aa r X i v : . [ m a t h - ph ] J a n On Relative Entropy and Global Index
Feng Xu ∗ Department of MathematicsUniversity of California at RiversideRiverside, CA 92521E-mail: [email protected]
Abstract
Certain duality of relative entropy can fail for chiral conformal net withnontrivial representations. In this paper we quantify such statement by defininga quantity which measures the failure of such duality, and identify this quantitywith relative entropy and global index associated with multi-interval subfactorsfor a large class of conformal nets. In particular we show that the duality holdsfor a large class of conformal nets if and only if they are holomorphic. The sameargument also applies to CFT in two dimensions. In particular we show thatthe duality holds for a large class of CFT in two dimensions if and only if theyare modular invariant. We also obtain various limiting properties of relativeentropies which naturally follow from our formula. ∗ Supported in part by NSF grant DMS-1764157. Introduction
In the last few years there has been an enormous amount of work by physicists con-cerning entanglement entropies in QFT, motivated by the connections with condensedmatter physics, black holes, etc.; see the references in [6] for a partial list of references.See [5], [12], [11], [13], [14] , [19],[22] and [23] for a partial list of recent mathematicalwork.This paper is motivated by a very simple fact about von Neumann entropy. Infinite dimensional case the von Neumann entropy of a pure state for a matrix algebra M and its commutant M ′ are equal, a simple exercise in linear algebra. In the caseof conformal net the algebra M is replaced by the algebra of observables localizedon disjoint union of intervals I denoted by A ( I ) . The vacuum state is a pure state.Hence one may expect that the von Neumann entropy of vacuum state for A ( I ) andits commutant are equal. But for type III factors von Neumann entropy is alwaysinfinity so this is not very interesting. By the work of [1] and [14] one can define aregularized von Neumann entropy (cf. Def. 2.9) for A ( I ) , denoted by G ( I ), which isfinite but not positive, yet verifies equations similar to von Neumann entropy in thefinite dimensional case. When the global dimension of A is one, A ( I ) ′ = A ( I ′ ) , onecan therefore ask if the regularized von Neumann entropy for A ( I ) and A ( I ) ′ = A ( I ′ )is the same. This is what we called a duality relation.It was observed in § A ( I )and A ( I ′ ) are different when the global dimension of A is greater than one, and it isnatural to conjecture that duality relation above holds if and only if the conformalnet has global index equal to 1. The only currently known example that verify sucha relation is the free fermion net for which we have explicit formulas for mutualinformation in general as in [14]. One of the goals of this paper is to prove thatthis conjecture is true for a large class chiral CFT (Cor. 2.16) and also CFT in twodimensions which are modular invariant (Cor. 3.7). For an example, it follows fromCor. 2.16 that such duality relation is true for conformal nets associated with anyeven positive unimodular lattices. The number of such lattices grow very fast as theirrank increase.To prove such results we are led to consider a quantity called deficit , which issimply the difference D A ( I ) = G ( I ) − G ( I ′ ), and conjecture (cf. 2.12) that D A ( I ) isequal to another quantity ˆ D A which is defined by using the data associated with theinclusion A ( I ) ⊂ A ( I ′ ) ′ (cf. [9]). Our key observation is Th. 2.13 that D A ( I ) − ˆ D A ( I )remain the same for a pair of conformal nets A ⊂ B with finite index. Recall that D A ( I ) − ˆ D A ( I ) for free fermion nets can be verified by explicit formulas of [14]. Itfollows that any conformal net A that is chain related to free fermion net A r , i.e.,there exists a sequence of conformal nets B , ..., B n such that B = A , B n = A r andeither B i ⊂ B i +1 or B i +1 ⊂ B i , ≤ i ≤ n − , and all inclusions are of finite indexmust verify our conjecture (cf. Cor. 2.14 and Cor. 3.6).To give the reader an idea what kind of equalities are proved in this paper let usconsider a special case of Cor. 2.14 for a conformal net A that is chain related to free2ermion net A r . Then for I = I ∪ I , I ′ = J ∪ J we have S ( ω, ω J ⊗ ω J ) − S ( ω, ω I ⊗ ω I ) − c η = S ( ω, ωF I ) −
