Entanglement Entropy from Spacetime Correlations DIAS STP Seminar - - PowerPoint PPT Presentation

Entanglement Entropy from Spacetime Correlations

DIAS STP Seminar

Yasaman K. Yazdi

Imperial College London

October 07, 2020

Outline

• General remarks on entanglement entropy (EE).
• Entropy of a chain of harmonic oscillators.
• The need for a spacetime definition of EE and the cutoff.
• Spacetime EE of a Gaussian scalar field and example

applications.

• Extensions to non-Gaussian and interacting theories.
• Future directions and summary.

Entanglement Entropy

Entanglement entropy is a measure of our limited access to a quantum system. For density matrix ρ on a spatial hypersurface Σ S = −Trρ ln ρ (1) If Σ is divided into complementary subregions A and B, then the reduced density matrix for subregion A is ρA = TrBρ (2) and its entanglement entropy with region B is SA = −TrρA ln ρA (3)

Complementarity and Area Laws

Complementarity of EE: we get the same answer whether we trace

• ut the degrees of freedom in region A or region B.

SA = −TrρA ln ρA = −TrρB ln ρB = SB (4) EE often obeys spatial area laws SA ∝ |∂A|/ℓd

UV

(5)

Is Black Hole Entropy Entanglement Entropy?

• R D Sorkin, On the Entropy of the Vacuum outside a Horizon,

(1983), arXiv:1402.3589. EE is a promising candidate for the origin of black hole and cosmological horizon entropy, which is still a major open question.

Applications of Entanglement Entropy (EE)

• Quantum Gravity: key insight to connect QM & GR
• Fundamental origin of (Bekenstein-Hawking) black hole and

cosmological horizon entropy. (Next slide)

• T. Jacobson, EE and the Einstein Equation, (2015).
• Lashkari, McDermott, Van Raamsdonk, Comments on QG and

Entanglement, (2013).

• Theorems in QFT (e.g. c-theorems) & AdS/CFT
• Ryu, Takayanagi, Holographic Derivation of EE from

AdS/CFT, (2006).

• Myers, Sinha, Seeing a c-theorem with Holography, (2010).
• Casini, Huerta, A c-theorem for the EE, (2007).
• Condensed Matter Physics: topological order & properties of Fermi

surfaces.

• Kitaev, Preskill, Topological EE, (2006).
• B. Swingle, EE and the Fermi Surface, (2010).
• Quantum Information: Teleportation & Firewalls
• Vidal, Werner, A Computable Measure of Entanglement,

(2001).

• AMPS, Black Holes: Complementarity or Firewalls?, (2012).

Numerous Papers on Entanglement Entropy

As mentioned, EE has been extensively studied in many different fields (QG, QFT, CMT, QI, ...) and it has had many important uses.

2005 2010 2015 2020 Year 50 100 150 200 250 300 350 # of Papers

Figure: The number of papers per year with “entanglement entropy” in the abstract, from arXiv.org.

Calculating Entanglement Entropy

SA = −TrρA ln ρA

• Numerically using mode expansions or lattice discretizations (eg. in

condensed-matter systems)

• Analytically in CFTs, e.g. via the Replica Trick:

SA = − lim

n→1 ∂ ∂n Tr (ρn A)

• Ryu-Takayanagi formula in holography, using areas of minimal

surfaces γ: SA = Area of γA

4G

• Using the Euclidean path integral (e.g. perturbatively in interacting

QFTs)

• Using spatial correlation functions φ(

x)φ( x′) (will review next)

• Using spacetime correlation functions φ(

x, t)φ( x′, t′) (focus of this talk)

Entropy of a 1d Chain of Harmonic Oscillators

Consider a chain of oscillators, with nearest-neighbour couplings. To find the EE associated to a subchain, we can do following1: The Lagrangian is L = 1 2  

Nmax

• N=1

ˆ ˙ q2

N − Nmax

• N,M=1

VMN ˆ qN ˆ qM   (6) = 1 2

Nmax

• N=1

[ˆ ˙ q2

N − m2ˆ

q2

N − k(ˆ

qN+1 − ˆ qN)2], (7) where k is the coupling strength between the oscillators, and in terms of the spatial UV cutoff a, k = 1/a2. C ≡ √ V (8)

1Bombelli, Koul, Lee, Sorkin, Quantum Source of Entropy for Black Holes,

PRD 34, 373, 1986.

