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Entanglement entropy:

From Field Theory to Condensed Matter

Based on collaboration with:

• J. Cardy, V. Alba, M. Fagotti, E. Tonni....

Pasquale Calabrese

SISSA-Trieste ERG 2016, Trieste, 23/9/2016

Why entanglement entropy?

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hep-th arXiv preprints with “entanglement” in the title

Why entanglement entropy?

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hep-th arXiv preprints with “entanglement” in the title

Many-body quantum systems

Free Fermions = metals Free Bosons = superfluids

When many particles do not interact, their properties follow straightforwardly from those of few Interactions dramatically change this paradigm especially in low dimensions

“More is different”

PW Anderson 1972

Interacting particles Mott insulators Topological states Unconventional Superconductors Spin-Charge separation

Non-abelian Statistics

The properties of many do not follow simply from those

• f few: “more is truly different!”

Interactions give rise to new phases of matter

Confined phases (QCD)

“The complexity frontier”

2N coefficients: too many for a classical PC

|Ψ⇤ = ⇤

si=±

As1s2...sN |s1, s2, . . . sN⇤

How to describe these many-body systems? Numerically? Too difficult, e.g. for a spin-chain

“The complexity frontier”

2N coefficients: too many for a classical PC

|Ψ⇤ = ⇤

si=±

As1s2...sN |s1, s2, . . . sN⇤

How to describe these many-body systems? Numerically? Too difficult, e.g. for a spin-chain

We need a criterion that sets physical states apart from the others

Entanglement is this criterion

Entanglement entropy

Consider a system in a quantum state |ψ〉 (ρ=|ψ〉〈ψ|)

H = HA ⇥ HB

Alice can measure only in A, while Bob in the remainder B Alice measures are entangled with Bob’s ones: Schmidt deco |Ψ =

• n

cn|ΨnA|ΨnB cn ⌅ 0,

• n

c2

n = 1

• If c1=1 ⇒ |ψ〉 unentagled
• If ci all equal ⇒ |ψ〉 maximally entangled

B

A

Entanglement entropy

Consider a system in a quantum state |ψ〉 (ρ=|ψ〉〈ψ|)

H = HA ⇥ HB

Alice can measure only in A, while Bob in the remainder B Alice measures are entangled with Bob’s ones: Schmidt deco |Ψ =

• n

cn|ΨnA|ΨnB cn ⌅ 0,

• n

c2

n = 1

• If c1=1 ⇒ |ψ〉 unentagled
• If ci all equal ⇒ |ψ〉 maximally entangled

B

A

A natural measure is the entanglement entropy (ρA =TrB ρ)

SA≡ -Tr ρA ln ρA = SB

= -∑ cn ln cn

2 2

basis independent

Area Law

SA∝ Area separating A and B If the Hamiltonian has a gap

[Srednicki ’93 +many more]

If |ψ〉 is the ground state of a local Hamiltonian

A

B

Entanglement in extended systems

Area Law

SA∝ Area separating A and B If the Hamiltonian has a gap

[Srednicki ’93 +many more]

If |ψ〉 is the ground state of a local Hamiltonian B B A

l

In a 1+1 D CFT Holzhey, Larsen, Wilczek ’94 This is the most effective way to determine the central charge

SA = c ln l

3

_

A

B

Entanglement in extended systems

Importance

Only a tiny fraction of states satisfy the area law (or small violations) If we can limit the search for the ground state to this small subset, the complexity of the problem is exponentially reduced

Full Hilbert space Area law states

SA gives the amount of classical information required to specify |Ψ⟩

One meaning of SA:

Tensor network states

MPS PEPS MERA “Alphabet soup of proposals” A new and powerful set of numerical methods based on entanglement content of quantum states

Subir Sachdev

|⇤ = X

s1,...,sN=±

Tr h A[1]

s1 . . . A[N] sN

i |s1 . . . sN.

• For each site there are two matrices A[i] of finite dimension χ×χ.

More entanglement can be stored as χ increases.

