ω)univ × [(−1)plog(`/δ)q] (3.29)where ` is the size parameter of the subsystem and for dodd : (p, q) = (d− 1)/2, 1)and for even dimension deven : (p, q) = (d−2)/2, 2), where the T (Ω) is a universalcoefficient completely related to the sharp angle Ω and has this property that forangle limiting to pi it is proportional to (pi − Ω)2.For d = 3, for the large N the defeat angle asymptotically behave like Ω ∼pi − 2piN . Therefore, we expect thatlimN→∞∑i=1%i(`i) ∼ ±N(2piN)2 = 0, (3.30)thus we expect that the contribution of corner’s divergent vanishes in this case in thecontinuous limit. If we continue our investigation for one more dimension, d = 4where the subregions are a perfect sphere, we realize that the intersection of spherescreates sharp angles and continued divergent crest, according to Fig.??. At the largeN limit, By approximating the corner’s angle with corners of a polyhedron, onemight estimate thatlimN→∞∑i=1%i(`i) ∼ ±N( 2√pi334√N+14pi4327× 3 13N 32+ · · · )2 = ±∞. (3.31)The similar argument for higher dimensions shows that the divergences from sharpedges for configuration suggested in Eq.3.28 are inevitable and the we should addextra terms to left side for equation to compensate the non-perfect spheres contri-bution. It’s important to notice that this contribution is universal and just depends283.1. Strong Subadditivity in Higher dimensions (d > 2)on the geometry of the subsystems, not the quantum states and associated densitymatrix, ρ.Up to here. we expect in the continues limit, the SSA inequlity converts to adiffrention equation with d+ d paramters:DSEE (`) ≥ %(`). (3.32)One might argue for d ≥ 0, the divergence of left hand side of above equationmakes the differential equation trivial, but hopefully it’s not the case. We can stillextract a meaning full information from this inequality if we focus on the renor-malized version of entanglement entropy. According to [14, 17, 18] the state cor-responds to regions anchored to light cone has a unique property; they are Markovstates which they have the strong super subadditivity property:SEE,γ1 + SEE,γ2 − SEE,γ1∩γ2 − SEE,γ1∪γ2 = 0, (3.33)where γ1,2 are the boundary of entangling surface of the associated subregion on thenull sheet .This implies the vacuum state anchored to ligh cone saturates the SSAinequality and this let us to get rid out of the %(`) by subtracting the vacuum statefrom the both side of Eq.3.28. It also possible to show that the modular Hamiltoniancan be written in a following form for region edge on the light sheets:∆Hσ =2pi~∫dd−2y∫ ∞γ(y)(λ− γ(y))Tλλdλ (3.34)associated to each region also follows from the similar relation:Hγ1 +Hγ2 −Hγ1∩γ2 −Hγ1∪γ2 = 0 (3.35)Therefore we may conclude that in general, for the regularized entanglement en-tropy, S := SEE,ρ − SEE,ρ0 , and relative entropy obeys form the following differ-ential inequalities:DS(x, y) ≥ 0, DS(ρ||ρ0) ≤ 0. (3.36)Similar to d = 2 case, depend on the initial configuration of subregion andthe desired subregion we will get sets of independent differential operators from293.1. Strong Subadditivity in Higher dimensions (d > 2)SSA. At the first step, we can start with simplest case, where the initial subregionis located on the fixed time slice. If we expand the entropy of a given subregion,with (c, l) coordinate system, we can expand the entropy for deformed subregionSEE (~c+~,~l+ ~δ) = SEE (~c,~l) + µ∂µSEE + · · · and sum up over all subsystem tofirst non-zero order of , δ and getdSdS ≥ −(d− 2)`S(`)′dSdS(ρ||ρ0) ≤ −(d− 2)`S(ρ||ρ0)′(3.37)It’s important to notice that the above differential equations are general and validfor all Lorantz invariant field theories