Entropy of a Chain of Harmonic Oscillators

Now consider the division of the chain into a subchain and its complement (Greek and Latin indices resp.). It is convenient to rewrite C in terms of blocks referring to these two subsets: CAB = Cab Caβ Cαb Cαβ

• and its inverse C AB =

C ab C aβ C αb C αβ

• ,

and the inverse of each block will be expressed with tildes (for example C ab is the inverse of Cab). ρred(qa, q′b) =

• det(

Cab) π e− 1

2 Cab(qaqb+q′aq′b)e 1 4

C αβCαaCβb(q+q′)a(q+q′)b

Finally, S = − Tr ρred ln ρred can be expressed in terms of the eigenvalues λn of the operator Λa

b ≡ C acCcα

C αβCβb, as S =

• n

{ln(1 2

• λn) +
• 1 + λn ln(
• 1 + 1/λn + 1/
• λn)}.

(9)

Spatial Area Laws in 1 + 1d

We can compare to CFT results for the EE between a shorter line-segment and a longer one containing it. S for m = 0 and a sub-interval with two boundaries takes the asymptotic form for a → 0 of2 S ∼ 1 3 ln[ L πa sin(πℓ L )] + c1 , (10) where a is the UV cutoff, ℓ and L are the lengths of the shorter and longer intervals, and c is a non-universal constant. In the limit ℓ

L → 0:

S ∼ 1 3 ln ℓ a

• + c1 .

(11)

• 2P. Calabrese and J. Cardy, Entanglement Entropy and Conformal Field

Theory, JPA: Mathematical and Theoretical 42 (2009), no. 50.

Periodic Boundary Conditions: q1 = qN+1

With finite mass (for IR regularity), 1/m2 = k = 106, the entropy

• beys the expected logarithmic scaling with the UV cutoff.3

5 10 15 20 25 30

• 4.6

4.8 5.0 5.2 5.4 S

Figure: S vs. ℓ/a, along with a fit to S = b1 ln(ℓ/a) + c1. The best fit parameters are b1 = 0.3337 and c1 = 5.9316.

3YKY, Zero Modes and EE, JHEP 04 (2017) 140.

Seeking a Spacetime Definition of EE

• EE requires a UV cutoff to render it finite. Need a covariant

cutoff in gravitational systems, esp. those with horizons.

• Quantum fields are too singular to always admit meaningful

restrictions to hypersurfaces. (Next slide)

• Causal sets, which provide a fundamental and covariant UV

cutoff, require a spacetime formulation of EE.

Quantum Fields Live more Happily in Spacetime

Consider the normal-ordered φ2 operator, and smear it with a test function with compact support on a time t′ = const slice, : φ2(t′, f ) :=

• d4xf (

x)δ(x0−t′)

• d3pd3k

2(2π)6E

pE k

• a

pa kei(p+k)·x + . . .

• where E

p =

• |

p|2 + m2. We then square the result and compute its expectation value in the Minkowski vacuum, 0| : φ2(t′, f ) :: φ2(t′, f ) : |0 = d3p d3k|˜ f ( k + p)|2 2(2π)12E

pE k

(12) ∝

• d3p′|˜

f ( p ′)|2

• d3k
• (

p ′ − k)2 + m2 k2 + m2 . where ˜ f is the Fourier inverse of f and p′ = p + k. The k-integral diverges linearly in the large | k| limit.