• The famous DMRG is a practical way to find a variational MPS
• At fixed χ, the maximum entanglement entropy of an MPS is lnχ
• 1D area is a number ⇒ entanglement entropy constant ⇒ an MPS

with finite χ can describe it

• In d dimensions, area law Nd−1 ⇒ χ needs to be χ∼exp(Nd−1)

Matrix Product States (MPS)

±

⇤Φ1(x)|ρA|Φ2(x)⌅ =

Entanglement entropy and path integral

PC, J Cardy 2004

⟨Φ1|ρ|Φ2⟩=

The density matrix at temperature β-1

= Z [dφ(x, τ)] Z Y

x

δ(φ(x, 0)φ2(x)) Y

x

δ(φ(x, β)φ1(x)) e−SE

R

The trace sews together the edges along τ = 0 and τ = β to form a cylinder

• f circumference β.

A = (u, v): ρA sews together only those points x which are not in A, leaving an

• pen cut along the τ = 0.

Replicas and Riemann surfaces

PC, J Cardy 2004

Trρn

A =

For n integer, Tr ρA is obtained by sewing cyclically n cylinders above. This is the partition function on a n-sheeted Riemann surface

n

Renyi EE: SA ≡ 1/(1-n) ln Tr ρA

n

SA = −TrρA log ρA = − lim

n→1

∂ ∂nTrρn

A

= cn|u − v|− c

6 (n−1/n)

w → = w−u

w−v ; → z = 1/n⇒ w → z =

• w−u

w−v

⇥1/n

is equivalent to the 2-point function of twist fields

Trn

A = ⌅Tn(u) ¯

Tn(v)⇧

⇤T T ⌅ ∆Tn = c 12 ⇧ n 1 n ⌃

with scaling dimension

Riemann surfaces and CFT

This Riemann surface is mapped to the plane by

SA = − lim

n→1

⇤ ⇤nTr⇥n

A = c

3 log ⌅

|u-v|= l Tr ρA

n

Tr ρA=

n

PC, J Cardy 2004

From CFT to cold atoms

ARTICLE

doi:10.1038/nature15750

Measuring entanglement entropy in a quantum many-body system

Rajibul Islam1, Ruichao Ma1, Philipp M. Preiss1, M. Eric Tai1, Alexander Lukin1, Matthew Rispoli1 & Markus Greiner1 Entanglement is one of the most intriguing features of quantum mechanics. It describes non-local correlations between quantum objects, and is at the heart of quantum information sciences. Entanglement is now being studied in diverse fields ranging from condensed matter to quantum gravity. However, measuring entanglement remains a challenge. This is especially so in systems of interacting delocalized particles, for which a direct experimental measurement of

spatial entanglement has been elusive. Here, we measure entanglement in such a system of itinerant particles using quantum interference of many-body twins. Making use of our single-site-resolved control of ultracold bosonic atoms in optical lattices, we prepare two identical copies of a many-body state and interfere them. This enables us to directly measure quantum purity, Rényi entanglement entropy, and mutual information. These experiments pave the way for using entanglement to characterize quantum phases and dynamics of strongly correlated many-body systems.

Figure 1 | Bipartite entanglement and partial measurements. A generic pure quantum many-body state has quantum correlations (shown as arrows) between different parts. If the system is divided into two subsystems A and B, the subsystems will be bipartite entangled with each other when there are quantum correlations between them (right column). Only when there is no bipartite entanglement present, the partitioned system | ψAB〉 can be described as a product of subsystem states | ψA〉 and | ψB〉 (left column). A path for measuring the bipartite entanglement emerges from the concept of partial measurements: ignoring all information about subsystem B (indicated as ‘Trace’) will put subsystem A into a statistical mixture, to a degree given by the amount of bipartite entanglement present. Finding ways of measuring the many-body quantum state purity of the system and comparing that of its subsystems would then enable measurements of entanglement. For an entangled state, the subsystems will have less purity than the full system.

