Entanglement in loop quantum gravity

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1 Entanglement in loop quantum gravity Eugenio Bianchi Penn State, Institute for Gravitation and the Cosmos [soap foam - microphotography by Pyanek] International Loop Quantum Gravity Seminar Tuesday, Nov 7th

2 Entanglement as a probe of locality - e.g. 1d fermionic chain Hilbert space: 2 N dimensional H = C 2 C 2 Geometric subsystem A =Tr B ( ih ) N A N B Entanglement entropy S A ( i) = Tr A ( A log A ) 1) Factorized basis states "i #i #i zero law Entanglement Entropy theoretical maximum "i #i #i zero-law entanglement theoretical maximum N A = subsystem size

3 Entanglement as a probe of locality - e.g. 1d fermionic chain Hilbert space: 2 N dimensional H = C 2 C 2 Geometric subsystem A =Tr B ( ih ) N A N B Entanglement entropy S A ( i) = Tr A ( A log A ) 1) Factorized basis states "i #i #i zero law 2) Ground state of a local Hamiltonian area law Entanglement Entropy theoretical maximum theoretical maximum ground state of local H area-law entanglement N A = subsystem size [Sorkin (10th GRG) 1985] [Srednicki, PRL 1993]

4 Entanglement as a probe of locality - e.g. 1d fermionic chain Hilbert space: 2 N dimensional H = C 2 C 2 Geometric subsystem A =Tr B ( ih ) N A N B Entanglement entropy S A ( i) = Tr A ( A log A ) 1) Factorized basis states "i #i #i zero law 2) Ground state of a local Hamiltonian area law 3) Typical state in the Hilbert space volume law - maximally entangled Entanglement Entropy theoretical maximum typical state maximally entangled theoretical maximum N A = subsystem size [Page, PRL 1993]

5 Entanglement as a probe of locality - e.g. 1d fermionic chain Hilbert space: 2 N dimensional H = C 2 C 2 Geometric subsystem A =Tr B ( ih ) N A N B Entanglement entropy S A ( i) = Tr A ( A log A ) 1) Factorized basis states "i #i #i zero law 2) Ground state of a local Hamiltonian area law 3) Typical state in the Hilbert space volume law - maximally entangled Entanglement Entropy theoretical maximum typical eigenstate theoretical maximum of a local Hamiltonian 4) Typical excited state of a local Hamiltonian have non-maximal ent. at finite fraction N A = subsystem size [E.B.-Hackl-Rigol-Vidmar, PRL 2017]

6 Entanglement as a probe of locality - e.g. 1d fermionic chain Hilbert space: 2 N dimensional H = C 2 C 2 Geometric subsystem A =Tr B ( ih ) N A N B Entanglement entropy S A ( i) = Tr A ( A log A ) 1) Factorized basis states "i #i #i zero law 2) Ground state of a local Hamiltonian area law 3) Typical state in the Hilbert space volume law - maximally entangled 4) Typical excited state of a local Hamiltonian have non-maximal ent. at finite fraction [E.B.-Hackl-Rigol-Vidmar, PRL 2017]

7 Building space from entanglement Plan: 1) Entanglement, mutual information and bosonic correlators 2) Gluing quantum polyhedra with entanglement 3) Entanglement and Lorentz invariance 4) Quantum BKL and entanglement in the sky

8 Defining entanglement entropy in loop quantum gravity Entanglement entropy S R ( i) = Tr( log ) [Ohya-Petz book 1993] characterizes the statistical fluctuations in a sub-algebra of observables A R A Two extreme choices of subalgebra: a) Determine the algebra of Dirac observables of LQG, difficult to use then consider a subalgebra b) Enlarge the Hilbert space of LQG to a bosonic Fock space, then consider a bosonic subalgebra [EB-Hackl-Yokomizo 2015] useful for building space Other choices: - In lattice gauge theory, trivial center sub-algebra [Casini-Huerta-Rosalba 2013 ] - Adding d.o.f. (on the boundary), electric center subalgebra [Donnelly 2012] [Donnelly-Freidel 2016] [Anza-Chirco 2016] [Chirco-Mele-Oriti-Vitale et al 2017] [Delcalp-Dittrich-Riello 2017] - Intertwiner subalgebra (at fixed spin) [Livine-Feller 2017 ] -