at arbitrary state, not only near to vacuum.3.1.1 General DeformationTo continue our investigation on the differential inequalities can be obtained fromthe strong subadditivity, We need to elaborate some concepts in the kinematic spacedictionary [19]. Let’s defineδH(xµ, yµ) ≡∫D(x,y)ddw( |K|2pi)∆O−d〈O(w)〉 (3.38)applying the conformal group SO(d, 2) generators,L ≡ Lp(x)+Lp(y) to the aboveintegral yields toL(δH(x, y)) =∫D(x,y)ddw( |K|2pi)∆O−d〈[L,O(w)]〉 (3.39)the commutator inside the integral doesn’t have simple form, for example for d = 2it looks like [Ln,O] = (n + 1)hzkO + zk+1∂zO. now applying the operatorsagain to the both side of equation gives us the Casimir operator equation, C =CpqLp(x)Lq(x). By recalling that [C,O] = Cpq[Lp, [Lq,O]] = ∆(d −∆)O, wewill have(L2K + ∆(d−∆))H = 0 (3.40)303.1. Strong Subadditivity in Higher dimensions (d > 2)we can generalized more the definition and considering operators with spin as wellδH(x, y;O) =∫Dddw((y − w)2(w − x)2−(y − x)2)∆−d2×Πi=li=1(y − x)2Kµi〈Oµ1···µl〉−2pi[−(y − w)2(w − x)2(y − x)2]l/2(3.41)and similarly(L2 + ∆(d−∆)− l(ld − 2))H = 0 (3.42)for given operators, O(u)and O(v), the product of them can be given by anexpansion called operator product exapnsion, OPE:O(u)O(v) =∑kCk(Ou, Ov,Ok)Bk(u, v,Ok) (3.43)where B is called OPE-block and made of infinte sereis of conformal descendants.It’s not hard to show that in general the OPE-blocks has the following formBk =∫D(u,v)ddw(u− v)d−∆Oi−∆u−∆v(u− w)∆Oi−d−∆u+∆v× (v − w)∆Oi−d+∆u−∆vOi(w)(3.44)for scalar operator, ∆u = ∆v = 0, the Bk takes a form similar to H that weintroduced. There is an intersting connectino between OPE-block and AdS fieldaccording to Kinematic space dictionary [19, 32]: at leading order the Bk is realtedto Radon transform of dual field in the AdS space up to normalization constant:Bk(u, v) = 1λ(∆k)R(φk)(B˜(u, v)) (3.45)where according to intertwining property,KR(φ) = −R(Adsφ). (3.46)We can relates H ( for scalar operators) to δS by noticing that according toWald’s formalizmδH =∫∞δQξ − ξ ·Θ(φ) =∫B¯δQξ − ξ ·Θ(φ)−∫Σ?(−2δEgabξb) = δSRT + δSBulk = δS(3.47)313.1. Strong Subadditivity in Higher dimensions (d > 2)Therefore for the near to vacuum states we have wave equation equality(K(d,d) + 2d)δS = 0, and (dSd + d)δS = 0. (3.48)For a given real Lie group G and closed subgroupsK and H(R · f)(g ·H) =∫Hf(gh ·K)dh (3.49)defines a integral transformation, called Radon transformR : Γ(G/K)→ C∞(G/H)if for all g ∈ G and f ∈ Γ(G/K)the integral converges. ,R is homomorphicunder G group, thus we expect R(Γ(G/K)) annihilates ubder any U(g), a gen-eral operator from G algebra. therefor we expect that functions in the range space,R(Γ(G/K)),follows from sets of partial differemtial equation on G/H . For G =SO(d, 1), we can also find set of constrain equation on f: C · f = 0, D · f = 0,where Cijrs and Dijrs define asCijrs = XijXrs +XriXjs +XjrXis for 1 ≤ i < j < r < s ≤ nDijl = XijYl +XliYj +XjlYi for 1 ≤ i < j < r ≤ n(3.50)whereXij ≡ eji−eij , Yl = el,n+1+en+1,l and eij = (δri ·δsj) for 1 ≥ i, j ≤ n+1.It would be interesting to find constraints on the Radon transform of f as well.We would review on the recent result on that in [21].In general, wemight expect that we have d+d different variable and therefore 2dindependent differential equation describing the dynamics of the quantities in ourproblem and that might sound completely complicated. The signature of kinematicspace is completely odd, and we have d time directions. However, it turns out thatthe evolution is allowed in a very narrow subgroup of phase space. Indeed we havemany constraints that putting together fulfill the extra information