12 ln µ A where S is the relative entropy, ω is the vacuum state, c is the central charge, µ A isthe global index of A , η = r J r J r I r I is a cross ratio, and F I : A ( J ∪ J ) ′ → A ( I ∪ I ) is theconditional expectation. Previously relations among relative entropies, central chargeand global index are given in asymptotic form in Th. 4.2 of [14]. The above relationis an identity. The duality condition as described above holds when the righthandside is 0.The rest of this paper is as follows: In § § D which is Deficit to measure the failure ofduality and we prove our main theorem Th. 2.13. We deduce Cor. 2.14, Cor. 2.15as consequences of Th. 2.13. In sections 2.4 and 2.5 we apply Th. 2.13 to study anumber of natural problems on relative entropy.In § § Let ψ be a normal state on a von Neumann algebra M acting on a Hilbert space H and φ ′ be a normal faithful state on the von Neumann algebra M ′ . The Connesspatial derivative, usually denoted by dψdφ ′ , is a positive operator (cf. [3]) . We will usethe simplified notation of [18] and write dψdφ ′ = ∆( ψφ ′ ) . If ψ is faithful , we have∆( ψφ ′ ) it m ∆( ψφ ′ ) − it = σ ψt ( m ) , ∀ m ∈ M, ∆( ψφ ′ ) it m ∆( ψφ ′ ) − it = σ φ ′ − t ( m ) , ∀ m ∈ M ′ where σ ψt , σ φ ′ − t are modular automorphisms.[ Dψ : ψ ] t := ∆( ψ φ ′ ) it ∆( ψ φ ′ ) − it is independent of the choice of φ ′ and is called Connes cocycle.Also if ψ ≥ ψ then ∆( ψ φ ′ ) ≥ ∆( ψ φ ′ ) . By Page 476 of [21] this is equivalent to11 + ∆( ψ φ ′ ) ≤
11 + ∆( ψ φ ′ )3s bounded operators.Suppose M acts on a Hilbert space H and ω is a vector state given by Ω ∈ H . Therelative entropy (cf. 5.1 of [18]) in this case is S ( ω, φ ) = −h ln ∆( φ/ω ′ )Ω , Ω i where ω ′ is the vector state on M ′ defined by vector Ω and ∆( φ/ω ′ ) := dφdω ′ is Connes spatialderivative. When Ω is not in the support of φ we set S ( ω, φ ) = ∞ .A list of properties of relative entropies that will be used later can be found in [18](cf. Th. 5.3, Th. 5.15 and Cor. 5.12 [18]): Theorem 2.1. (1) Let M be a von Neumann algebra and M a von Neumann sub-algebra of M. Assume that there exists a faithful normal conditional expectation E of M onto M . If ψ and ω are states of M and M , respectively, then S ( ω, ψ · E ) = S ( ω ↾ M , ψ ) + S ( ω, ω · E ); (2) Let be M i an increasing net of von Neumann subalgebras of M with the property ( S i M i ) ′′ = M . Then S ( ω ↾ M i , ω ↾ M i ) converges to S ( ω , ω ) where ω , ω are twonormal states on M ;(3) Let ω and ω be two normal states on a von Neumann algebra M . If ω ≥ µω, then S ( ω, ω ) ≤ ln µ − ;(4) Let ω and φ be two normal states on a von Neumann algebra M , and denoteby ω and φ the restrictions of ω and φ to a von Neumann subalgebra M ⊂ M respectively. Then S ( ω , φ ) ≤ S ( ω, φ ) ;(5) Let φ be a normal faithful state on M ⊗ M . Denote by φ i the restriction of φ to M i , i = 1 , . Let ψ i be normal faithful states on M i , i = 1 , . Then S ( φ, ψ ⊗ ψ ) = S ( φ , ψ ) + S ( φ , ψ ) + S ( φ, φ ⊗ φ )Let E : M → N be a normal faithful conditional expectation onto a subalgebra N . E − : N ′ → is in general an operator valued weight which verifies the followingequation: for any pair of normal faithful weights ψ on N and φ ′ on M ′ we have∆( ψEφ ′ ) = ∆( ψφ ′ E − )Kosaki (cf. [8]) defined index of E , denoted by Ind E to be E − (1). When 1 is in thedomain of E − , we say that E has finite index. When both N, M are factors and E has finite index, we have the (cf. [20]) Pimsner-Popa inequality E ( m ) ≥ λm, ∀ m ∈ M + , where λ = (Ind E ) − . The action of the modular group σ ψEt on N ′ ∩ M is independentof the choice of ψ . When E is the minimal conditional expectation such action istrivial on N ′ ∩ M . Also the compositions of minimal conditional expectations areminimal (cf. [10]). Lemma 2.2.
Let
A, B be positive unbounded operators on a Hilbert space such that A ≥ B, and Ω is a unit vector such that B Ω = c Ω where c > is a constant, h A Ω , Ω i =4 . Let m A be the spectral measure of A associated with Ω . Then R ∞ (ln λ ) dm A ( λ ) < ∞ . Proof.
