quantum stationary states

Vladimir Bazhanov Australian National University

(with Sergei Lukyanov (Rutgers), arxiv:1310.4390, 1310.8082)

1

Connection between quantum and classical systems.

•Quasiclassical approximation, when Planck constant→0 Quantum theory =⇒ Classical theory

•New type of (mathematical) connection for finite values of, Integrable Quantum Field Theory

⇐⇒

Integrable Classical Field Theory

2

• Integrable Quantum Field Theory (QFT), Integrals of Motion – Conformal Field Theory (CFT), Infinite-dimensional algebra

of (extended) conformal symmetry

– Bethe Ansatz, functional relations for commuting transfer matrices

• Theory of differential equations

– Scattering problem for ODE, connection coefficients, Stocks multipliers, . . .

– monodromy group, monodromy-free singular points – What’s the meaning of the number18in the theory

of the hypergeometric equation?

– second order PDE, arising as “zero-curvature condition”

for multivalued flat connections on the punctured Riemann sphere

• Set of stationary states in the Hilbert space of massive integrable QFT (Fateev model) ⇐⇒

Set of singular solutions of the modified sinh-Gordon equation on the punctured Riemann sphere

3

Local Integrals of Motion (IM) in CFT

(Sasaki-Yamanaka 1988, Eguchi-Yang 1989, VB-Lukyanov-Zamolodchikov (BLZ) 1994)

LetV irbe the Virasoro algebra generated byLn∈V ir, [Lm, Ln] = (m−n)Lm+n+ c

12(n³−n)δm+n,0

To construct the Cartan-Weyl style representation theory we need to first diagonalize the maximal Abelian subalgebra, which is a set of mutually commuting operators from the universal enveloping algebra ofV ir:

Is∈U(V ir) : [Is,Is] = 0.

We need to find the spectrum ofIsin thehighest weight representation ofV ir: VΔ,c: Ln|Δ= 0, n >0; L0|Δ= Δ|Δ First, make some assumptions about this Abelian subalgebra.

4

•It would be natural to includeL0in the commuting set;L0splitsVΔ,c

on the finite dimensional level subspaces:

L0V_Δ,c^(L)= (Δ +L)V_Δ,c^(L) dim

V_Δ,c^(L) <∞.

Therefore, the problem is reduced to a finite dimensional spectrum problem inV_Δ,c^(L).

•LetT(x),x∈S¹(x∼x+R) be the holomorphic component of the stress-energy tensor,

T(x) =−c

24+

n∈Z

e^2πinx/RL−n

• Locality condition: We assume thatIsare given by integrals over local densitiesbuild from the fieldT(x), for example,

I1=

T=R 2π

L0− c 24

•The quadratic inLnoperator is defined up to an overall normalization by the locality requirement

I3=

T²= R

2π 3

2 ∞ n=1

LnLn+L²0−c+ 2

12 L0+c(5c+ 22) 2880

5

All other operatorsIsare defined (up to overall factors) by the commutativity condition.

For example

I5= T³+c+ 2 12 (T)²

There exists an infinite set{I2n−1}^∞_n=1which first representatives are given by the above formulas. They are the so called localIntegrals of Motion(IM).

The odd-integers2n−1stand for the values of the Lorentz spin.

We’ll focus on the highest vector eigenvalues:

I_2n−1^(vac)(Δ, c) : I2n−1|Δ=R 2π

2n−1

I_2n−1^(vac)|Δ, which are certain polynomials in Δ andc:

I1^(vac)= Δ− c

24, I^(vac)3 = Δ²−c+ 2

12 Δ +c(5c+ 22) 2880 , . . . CFT integrals of motion — quantum analogs of conserved quantities in KdV theory

T(x)→ −c

6U(x), ∂tU=UUx−6Uxxx, c→ ∞ 6

79

Functional relations for generating functions of IM

•Transfer matricesTj(μ) (quantum analogs of traces of monodromy matrices for mKdV) satisfy thefusion relations

Tj(qμ)Tj(q⁻¹μ) = 1 +T_j+1 2(μ)T_j−1

2(μ),

q= e^iπβ2, c= 1−6 (β−β⁻¹)²

•Tjcan be regarded as generating function for the local IM logTj∼^∞

n=0

c^(j)n I2n−1κ¹⁻²ⁿ κ=μ^2(1−β2)1

•Asβ²=_p^pthe functional relations are truncated. In this case the vacuum eigenvalues, Tj(μ)|Δ=tj(μ)|Δ

satisfy a finite set of integral equations (TBA equations). Numerical values of the vacuum eigenvaluesI_2n−1^(vac)can be extracted from the solutions of the TBA equations.