required to solvethe differential equation correctly. We can realize some set of constraints in ourproblem which is involved with SO(2, d) conformal group by noticing that thereare many operators that annihilate the state in CFT and they are not trivial. If Jij bethe generator of the SO(2, d) group, then any states annihilates with CijklO(z) ≡J[ijJkl]O(z) = 0. Thus we expect that in generalCijklf(x, y) ≡ Cijkl∫D(x,y)ddwI(x, y, w)O(w) = 0 (3.51)323.1. Strong Subadditivity in Higher dimensions (d > 2)whereCijkl(x, y) ≡ J[ij(x)Jkl(x) + J[ij(y)Jkl](y), (3.52)with constraints (d+ 2)!/4!(d− 2)! distinct operator. This number is more thansome equations we need. Thus there should be a subgroup of non-trivial operators.There is a well know constraint called John’s equation which states the SO(2, 2)subgroup of above equation gives useful constraintsEjk ≡ 12C1jkl = 2{Mjk, D} − {Pj , Qk}+ {Qj , Pk} (3.53)whereMij translation operator, Pj is boost operator andQ is special transformationof conformal group. It turns out studying E0i’s are enough to extract the wholeinformation we need. According to first law of entanglement δH = δS and δH aswe saw in past paragraph can seen as o Radon transform and OPE-block integralform, thus we can get a non-trivial differential equation that entanglement entropyshould obeys:E0jδS = 0 (3.54)To see the impact of this argument, let’s do the calculation explicitly for CFT2. Intwo dimension we just have one set of equation which impliesE01δS = (dSL,2 −dSR,2)δS = 0 for d = 2. (3.55)By recalling that K(2,2) = dSL,2 −dSR,2 we realize that the12K(2,2)δS = dSL,2δS = dSR,2δS = 0 (3.56)this case study shows that there is no more information on by applying the generaldeformation of subregion in strong subadditivity scenario rather than consideringthe subregion on fixed time slice. The extension of this observation to higher di-mension is not fully understood yet. For example for CFT3 if we start with an ar-bitrary subregion( not necessary lies the fixed time slice) and then generating otherdeformed region with (cµ, lµ)→ (c+µ/2, l−µ/2) aligned the light cone and ap-plying the strong subadditvity to them with get the following differential operator:333.2. QNEC From Convexity of Relative EntropyDS(cµ, lµ) = (l20 − l21 − l22)[−∂2∂l20− ∂2∂c20+∂2∂l21+∂2∂c21+∂2∂l22+∂2∂c22+ 2∂∂c0∂∂l0− 2 ∂∂c1∂∂l1− 2 ∂∂c2∂∂l2]+ 2l0(∂∂l0− ∂∂c0) + 2l1(∂∂l1− ∂∂c1) + 2l2(∂∂l2− ∂∂c2),(3.57)despite it’s different from dSd and K(d,d), the combination of non-trivial equa-tion, E01, E02, E12, might helps to simplify the differential equation and checkwhether it has a new information about the structure of entanglement entropy ornot.3.2 QNEC From Convexity of Relative EntropyIn [27] stated there is a deep connection between relative entropy and quantumnull energy condition. The monotonicity of relative entropy implies Average NullEnergy Condition (ANEC) and it seems the pseudo convexity of relative entropy,S(ρ||ρ0)′′ ≥ 0, implies QNEC. To see this connection, let’s recall the argument in[27] that the general form of QNEC given by∫dd−2y〈Tkk〉V˙ (y)2 ≥ ~2pid2Sdλ2, (3.58)and for the special selection of V˙ (y) = δ(y − y′), the previous statement turns to〈Tkk〉 ≥ ~2piS′′. (3.59)Now, by taking second order derivate to the first law of entanglement, we getd2S(ρ)dλ2− d2S(ρ0)dλ2= Trδρd2Hρ0dλ2+O(δρ2) (3.60)where λ boundary parameter of a subregion and define the size of the subsytemin our problem. If we considering the state ρ0 such that saturates the QNEC, thencombination of Eq.??, 3.34and 3.60 impliesH ′′ρ0 =~2piTkk (3.61)343.2. QNEC From Convexity of Relative EntropyNow, by recalling the definition of relative entropy