Entangled state Product state

• A

B A B Trace Pure Trace Mixed |= |A ⊗ | B |≠ |A ⊗ | B

Figure 2 | Measurement of quantum purity with many-body bosonic interference of quantum twins. a, When two N-particle bosonic systems that are in identical pure quantum states are interfered on a 50%–50% beam splitter, they always produce output states with an even number

• f particles in each copy. This is due to the destructive interference of

a b

Even particle number in Output 2 Even particle number in Output 1 Odd particle number Odd particle number Two identical N-particle bosonic states |N |N Pi Average parity Tr (1 2) Quantum state

• verlap

Tr (2) Purity 1 2 P1 P2 1 = 2 pi Pi = =

∏

• Nature 528, 77 (2015)

From CFT to cold atoms

II 0.5 1 1.5 2 2.5 3 3.5 0.2 0.4 0.6 1 2 3 4 0.5 1 1.5 2 System A size 1 2 3 1 2 Boundary size 1 2 3 0.5 1 Distance, d S2(A) S2(B) S2(A) S2(B) II I U/J 1.2 2.7 39 12 7.2 A B A B d Mutual Information, IAB IAB

a b

Mott insulator Superfluid Adiabatic melt A B A B A B

c

A B A A B A A B B + + + A B I Superfluid Mott Renyi entropy, S2 Renyi entropy, S2 Renyi entropy, S2 Superfluid Mott insulator II Superfluid 100 101 102 U/Jx IAB IAB IAB

A look to a more difficult problem

A = Disconnected regions: More complex Riemann surface: Rn,2 of genus (n − 1)

[Rn,N has genus (n − 1)(N − 1)] PC, J Cardy, E Tonni 2009/10

Tr ρA, SA ??

n

Disjoint intervals

Trρn

A = c2 n

„ |u1 − u2||v1 − v2| |u1 − v1||u2 − v2||u1 − v2||u2 − v1| « c

6 (n−1/n)

Fn(x)

„ | − || − x = (u1−v1)(u2−v2)

(u1−u2)(v1−v2) = 4 − point ratio

Can we get Fn(x) for some explicit models??

PC, J Cardy, E Tonni 2009/10

Fn(x) is a calculable function depending on the full operator content

Tr ⇥n

A = c2 n(⌃1⌃2)− c

6 (n− 1 n ) ⇧

{kj}

⇤ ⌃1⌃2 n2r2 ⌅P

j(∆j+∆j)

⇥

n

⌃

j=1

⇤kj

• e2πij/n⇥

⇤2

C

The compactified boson

Using old results of CFT

• n orbifolds Dixon et al 86

Fn(x) = Θ

• 0|ηΓ

⇥ Θ

• 0|Γ/η

⇥ [Θ

• 0|Γ

⇥ ]2

Γ is an (n − 1) × (n − 1) matrix Γrs = 2i n

n−1

X

k = 1

sin „ π k n « β k

n cos

» 2π k n (r − s) – with βy = Hy(1 − x) Hy(x) , Hy(x) =

2F1(y, 1 − y; 1; x)

Riemann theta function Θ(z|Γ) ≡ X

m ∈ Zn−1

exp ˆ iπ m · Γ · m + 2πim · z ˜

PC, J Cardy, E Tonni 2009/2010

Nowadays generalized to many other cases: Ising (PC, Cardy, Tonni), Askhin-Teller (Alba, Tagliacozzo, PC), Fusion-twist (Rajabpour, Gliozzi), merged models (Fagotti).....

Does it work?

M Fagotti, PC, 2010

The RDM of two intervals is not trivial because of JW string Igloi-Peschel

−

Reviews (up to 2009)

Further developments

◉ Detect and characterize quantum criticality

In random quantum spin chains SA ∝ ln l

Refael and Moore, Laflorencie, Santachiara, Jacobsen, Saleur ...

◉ Topological entanglement entropy

SA =αL-γ γ is the topological charge

Kitaev and Preskill, Levin and Wen, Fradkin and Moore, Schoutens et al....

◉ Entanglement spectrum

Universal corrections to the scaling

PC, Essler, Cardy, Ravanini, Franchini,Ercolessi, Alcaraz.... Haldane, Regnault, Read, Ludwig, Bernevig, Poilblanc, Rezayi, Haque.........

Eigenvalues of ρA

1 1 2 3 5

Further developments (II)

◉ Holography: SA=length of the geodesic in the AdS bulk ◉ c-theorem analogues with SA

Casini and Huerta, Myers,

◉ Entanglement out of equilibrium (quenches)

Ryu and Takayanagi. Headrick, Maldacena, Myers PC and Cardy, Vidal, Schollwoeck, Kollath, Eisert, Cirac....

◉ Other measures of entanglement (eg mixed states)

Fazio, Amico, Vidal....