By Page 476 of [21] we have that /n + A ≤ /n + B , ∀ n > Z ∞ /n + λ dm A ( λ ) ≤ /n + c , ∀ n > n goes to infinity and by Monotone convergence theorem we have Z ∞ λ dm A ( λ ) ≤ c , ∀ n > λ ) is bounded by a constant times 1 /λ on (0 , λ on [1 , ∞ ). Since by assumption R ∞ λdm A ( λ ) = 1 , we have shown that R (ln λ ) dm A ( λ ) < ∞ , R ∞ (ln λ ) dm A ( λ ) < ∞ , and the proof is complete. (cid:4) Lemma 2.3.
Let A be a self adjoint operator on a Hilbert space, and Ω be a vector inthe domain of A . Let f ( t ) be a strong operator continuous function in a neighborhoodof with value in the space of bounded operators such that f (0) is identity. Then lim t → − it h ( e itA − f ( t )Ω , Ω i = h A Ω , Ω i Proof.
By assumption it is enough to check thatlim t → − it h ( e itA − f ( t ) − , Ω i = 0We note that || − it ( e itA − || = Z | t ( e itλ − | dm A ( λ ) ≤ Z | λ | dm A ( λ ) < ∞|| ( f ( t ) − f (0)) ω || goes to 0 as t goes to 0, and the lemma is proved. (cid:4) Proposition 2.4.
Let M be a factor and ω a normal faithful state on M actingon the standard representation space H , and Ω the corresponding vector such that h m Ω , Ω i = ω ( m ) , ∀ m ∈ M. We shall use the same notation ω to denote the vectorstate on B ( H ) and its restriction to subalgebras of B ( H ) .Let E : M → M , E : M ′ → M be normal conditional expectation with finiteindex, where M , M are also factors. Then S ( ω, ωE ) − S ( ω, ωE ) = S ( ω, ωE E − ) and this equation can also be written as S ( ω, ωE ) + S ( ω, ωE − ) = S ( ω, ωE E − )5 roof. Ad (1): By definition we have S ( ω, ωE ) − S ( ω, ωE ) = lim t → − it h (∆( ωE ω ) it − (∆( ωE ω ′ ) it )Ω , Ω i We note that∆( ωE ω ′ ) it Ω = ∆( ωE ω ′ ) it ∆( ωω ′ ) − it Ω = [
DωE : Dω ] t Ω∆( ωE ωE ) it ∆( ωE ω ) it = ∆( ωE ω ′ ) it ∆( ωω ′ ) − it It follows that S ( ω, ωE ) − S ( ω, ωE ) = lim t → − it h (∆( ωE ωE ) − it − ωE ω ′ ) it Ω , Ω i Note that ∆( ωE ωE ) = ∆( ωE E − ω ′ ) ≥ µ ∆( ωω ′ ), for some µ >
0. Here the spatialderivative ∆( ωω ′ ) is determined by state ω on M ′ and M respectively.By Lemma 2.2 and Lemma 2.3 we have proved the first equation. Apply thisequation with E equal to identity we get S ( ω, ω ) − S ( ω, ωE ) = S ( ω, ωE − )and the second equation follows. (cid:4) It is convenient to formulate the second equation of the above Prop. in the fol-lowing form:
Corollary 2.5.
Let N ⊂ N ⊂ N be factors on a Hilbert space H and ω is a vectorstate on B ( H ) given by a vector Ω ∈ H . Let F i , N i → N i +1 , i = 1 , be conditionalexpectation with finite index. Assume that Ω is cyclic and separating for N . Then S ( ω, ωF F ) = S ( ω, ωF ) + S ( ω, ωF ) Proof.
This is just a reformulation of the second equation of Prop. 2.4 by noting thatwe can rename N = M ′ , N = M, N = M , F = (Ind E ) − E − , F = E . (cid:4) Remark 2.6.
Under the conditions of the above Cor. S ( ω, ωF ) is additive undercompositions of conditional expectations, just like ln Ind E . But of course S ( ω, ωF ) also depends on the state ω. This fact plays important role in the proof of Th. 2.13and Th. 2.20 in the following. .3 Chiral CFT case Let A be a conformal net (cf. [9] and [14]) . It is always split (cf. [16] ). Let PI bethe set whose elements are disjoint union of intervals. If I is an interval on the circlewith two end points a, b , r I := | b − a | is called the length of I .For any I ∈ PI , ω I denotes the restriction of ω to A ( I ). It follows that ω I ⊗ ... ⊗ ω I n is a normal state on A ( I ).Since we will be concerned with relative entropy of various states, we introducesome definitions to simplify notations. For I = I ∪ I ... ∪ I n ∈ PI where I i aredisjoint intervals, ω ⊗ := ω I ⊗ ω I ⊗ ... ⊗ ω I n . A state ψ on A ( I ) is said to be related to vacuum state ω if we can partition I intodisjoint union I = J ∪ J ... ∪ J m , J i ∈ PI , ≤ i ≤ m, such that ψ = ω J ⊗ ω J ⊗ ... ⊗ ω J m . We shall consider conformal net whose mutual information for vacuum state arealways finite.