9 Bosonic formulation of LQG on a graph [also known as the twistorial formulation] - Two oscillators per end-point of a link spin from oscillators j, mi = (a0 ) j+m (a 1 ) j m p p 0i (j + m)! (j m)! [Schwinger 1952] - Hilbert space of LQG and the bosonic Hilbert space L 2 (SU(2) L /SU(2) N ) H bosonic i = 1X n i =1 c n1 n 4L n 1,...,n 4L i [Girelli-Livine 2005] [Freidel-Speziale 2010] [Livine-Tambornino 2011] [Wieland 2011] [EB-Guglielmon-Hackl-Yokomizo 2016] - The bosonic Hilbert space factorizes over nodes: easy to define and compute the entanglement entropy - Geometric operators in a region R of the graph generate a subalgebra A LQG R A bosonic R

10 Entanglement entropy of a bosonic subalgebra A - Spin-network state,j l,i n i factorized over nodes no correlations, zero entanglement entropy in A A - Coherent states P zi = Pe zi A aa i 0i not factorized over nodes only because of the projector P B exponential fall off of correlations area law from Planckian correlations only - Squeezed states P i = Pe ij AB aa i a B j 0i not factorized over nodes because of the projector P and because of off-diag. terms in long-range correlations from ij AB ij AB efficient parametrization of a corner of the Hilbert space characterized by correlations zero-law, area-law, volume-law entanglement entropy depending on ij AB

11 Area law: formulation relevant for semiclassical physics and QFT Entanglement between region A and B with buffer zone C S geom = 1 2 S( AB A B ) A = 1 2 Tr AB log AB AB log( A B ) Area(@A) C B Area law for squeezed state with long-range correlations encoded in ij AB The area-law is non generic in loop quantum gravity, property of semiclassical states The entanglement entropy provides a probe of the architecture of a quantum spacetime

12 Entanglement and the architecture of a spacetime geometry - Entanglement entropy as a probe of the architecture of spacetime Area-law not generic, property of semiclassical states [EB-Myers 2012] On the Architecture of Spacetime Geometry S A ( 0i) = 2 Area(@A) L 2 Planck +... Arguments from: Black hole thermodynamics (Bekenstein, Hawking, Sorkin, ) Holography and AdS/CFT (Maldacena,.. Van Raamsdonk,.. Ryu, Takayanagi, ) Entanglement equilibrium (Jacobson) Loop quantum gravity (EB)

13 Long-range correlations and the bosonic mutual information - Macroscopic observables in region A and B - Correlations bounded by relative entropy of A, B ho A O B i 2 ho A iho B i 2 ko A k 2 ko B k 2 apple I(A, B) A where I(A, B) S( AB A B ) = S A + S B S AB C B The bosonic formulation is useful because it allows us to define and compute the mutual information I(A, B) This quantity bounds from above the correlations of all LQG-geometric observables in A and B

14 Building space from entanglement Plan: 1) Entanglement, mutual information and bosonic correlators 2) Gluing quantum polyhedra with entanglement 3) Entanglement and Lorentz invariance 4) Quantum BKL and entanglement in the sky

15 Gluing quantum polyhedra with entanglement - Fluctuations of nearby quantum shapes are in general uncorrelated: twisted geometry [Dittrich-Speziale 2008] [EB 2008] [Freidel-Speziale 2010] [EB-Dona-Speziale 2010] [Dona-Fanizza-Sarno-Speziale 2017] - Saturating uniformly the short-ranged relative entropy where ho A O B i 2 ho A iho B i 2 ko A k 2 ko B k 2 apple I(A, B) A B I(A, B) S( AB A B ) = S A + S B S AB correlates fluctuations of the quantum geometry State with max X I(A, B) Glued geometry from entanglement [EB-Baytas-Yokomizo, to appear] ha,bi

16 Building space from entanglement Plan: 1) Entanglement, mutual information and bosonic correlators 2) Gluing quantum polyhedra with entanglement 3) Entanglement and Lorentz invariance 4) Quantum BKL and entanglement in the sky