•The TBA equations are expecially simple in the case β²= 1

N+ 1, N= 1,2, . . . Δ =1−4N² 6(N+ 1). Related to RSOS models(Andrews-Baxter-Forrester ’84, Baxter-Pearce ’87).

7

ODE/IM correspondence

Let us consider the anharmonic potential −d²

dy²+y^2N−E Ψ = 0.

The WKB spectrum can be determined by means of the WKB approximation.

E1

E2

WKB spectra {En}^∞n=1 =⇒ dy

En−U(y) = 2π(n+...) y^2N

• Voros (1992)derived the exact Exact Bohr-Sommerfeld quantization condition.

• Dorey-Tateo (1998)observed that forβ²=_N+1¹ the TBA equations in CFT are exa the same as the Voros quantization conditions.

•Generalization and proofBLZ (1998)

In the general case the vacuum eigenvalues ofTj(μ), i.e.,tj(μ),(j=¹₂,1, . . .) 8

coincide with certain monodromy coefficients for the ODE − d²

dz²+l(l+ 1) z² +κ²p(z)

Ψ = 0, p(z) =z^2α−1. One can reformulate this result in terms of the vacuum eigenvaluesI2n−1^(vac);

w=

dz p(z) :

− d²

dw²+ ˆu(w) +κ² Ψ = 0˜

cn=^(2n−3)!!₂nn!

Ψ(˜w)∼e^F(w)exp

−κw+_∞

n=1κ¹⁻²ⁿcn wdw Un[ˆu] F(w) =

∞ n=1

κ⁻²ⁿFn[ˆu(w)] Fn[ˆu]−differential polinomials in ˆu . AlsoUn[ ˆu] are homogeneous (grade(ˆu) = 2,grade(∂) = 1,grade(Un) = 2n) differential polynomials in ˆuof degreen(known as the Gel’fand-Dikii polynomials):

U1= ˆu , U2= ˆu²−1 3ˆu...

9

Hence the monodromy coefficients are given by logt¹₂(μ)∼

n

cnκ¹⁻²ⁿq_2n−1, q_2n−1=

Cw

dw Un[ˆu(w)]

We may now return to the original variablez

w→z , Un[ˆu(w)]→U˜n(z)

z

0 1 π C α

q_2n−1=

CdzU˜n(z) (p(z) =z^2α−1)

The ODE/IM correspondence : I2n−1^(vac)=dnq2n−1

Herednare some (known) constants which depend on normalization conventions for q_2n−1andI2n−1, whereas the parameters are identified as follows:

c= 1− 6α²

α+ 1, Δ =(2l+ 1)²−4α² 16(α+ 1) .

10

Virasoro CFT summary

•Problem of diagonalization of the Abelian subalgebra ofV irspanned by the local IM{Is}.

•Generating function for the vacuum eigenvalues coincides with some connection coefficient of the ODE

− d² dz²+l(l+ 1)

z² +κ²p(z)

Ψ = 0, p(z) =z^2α−1.

•A similar correspondence exists for the higher (excited) eigenstates:

each state corresponds to its own ODE.BLZ (2003)

•Difficulty: the above ODE hasan essential singularityatz=∞

•Resolution:consider more a complicated case, related to the Fateev model

11

Fateev model (1996)

L = 1 16π

3 i=1

(∂tϕi)²−(∂xϕi)²

+ 2μ

e^iα3ϕ3 cos(α1ϕ1+α2ϕ2) + e^−iα3ϕ3 cos(α1ϕ1−α2ϕ2)

Hereαiare coupling constants subject to a single constraint α²1+α²2+α²3=1

2. α²₁>0, α²₂>0, α²₃>0.

