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Von Neumann entropy is the basic concept in quantum information and extends the classical Shannon’s information entropy notion to the non commutative setting. As is well known, a state ω on a matrix algebra M is given by a density matrix ρ, namely ω(T ) = Tr(ρT ), T M. The von Neumann entropy of ω is given by ∈ S(ω)= Tr(ρ log ρ) . − Entanglement entropy is a measure of quantum information by the degree of quantum entanglement of a quantum state. Let’s consider a bipartite, finite-dimensional quantum system M = A B, where A and ⊗ B are matrix algebras. Given a pure state ω on the matrix algebra M, let ρA and ρB the density matrices associated with the restrictions of ω to A and B respectively on A and B. The entanglement entropy of ω is defined as

Tr(ρ log ρ )= Tr(ρ log ρ ) . (1) − A A − B B The above definition directly extends to the case M = A B is a direct sum, where k k ⊗ k Ak, Bk are matrix algebras and the restrictions ω A ⊗BLare pure (not normalised). It also | k k extends to the infinite-dimensional case where Ak, Bk are type I factors; in this last case, however, the entanglement entropy may be infinite. Entanglement is certainly one of the main feature of quantum physics and there is a famous, long standing debate on its interpretation (EPR paradox, Bell’s inequalities, etc.). An overview of the matter lies beyond the purpose of this introduction. The role of entanglement in Quantum Field Theory is more recent and increasingly important; it represents a piece of the quantum information framework in this subject. It appears in relation with several primary research topics in theoretical physics as area theorems, c-theorems, quantum null energy inequality, etc. (see for instance [5, 6, 32] and refs. therein). Despite the rich physical literature on the subject, the rigorous definition of entanglement entropy in QFT is however not obvious. The point is that the von Neumann algebra (O) A associated with a double cone spacetime region O is typically a factor of type III, so no trace exists on (O) and one cannot naively extends the definition (1) as one would do with A A = (O), B = (O′), where O is a double cone and O′ is its causal complement and ω the A A vacuum state. Due to ultraviolet divergence, such a measure of the vacuum entanglement would always result to be infinite. By Haag duality, that holds in much generality, (O′) A is the commutant (O)′ of (O) on the vacuum Hilbert space , so the von Neumann A A H (O) (O′) generated by (O) and (O′) is equal to B( ) and cannot be naturally A ∨A A A H isomorphic to the von Neumann tensor product (O) (O′). A ⊗A To get rid of short distance divergences, on may however consider a slightly larger double cone O O, namely the closure of O is contained in the interior of O. The split property ⊂ states that there is a natural isomorphism of von Neumann algebras e e (O) (O′) (O) (O′) , A ∨A ≃A ⊗A that identifies (O) with (O) 1 and e (O′) with 1 e(O′). A A ⊗ A ⊗A The split property expresses the statistical independence of (O) and (O′); it was A A verified for the free, neutral Boson QFT case in [3]. It was studied in [10] and led to e

2 important structural features both in Mathematics and in Physics. It follows under natural, general physical requirements [4]. It holds automatically in chiral conformal QFT [26]. (See [16] for a discussion of its validity in topologically non trivial spacetimes). Approaches to the entanglement entropy by means of the split property are studied in [27, 7, 25, 32, 17, 11]. In particular, [17] contains a rigorous definition via separable states and [25] considers essentially our definition below. The split property is a local property, in fact it is equivalent to the existence of an intermediate type I factor between (O) and (O) F A A (O) (Oe) . (2) A ⊂F⊂A A type I factor is a von Neumann algebra isomorphice to B( ), the algebra of all bounded F K linear operators on some Hilbert space . K We may then define the entanglement entropy of the net associated with the double A cones O O as the vacuum von Neumann entropy associated with the as in (1); here ⊂ F the global systems is B( ), the factorisation is given by , namely A = , B = ′ with a e H F F F tensor product decomposition