► Entanglement negativity PC, Cardy, Tonni 2012/13 ► Shannon information Stephan, Pasquier, Oshikawa Alcaraz ◉ Too many more to be mentioned here

Further developments (II)

◉ Holography: SA=length of the geodesic in the AdS bulk ◉ c-theorem analogues with SA

Casini and Huerta, Myers,

◉ Entanglement out of equilibrium (quenches)

Ryu and Takayanagi. Headrick, Maldacena, Myers PC and Cardy, Vidal, Schollwoeck, Kollath, Eisert, Cirac....

◉ Other measures of entanglement (eg mixed states)

Fazio, Amico, Vidal....

► Entanglement negativity PC, Cardy, Tonni 2012/13 ► Shannon information Stephan, Pasquier, Oshikawa Alcaraz ◉ Too many more to be mentioned here

Many particles out of equilibrium

Quantum Quench idea:

1) prepare a many-body system in a pure state that is not an eigenstate

• f the Hamiltonian

2) let it evolve according to QM laws (no coupling to environment)

|⇧(t)⇤ = e−iHt|⇧0⇤, t

Questions:

• How can we describe the dynamics?
• Does it exist a stationary state? In which sense?

How do the Gibbs distribution emerge in QM?

von Neumann in 1929 posed the question [1003.2133]

|Ψ(t)⟩ remains pure for any t

The Reduced density matrix helps!

Reduced density matrix: ρA(t)=TrB ρ(t) |Ψ(t)⟩ time dependent pure state The expectation values of all local

• bservables within A are

⟨Ψ(t)|OA(x) |Ψ(t)⟩ = Tr[ρA(t) OA(x)] Stationary state: if exists the limit

lim ρA(t) = ρA(∞)

t→∞

B

A

Infinite system (AUB ) ◉ ◉ ◉ A finite

The Reduced density matrix helps!

Reduced density matrix: ρA(t)=TrB ρ(t) |Ψ(t)⟩ time dependent pure state The expectation values of all local

• bservables within A are

⟨Ψ(t)|OA(x) |Ψ(t)⟩ = Tr[ρA(t) OA(x)] Stationary state: if exists the limit

lim ρA(t) = ρA(∞)

t→∞

B

A

Infinite system (AUB ) ◉ ◉ ◉ A finite

Thermalization vs Generalized Gibbs

ρT= e-βeffH/Z

ρGGE= e-∑ λm Im /Z

but this is another story/talk....

vs

In a CFT (i.e. exactly linear dispersion relation E=vk up to a cutoff)

Entanglement after a quench

l2/2

SA(l,t)

l1/2 l2 l1<l2

vt SA(l,t)=

         πct 6 πc ℓ 12

v

2vt<l 2vt>l

PC, Cardy 2005

In a CFT (i.e. exactly linear dispersion relation E=vk up to a cutoff)

Entanglement after a quench

l2/2

SA(l,t)

l1/2 l2 l1<l2

vt SA(l,t)=

         πct 6 πc ℓ 12

v

2vt<l 2vt>l

PC, Cardy 2005

SA(l,t)

Exact results for the Ising model Curvature: effect of the non linear dispersion

Fagotti, PC 2008

Physical explanation

2t 2t < l A B

vk = dΩk

dk

vmax exists

PC, Cardy 2005

• |ψ0〉 has large energy: source of quasi-particles
• Pairs of quasi-particles move in opposite

directions with velocity ± vk

• Particles emitted from the same point are

entangled

Physical explanation

2t 2t < l A B

vk = dΩk

dk

vmax exists

PC, Cardy 2005

correlations form at t = l/2vmax

Slower particles change entanglement and correlations after t = l/2vmax: large t is driven by slowest particles

• Light cone: Points at separation l become entangled when left- and

right-movers originated from the same point reach them

• If all particles move at the same speed, entanglement and correlations

are frozen for t>l/2v

• |ψ0〉 has large energy: source of quasi-particles
• Pairs of quasi-particles move in opposite

directions with velocity ± vk

• Particles emitted from the same point are

entangled

Is it true?