The parameterμin the Lagrangian sets the mass scale,μ∼[ mass ]. We shall consider the theory in finite-size geometry, with the spatial coordinatexinϕi=ϕi(x, t) compactified on a circle of circumferenceR, with the periodic boundary conditions

ϕi(x+R, t) =ϕi(x, t).

x~x+R

Aμ=ACF T+μ d²xΦ

12

80

Perturbed Hyper-Geometric Equation

−∂z²+T0(z)

ψ= 0, T0(z) =−³

i=1

δi

(z−zi)²+ ci

z−zi

δi=1

4−p²i, i= 1,2,3.

zj

zi

zk

ψ1(γ◦z), ψ2(γ◦z)

=

ψ1(z), ψ2(z) M(γ). Monodromy group

M : π1 CP¹\{zi}

→SL(2,C),

Conditions on the monodromy matrices (six equations for nine unknowns) Tr

M⁽ⁱ⁾

=−2 cos(2πpi), M⁽³⁾M⁽²⁾M⁽¹⁾= 1

13

D(λ) =−d²

dz²+T0(z) +λ²P(z), P(z) = (z3−z2)^a1(z1−z3)^a2(z2−z1)^a3 (z−z1)^2−a1(z−z2)^2−a2(z−z3)^2−a3

and parameters 0< ai<2 satisfy the constrainta1+a2+a3= 2.

Define bases of linearly independent solutions and connection matrices χ⁽ⁱ⁾±→(z−zi)¹2±pi

1 +O((z−zi)^ai/2)

, 0< pi< ai/4 χ⁽ⁱ⁾= (χ⁽ⁱ⁾−, χ⁽ⁱ⁾+), χ⁽ⁱ⁾=χ^(j)S^(j,i)(λ), i= 1,2,3 det

S^(j,i)(λ)

= 1, S^(i,k)(λ)S^(k,i)(λ) =I, S^(i,k)(λ)S^(k,j)(λ)S^(j,i)(λ) =I, Symmetry properties

Ωi: z→γi◦z , λ→q⁻¹i λ qi=e^iπai Ω₃Ω₂Ω₁= 1

lead to

S^(i,k)(λ) e^−2πipiσ3S^(k,j)(λ qk⁻¹) e^−2πipjσ3S^(j,i)(λ qi) e^−2πipiσ3=−I. Asymptotic expansion of these coefficients at largeλcan be connected to vacuum eigenvalues of the local IM in some CFT with extended conformal symmmetry, related to the Fateev model.

14

Further generalizations: higher eigenstates

−∂z²+TL(z)

ψ= 0, TL(z) =−

L+3

i=1

δi

(z−zi)²+ ci

z−zi

with{zi}={z1, z2, z3, x1, . . . , xL}and δi=1

4−p²i, i= 1,2,3; δa+3=−2, a= 1,2, . . . , L Condition: pointsx1, . . . , xLare monodromy-free

TL(z) =−la(la+ 1) (z−xa)²− ca+3

z−xa−^+∞

k=0

t^(a)k (z−xa)^k, a= 1, . . . , L

(ca+3)³−4ca+3t^(a)0 + 4t^(a)1 = 0. For fixedpi, the only free parameters are the positionsx1, . . . , xL.

The monodromy group for this equation will be the same as for the hypergeometric equation, corresponding toL= 0

15

D(λ) =−d²

dz²+TL(z) +λ²P(z), P(z) = (z3−z2)^a1(z1−z3)^a2(z2−z1)^a3 (z−z1)^2−a1(z−z2)^2−a2(z−z3)^2−a3 Monodromy free conditions give additionalLequations

ca+3=−∂zlogP(z)

z=xa= 3 i=1

2−ai

xa−zi, a= 1, . . . L . number of solutionsNL=p3(L) = 3,9,22, . . .. (dimensions of level subspaces in a W-algebra, realized with 3 Bose fields, arizing in the conformal limit of the Fateev model).

The perturbed equation possesses the same symmetry Ω_i: z→γi◦z , λ→q⁻¹i λ qi=e^iπai

Ω₃Ω₂Ω₁= 1

Connection coefficient satisfy the same functional equations as forL= 0.

det S^(j,i)(λ)

= 1, S^(i,k)(λ)S^(k,i)(λ) =I, S^(i,k)(λ)S^(k,j)(λ)S^(j,i)(λ) =I, S^(i,k)(λ) e^−2πipiσ3S^(k,j)(λ qk⁻¹) e^−2πipjσ3S^(j,i)(λ qi) e^−2πipiσ3=−I.

16

Monodromy matrix for the Pochhammer loop

zj

zi

zk

z1

z2

z3

γ_P

W(λ) = TrM(γP) = 2

2 +c(4p1) +c(4p2) +c(4p3) +c(2p1+ 2p2+ 2p3) +c(2p1+ 2p2−2p3) +c(2p1−2p2+ 2p3) +c(−2p1+ 2p2+ 2p3)

+O(λ²) wherec(x) = cos(πx). Forpi= 0 the constant term equals18.

WHY? What does it mean for the hypergeometric equation?