= , A B( ) 1, B 1 B( ) , H HA ⊗HB ≃ HA ⊗ ≃ ⊗ HB and the pure state is the vacuum state. This definition however depends on the choice of . Actually, if the split property holds, F there are infinitely many intermediate type I factors in (2). Yet, as shown in [10], there is F a canonical intermediate type I factor , associated with the O, O and the vacuum vector F Ω, given by the formula e = (O) J (O)J = (O) J (O)J (3) F A ∨ A B ∩ B (if the local von Neumann algebras are factors), withe J is thee modular conjugation of the relative commutant von Neumann algebra (O)′ (O) associated with Ω. A ∩A We then define the (canonical) entanglement entropy of with respect to O, O as e A S (O, O)= Tr(ρ log ρ ) , e (4) A − F F where is the canonical intermediatee type I factor (3). Here Tr is the trace of (namely F F = B( ) 1 and Tr corresponds to the usual trace on B( )) and ρ is the vacuum F HA ⊗ HB HA F density matrix relative to . F The above definition concerns a local net . If if a Fermi net, graded locality rather A A than locality holds. In this case, the split property is still defined by (2) and the entanglement entropy by (4). However, the canonical intermediate type I factor is to be defined by a twisted version of formula (3), cf. (50). A main result in this paper is that above defined canonical entanglement entropy is finite for the chiral conformal net generated by a complex free fermion on S1. Here, double M cones are intervals I I of S1. ⊂ is a second quantisation net and we first provide an abstract analysis on the one M e particle Hilbert space . Let F be a closed, real linear subspace of . We define H0 ⊂ H0 H0 the entropy of H as the von Neumann entropy

S(F )= S(σ) ,

3 ′ with σ = PF PF ′ PF , where PF is the real orthogonal projection onto F and F is the ′ symplectic complement of F . Note that PF ′ PF is an “angle operator” (cf. [9, 14, 30]). σ is a bounded, real linear operator. If S(σ) < , then σ has to be a (real) trace class operator; ∞ with λ the (positive) proper values of σ, then S(σ)= λ log λ . { n} − n n n It turns out that P S(σ) < S(ρ ) < , ∞ ⇐⇒ F ∞ where S(ρ ) is the vacuum von Neumann entropy relative to the type I factor associated F F with F in second quantisation. We shall choose a canonical type I subspace F , intermediate between the subspaces real H(I) H(I) of associated with the interval I I, so that ⊂ H0 ⊂ S(σ) will be the canonical entanglement entropy (4). e e In the second part of this paper, we provide model analysis in order to derive the finite- ness of S(σ), in the one complex free fermion case. This is the first case where the entan- glement entropy (4) is proved to be finite, a problem that is explicit or implicit in various papers, [27, 25]. The same finiteness result then follows for other related nets, as in the r-fermion case or in the case of a real fermion. Now, there is another natural definition of entanglement entropy:

SA(I, I) = inf S( ) , F F e the infimum of the the von Neumann entropy S( ) = Tr(ρ log ρ ) over all the inter- F − F F mediate type I factors or, more generally, over all the intermediate type I discrete von F Neumann algebras . Of course, the lower entanglement entropy S (I, I) is bounded by F A the canonical entanglement entropy S (I, I). A e We shall infer form our results that e S (I, I) < A ∞ also in the case of the local net generated byer free bosons on S1 (r copies of the current algebra net). We expect the canonical entanglement entropy to be finite in this case too, but this remains unproven in this paper.

2 Entropy of standard subspaces

In this first part, we discuss basic, abstract aspects concerning entropy in first and second quantisation. We go slightly beyond what is strictly needed in the second part, with the purpose of clarifying the general picture, that gives motivation for future work.

2.1 Trace and determinants in second quantisation Let be an Hilbert space and Γ( ) (resp. Λ( )), the Bose (resp. Fermi) Fock Hilbert H H H space over H ∞ ∞ Γ( )= Γn( ) , Λ( )= Λn( ) . (5) H H H H Mn=0 Mn=0 If A B( ) and A 1 the second quantisation of Γ(A) (resp. Λ(A)) is the linear ∈ H || || ≤ contraction on Γ( ) (resp. Λ( )) defined by H H 1 A (A A) (A A A) ⊕ ⊕ ⊗ ⊕ ⊗ ⊗ ⊕···