Definition 2.7.
A conformal net A is said to have finite mutual information if S ( ω, ω ⊗ I ) < ∞ , ∀ I ∈ PI Suppose
A ⊂ B is an inclusion of conformal nets with finite index. We shall denoteby E I : B ( I ) → A ( I ) the unique conditional expectation which preserves the vacuumstate when I is an interval. When I = I ∪ I ∪ ... ∪ I n is a disjoint union of n intervals,we shall use E I to denote E I ⊗ ... ⊗ E I n which is the unique conditional expectationfrom B ( I ) to A ( I ) which preserves ω I ⊗ ... ⊗ ω I n . Lemma 2.8. (1) If A has finite mutual information, then S ( ω, ψ ) < ∞ for all ψ on A ( I ) that is related to vacuum state ω .(2) If A ⊂ B and B has finite mutual information, then A also has finite mutualinformation;(3) If A ⊂ B has finite index and A has finite mutual information, then A alsohas finite mutual information.Proof. By (5) of Th. 2.1 we have S ( ω, ω ⊗ I ∪ J ) = S ( ω, ω ⊗ I ) + S ( ω, ω ⊗ J ) + S ( ω, ω I ⊗ ω I )and S ( ω, ψ I ⊗ φ J ) = S ( ω, ψ I ) + S ( ω, φ J ) + S ( ω, ω I ⊗ ω I )It follows that any S ( ω, ψ ) can be expressed as linear combination of S ( ω, ω ⊗ J ) forsuitable intervals J ⊂ I and (1) is proved.(2) follows from definition and monoticity of relative entropy in Th. 2.1.By Th. 2.1 S B ( ω, ω ⊗ I ) − S A ( ω, ω ⊗ I ) = S ( ω, ωE I ) . Since S ( ω, ωE I ) ≤ ln(Ind E I ) , (3)is proved. (cid:4) It is proved on Page 13 of [23] that essentially all known conformal net (andprobably all) has finite mutual information.7 conformal net is called rational if for some I = I ∪ I , ¯ I ∩ ¯ I = ∅ where the A ( I ) ⊂ A ( I ′ ) ′ has finite index which is called Global index and is denoted by µ A .Two conformal nets A and B are said to be chain related if there exists asequence of conformal nets B , ..., B n such that B = A , B n = B and either B i ⊂ B i +1 or B i +1 ⊂ B i , ≤ i ≤ n − , and all inclusions are of finite index. See § A with central charge c and finite mutual information, wedefine: Definition 2.9.
The regularized von Neumann entropy of vacuum state for A ( I ) , I ∈PI is defined as follows: For an interval I we let G ( I ) := c/ r I , r I is the length ofinterval I , and G ( I ∪ I ∪ ... ∪ I n ) = G ( I ) + ... + G ( I n ) − S ( ω, ω I ⊗ ω I ⊗ ... ⊗ ω I n )Note that von Neumann entropy for type III factors are always infinity, and regu-larized von Neumann entropy as defined are motivated by the results of [1] and § I .When µ A = 1, A ( I ) = A ( I ′ ) ′ , ∀ I ∈ PI , and the vacuum state ω is a pure vectorstate, we expect that the von Neumann entropy of ω for A ( I ) , I ∈ PI and A ( I ′ ) , I ∈PI should be the same. Of course both are infinity, but what is more interesting isto conjecture that G ( I ) = G ( I ′ ) , ∀ I ∈ PI if µ A = 1. In § G ( I ) = G ( I ′ )if µ A > . Hence we expect that G ( I ) = G ( I ′ ) , ∀ I ∈ PI if and only if µ A = 1. At present the only known example which verifies µ A = 1 and G ( I ) = G ( I ′ ) , ∀ I ∈ PI is the free fermion net (cf. § G ( I ) , ∀ I ∈ PI is known. To investigatethe general cases we define the following Definition 2.10.