~ ~ position time b d = v t a

quench

• FIG. 1. Spreading of correlations in a quenched atomic

Mott insulator. a, A 1d ultracold gas of bosonic atoms (black balls) in an optical lattice is initially prepared deep in the Mott-insulating phase with unity filling. The lattice depth is then abruptly lowered, bringing the system out of

• equilibrium. b, Following the quench, entangled quasiparticle

pairs emerge at all sites. Each of these pairs consists of a doublon (red ball) and a holon (blue ball) on top of the unity- filling background, which propagate ballistically in opposite

• directions. It follows that a correlation in the parity of the

site occupancy builds up at time t between any pair of sites separated by a distance d = vt, where v is the relative velocity

• f the doublons and holons.
• M. Cheneau et al, Nature 2012

Light-cone spreading of entanglement entropy

PC, J Cardy 2005

• The entanglement entropy of an interval A of length l !is proportional to

the total number of pairs of particles emitted from arbitrary points such that at time t, x ∈ A and x’ ∈ B

• Denoting with f(p) the rate of production of pairs of momenta ±p and their

contribution to the entanglement entropy, this implies

SA(t) ⇡ Z

x02A

dx0 Z

x002B

dx00 Z 1

1

dx Z f(p)dp

• x0 x vpt
• x00 x + vpt
• / t

Z 1 dpf(p)2vp✓(` 2vpt) + ` Z 1 dpf(p)✓(2vpt `) (

• When vp is bounded (e.g. Lieb-Robinson bounds) |vp|<vmax, the second

term is vanishing for 2 vmax t<l and the entanglement entropy grows linearly with time up to a value linear in l !

One example

PC, J Cardy 2005

Transverse field Ising chain

S(t) = t Z

2|✏0|t<`

d' 2⇡ 2|✏0|H(cos ∆')+ ` Z

2|✏0|t>`

d' 2⇡ H(cos ∆') (2)

Analytically for t, l ⨠ 1 with t/l constant

M Fagotti, PC 2008

H(x) = 1 + x 2 log 1 + x 2 1 x 2 log 1 x 2

= 1 cos '(h + h0) + hh0 ✏'✏0

'

.

cos ∆' = ( contains al

Physical interpretation at t=∞

l2/2

SA(l,t)

l1/2 l2 l1<l2

vt

The extensive value at t=∞ is the thermodynamic entropy in the mixed state because

lim ρA(t) = ρA(∞)

t→∞

For large time the entanglement entropy becomes thermodynamic entropy

Understood even in more complicated situations

What about experiments?

• e
• e
• u-

ge s s ,

• g

s e c s s al

• t

T0

global unitary dynamics local thermalization

1 2 3 4 5 6 0.0 0.2 0.4 0.6 0.8 1.0 Observable A P(A)

pure state pure state

quantum quench

1 2 3 4 5 6 0.0 0.2 0.4 0.6 0.8 1.0 Observable A P(A)

T>0

• FIG. 1.

Schematic of thermalization dynamics in closed systems. An isolated quantum system at zero tem- perature can be described by a single pure wavefunction |Ψi. Subsystems of the full quantum state appear pure, as long as the entanglement (indicated by grey lines) between sub- systems is negligible. If suddenly perturbed, the full system evolves unitarily, developing significant entanglement between all parts of the system. While the full system remains in a pure, zero-entropy state, the entropy of entanglement causes the subsystems to equilibrate, and local, thermal mixed states appear to emerge within a globally pure quantum state.

Quench

1

• 1

1

• 1
• 1
• 1

Expand and Measure Local and Global Purity

1

• 1

1

• 1

1 1

Expand and Measure Local Occupation Number

1 2 2 1 2 1 1 2

~ 50 Sites ~ 50 Sites Mott insulator Even Odd

680 nm

Initialize Many-body interference 45 Er 6 Er Global thermal state purity Locally thermal Locally pure Globally pure On-site Statistics Particle number time after quench (ms) 0.2 0.4 0.6 0.8 1 1 2 3 4 5 6 Entropy: -Log(Tr[2]) 100 10-2 4.6 2.3 10 20

A B C

y x Initial state

quench

Purity: Tr[2] 10-1 P(n) 0.2 0.4 0.6 0.8 1 P(n) Particle number 1 2 3 4 5 6 On-site Statistics Many-body purity t=0 ms t=16 ms

Quantum thermalization through entanglement in an isolated many-body system

Adam M. Kaufman, M. Eric Tai, Alexander Lukin, Matthew Rispoli, Robert Schittko, Philipp M. Preiss, Markus Greiner*

http://science.sciencemag.org/ Downloaded from

Science 353, 794 (2016)

What about experiments?