4 where the A A acts on the symmetric part Γk( ) (resp. anti-symmetric part Λk( )) ⊗···⊗ H H of . H⊗···⊗H We recall the following lemma, see [19]. Lemma 2.1. If A is selfadjoint, 0 A< 1, then ≤ Tr Γ(A) = det(1 A)−1, Tr Λ(A) = det(1 + A) , (6) − log Tr Γ(A) =
Tr log(1 A) , log Tr Λ(
A) = Tr log(1 + A) . (7) − −

Proof. We may assume that A has discrete spectrum, otherwise all quantities in (6), (7) are infinite. Suppose first that is one-dimensional, thus A = λ is a scalar 0 λ< 1. In n H ≤ ∞ n the Bose case, Γ ( ) is then one-dimensional for all n, thus we have Γ(A) = n=0 λ , so ∞ H Tr Γ(A) = λn = (1 λ)−1. L n=0 − For a
generalP A, we may decompose = n so that dim n = 1 and A = H n H H λ . Then Γ( ) = {Ωn} Γ( ), where ΩLis the vacuum vector of Γ( ), and n n H n Hn n Hn LΓ(A)= n Γ(An). It followsN that N Tr Γ(A) = Tr Γ(A ) = (1 λ )−1 = det(1 A)−1. n − n −
Yn
Yn In the Fermi case, if is one-dimensional then Λn( )= 0 if n 2 and is one-dimensional H H { } ≥ if n = 0, 1; if A = λ we then have Λ(A) = 1 λ so Tr Λ(A) =1+ λ. Since, also in the ⊕ Fermi case, there is a canonical equivalence between Λ A B
and Λ(A) Λ(B), we have ⊕ ⊗
Tr Λ(A) = Tr Λ((λn) = (1 + λn) = det(1+ A) ,
Yn
Yn where A = n λn. ConcerningL formulas (7), notice that det A = eTr log A, hence by (7) we have

log Tr Γ(A) = log det(1 A)= Tr log(1 A) − − − −

and log Tr Λ(A) = log det(1 + A) = Tr log(1 + A) .



As a consequence Tr(A) < Tr Γ(A) < Tr Λ(A) . (8) ∞⇔ ∞⇔

2.2 Angle between subspaces and type I property Let be a complex Hilbert space and T a (complex) linear operator on . We shall say H H that T p, p (0, ), if there exist orthonormal sequences of vectors e , f of and ∈L ∈ ∞ { k} { k} H a sequence of complex numbers λ with λ p < , i.e. λ ℓp, such that k k | k| ∞ { k}∈ P Tξ = λ (e ,ξ)f , ξ , (9) k k k ∈H Xk

5 p p n thus T iff Tr( T ) < and we have T Tn 0 with Tn k=1(ek, )fk. We ∈ L | | ∞1 || 1− || → ≡ · set as usual T Tr( T p) p = ( ∞ λ p) p . 1-operators are alsoP called trace class || ||p ≡ | | k=0 | k| L operators and 2-operators Hilbert-SchmidtP operators. L One can define p real linear operators between real Hilbert spaces analogously as in L (9). We may view the complex Hilbert space as a real Hilbert with scalar product (ξ, η)R H ≡ (ξ, η), and denote it by R. A complex linear operator T : is also a real linear ℜ H H→H operator T : R R, that we may also denote by TR if we want to emphasise that T is H → H regarded as a real linear operator.

Lemma 2.2. T : is p as a complex linear operator iff TR : R R is p as a H→H L H → H L real linear operator.