We define the deficit for A ( I ) , I ∈ PI to be D A ( I ) := G A ( I ) − G A ( I ′ ) . Let F I : A ( I ′ ) ′ → A ( I ) be the condition expectation of index µ n − A (cf. [9]). Whenthere are a pair of nets involved we shall use the notation F I, A to avoid confusions. Definition 2.11.
Let I ∈ PI be a disjoint union of n intervals, define ˆ D A ( I ) := S ( ω, ωF I ) − n −
12 ln µ A . Conjecture 2.12.
For a rational conformal net D A ( I ) = ˆ D A ( I )Note that when µ A = 1 , the above conjecture implies that G ( I ) = G ( I ′ ) , ∀ I ∈ PI Suppose
A ⊂ B is an inclusion of conformal nets with finite index. Recall that E I : B ( I ) → A ( I ) is the unique conditional expectation which preserves the vacuumstate when I is an interval. When I = I ∪ I ∪ ... ∪ I n is a disjoint union of n intervals, E I denotes E I ⊗ ... ⊗ E I n which is the unique conditional expectation from B ( I ) to A ( I ) which preserves ω I ⊗ ... ⊗ ω I n . We will prove Conj. 2.12 for a large class of conformal nets. The idea is thefollowing : Since we have an important example of free fermion net A r for which wealready know D A r ( I ) = ˆ D A r ( I ), and there are many conformal nets that are chain related to A r , if we can show thatfor a pair of conformal nets A ⊂ B with finite index that D A ( I ) − ˆ D A ( I ) = D B ( I ) − ˆ D B ( I ), then it follows that Conj. 2.12 is true for conformal nets that are chain related to A r . To state the theorem in more general terms, we note that assuming that all thequantities involved on the left hand side are finite, then D A ( I ) − ˆ D A ( I ) = D B ( I ) − ˆ D B ( I )is equivalent to S ( ω, ωE I ) − S ( ω, ωE I ′ ) = ˆ D A ( I ) − ˆ D B ( I )Then the following Th. does exactly this: Theorem 2.13. (1) Let
A ⊂ B be rational conformal nets with finite index, then S ( ω, ωE I ) − S ( ω, ωE I ′ ) = ˆ D A ( I ) − ˆ D B ( I ) (2) (1) also holds when B is free fermion net A r .Proof. Fix I ∈ PI which is a disjoint union of n intervals.Ad (1): Let E := (Ind E I ′ Ind F I ′ , B ) − E I F − I ′ , B E − I ′ be the condition expectation from A ( I ′ ) ′ → A ( I ) . Set E := E I F − I ′ , B E − I ′ . Let us compute S ( ω, ωE I ) − S ( ω, ωF I ′ , B ) − S ( ω, ωE I ′ ). Note that Ω is separatingand cyclic for B ( I ) ′ . By Prop. 2.4 we have S ( ω, ωE I ) − S ( ω, ωF I ′ , B ) − S ( ω, ωE I ′ ) = S ( ω, ωE )9y § E restricts to trace on A ( I ) ′ ∩ A ( I ′ ) ′ . Let P A be the projectionin A ( I ) ′ ∩ A ( I ′ ) ′ which projects onto the closure of A ( I )Ω. Then we have∆( ωEω ′ ) it P A ∆( ωEω ′ ) − it = P A , ∀ t where ω ′ is the state on A ( I ′ ) given by Ω. It follows that ∆( ωEω ′ ) commutes with P A .We note that when restricted to P A A ( I ′ ) ′ P A , ωE is given by E ( P A ) ωE P A where E P A : P A A ( I ′ ) ′ P A → P A A ( I )is the unique conditional expectation and can be identified with F I, A : A ( I ′ ) ′ → A ( I )where the algebras are on P A H B = H A . Note that E ( P A ) = [ B : A ] − = µ / B µ / A . Hence h ln ∆( ωEω ′ )Ω , Ω i = ln E ( P A ) + h ln ∆( ωE P A ω ′ )Ω , Ω i = ln E ( P A ) + h ln ∆( ωF I, A ω ′ )Ω , Ω i Note that Ind E I = ( µ A µ B ) n/ , Ind F I ′ , B = µ n − B Putting the above pieces together we have shown that S ( ω, ωE I ) − S ( ω, ωF I ′ , B ) − S ( ω, ωE I ′ ) = S ( ω, ωF I, A ) − n −
12 (ln µ A + ln µ B )Finally by Prop. 2.4 we have − S ( ω, ωF I, B ) = S ( ω, ω ) − S ( ω, ωF I, B ) = S ( ω, ωF − I, B ) = S ( ω, ωF I ′ , B ) − ( n −