Causal Set Theory: Spacetime is Fundamentally Discrete4

A causal set is a locally finite partially ordered set. It is a set C along with an ordering relation that satisfy:

• Reflexivity: for all X ∈ C, X X.
• Antisymmetry: for all X, Y ∈ C, X Y X implies X = Y .
• Transitivity: for all X, Y , Z ∈ C, X Y Z implies X Z.
• Local finiteness: for all X, Y ∈ C, |I(X, Y )| < ∞, where | · |

denotes cardinality and I(X, Y ) is the causal interval defined by I(X, Y ) := {Z ∈ C|X Z Y }. Order + Number = Geometry

4Bombelli, L., Lee, J. H., Meyer, D. and Sorkin, R. D., 1987, Space-Time as

a Causal Set, Phys. Rev. Lett. 59, 521.

Hasse Diagrams for a Sample of 6 Element Causal Sets

Figure: Relations not implied by transitivity are drawn in as lines. Time goes upwards.

Causal Set Sprinklings

Sprinkling: generates a causal set from a given Lorentzian manifold M, by placing points at random in M via a Poisson process with “density” σ, such that P(N) = (σV )N

N! e−σV . N ∼ V

Lorentz invariant and nonlocal.

Spacetime Definition of S for Gaussian Theory5

Express S directly in terms of the spacetime correlation functions. The entropy can be expressed as a sum over the solutions λ of the generalized eigenvalue problem W v = iλ ∆ v, (∆v = 0) (13) as S =

• λ

λ ln |λ| . (14)

W and i∆ are the Wightman W (x, x′) = 0|φ(x)φ(x′)|0 and Pauli-Jordan i∆(x, x′) = [φ(x), φ(x′)] functions. Also i∆ = i(GR − GA) = 2 Im(W ).

5Sorkin, Expressing Entropy Globally inTerms of (4D) Field Correlations,

2012, arXiv:1205.2953.

Spacetime Definition of S for Gaussian Theory

First we calculate the entropy with the replica trick. The field theory entropy can be broken up into a sum of entropies of single degrees of freedom {q, p}. ρ(q, q′) ≡ q|ρ|q′ = N e−A/2(q2+q′2)+iB/2(q2−q′2)−C/2(q−q′)2, (15) Replica trick: S = − Tr (ρ log ρ) = − lim

n→1

∂ ∂n Tr (ρn) , (16) Tr (ρn) = Nn

• dq1...dqn ρ(q1, q2)ρ(q2, q3)...ρ(qn, q1)

= Nn

• dnq exp
• −(A + C)

n

• i=1

q2

i + C n

• i=1

qiqi+1

• = |1 − µ|n

|1 − µn| where µ = √

1+2C/A−1

√

1+2C/A+1. We insert this into (16) and take the limit to

• btain the entropy.

Spacetime Definition of S for Gaussian Theory6

S = − lim

n→1

∂ ∂n Tr (ρn) = − lim

n→1

log(1 − µ)(1 − µ)n 1 − µn + log(µ)µn(1 − µ)n (1 − µn)2 = −µ log µ + (1 − µ) log(1 − µ) 1 − µ (17) Now the spacetime formulation with one degree of freedom: ∆ = 2 Im

• qq

qp pq pp

• =
• 1

−1

• ,

(18) W = Re qq qp pq pp

• + i∆/2 =

1/(2A) i/2 −i/2 A/2 + C

• .

(19) Since µ = √

1+2C/A−1

√

1+2C/A+1, with some algebra we see (17) is equivalent to

S =

λ λ ln |λ| with λ given by W v = iλ ∆ v.

6Chen, Hackl, Kunjwal, Moradi, YKY, Zilh˜

ao, Towards Spacetime Entanglement Entropy for Interacting Theories, arXiv:2002.00966.

The Sorkin-Johnston Vacuum7,8

Write i∆(X, X ′) in terms of its positive (uk) and negative (vk) eigenfunctions: i∆(X, X ′) =

• k
• λkuk(X)u†

k(X ′) − λkvk(X)v † k (X ′)

• .