Proof. Let T : be p as a complex linear operator. For some orthonormal families H→H L e and f in and a ℓp-sequence λ (we may assume that λ 0) we have { k} { k} H { k} k ≥ Tξ = λ (e ,ξ)f = λ (e ,ξ)f + i λ (e ,ξ)f k k k kℜ k k kℑ k k Xk Xk Xk = λ (e ,ξ)f + λ (ie ,ξ)if , kℜ k k kℜ h k Xk Xk p p thus TR p = 2 T p. || || || || Conversely, assume that TR p. For some orthogonal families e and f in R we ∈L { k} { k} H have Tξ = λ (e ,ξ)f , Tiξ = λ (e , iξ)f = λ (e ,ξ)f , kℜ k k kℜ k k − kℑ k k Xk Xk Xk thus 1+ i Tξ = (1 i)−1(Tξ iT ξ)= λ (e ,ξ)f − − 2 k k k Xk and we have T √2 TR .  || ||p ≤ || ||p Let be a complex Hilbert space and H a standard subspace, i.e. H is a closed, real H ⊂H linear subspace of with H iH = 0 and H + iH dense in . S , ∆ , J denote the H ∩ { } H H H H usual operators associated with H by the modular theory, see [21]. We denote by EH (Z) the spectral projection of ∆ associated with the Borel subset Z R, and by (Z) the F ⊂ HF corresponding spectral subspace. We shall say that H is factorial if H H′ = 0 , with H′ the real orthogonal of iH. Note ∩ { } that H is factorial iff 1 is not an eigenvalue of ∆H . This is equivalent to the Bose (resp. Fermi) second quantisation von Neumann algebra (H) (resp. (H)) to be a factor. R M We shall say that a factorial standard subspace H is of type if (H) is a type I ⊂ H R factor; as we shall see in Corollary 2.9, H is of type I iff (H) is a type I factor. M The following lemma is proved in [12] in the Bose case. Proposition 2.3. H is of type I iff ∆ E (0, 1) is 1, with E (0, 1) the spectral projection H H L H of ∆H associated with the interval (0, 1).

Proof. Immediate by Corollary 2.10. 

6 We denote by P the real linear orthogonal projection from R onto H. Note that H H iP = P i , P ′ = 1 P . (10) iH H H − iH The following relation holds [12]:

1/2 1 ∆H PH = + JH . (11) ∆H + 1 ∆H + 1

To get the above formula, one easily checks that PH ξ = ξ if SH ξ = ξ and that SH PH ξ = PH ξ. −1 Now, taking into account that ∆H′ =∆H = JH ∆H JH , we have

1/2 ∆H ∆H PH′ = + JH . (12) ∆H + 1 ∆H + 1 Thus the angle operator between H and H′ is given by

1/2 ∆H ∆H ′ PH PH = 2 2 + 2 2 JH (13) (∆H + 1) (∆H + 1) ∆H ∆H = 2 2 + 2 2 SH . (14) (∆H + 1) (∆H + 1)

∆H Lemma 2.4. [12] Set AH 4 2 . We have ≡ (∆H +1)

′ PH PH H = AH H . (15)

p p As a consequence, P P ′ is iff A is . H H |H L H L Proof. Formula (15) follows by (14) and the above discussion. p p If AH is as a complex linear operator, then AH is as a real linear operator, hence L p L P P ′ = A is . H H |H H |H L Conversely, if P P is p, then A : H H is p as real linear operator; choose H H |H L H |H → L an orthonormal basis e and f for H w.r.t. the real part of the scalar product and { k} { k} λ ℓp such that A ξ = λ (e ,ξ)f , ξ H. Since A is complex linear { k}∈ H k kℜ k k ∈ H P A (ξ + iη)= A ξ + iA η = λ (e ,ξ)f + i λ (e , η)f ; H H H kℜ k k kℜ k k Xk Xk as H + iH is dense in the Hilbert space, A is thus p as real linear operator, hence as H L complex linear operator. 

In the following, H is a factorial standard subspace of . H Proposition 2.5. [1] H is of type I iff AH is trace class.

−1 1 −1 1 Proof. Since JH ∆H JH = ∆H , we have that ∆H EH (0, 1) is iff ∆H EH (1, ) is , ∆H 1 L ∞ L thus iff 2 is . The proposition then follows by Prop. 2.3.  (∆H +1) L

Corollary 2.6. The following are equivalent:

7 H is of type I, • 2 P P ′ is , • H H L 1 P P ′ is , • H H |H L 1 P P ′ P is , • H H H L [P , i] is 2, where [P , i] P i iP , • H L H ≡ H − H

′ The eigenvalues of PH PH PH and ∆H HH (0,1) have equivalent asymptotic, in particular p 2p | ∆ is iff P P ′ is , p> 0. H |HH (0,1) L H H L 1 1 Proof. H is of type I iff AH is iff PH PH′ H is by Lemma 2.4. On the other hand, 1 L1 | L 2 ∗ P P ′ is iff P P ′ P is , thus iff P P ′ is as P P ′ P = (P ′ P ) P ′ P . H H |H L H H H L H H L H H H H H H H So the equivalence among the first four properties follows. Concerning the last property, it suffices to note that, by (11), we have

1/2 ∆H 1/2 [PH , i] = 2JH = JH AH . ∆H + 1 The last assertion follows by Lemma 2.4. 