1) ln µ B and the proof of the theorem is complete.Ad (2): Note that in this case F I, B is identity, so we only need to evaluate S ( ω, ωE I ) − S ( ω, ωE I ′ )Note that E − I ′ : A ( I ′ ) ′ → A r ( I ′ ) ′ = k A r ( I ) k − where k is the Klein transform.Let us define ˆ E I ( kak − ) = E I ( a ) , ∀ a ∈ A r ( I )Since k Ω = Ω , it follows that ω ( ˆ E I ( kak − )) = ω ( E I ( a )) , ω ( kak − ) = ω ( a )and S ( ω, ωE I ) = S ( ω, ω ˆ E I ) . Hence by (2) of Prop. 2.4 S ( ω, ωE I ) − S ( ω, ωE I ′ ) = S ( ω, ω ˆ E I ) − S ( ω, ωE I ′ ) = S ( ω, ω ˆ E I E − I ′ )The rest of the proof is the same as in (1) above. (cid:4) By Th. 2.13 we immediately have 10 orollary 2.14. If A is chain related to A r , then Conj. 2.12 is true for A . We also have
Corollary 2.15. If A is chain related to A r , then D A = 0 if and only if µ A = 1 .Proof. If µ A = 1, then D A = 0 by Cor. 2.14. Now suppose that D A ( I ∪ I ) = 0 . By(2) of Th. 4.2 in [14], it follows that µ A = 1 . (cid:4) Corollary 2.16.
Conj. 2.12 is true for conformal nets associated with even positivedefinite lattices.Proof.
First we prove this for rank one lattices. Let A U (1) a be the conformal netassociated with rank one lattice with a a positive even integer. Denote by D ( a ) := D A U (1) a ( I ) − ˆ D A U (1) a ( I ) . We prove by induction on k that D ( ka ) = D ( a ) , ∀ k ≥ k = 1 this is trivial. Assume the above equation is true for k . Consider thefollowing finite index inclusions: U (1) ( k +1) a × U (1) ( k +1) ka ⊂ U (1) ka × U (1) a where U (1) ( k +1) a is diagonally embedded in U (1) ka × U (1) a and its commutant in U (1) ka × U (1) a is U (1) ( k +1) ka . By Th. 2.13 and induction hypothesis we have2 D (( k + 1) a ) = 2 D ( a )and it follows by induction we have proved D ( ka ) = D ( a ) , ∀ k ≥ . Now from the inclusion U (1) × U (1) ⊂ U (1) × U (1) and Th. 2.13 we conclude that D (2) = 0. It follows that D ( a ) = 0 for all even a . Now assume that the Corollary is proved for all rank k lattices. If L is an evenpositive definite lattice, choose a nonzero element e ∈ L and consider sublattices L = Z e of L and L of L which is orthogonal to L with rank equal to k . Apply Th.2.13 to the finite index inclusions A L ⊗ A L ⊂ A L and induction hypothesis, we have proved the Corollary. (cid:4) .4 Some continuous properties Let us first fix a rational conformal net A with finite mutual information.By (2) of Th. 2.1 relative entropies are continuous from “inside”. As an applicationof Th. 2.13, we will prove that relative entropies in Th. 2.13 are also continuous from“outside”. First we have: Lemma 2.17. If I ⊂ J, I, J ∈ PI , then F J restrict to F I on A ( I ) and hence S ( ω, ωF I ) increase with I ;Proof. This is proved in § n = 2, but the same argument works for any n . (cid:4) Corollary 2.18.
Let
A ⊂ B be as in Th. 2.13. Then S ( ω, ωE I ) is continuous from“outside”, i.e., if I n is a decreasing sequence of intervals such that ∩ I n = I, and E I ′ restrict to E I ′ n , then lim n →∞ S ( ω, ωE I n ) = S ( ω, ωE I ) Proof.
This follows from Th. 2.13 and Lemma 2.17. (cid:4)
It is usually an interesting problem to study the limiting properties of relative entropieswhen intervals get close together. One can find such studies in § § S ( ω, ωF I ) fora conformal net A .The following Theorem is a reformulation of Proposition 3.25 of [14]: Theorem 2.19.
Assume that M n is an increasing sequence of factors act on a fixedHilbert space, N n ⊂ M n are subfactors and ω is a vector state associated with a vector Ω . Suppose that E n : M n → N n , n ≥ is a sequence of conditional expectations suchthat when restricting to M n , E n +1 = E n , n ≥ , and Ind E n = λ is a positive realnumber independent of n . If strong operator closure of ∪ n N n contains M , then lim n →∞ S ( ω, ωE n ) = ln λ Proof.