1 2 3 4

Renyi entropy S2

5 10 15 20 1 2 3 4

time (ms) Renyi entropy S2

5 10 15 20

time (ms)

1 2 3 0.5 1 1.5 Subsystem size slope (ms-1)

• FIG. 3. Dynamics of entanglement entropy.

Starting from a low-entanglement ground state, a global quantum quench leads to the development of large-scale entanglement between all subsystems. We quench a six-site system from the Mott insulating product state (J/U ⌧ 1) with one atom per site to the weakly interacting regime of J/U = 0.64 and measure the dynamics of the entanglement entropy. As it equilibrates, the system acquires local entropy while the full system entropy remains constant and at a value given by measurement imperfections. The dynamics agree with exact numerical simulations with no free parameters (solid lines). Error bars are the standard error of the mean (S.E.M.). For the largest entropies encountered in the three-site system, the large number of populated microstates leads to a significant statistical uncertainty in the entropy, which is reflected in the upper error bar extending to large entropies or being unbounded. Inset: slope of the early time dynamics, extracted with a piecewise linear fit (see Supplementary Material). The dashed line is the mean of these measurements.

For large time the entanglement entropy becomes thermodynamic entropy

Idea: We could use the knowledge of the entropy in the stationary state to go backward in time for the entanglement entropy.

Alba & PC, 2016

SY Y = L

∞

X

n=1

Z dλ[ρn,t(λ) ln ρn,t(λ)−ρn,p(λ) ln ρn,p(λ)−ρn,h(λ) ln ρn,h(λ)]

Making a long story very short: after a quench in a Bethe ansatz integrable model, the TD entropy has the Yang-Yang form:

S(t) = X

n

h 2t Z

2|vn|t<`

dvn()sn() + ` Z

2|vn|t>`

dsn() i ,

sn(λ)

Assuming that the Bethe excitations are the entangling quasi-particles: Warning: Determining vn(λ) is non-trivial conjecture:

For large time the entanglement entropy becomes thermodynamic entropy

S(t) = X

n

h 2t Z

2|vn|t<`

dvn()sn() + ` Z

2|vn|t>`

dsn() i ,

conjecture:

Check for quenches in the XXZ spin-chain

2 4 6 8

time t

0.25 0.5 0.75

S’

∆=2 ∆=4 2 4 6 8 10 12 0.2 0.4

S’

ϑ=0 ∆=1 ϑ=0 ∆=2 ϑ=0 ∆=4 ϑ=0 ∆=10 ϑ=π/6 ∆=2 ϑ=π/9 ∆=8 4 8 12 16

• 0.2

0.2 0.4 0.6

S’

ϑ=π/3 ∆=4 ϑ=π/6 ∆=4 ϑ=π/3 ∆=2

(a) Neel (b) Ferromagnet (c) Dimer (MG)

0.5 1 1.5 2

vMt/ℓ

0.1 0.2 0.3 0.4

S/ℓ

tDMRG Conjecture Extrapolations 8

(c)

ℓ=5 - 20

Conclusions

The entanglement entropy is a very useful concept for many- body systems:

• It encodes all universal properties of critical 1D many-body

systems (i.e. central charge, operator content, etc.).

• Note: the entanglement in the vacuum encodes all this.

There is a lot in the vacuum (ground-state)

• It is a tool to design better performing numerical algorithms
• It provides a mechanism for thermalization
• Many other things that do not fit in one hour

Entanglement of non-complementary parts

gives an upper bound on the entanglement between A1 and A2 SA1∪A2 gives the entanglement between A and B The mutual information SA1 + SA2 - SA1∪A2

A1

B

A2

Entanglement of non-complementary parts

gives an upper bound on the entanglement between A1 and A2 A computable measure of entanglement exists: the logarithmic negativity What is the entanglement between the two non-complementary parts A1 and A2?