2.3 Type I density matrix Let be a complex Hilbert space, Γ( ) the Bose Fock space over and F be a H H H ⊂ H standard subspace. If the von Neumann algebra (F ) on Γ( ) associated with F (i.e. R H generated by the Weyl unitaries W (h) with h F , see [20]) is of type I, the restriction of ∈ the vacuum state ω of B Γ( ) to the type I factor (F ) gives a density matrix ρ (F ) H R F ∈ R with respect to the trace Tr of
(F ) F R ω = Tr (ρ ) , |R(F ) F F · that we aim to analyse. ′ Similarly, in the Fermi case, ρF denotes the density matrix associated with the restriction of ω to (F ), where (F ) is the von Neumann algebra on Λ( ) associated with F (see M M H [13] or Sect. 3.2), assuming it to be a type I factor. We shall consider also the norm one normalisations ρ ρ′ ρˆ F , ρˆ′ F ; F ≡ ρ F ≡ ρ′ || F || || F || ′ ρˆF ′ ρˆF thus ρF Tr (ˆρ ) and ρF ′ . ≡ F F ≡ TrF (ˆρF ) Lemma 2.7. Let be a 2-dimensional complex Hilbert space and F F a standard H ≡ λ ⊂H subspace with sp(∆ )= λ, λ−1 , λ (0, 1). We have: F { } ∈ Bose case: a) (F ) is a factor of type I . R ∞

8 b) sp(ˆρ )= λn,n = 0, 1, 2 . . . − and each λn, n> 1, is an eigenvalue with multiplicity F { } 1, while 0 is not in the point spectrum. c) Tr (ˆρ )= 1 , with Tr the trace of (F ). F F 1−λ F R Fermi case: a) (F ) is a factor of type I (2 2 matrix algebra). M 2 × b) sp(ˆρ′ )= 1, λ . F { } c) Tr (ˆρ′ )=1+ λ, with Tr the trace of (F ). F F F M Proof. (Bose case). is the direct sum of two one-dimensional complex Hilbert spaces, H = , with ∆ξ = λ±1ξ, ξ and J(ξ ξ ) = ξ¯ ξ¯ (with ∆ = ∆ , H H1 ⊕H−1 ∈ H±1 1 ⊕ −1 −1 ⊕ 1 F J = J ). Then ξ ξ H iff Sξ ξ = ξ ξ , thus iff F 1 ⊕ −1 ∈ 1 ⊕ −1 1 ⊕ −1 ξ ξ = λ−1/2ξ¯ λ1/2ξ¯ , 1 ⊕ −1 −1 ⊕ 1 so F = ξ λJξ, ξ . { ⊕ ∈H1} The real linear map T : F , Tξ (1 + λ2)−1/2(ξ λJξ) satisfies H1 → 7→ ⊕ T λitξ =∆itTξ and, by the uniqueness of the canonical commutation relations, we have an isomorphism Φ : ( ) (F ) with Φ W (ξ) = W (Tξ) because R H1 → R
(Tξ,Tη)=(1+ λ2)−1 (ξ λJξ, η λJη)=(1+ λ2)−1( (ξ, η)+ (λJξ, λJη)) ℑ ℑ ⊕ ⊕ ℑ ℑ = (1+ λ2)−1( (ξ, η)+ λ2 (λη, λξ)) = (ξ, η) . ℑ ℑ ℑ Now Ad∆it is the inner automorphism implemented by the unitary ρit where ρ (F ) |R(F ) ∈ R is the positive selfadjoint elementρ ˆF which is unitary equivalent to Γ(λ). The rest is now clear. (Fermi case.) In this case Λ( )= C C is 4-dimensional, so (F ) is isomorphic to H ⊕H⊕ M a 2 2 matrix algebra. Clearly sp(∆ )= 1, λ, λ−1 , so sp(ˆρ′ )= 1, λ and Tr (ˆρ′ )= × M(F ) { } F { } F F 1+ λ. 