17 Lorentz invariance in LQG - Discrete spectra are not incompatible with Lorentz invariance [Rovelli-Speziale 2002] - Lorentz invariant state in LQG? 1) Minkowski geometry as expectation value 2) Lorentz-invariant 2-point correlation functions, 3-point - Homogeneous and isotropic states in LQG? similarly (1), (2) Strategy: double-scaling encoded in the state - use squeezed states defined in terms of 1- and 2-point correlations - graph, e.g. cubic lattice with N nodes ij - choose the diagonal entries of the squeezing matrix to fix the expectation value of the spin ij AB AB - choose the off-diagonal entries of to fix the correlation function C = ho A O B i ho A iho B i at a lattice distance hji Escher 1953 n 0 - the correlation function can be expressed in terms of the physical length ` n 0 p hji i hji!0 n 0!1 - take the limit of the squeezed state such that, with C(`) fixed V N ( p hji ) 3 - the limit can be studied at fixed physical volume, with symmetries imposed on C(`) Toy model: 1d chain of quantum cubes with long-range entanglement and translational invariance [EB-Dona]

18 Building space from entanglement Plan: 1) Entanglement, mutual information and bosonic correlators 2) Gluing quantum polyhedra with entanglement 3) Entanglement and Lorentz invariance 4) Quantum BKL and entanglement in the sky

19 Correlations at space-like separation A r B - In quantum field theory Fock space F = C H S(H H) contains {(i) states with no space-like correlations (ii) states with specific short-ranged correlations (e.g. Minkowski vacuum) crucial ingredient for quantum origin of cosmological perturbations - In loop quantum gravity Hilbert space H = L 2 (SU(2) L /SU(2) N ) contains {(i) states with no space-like correlations (ii) states with long-range space-like correlations (e.g. spin-networks) (e.g. squeezed vacua)

20 The Vacuum State of a Quantum Field

21 The Vacuum State of a Quantum Field

22 The vacuum state of a quantum field is highly entangled

23 The vacuum state of a quantum field is highly entangled

24 The vacuum state of a quantum field is highly entangled

25 The vacuum state of a quantum field is highly entangled

26 Planck Scale Inflation Hot Big Bang CMB

27 The Vacuum State of a Quantum Field No particles a( ~ k) 0i = 0 Vanishing expectation value h0 '(~x) 0i = 0 but non-vanishing fluctuations Uncorrelated momenta h0 '( ~ k) '( ~ k 0 ) 0i = P ( ~ k )(2 ) 3 ( ~ k + ~ k 0 ) with power spectrum P (k) = 1 2k Non-vanishing correlations at space-like separation h0 '(~x) '(~y) 0i = Z 1 0 k 3 P (k) sin(k ~x ~y ) 2 2 k ~x ~y dk k = 1 (2 ) 2 1 ~x ~y 2 Fluctuations of the field averaged over a region of size R ( ' R ) 2 h0 ' R ' R 0i (h0 ' R 0i) 2 1 R 2

28 The vacuum of a quantum field after inflation Minkowski P (k) = 1 2k

29 The vacuum of a quantum field after inflation Minkowski P (k) = 1 2k de Sitter P (k) = 1 2k e 2H 0t + H2 0 2 k 3

30 The vacuum of a quantum field after inflation Mukhanov-Chibisov (1981) Minkowski P (k) = 1 2k de Sitter P (k) = 1 2k e 2H 0t + H2 0 2 k 3 Inflation (quasi-de Sitter) P s (k) 2 2 A s k 3 k k! ns 1 Planck 2015 A s (k )= n s (k )=0.96 with A s G~ H2 "

31 Mechanism: amplification of vacuum fluctuations by instabilities Harmonic oscillator with time-dependent frequency H(t) = 1 2 p k2 f(t) q 2

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37 The anisotropies of the Cosmic Microwave Background as observed by Planck

38 n s =0.965 ± ` Planck Collaboration, arxiv.org/abs/ ``Planck 2015 results. XX. Constraints on inflation''

39 A distinguishing feature of loop quantum gravity: existence of states with no correlation at space-like separation Scenario uncorrelated initial state and its phenomenological imprints