[Vidal-Werner 02]

SA1∪A2 gives the entanglement between A and B The mutual information SA1 + SA2 - SA1∪A2

A1

B

A2

Entanglement negativity

Let us denote with

case is y |e(1)

i ⌥

d H of

d |e(2)

j ⌥

ach part,

and two bases in A1 and A2 The partial transpose is

⌃e(1)

i e(2) j |⇥T2|e(1) k e(2) l

⌥ = ⌃e(1)

i e(2) l

|⇥|e(1)

k e(2) j ⌥

logarithmic negativity as

And the logarithmic negativity

⇤T T ⌅ T T Tr|⇥T2| =

i

|i| =

λi>0

i

λi<0

i

• ⌥

It measures “how much” the eigenvalues of ρT2 are negative because Tr (ρT2)=1 ρ is the density matrix of A1∪A2 , not pure

ℰ≡ ln|| ρT2 ||= ln Tr |ρT2|

A1

B

A2

A replica approach to negativity

Let us consider traces of integer powers of ρT2

Tr(⇥T2)ne =

• i

ne

i

=

• λi>0

|i|ne +

• λi<0

|i|ne , Tr(⇥T2)no =

• i

no

i

=

• λi>0

|i|no

• λi<0

|i|no

The analytic continuations from ne and no are different

alytic continuation of the . E = lim

ne→1 ln Tr(⇥T2)ne,

normalization Tr = 1.

For a pure state ρ=|ψ〉〈ψ|

⇤T T ⌅ lim

no→1 Tr(T2)no = TrT2 = 1

• Tr (T2)n =

⌥ Tr n

2

n = no odd (Tr n/2

2

)2 n = ne even

• ne even

no odd

PC, J Cardy, E Tonni 2012

Negativity and QFT

A1 A2 B B B u1 v1 u2 v2

Tripartion:

Trn

A = ⌦Tn(u1) ¯

Tn(v1)Tn(u2) ¯ Tn(v2)↵

Negativity and QFT

Tr(T2

A )n =

The partial transposition exchanges two twist operators

)n = ⌦Tn(u1) ¯ Tn(v1) ¯ Tn(u2)Tn(v2)↵

Pure States in QFT

Tr(T2

A )n = ⌦T 2 n (u2) ¯

T 2

n (v2)↵

For n=no odd, the no-sheeted surface remains no-sheeted For n=ne even, the R-surface decouples in two ne/2 surface

τn τ

u1 v1 u2 v2

τn τn τn

_

τn

_

u2 v2

τn

2

τn

2

_ connects the j-th sheet with the (j+2)-th one:

τn

2

Tr(T2

A )ne = (⌦Tne/2(u2) ¯

Tne/2(v2)↵)2 = (Trne/2

A2

)2 Tr(T2

A )no = ⌦Tno(u2) ¯

Tno(v2)↵ = Trno

A2 ,

PC, J Cardy, E Tonni 2012

Tr(T2

A )ne = (⌦Tne/2(u2) ¯

Tne/2(v2)↵)2 = (Trne/2

A2

)2 Tr(T2

A )no = ⌦Tno(u2) ¯

Tno(v2)↵ = Trno

A2 ,

A

⇤T T ⌅ T 2

no has dimension ∆T 2

no = c

12 ⇧ no 1 no ⌃ , the same as Tno n ⌥ ⇤T T ⌅ T 2

ne has dimension ∆T 2

ne = c

6 ⇧ne 2 2 ne ⌃

• ⌥

Pure States in CFT

Tr(T2

A )n = ⌦T 2 n (u2) ¯

T 2

n (v2)↵

u1 v1 u2 v2

τn τn τn

_

τn

_

u2 v2

τn

2

τn

2

_

Thus in a CFT:

ε(y)-(ln L)/4

1 4 ln sin( ⇥⇤1

L ) sin(⇥⇤2 L )

sin ⇥(⇤1+⇤2)

L

+ cnst

• Example:Two adjacent intervals
• l1

l2

Tr(⇤T2

A )n = ↵Tn(1) ¯

T 2

n (0)Tn(2)

⌅ ||⇤T2

A || ⌃

✓ 12 1 + 2 ◆ c

4

⇧ E = c 4 ln 12 1 + 2 + cnst

PC, J Cardy, E Tonni 2012

Several other universal results can be similarly derived