17

ODE/IM correspondence for massive integrable QFT

Now we consider the CFT perturbed by a relevant operator in the bulk

x~x+R

Aμ=ACF T+μ

d²xΦ (dΦ= 2Δ_Φ<2)

In general one expects that the perturbation leads to the massive QFT Ma∼μ^2−d1Φ

In the case of integrable perturbation the theory possesses an infinite set of local IM Is|μ→0=I^{(CF T}s ⁾, ¯Is|μ→0= ¯I^{(CF T)}s

LetI2n−1= ¯I2n−1be the vacuum eigenvalues ofIsand ¯Is.

Is it possible to relateI2n−1(μ)to monodromic characteristics of some ODE?

18

81

During the decade 1998-2008, all attempts to incorporate massive integrable QFT in the ODE/IM correspondence have failed.

• Gaiotto, Moore and Neitzke (2008): TBA-like equations for the Hitchin systems

• Alday, Maldacena (2009): Strong coupling amplitudes in ADS/CFT

• Zamolodchikov, Lukyanov (2010): ODE/IM for the vacuum state in sin(h)-Gordon model

• Lukyanov (2013): ODE/IM for the vacuum state of the Fateev model

19

CMC embedding of a 3-punctured sphere inAdS3 Let Σg,nbe a compact Riemann surface withnmarked points (“punctures”) and a1, a2, . . . anbe positive numbers such that2χ(Σ_g) +n

i=1(ai−2) = 0. Then there exists aflatmetric on Σ_g,nwith conical singularities of angleπaiat thei^thpuncture.

The metric is unique up to homothety.

Conical Punctures

In the case Σ_0,3=S²/{P1, P2, P3} : a1+a2+a3= 2

Introduce a complex coordinatezand define a holomorphic differentialp(z) (dz)²on the universal cover of Σ_0,3:

p(z) =ρ² (z3−z2)^a1(z1−z3)^a2(z2−z1)^a3

(z−z1)^2−a1(z−z2)^2−a2(z−z3)^2−a3 : (ds)²₀=

p(z)¯p(¯z) dzd¯z Hereρstands for the homothety parameter andzilabels the punctures.

20

Consider now the problem of constant mean curvature embedding of Σ_0,3intoAdS3. In this case, the Gauss-Peterson-Codazzi equation can be brought to the form of the modified Sinh-Gordon(MShG) equation

∂z∂¯zη−e^2η+p(z)¯p(¯z) e^−2η= 0, where the fieldηdefines the induced metric

(ds)²_cmc= 4 1 +H²

e^2η p(z)¯p(¯z)(ds)²₀

andH=conststands for the mean curvature. A suitable solution should be real and smooth asz=zi, and, if we want to preserve the amount of the Gaussian curvature localized at the punctures, it should satisfy the conditions

η−¹₄log

p(z)¯p(¯z) ) =O(1) at z→zi (i= 1,2,3) and∞. Generalized problem: η=

−2 log|z|+O(1) at z→ ∞ 2milog|z−zi|+O(1) at z→zi

If 0< ai<2 and −1

2< mi≤ −1 4(2−ai) then the solution of the generalized problem exists and is unique.

21

The MShG equation is the compatibility condition of the linear problem D(λ)Ψ= 0, D¯(¯λ)Ψ= 0.

D(λ) =∂z−Az, D¯(¯λ) =∂z¯−Az¯, λ=ρe^θ, ¯λ=ρe^−θ Az = −¹₂∂zη σ3+λ

σ+e^η+σ−P(z) e^−η Az¯ = ¹₂∂z¯η σ3+ ¯λ

σ−e^−η+σ+P(¯¯z) e^η . Additional monodromy-free punctures

e^−η∼¯z−x¯a

z−xa, (a= 1, . . . L), e^−η∼z−yb

z¯−y¯b, (b= 1, . . .L¯). satisfy the conditions

∂zη= 1 z−xa

+1

2γa+o(1), ∂z¯η=− 1 z¯−x¯a

+o(1), a= 1, . . . L and

γa=∂zlogP(z)|z=xa

and similarly foryb.

22

The MShG equation is a flatness condition forsl(2)-valued connectionA=Azdz+ ¯A¯zd¯z. The connection is not single-valued on the punctured sphere. However, it does return to the original branch after a continuation along the non-contractible loopC

z2

z1

z3 C

Therefore the Wilson loop

W(θ) = Tr

Pexp

CA

does not depend on the precise shape of the cycle used. It can be regarded as generating functions for the conserved charges

logW(θ)∼ −q₀e^θ+ ∞ n=1

cnq_2n−1e^{−(2n−1)θ} as e(θ)→+∞, |m(θ)|<π 2 herecn=⁽⁻¹⁾_2n!^nΓ(n−√π¹²⁾.