Set φ n := ωE n . It is sufficient to prove the following as in Proposition 3.25 of [14]: Given any ǫ >
0, we need to find e ∈ M n for sufficiently large n , such that | ω ( e ) − | < ǫ, | ω ( e ∗ ) − | < ǫ, | ω ( e ∗ e ) − | < ǫ, | φ n ( ee ∗ ) − λ | < ǫ . Let e ∈ M be the Jones projection for E : M → N , and v ∈ N be theisometry such that λv ∗ e v = 1 . By assumptions we can find a sequence of elements e n ∈ N n , n ≥ e . Now choose x n =12 − v ∗ e e n v. Then x n → ω ( x n ) , ω ( x n x ∗ n ) convergesto 1. On the other hand by definition E n ( x ∗ n x n ) = v ∗ e ∗ n e n v converges to v ∗ e v = λ − strongly. Hence given any ǫ >
0, we can choose n sufficientlylarge such that if we set e = x ∗ n , then e ∈ M n , and | ω ( e ) − | < ǫ, | ω ( e ∗ ) − | < ǫ, | ω ( e ∗ e ) − | < ǫ, | φ n ( ee ∗ ) − λ | < ǫ . (cid:4) Let I = I ∪ I ∪ ... ∪ I n ∈ PI and I ′ = ˆ I ∪ ˆ I ∪ ... ∪ ˆ I n . Let us arrange indices suchthat ˆ I i share end points with I i , I i +1 , ≤ i ≤ n − . We are interested in shrinking I ′ . Let us first introduce some terminology. By a contraction of I along ˆ I we meankeep I ∪ ˆ I ∪ I := I fixed and let the length of ˆ I go to 0. We will use a sequence I ( k ) , ˆ I ( k ) , I ( k ) such that ˆ I ( k ) is decreasing to describe such a process. Such asequence is called a contraction sequence along ˆ I . Let C ( I ) = I ∪ I ... ∪ I n ∈ PI . Theorem 2.20.
Choosing a contracting I ( k ) , ˆ I ( k ) , I ( k ) sequence along ˆ I . Then lim k →∞ S ( ω, ωF I ) = S ( ω, ωF C ( I ) ) + ln µ Proof.
Observe that when restricting F C ( I ) to A ( I ′ ) ′ , we get a conditional expectationsimply denoted only in the proof by F k : A ( I ′ ) ′ → A ( C ( I )) ∩ A ( ˆ I ) ′ . Let E k : A ( C ( I )) ∩ A ( ˆ I ) ′ → A ( I ) be the conditional expectation such that E k restricts toidentity on A ( I ∪ ... ∪ I n ), and on A ( I ) ∩A ( ˆ I ) ′ is the unique conditional expectationonto A ( I ∪ I ) . Note that the index of E k is µ . Notice that Ω is cyclic and separatingfor A ( C ( I )) ∩ A ( ˆ I ) ′ . By Cor. 2.5 we have S ( ω, ωF I ) = S ( ω, ωE k F k ) = S ( ω, ωE k ) + S ( ω, ωF k )By (2) of Th. 2.1 we have lim k S ( ω, ωF k ) = S ( ω, ωF C ( I ) ) . To finish the proof it issufficient to show that lim k S ( ω, ωE k ) = ln µ. This follows from Th. 2.19 since ∪ k I ( k ) ∪ I ( k ) is equal to I minus a point. (cid:4) We note that we can apply Th. 2.20 a few times to shrink intervals ˆ I , ..., ˆ I n − successively. This way we see thatlim k S ( ω, ωF I k ) = n −
12 ln µ A where one take an increasing of disjoint intervals I k , each one is a disjoint union n intervals such that ∪ k I k is equal to S minus finitely many points. This can of coursebe proved directly using Th. 2.19.Now consider the case of A ⊂ B with finite index.13 emma 2.21.
Choosing a contracting I ( k ) , ˆ I ( k ) , I ( k ) sequence along ˆ I . Then lim k →∞ S ( ω, ωE I ) = 1 / µ A − ln µ B ) + S ( ω, ωE C ( I ) ) Proof.
For the ease of notations we set ω := ω I ⊗ ... ⊗ ω I n . By (5) of Th. 2.1 S ( ω, ω I ⊗ ω I ⊗ ω ) = S ( ω, ω I ⊗ ω I ) + S ( ω, ω ) + S ( ω, ω I ∪ I ⊗ ω )We note that as k goes to infinity, I ∪ I increase to I , hencelim k S ( ω, ω I ∪ I ⊗ ω ) = S ( ω, ω I ⊗ ω )Hence lim k S ( ω, ωE I ) = lim n S ( ω, ωE I ∪ I ) + S ( ω, ωE C ( I ) )The lemma now follows from Th. 4.4 of [14]. (cid:4) Proposition 2.22.