(20) Restrict to positive eigenspace: WSJ(X, X ′) ≡ Pos(i∆) =

• k

λkuk(X)u†

k(X ′).

(21) When W = WSJ, purity follows by λ = {0, 1} in W v = iλ ∆ v and S =

λ λ ln |λ|.

1 commutator:

i∆(X, X ′) = WSJ(X, X ′) − W ∗

SJ(X, X ′)

2 positivity:

• M dV
• M dV ′f ∗(X)WSJ(X, X ′)f (X ′) ≥ 0

3 orthogonal supports:

• M dV ′ WSJ(X, X ′) WSJ(X ′, X ′′)∗ = 0,
• 7R. D. Sorkin, J. Phys. Conf. Ser. 306 (2011) 012017, arXiv:1107.0698.
• 8S. P. Johnston, Quantum Fields on Causal Sets, 2010, arXiv:1010.5514.

1+1 Continuum Setup9

In Minkowski lightcone coordinates u = t+x

√ 2 and v = t−x √ 2 ,

∆(u, v; u′, v′) = −1 2[θ(u − u′) + θ(v − v′) − 1] . (22) W = − 1 4π ln |∆u∆v|− i 4sgn(∆u+∆v)θ(∆u∆v)− 1 2π π √ 2 4L +ǫ+O δ L

• where ǫ ≈ −0.063.

δ collectively denotes the coordinate differences u − u′, v − v′, u − v′, v − u′. We set ℓ

L = .01.

ℓ L 𝑦 𝑢

• 9N. Afshordi, M. Buck, F. Dowker, D. Rideout, R. D. Sorkin, YKY,

JHEP10(2012)08, arXiv:1207.7101

1+1 Continuum Calculation10

We represent W and i∆ as matrices in the eigenbasis of i∆: fk(u, v) := e−iku − e−ikv, with k = nπ ℓ , n = ±1, ±2, . . . gk(u, v) := e−iku + e−ikv − 2 cos(kℓ), with k ∈ K, (23) where K = {k ∈ R | tan(kℓ) = 2kℓ and k = 0}. The eigenvalues are ˜ λk = ℓ/k. For large k: ˜ λk = ℓ2/nπ. The L2-norms are ||fk||2 = 8ℓ2 and ||gk||2 = 8ℓ2 − 16ℓ2cos2(kℓ).

• 10M. Saravani, R. D. Sorkin, YKY, Class. Quantum Grav. 31 (2014) 214006,

arXiv:1311.7146

1+1 Continuum Calculation: The Cutoff

For the representation of W , we compute fk|W |fk′ and gk|W |gk′. The terms fk|W |gk′ vanish, making W block diagonal in this basis.

Cutoff:

We truncate the matrices representing W and ∆ by retaining only a finite number of eigenfunctions fk and gk up to a maximum value kmax of k. Ker( − m2) = Im(∆). With the nmax modes retained, we can expand Cauchy data of wavelengths longer than ∼ 1/kmax. It is therefore natural to equate the cutoff to 1/kmax. In the calculations, we keep ℓ/L fixed and vary kmax.

1+1 Continuum Results

The obtained values of S are fit almost perfectly by the curve S = b ln ℓ a

• + c

(24) with b = 0.33277 and c = 0.70782.

500 1000 1500 2000 2500

• a

2.8 3.0 3.2 S

Figure: Data points represent calculated values of S =

λ λ ln |λ|. The

cutoff a = 1/kmax

Causal Set Setup in Nested Diamonds

∆(X, X ′) := GR(X, X ′)−GR(X ′, X) For m = 0, we have that GR = 1

2C, where C is the causal

matrix: Cxy :=

• 1,

if x y. 0,

• therwise

For W , we choose WSJ.

• !

!

• N. Afshordi, M. Buck, F. Dowker, D. Rideout, R. D.