23

Fateev model (1996)

L = 1 16π

3 i=1

(∂tϕi)²−(∂xϕi)²

+ 2μ

e^iα3ϕ3 cos(α1ϕ1+α2ϕ2) + e^−iα3ϕ3 cos(α1ϕ1−α2ϕ2) Hereαiare coupling constants subject to a single constraint

α²₁+α²₂+α²₃=1 2. α²1>0, α²2>0, α²3>0.

The parameterμin the Lagrangian sets the mass scale,μ∼[ mass ]. We shall consider the theory in finite-size geometry, with the spatial coordinatexinϕi=ϕi(x, t) compactified on a circle of circumferenceR, with the periodic boundary conditions

ϕi(x+R, t) =ϕi(x, t).

x~x+R

Aμ=ACF T+μ d²xΦ

24

82

Due to the periodicity of the potential term inϕi, L = 1

16π 3 i=1

(∂tϕi)²−(∂xϕi)²

+ 2μ

e^iα3ϕ3 cos(α1ϕ1+α2ϕ2) + e^−iα3ϕ3 cos(α1ϕ1−α2ϕ2) the space of statesHsplits on the orthogonal subspacesHk1,k2,k3characterized by the three “quasimomentums”ki:

ϕi→ϕi+ 2π/αi : |Ψk₁,k₂,k₃ →e^2πiki|Ψk₁,k₂,k₃. With realαithe model is non-unitary.

Sigma-model description

In the unitary regime (α3=ib) the model admits a dual description in terms of the action

S=

d²x Gμν(X)∂aX^μ∂aX^ν,

whereGμνis a certain two-parameter families of metric on the topological three-sphere which possesses twoU(1) Killing vector fields.

25

The Fateev model is integrable, in particular it has infinite set of commuting local IM I⁽⁺⁾2n−1,I⁽⁻⁾2n−1, 2n= 2,4,6, . . .being the Lorentz spins of the associated local densities

I^(±)2n−1= R

0

dx

2π _i+j+k=nC⁽ⁿ⁾ijk(∂±ϕ1)²ⁱ(∂±ϕ2)^2j(∂±ϕ3)^2k+. . .

where∂±=¹₂(∂x∓∂t) and. . .stand for the terms involving higher derivatives ofϕi, as well as the terms proportional to powers ofμ. The constantC⁽ⁿ⁾ijkis known (Zamolodchikov, Lukyanov, 2012)

Cijk⁽ⁿ⁾= n! i!j!k!

2α²₁(1−2n)

n−i

2α²₂(1−2n)

n−j

2α²₃(1−2n)

n−k

(2n−1)³(4α²1)¹⁻ⁱ(4α²2)^1−j(4α²3)^1−k , where (x)nis the Pochhammer symbol. The displayed terms with the given Cijk⁽ⁿ⁾set the normalization ofI^(±)_2n−1unambiguously.

26

Of primary interest are the eigenvalues

I2n−1=I2n−1⁽⁺⁾({ki} |R) =I2n−1⁽⁻⁾ ({ki} |R) especially thek-vacuum energy

E= 2I1.

In the large-Rlimit all vacuum eigenvaluesI2n−1vanish exceptI1. The vacuum energy is composed of an extensive part proportional to the length of the system,

E=RE0+o(1) at R→ ∞ Specific bulk energy (Fateev, 1996)

E0=−πμ²³

i=1

Γ(2α²i) Γ(1−2α²i).

27

ODE/IM correspondence

The eigenvalues of the local IM in the Fateev model can be expressed in terms of the classical conserved chargesq_2n−1:

μ⁻¹

I1−¹₂RE0

= d1q₁

μ¹⁻²ⁿI2n−1 = dnq_2n−1 (n= 2,3, . . .). Herednare constants, independent ofkiandR. With the normalization conditions forq_2n−1andI^(±)_2n−1described above,dnreads explicitly as

dn= (2π)²ⁿ⁻¹ (−1)ⁿ⁻¹ 16π²

3 i=1

Γ

2 (2n−1)α²i

.