Let F be a factorial standard subspace and assume that the spectrum of the modular ⊂H operator ∆ is discrete. Let λ be the eigenvalues of ∆ (with multiplicity). Then F k F |HF (0,1) is the direct sum of the 2-dimensional complex Hilbert spaces with corresponding H Hλk direct sum decomposition F = k Fλk as in Lemma 2.7. Then L ρˆ = ρˆ , F Fλk Ok where the infinite tensor product (w.r.t. the vacuum vectors) is convergent iff (F ) is a R type I factor. In this case

−1 TrF (ˆρF )= TrF (ˆρF )= (1 λk) . (16) λk λk − Yk Yk Similarly, ′ ′ TrF (ˆρ )= TrFλ (ˆρ )= (1 + λk) . (17) F k Fλk Yk Yk

9 Corollary 2.8. sp(∆R(F )) and sp(∆M(F )) are equal to the closure of the multiplicative group generated by sp(∆ ) r 0 . If (F ) (resp. (F )) is of type I, we have sp(ˆρ ) F { } R M F ⊂ sp(∆ ) [0, 1] (resp. sp(ˆρ′ ) sp(∆ ) [0, 1]. R(F ) ∩ F ⊂ M(F ) ∩ Proof. Immediate by the above discussion. 

Corollary 2.9. (F ) is a type I factor iff (F ) is a type I factor and this is the case iff R M ∆ E (0, 1) is 1. F F L Proof. By (16), (17), we have

Tr (ˆρ ) < λ < Tr (ˆρ′ ) < F F ∞ ⇐⇒ k ∞ ⇐⇒ F F ∞ Xk  and the corollary follows because Tr ∆F EF (0, 1) = k λk.
P We make explicit the following direct consequence.

Corollary 2.10. Let F standard subspace with (F ) a type I factor. If we identify ⊂ H R (F ) with B( ) with a Hilbert space, then ρˆ is unitarily equivalent to Γ(∆ ) on R K K F F |HF (0,1) Γ( (0, 1)). Similarly, ρˆ′ is unitarily equivalent to Λ(∆ ). HF F F |HF (0,1)

Proof. As in Lemma 2.7, Γ(∆F HF (0,1))= k Γ(λk) which is unitarily equivalent toρ ˆF = |  k ρˆFk . The Fermi case is analogous. N N

2.4 Second quantisation entropy With ρ a positive, non-singular selfadjoint linear operator, the von Neumann entropy of ρ is defined by S(ρ) Tr(ρ log ρ) . (18) ≡− Here, ρ is not assumed to have trace one, note however that for λ> 0

S(ρ) < S(λρ) < . ∞ ⇐⇒ ∞ The following proposition (Fermi case) is equivalent to [8, Lemma 3.3].

Proposition 2.11. Let A be operator, 0 A< 1. The von Neumann entropy of the Bose ≤ and Fermi second quantisation of A is given respectively by

A A S Γ(A) = Tr log A Tr Γ(A) = Tr log A det(1 A)−1 , (19) − 1 A  − 1 A  −
−
− A A S Λ(A) = Tr log A Tr Λ(A) = Tr log A det(1 + A) . (20) − 1+ A  − 1+ A 

10 Proof. We use the formula d S(ρ)= Tr(ρα) , − dα α=1 hence we may compute S(ρ) by the logarithmic derivative

d S(ρ)= log Tr(ρα) Tr(ρ) . − dα α=1

Since Γ(Aα)=Γ(A)α, we have

d d S Γ(A) = log Tr Γ(Aα) Tr Γ(A) = Tr log(1 Aα) Tr Γ(A) − dα α=1 dα − α=1




d Aα = Tr log 1 Aα) Tr Γ(A) = Tr log A Tr Γ(A) dα − α=1 − 1 Aα α=1  
 
A − = Tr log A Tr Γ(A) ; − 1 A  − 
the second equation in (19) follows by (6). Similarly,

d d S Λ(A) = log Tr Λ(Aα) Tr Λ(A) = Tr(log(1 + Aα)) Tr Λ(A) − dα α=1 − dα α=1



d Aα = Tr log 1+ Aα) Tr Λ(A) = Tr log A Tr Λ(A) − dα α=1 − 1+ Aα α=1  
 
A = Tr log A Tr Λ(A) . − 1+ A  
The second equations in (19), (20) follow by (6). 