S(ρ||σ) = 〈Hσ〉 −∆S werealize that convexity of relative entropy is equal to quantum null energy condition:S(ρ||ρ0)′′ ≥ 0⇐⇒ QNEC (3.62)In addition, the monotonicity of relative entropy implies∂RS(ρ(λ)||σ(λ)) ≥ 0 (3.63)By recalling thatdSdS(ρ||ρ0) = [−∂2R +∇2]S(ρ||ρ0) ≤ −(d− 2)RS(ρ||ρ0)′ (3.64)We realize thatS(ρ||ρ0)′′ ≥ 0 (3.65)This is equal to say that quantum null energy condition is a valid condition inquantum field theories. It’s important to notice the above convexity is valid forgeneral state ρ and it’s not bound to near vacuum states only.35Chapter 4Intertwinement and RadonTransformation4.1 Radon Transform and IntertwinementIn this chapter we would like to investigate more on the connection of dynamicsin CFT and OPE blocks space and dynamics in classical gravity theories. Thisconnection recently elaborated by the context of Kinematic space in [19–21, 32].According to RT formula, there is a duality between the entropy of a subregion,D,at the boundary and minimal surface area in the bulkSEE (D) =extd∼DA(d)4kc3G~. (4.1)For simplicity we set k = c = 1 for later calculations. This simple relation hasbeen a key element on many investigations in gravity and QFT side of the duality.In the previous sections, we realized that the entanglement entropy follows fromsecond order differential inequality. The RT motivates us to see what the aspectof this differential equation in the gravity side, bulk, of the theory by consideringthe variation of the minimal surface under changing the physical states in the CFTsystem. For the vacuum states in the CFT, the metric in bulk is pure AdS space.Modification of the density matrix from the ground state, ρvac, to a state near toground state, ρvac + δρ, is equal to variation of the background metric from g0 =gAdS,αβ to g0+δgαβ . Under this perturbation, the first order variation of theminimalsurface for the ball shape region will be given byδA = 12∫B˜hαβδgαβ (4.2)364.1. Radon Transform and Intertwinementwhere γµν is induced metric on the minimal surface, B˜. Here we observe thatthe quantities which are integrating over the geodesics or the minimal surface aretensors; thus it is useful to extend the definition of the Radon transform beyond thescalars and define the δA as a new kind of Radon transformation. For our purpose,we use the definition of parallel and transverse Radon transform. For any rank-2tensor Kαβ one can define [20, 32]:R‖[Kαβ]B˜ =∫B˜hαβKαβR⊥[Kαβ]B˜ =∫B˜(gαβ − hαβ)Kαβ.(4.3)such that the summation of parallel and transverse Radon transform are simply thenormal Radon transform:R‖[Kαβ]B˜ +R⊥[Kαβ]B˜ = RB˜[TrK] (4.4)By these definitions, the variation of surface area can be equal toδA = 12R‖[δgαβ], (4.5)It’s also sometime useful to rewrite the R‖ and R⊥ in terms of each other:R‖[Kαβ]B˜ = −R⊥[Kαβ −12gαβTrK]B˜ , (4.6)and in the same manner:R⊥[Kαβ]B˜ = −R‖[Kαβ −1d− 1gαβTrK]B˜, (4.7)here d+ 1 is the dimension of space time. Applying these identities to δg gives It’suseful to notice that according to Eq.4.7R⊥[δgαβ] = −R‖[δgαβ −1d− 1gαβTrδg]= −R‖[δgαβ −gαβgαβδgαβ]d− 1 =2d− 1R‖[δgαβ](4.8)R⊥[δgαβ] =2d− 1R‖[δgαβ] (4.9)374.1. Radon Transform and IntertwinementTo find the relation between δA and the energy-momentum tensor of the bulk,it’s nice to see the application of Radon transform over the Einstein’s equationRαβ − 12gαβR = 8piGNTαβ. (4.10)The similarity between Einstein’s equation and Eq.4.6 lead to the simple rela-tionR‖[Rαβ] = −8piGNR⊥[Tαβ]. (4.11)Now if we perturb the metric around the pure AdS, the Einstein equation ex-pands as a followingδRαβ − 12δgαβR− 12gαβδR = Λδgαβ + 8piGNδTαβ. (4.12)Recalling that for AdSd+1 the curvature given byR = d(d+1)/L2 and cosmo-logical constant byΛ = d(d− 1)/2L2 where L is the AdS radius which we expectit be related to the central charge of the theory at the boundary. Now by noticingthat, Substituting the R and Λ. Therefore:δRαβ − 12gαβTrδR = δgαβ[Λ +12R] +12gαβδgαβRαβ + 8piGNδTαβ= δgαβ[Λ +12R]− 12gαβδgαβRαβ + 8piGNδTαβ= δgαβΛ + 8piGNδTαβ=d(d− 1)2L2δgαβ + 8piGNδTαβ(4.13)thusR⊥[δRαβ − 12gαβTrδR] = −R‖[Rαβ] = R⊥[δgαβ]d(d− 1)2L2+ 8piGNR⊥[δTαβ]=2d2L2R‖[δgαβ] + 8piGNR⊥[δTαβ]=2dL2δA+ 8piGNR⊥[δTαβ](4.14)δA = −L22d(R‖[δRαβ] + 8piGNR⊥[δTαβ]) (4.15)384.1. Radon Transform and IntertwinementAfter defining the Radon transform for the tensors, we are ready to investigatemore on the action of Casimir operator of the SO(d, 2) and SO(d, 1) on δA andgain from the intertwinement property. To find the equation of motion of Radontransform, it’s useful to recall the intertwining features which connects the evolutionof radon transforms in kinematic space to the dynamic in the hyperbolic space[19–21, 32]:L2KRˆ[Xµν ](B˜) = −Rˆ[L2HdXµν ](B˜) (4.16)the representation of tensor order l is related to SO(d − 1, 1) group and theassociate representation given byL2AdS = −(∇2 + `(`+ d− 1)) (4.17)We can decompose the general tensor to full trace and tracelss mode as a linearcombination:Xµν = gµνTrXd+ 1+ XTracelessµν , (4.18)and as a result the intertwinement impliesKRˆ[Xµν ](B˜) = −Rˆ[∇2TrX+ 2(d+ 1)XTracelessµν ]= −Rˆ[(∇2 + 2(d+ 1))Xµν − 2gµνTrX].(4.19)If we apply the above identity for Xµν = δgµν and comparing the result withEq.4.5 we getKδA = −12R‖[∇2δgαβ + 2(d+ 1)δgαβ − 2gαβTrδg]. (4.20)in the other hand, the variation of Ricci tensor for AdSd+1 given by2δRµν = ∇µ[gλσ∇λδgνβ ] +∇ν [gλσ∇λδgµσ]−∇µ∇νTrδg − (∇2 + 2(d+ 1))δgµν + 2gµνTrδg.(4.21)Considering this fact that the parallel Radon transform for the total derivative formsvanishes R‖[∇αTβ] = 0, we can eliminate three terms is Eq.4.21 and obtain verysimple relation:KδA = R‖[δRαβ]B˜. (4.22)394.1. Radon Transform and IntertwinementIn the following, we would like to check the action of Casimir operator SO(d, 1)over the perturbed area functional. Thus we need to the relation similar to theEq.4.19. Let’s denote the induced metric in the extremal surface for the fixed timeslice case by hαβ , and the perturbed one with δhαβ , then the area term similar toEq.4.2 will give byδAΣ = 12∫Σthαβδhαβ =12R‖[δhαβ], (4.23)and according to the intertwinement relation for fixed time slice case, the dynamicsof δhαβ with an associated Ricci tensor rαβ , given bydSR‖[δhαβ] = −R‖[∇2δhαβ + 2dδhαβ − 2gαβTrhαβ] (4.24)Combination of these two identity is enough to show thatdSδA = 12R‖[δrαβ] (4.25)It would be great to express the right-hand side of the equation regarding energy-momentum tensor. For this reason, we need to review deeper the geometric struc-ture of space time on fixed Cauchy time slice. Let’s consider the non-null Hyper-surface with intrinsics metric hab such that habhbc = δac on the surface. There aremany different ways to define intrinsic metric respect to the general metric of spacetime, for example in ADM coordinate:g = −(αdt)2 + hab(βadt+ dxa)(βbdt+ dxb) (4.26)It’s would be useful to define a unit normal vector, nµ, to our Cauchy time slice, Σ.Then by this convention, the extrinsic curvature can define byχµν = hλµhρν∇λnρ =12hλµhρνLngλρ =12Lnhµν . (4.27)Now if we assume normal vector to Cauchy slice, be timelike then we can fixthe α and β and and the extrinsic curvature becomes simplerχab =12∂thab. (4.28)404.1. Radon Transform and IntertwinementTo see the constraints and relationships between elements in Einstein equation, it’sgood to notice that Hamiltonian constraint implies0 = 2ZH := nµnν(Gµν + Λgµν − 8piGNTµν),= r − χabχab + (χaa)2 − 2Λ− 2piGNρ.