40 Emergence of space-like correlations in loop quantum gravity States with no space-like correlations: allowed in quantum gravity A r B BKL conjecture (Belinsky-Khalatnikov-Lifshitz 1970) In classical General Relativity, the spatial coupling of degrees of freedom is suppressed in the approach to a space-like singularity Quantum BKL conjecture (E.B.-Hackl-Yokomizo 2015) In quantum gravity, correlations between spatially separated degrees of freedom are suppressed in Ĥ [g ij (x), '(x)] = 0 the approach to a Planck curvature phase {lim a!0 [a,, g ij (x), '(x)] = Y ~x, g ij (x), '(x) Scenario: the correlations present at the beginning of slow-roll inflation are produced in a pre-inflationary phase when the LQG-to-QFT transition takes place

41 Background dynamics and pre-inflationary initial conditions Ḣ Quasi-deSitter phase: H Hubble rate 1 (t) = Ḣ H 2 1 Pre-inflationary phase R µ R µ = 1 L (G~) 2 HIC SVNT LOOPS

42 Perturbations and pre-inflationary initial conditions Ḣ Quasi-deSitter phase: H Hubble rate 1 (t) = Ḣ H 2 1 Pre-inflationary phase R µ R µ = 1 L (G~) 2 HIC SVNT LOOPS

43 Scalar power spectrum with LQG-to-QFT initial conditions - Mpc 1 A s 10 9 Bunch-Davies state = A s (k )= n s (k )=0.96 Initially uncorrelated state / k =0.002 Mpc 1 Power suppression at large angular scales E.B.-Fernandez 2017

44 Tensor power spectrum with LQG-to-QFT initial conditions A t Mpc 1 Bunch-Davies state = r = A t(k ) A s (k ) = Initially uncorrelated state / k =0.002 Mpc 1 Power suppression at large angular scales E.B.-Fernandez 2017

45 Scenario for the emergence of primordial entanglement in loop quantum gravity Planck Scale pre-infl. phase Inflation Hot Big Bang CMB

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48 Inflation and spinfoams - Effective spinfoam action S[e I,! IJ,r, IJ ]= Z (1 + 2 r)b IJ ^ F IJ r ! IJKLB IJ ^ B KL + B IJ ^r IJ where B IJ = 1 8 G 1 2 IJKL e K ^ e L 1 ei ^ e J = Barbero-Immirzi parameter r = 0-form, effective Ricci scalar at a coarse-graining scale = coupling constant dimensions of Area - It provides an embedding in spinfoams of the Starobinsky model (1979) S[g µ ]= 1 16 G Z d 4 x p g R + R 2 G µ + H µ = 8 G T µ Friedman eq: H H 2 Ḣ Ḣ 2 +2HḦ =0 gravity-driven inflation E.B.-Fernandez-Rincon 2017

49 Primordial spectra from adiabatic vacuum in the quasi de-sitter phase - Scalar perturbations A s k3 P s (k,t ) 2 2 G~ H2 2 1 n s 1+k d dk log k 3 P s (k, t ) - Tensor perturbations A t k3 P t (k,t ) 2 2 n t k d dk log k 3 P t (k, t ) r A t A s ? G~ H2 2 k=k? k=k? H 2 = 1 PLANCK 2015 ( ) k =0.002 Mpc 1 A s =(2.474 ± 0.116) 10 9 n s = ± r<0.11 N = Z tend t H(t)dt = 18 H G~ H p G~ r {N 70

50 Background dynamics and pre-inflationary initial conditions Ḣ Friedman eq: H H 2 Ḣ Ḣ 2 +2HḦ =0 Quasi-deSitter phase: H 1 (t) = Ḣ = 1 36 Ḣ H 2 1 Pre-inflationary phase H(t) 1 2t R µ R µ = 1 L (G~) 2 HIC SVNT LOOPS

51 Pre-inflationary initial conditions: scalar and tensor modes Ḣ Pre-inflationary phase H(t) 1 2t log a(t)h(t) +a(t) Q s (t) 3Q s (t)! H H In the pre-inflationary phase HIC SVNT LOOPS t/t P both scalar and tensor perturbations satisfy ü(k, t)+ 1 t u(k, t)+ k2 H c t u(k, t) =0 adiabatic vacuum u 0 (k, t) = r 2 J 0 2k p t/h c iy 0 2k p t/h c vanishing correlations in the limit t! 0, Bunch-Davies like correlations produced before the quasi-de Sitter phase