With A as above, note that Tr(A) is finite iff det(1 + A) is finite and iff det(1 A)−1 is − finite. So we have:

Corollary 2.12. S(A) is finite iff S Γ(A) is finite and iff S Λ(A) is finite.

Proof. Clearly, S(A) < Tr(A) < . ∞⇒ ∞ Then by (19) we have

A S Γ(A) < Tr log A < S(A) < . ∞⇒− 1 A  ∞⇔ ∞
− The converse implication S(A) < S Γ(A) < follows again by (19) and by (8). ∞⇒ ∞ The equivalence S(A) < S Λ(A) <
is analogously obtained by (19) and (8). ∞ ⇔ ∞


11 Let F be a factorial, type I standard subspace of . We define the entropy S(F ) of ⊂ H H F as S(F )= S(PF PF ′ PF ) (21) and the Bose (resp. Fermi) entanglement entropy of F to be the von Neumann entropy ′ S(ρF ) (resp. S(ρF )). As above, ρF is the density matrix of the restriction of the vacuum state to (F ) (resp. (F )). We have R M ′ ′ S(ρF )= S(ρF ′ ) , S(ρF )= S(ρF ′ ) . Corollary 2.13. S(ρ ) < S(ρ′ ) < S(F ) < . F ∞ ⇐⇒ F ∞ ⇐⇒ ∞

Proof. By Corollary 2.10,ρ ˆF is unitarily equivalent to Γ(PF PF ′ PF ) (after an identification ′ of (F ) with B( )); and an analogous unitary equivalence ofρ ˆ with Λ(P P ′ P ) holds. R K F F F F So the statement follows by Cor. 2.6 and Cor. 2.12. 

2.5 Split inclusions Recall that an inclusion of von Neumann algebras is said to be split if there exists N1 ⊂N2 an intermediate type I factor , so [10], see this reference for more on these F N1 ⊂F⊂N2 inclusions. In this section we begin to study split inclusions of standard subspaces. Let be a Hilbert space. With K a closed, real linear subspace, we denote as H ⊂ H above by P the real orthogonal projection onto K. Let K H be an inclusion of standard K ⊂ subspaces of . We shall say that K H is split if the corresponding inclusion of von H ⊂ Neumann algebras (K) (H) on the Bose Fock space is split. R ⊂ R Proposition 2.14. Let K H be an inclusion of standard factorial subspaces of with ⊂ H K′ H standard. The following are equivalent:’ ∩ (i) K H is split, ⊂ (ii) There exists an intermediate type I standard space F : K F H, ⊂ ⊂ (iii) F K + J ′ K = H J ′ H is a canonical type I intermediate subspace. ≡ K ∩H ∩ K ∩H Proof. Clearly (iii) (ii) (i) and we have to show that (i) (iii). Assuming then ⇒ ⇒ ⇒ (i), (K) (H) is split by definition. The vacuum vector Ω is cyclic for the relative R ⊂ R commutant (K)′ R(H)= (K′) (H) = (K′ H) because K′ H is standard by R ∩ R ∩ R R ∩ ∩ assumption. By [10], the von Neumann algebra

= (K) J (K)J = (H) J (H)J (22) F R ∨ R R ∩ R is a canonical intermediate type I factor. Here J is the modular conjugation of (K′) (H) R ∩R w.r.t. Ω. Now, J = Γ(JK′∩H ), the second quantisation of the modular conjugation JK′∩H of K′ H. Thus ∩ = (K) J (K)J = R(K) (J ′ K)= (F ) , (23) F R ∨ R ∨ R K ∩H R

12 where F = K + JK′∩H K is an intermediate type I factor between K and H, and F if of type I because (F )= is of type I.  R F

Let K H be a split inclusion of standard subspaces of with K′ H standard. By Prop. ⊂ H ∩ 2.14, there is a canonical intermediate type I factor (F ) between (K) and (H), with R R R F given by (23). This is indeed the canonical intermediate type I factor for (K) (H) R ⊂ R associated with the vacuum vector. By Corollary 2.9, the von Neumann algebra (F ) in the Fermi quantisation of F is also M a canonical intermediate type I factor between (K) and (H), that we shall study in M M Section 3, where the standard subspaces are associated with intervals of S1 (free Fermi net in first quantisation). This is a of canonical intermediate type I factor for (K) (H) M ⊂ M associated with the vacuum vector, here constructed by a graded version (50) of formula (22), where the relative commutant is replaced by the graded relative commutant. Before ending this section, we note the following proposition.