Let
A ⊂ B be as in Th. 2.13. Choosing a contracting I ( k ) , ˆ I ( k ) , I ( k ) sequence along ˆ I . Then lim k →∞ S ( ω, ωE I ′ ) = S ( ω, ωE C ( I ′ ) )This follows from Th. 2.13, Th. 2.20 and Lemma 2.21. (cid:4) The above Cor. can be phrased as follows: Let I k = I k ∪ I ∪ .. ∪ I n ∈ PI be suchthat I k is a decreasing sequence such that the length of I k tends to 0 as n goes toinfinity. Then lim k →∞ S ( ω, ωE I k ) = S ( ω, ωE I ∪ .. ∪ I n )It follows that if either A or B has the property thatlim k →∞ S ( ω, ω ⊗ I k ) = S ( ω, ω ⊗ I ∪ ... ∪ I n )then the other net also has this property. In particular all conformal nets that arechain related to free fermion nets have this property since free fermion nets verifysuch property. It will be interesting to see if this can be proved under more generalconditions. For a formulation of CFT in two dimensions we refer to § C is defined to be I × J where I, J are intervals on the circle S ,and we consider C to be a subset of S × S . Denote by PC the set which consists offinite disjoint union of double cones. We shall use C ′ to denote the casual complementof C .We will consider the case A ⊂ B where A ( I × J ) = A L ( I ) × A R ( J ) , both A L and A R are rational, and A ⊂ B has finite index. Denote by c L , c R the central charges of A L and A R . 14 efinition 3.1. For a double cone C = I × J we let G ( C ) := c L / r I + c R / r J , and G ( C ∪ C ∪ ... ∪ C n ) = G ( C ) + ... + G ( C n ) − S ( ω, ω C ⊗ ω C ⊗ ... ⊗ ω C n ) Definition 3.2.
We define the deficit for B ( C ) , C ∈ PC to be D B ( C ) := G B ( C ) − G B ( C ′ ) . Note that when the two dimensional net is tensor product A L ⊗ A R , and C = C ∪ C ∪ ... ∪ C n , C i = I i × J i , ≤ i ≤ n, we have G A L ⊗A R ( C ) = G A L ( I ∪ I ∪ ... ∪ I n ) + G A L ( J ∪ J ∪ ... ∪ J n ) . Let F C : B ( C ′ ) ′ → B ( C ) be the condition expectation of index µ n − B . Definition 3.3.
When C is a disjoint union of n double cones, define ˆ D B ( C ) := S ( ω, ωF C ) − n −
12 ln µ B . The Conj. 2.12 for B is now Conjecture 3.4.
For a rational two dimensional conformal net D B ( C ) = ˆ D B ( C )The proof of Th. 2.13 applies verbatim to the case of two dimensional conformalnets A ⊂ B , and we have the following
Theorem 3.5. (1) Let
A ⊂ B be rational two dimensional conformal nets with finiteindex, then S ( ω, ωE C ) − S ( ω, ωE C ′ ) = ˆ D A ( C ) − ˆ D B ( C ) Corollary 3.6.
Suppose B is chain related to A L ⊗ A R , where both A L and A R arechain related to A r , then Conj. 3.4 is true for B . We also have
Corollary 3.7. (1) Suppose B is chain related to A L ⊗ A R , where both A L and A R are chain related to A r then D B = 0 if and only if µ B = 1 ;(2) Suppose that A L ⊗ A R ⊂ B , and both A L and A R are chain related to A r , then D B = 0 if and only if B is modular invariant.Proof. The proof of (1) is the same as the proof of (1) of Cor. 2.15. (2) follows fromTh. 4.2 of [15]. (cid:4)
A large class of examples with µ B = 1 can be obtained as follows: take anyconformal net A which is chain related to free fermion net and take the Longo-Rehrentwo dimensional net (which corresponds to identity modular invariant), it follows bythe above corollary that such net verifies D B = 0.15 emark 3.8. The computation of entropies in physics literature is usually done (cf.[2]) with replica trick using path integrals, and when the underlying CFT can be de-scribed by a Lagrangian it is usually assumed that the CFT is modular invariant. Incases where such computations are done, one finds that the deficit vanishes. Hence(2) of the above Cor. is a rigorous formulation of such intuitions.
Finally we note that the results of sections 2.4 and 2.5 apply to two dimensionalconformal nets as well, with essentially the same proof.
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