(4.29)In addition, the momentum constriants lead to0 = ZMν := nµhρν(Gµρ + Λgµρ − 8piGNTµρ)0 = ZSµν := hλµhρνZλρ(4.30)For equal time slice of AdS we can set χab = 0 and thusr = 2Λ + 16piGNρ = 2Λ + 16piGNT00 (4.31)To fully understand the physical property of the space time in another time slice,we need extra information such as extrinsic curvature or momentum and variationof metric in the time direction. It’s not hard to show that Einstein equation basedon intrinsics and extrinsic metric get the following fromZµν = ZµνS − nνZµM − nµZνM + nµnνZH= Gµν + Λgµν − 8piGNTµν .(4.32)For the fixed time slice case the above Einstien equation reduces to∂thab = 2αχab + Lβhab∂tχab = −α(rab + χccχab − 2χacχcb) +DaDbα+ βχab + α(Λhab + 8piGN (Tab − 12Thab)).(4.33)Working with static background metric guaranty that ∂tχab = 0 and ∂thab = 0,thus Einstien equation, but now over the hypersurface find a simple form for generaldimensionrab =2d− 2Λhab + 8piGN (Tab −1d− 2Thab) (4.34)414.1. Radon Transform and IntertwinementSimilar to Eq.4.12, if would be useful to study the perturbation of above equationof the Cauchy slice.δrαβ − 12δhαβr − 12hαβδr = Λδhαβ + 8piGNδTαβ (4.35)denoting the r′αβ in the time slice of space time with one dimension lower than theglobal one leads to vanishing the term contains r in the left-hand side and termproportional to δhαβ on the right-hand side.Thus the perturbed equation of motion reduces toδrµν − 12hµνδr = 8piGNδTµν . (4.36)Applying the Radon transform to both side and substitutingdSdδA = −12R⊥[8piGNδTµν ] (4.37)Recall that for non-null hypersurfaces we can connect the global and intrinsicmetric by normal vector gαβ−hαβ = −nαnβ and then decomposing to null vectors.Applying the result to differential inequality governed by strong subadditivity andreplacing the entanglement entropy by area according to RT formula leads todSdδS =2pi~∫B¯uαuβδTαβ ≥ −(d− 2)`S ′(`) (4.38)The above result can have a profound interpretation. First of all, we realize that,at least for states near to ground state, the Strong subadditivity of entanglement en-tropy is equal to energy condition in the bulk. Recently, this inviolable energy con-dition was checked specifically in d = 2. As we can see the right-hand side of aboveenergy condition is zero for two-dimensional Conformal field theories and for thisreason that was interpreted as an Averaged Null Energy Condition (ANEC)[8, 30].However, for the higher dimensions, the right hand side of the inequality becomesnon-trivial. If we rewrite the above relation again∫B¯〈δTkk〉 ≥ −(d− 2) ~2pi`S ′(`), (4.39)the ~ coefficient indicates that the above averaged energy condition is a quantumeffect and the lack of GN shows despite the integration is the bulk, which is the424.1. Radon Transform and Intertwinementgravitational system, there is no gravitational effect on this bound. This might beinterested as quantum bound in the curve space. surprisingly, the form of energybound which we found has a similar form of quantum null energy condition. In theintegral form of QNEC, we have the first order derivative of entropy respect to λ,an affine parameter in the null direction of deformation. But here we have a firstorder derivation of the entropy of ball shape region respect to its size. In additionwe have a extra (d− 2)× ` which makes this connection more mysterious.43Chapter 5ConclusionsStrong subadditivity states that "More strongly a quantum system A is entangledwith a system B, the less strongly it can be entangled with another system C." Thisquite simple constraint in nature, leads to the profound