Sorkin, YKY, JHEP10(2012)08, arXiv:1207.7101.

EE for Nested Diamonds in 1 + 1 Dimensions11

The EE fits S = b ln(√Nℓ/4π) + c1 with b = 0.346 ± 0.028 and c1 = 1.883 ± 0.035. This result is consistent with the expected b = 1/3 coefficient. Figure: S vs. √Nℓ/4π. ℓ/L = 1/2 in this example.

• 11R. D. Sorkin, and YKY, Entanglement Entropy in Causal Set Theory,

(2018), Class. Quantum Grav. 35 074004, arXiv:1611.10281.

de Sitter Sprinklings into ds2 =

1 cos2 T

• −dT 2 + dΩ2

d−1

• 1

1 T

• 1

1 1

• 2

2 X2

• 2

2 X0

• 2

2 X1 Figure: A sprinkling of N = 10000 elements into the de Sitter slab for the time interval −1.2 < T < 1.2. Global de Sitter has −π/2 < T < π/2. S Surya, Nomaan X, YKY JHEP 1907 (2019) 009, arXiv:1812.10228.

EE of Entangled Wedges in dS

South Pole North Pole I− I+ H

+

H

−

H

+

H

−

θ = π θ = 0 time a 3.02193 b -1.89342 0.0677285 6.77971 a 3.06282 b -8.35944 0.0599755 6.00362 a x + b 5000 10000 15000

N

100 200 300 400

S

The example above is for entangled wedges in 3 + 1 dimensional dS, with cutoff ∝ N1/4 → |∂A| ∝ √

• N. Again, a spatial area law is
• btained as expected.
• S. Surya, Nomaan, YKY arXiv:2008.07697

Interacting and non-Gaussian Scalar Theories12

W no longer specifies the whole theory and we need all higher order n-point functions. It turns out that S =

λ λ ln |λ| with λ given by

W v = iλ ∆ v captures the EE up to first order in perturbation theory even for interacting and non-Gaussian theories. The first law of entanglement entropy states (primes indicate

d dǫ):

Sǫ = − Tr(ρǫ log ρǫ) = S0 + ǫ S′

0 + ǫ2 S′′ 0 + O(ǫ3) .

(25) The first order perturbation S′

0 of a Gaussian state (where ˆ

H0 is quadratic) will only depend on the change of W : S′

ǫ = − Tr

• ρ′

ǫ log ρǫ + ρǫρ−1 ǫ ρ′ ǫ

• = −tr(ρ′

ǫ log ρǫ) = tr(ρ′ ǫ ˆ

Hǫ) . (26) At second order, non-Gaussian contributions enter (second term below) S′′

ǫ = −tr(ρ′′ ǫ log ρǫ + ρ′ ǫρ−1 ǫ ρ′ ǫ).

(27)

12Chen, Hackl, Kunjwal, Moradi, YKY, Zilh˜

ao, Towards Spacetime Entanglement Entropy for Interacting Theories, arXiv:2002.00966.

Future Directions

• Gaussian Theories:
• An ultimate goal is to understand black hole entropy. EE

calculations in the continuum are challenging, and the expression for W in a causal set black hole is not currently known.

• Fermions. Current work with Ian Jubb.
• Interacting Theories:
• At least up to first order, apply S from W formula to theories

in the continuum and causal set.

• Further work on higher order contributions in terms of

spacetime correlations.

• Find spacetime formulation of EE not tied to a fixed

background, possibly using Lorentzian path integral methods.

Conclusions

• Entanglement entropy has many important applications in

various fields of physics.

• In QFT and gravitational settings especially, it is desirable to

have a spacetime formulation of EE.

• There exists a spacetime formulation of EE in terms of the

spacetime two-point correlation function. It captures the EE fully for Gaussian theories and up to first order in perturbation theory for all other theories.

• Future directions include the EE of black holes as well as

further studying EE of interacting theories and casting the higher order contributions in terms of spacetime correlators.