The parameters of the quantum and classical problems are identified as follows:

α²i = ai

4 (i= 1,2,3)

|ki| = 1 ai

(2mi+ 1) μR = 2ρ

28

Conclusion

•There a connection between the theory ofIntegrable Modelsin two dimensions and the spectral theory ofOrdinary Differential Equations.

• Classical conserved charges = Eigenvalues of IM in the integrable QFT

• Eigenvalues of transfer matrices = connection coefficients between different bases solutions of ODE.

• We considered a class of “Perturbed Fuchsian differential equations”

• Multivalued flat connections on the punctured sphere and monodromy properties of associated singular differential operators

• Future tasks: Apply “Quantum Inverse Problem Method”

to the Fateev model (Yang-Baxter structure, lattice regularization, etc.)

• What is 18? (Mininal dimension of representation of the quantized exceptional affine superalgebraUq(D(2,1,;α)))

29

83

RIKKYOMathPhys 2014,Jan.1113th,2014

QuantumEntanglement ofLocalOperators inConformalFieldTheories

TadashiTakayanagi

YukawaInstituteforTheoreticalPhysics(YITP),KyotoUniversity

MainlyBasedon

(i) arXiv:1401.0539withMasahiroNozaki(YITP,Kyoto)and Tokiro Numasawa (YITP,Kyoto) эThesecondlimitofEE

Seealso

(ii) Phys.Rev.Lett.110(2013)9,091602[arXiv:1212.1164]

withJyotirmoyBhattacharya(Kavli IPMU,Tokyo), MasahiroNozaki(YITP,Kyoto)and

TomonoriUgajin (Kavli IPMU/YITP) эThefirstlimitofEE

䐟Introduction

Consideranarbitraryconformalfieldtheory(CFT).

WewouldliketocharacterizelocaloperatorsinaCFT.

эFindamap

Wellknownexamples

(i)f=conformaldimension (ii)f=somecharge

Butnomorethanthesesofar….

R ) ( )

( :

operator local a

 o f O x

O f

'

O

QO

Themainpurposeofthistalkistointroduce anewinterestingclassofsuchquantities:

> @ > @ > @

. vac :

state Excited

,

) vac

( )

( )

(

O(x) O

S O S O

S_Aⁿ _Aⁿ _Aⁿ {

'

> @

<

<

state for the

Entropy Renyi

th -nt Entangleme

)

( n

S_Aⁿ

A=asubregion(subsystem) onatimeslice

~Lossofinformationwhenweassume thattheregionBisinvisible.

~``degreesoffreedom’’oftheoperatorO.

ThismaybeanalogoustocentralchargesfortheCFTitself.

(Indeed,EEleadstoproofsofcth andFth[CasiniHuerta].) A

B

) (n

SA

> @

^O

S_A⁽ⁿ⁾ '

Twolimits

(1)

Inthiscase,wefindthepropertyanalogousto thefirstlawofthermodynamics

(2)[NozakiNumasawaTT14]

Thisleadstoatotallynewentropicquantity!

эThemainpurposeofthistalk.

> @

^{Not New!}

)

( v |' o

' O O

n

A O E

S

[BhattacharyaNozakiUgajinTT12]

A B

84

Contents

䐟 Introduction

䐠 WhatisEntanglementEntropy(EE)?

䐡 ThefirstlimitofEE(`1stlawofthermodynamics’) 䐢 ThesecondlimitofEE(somenewquantities) 䐣 Conclusions

Whatisthequantumentanglement?

Inquantummechanics,

aphysicalstate=avectorinHilbertspace.

Consideraspinofanelectron,anystateisdescribed byalinearcombination:

. 1

|

|

|

|

, ^{2 2}

p n

< a b a b

䐠 WhatisEntanglementEntropy?

Considerthefollowingstates intwospinsystems:

(i)Adirectproductstate(unentangled state)

(ii)Anentangledstate(EPRpair)

> @ > @

^.

2 1

B B A

Ap … n p

n

<

>

n …p p …nB

@

^{/ 2.}

< _{A B A}

Independent

Onedeterminestheother!

ѩNonlocalcorrelation

DivideaquantumsystemintotwosubsystemsAandB.

Definethereduceddensitymatrix by

Theentanglemententropyisnowdefinedby (vonNeumannentropy)

U

A B

.

A

tot

H H

H …

Ameasureofquantumentanglementisknownas entanglemententropywhichisdefinedasfollows.

> @ > @

> @

₂¹

> @ > @

^.

Tr 2 (i) 1

B

A A A A A

B B A A

p n

˜ p n

<

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