result and bound in physics.There is enough evidence that Strong subadditivity is responsible for having a boundfrom the below on energies in theories of physics. In this project, we investigate onthe aspects of strong subadditivity in quantum field theories and also their holo-graphic dual which are gravitational systems.In this thesis, we showed the strong subadditivity of entanglement entropy withmonotonicity of relative entropy leads to the pseudo convexity of the relative en-tropy. In the other hand, this property of relative entropy is equivalent to havingquantum null energy conditions in theory. After proofing this connection to QNECat boundaries, we investigate on the aspect of the strong subadditivity in bulk. Inter-twinement of radon transform helped us to translate the strong subadditivity state-ment to a geometrical constraint. We found having the strong subadditivity at theboundary is equal to having an averaged energy condition simulated to QNEC atthe bulk.44Bibliography[1] Curtis T. Asplund, Nele Callebaut, and Claire Zukowski. Equivalence ofEmergent de Sitter Spaces from Conformal Field Theory. 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The strong subadditivity of holographic entanglement entropy ; from boundary to bulk Rad, Ali I. 2017
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Title | The strong subadditivity of holographic entanglement entropy ; from boundary to bulk |
Creator |
Rad, Ali I. |
Publisher | University of British Columbia |
Date Issued | 2017 |
Description | One decade ago, Ryu-Takayanagi explicitly introduced a formula that relates the entropy of a subregion in CFT system to a geometrical quantity which is called minimal surface in hyperbolic space. This formula extended to the idea of connection of gravy to quantum mechanics of gauge/gravity duality. This duality which can help us to learn a more interesting feature of each side from the other. Quantum systems obey from some constraints come from the quantum information theory. I would be interesting to find out what is the dual of this constraint in the gravitation system. Dual to the specific class of quantum theories which is called conformal field theories. One of the most significant constraint that QFTs should obey is the strong subadditivity of entanglement entropy. These constraints let the theories have bound on the energy spectrum from the below; Recently there has been the development that the combination of monotonicity of relative entropy and the strong subadditivity of entanglement entropy is equal to have a specific bound on the energy momentum tensor, called quantum null energy condition. In this thesis, we re-look to this argument by introducing the entanglement density and obtain a differential operator from the strong subadditivity and exploiting from the Markov property of the vacuum of CFT. In the next step, by using from the Ryu-Takayangi, we rewrite the strong subadditivity inequality regarding geometrical quantities. By using from the kinematic languages and intertwinement, we realize that the strong subadditivity at the boundary implies new bound on averaged energy condition which has some common feature with the quantum null energy condition statement. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2017-08-31 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0355251 |
URI | http://hdl.handle.net/2429/62925 |
Degree |
Master of Science - MSc |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2017-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
AggregatedSourceRepository | DSpace |
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