Proposition 2.15. Let K H be an inclusion of standard factorial subspaces of with ′ ⊂ 2 H K H standard. If K H is split then P P ′ and P JP ′ are , with J = J ′ . ∩ ⊂ K H K H L K ∩H

Proof. Let F be an intermediate type I subspace. By Cor. 2.6 PF PF ′ PF is trace class. As PH′ < PF ′ , we have PF PH′ PF < PF PF ′ PF , so PF PH′ PF is trace class. By multiplying the latter by PK from the left and from the right we see that PKPH′ PK is trace class. 2 Equivalently, P ′ P is . H K L With F is the canonical intermediate type I subspace, we have JK F , so JK H 2 ⊂ ⊂ is split. By the above argument, PJK PH′ is thus . As PJK = JPKJ, we then have that 2 L JP JP ′ , hence P JP ′ , is .  K H K H L

Let K H be an inclusion of standard factorial subspaces of with K′ H standard. ⊂ H p ∩ p By the above proposition, and its proof, we might expect that P P ′ is iff P P ′ is , K H L F F L where F is the canonical intermediate type I subspace. A relation of this kind would be useful for entropy computations.

3 Entanglement entropy for conformal nets

In this second part, we are dealing with QFT nets of standard subspaces and of von Neumann algebras on S1 and we state our definitions in this setting, although they directly extends to the case of net on higher dimensional spacetimes.

3.1 Free Quantum Field case Let H : I S1 H(I) be a SL(2, R)-covariant net of standard subspaces of a Hilbert ⊂ 7→ space on the interval of S1. We assume that H is local or twisted-local. H Given intervals I I of S1, with I¯ contained in the interior of I, and a factorial, type ⊂ I standard subspace F with H(I) F H(I), we set e ⊂ ⊂ e e SH (I, I; F ) := S(F ) , e

13 with S(F ) the von Neumann entropy of F defined in (21). As seen in Sect. 2.5, there is a canonical choice for F . We set

SH (I, I)= SH (I, I; F ) , with F the canonical intermediate type eI factorial standarde subspace. Analogously, let be SL(2, R)-covariant net of von Neumann algebras on S1. We A assume that is either local or twisted-local and the split property holds. A Let be an intermediate type I factor between (I) and (I), and ρ be the F A A F ∈ F density matrix (with trace one) associated with the restriction of the vacuum state to . e F We set S (I, I; ) := S(ρ ) . (24) A F F Now, consider the net of von Neumann algebrase (I) associated with the Bose (resp. Fermi) A second quantisation of H, and set

S (I, I) := S (I, I; ) , A A F with the type I factor associated withe the Bose (resp.e Fermi) second quantisation of the F canonical intermediate type I standard subspace F . Then, by Corollary 2.13, we have

S (I, I) < S (I, I) < . (25) A ∞ ⇐⇒ H ∞ We may also consider the lower entropye e

SA(I, I) := inf SA(I, I; ) , F F e e where the infimum is taken over all intermediate type I factor or, more generally, over F all intermediate, discrete type I von Neumann algebras (countable direct sum of type I factors). Of course S (I, I) S (I, I) . A ≤ A In some case, it may be easier to get a bounde for SA(eI, I) rather than for SA(I, I), cf. Sect. 3.5. e e 3.2 Free Fermi nets We refer the reader to Section 3 of Chapter I and Section 13 of Chapter II of [31] for more details. Let be a complex Hilbert space and K a standard subspace of . If ξ , we H ⊂H H ∈ H denote by a(ξ) the associated exterior multiplication operator on the Fermi Fock space over (5) (creation operator) and c(ξ)= a(ξ)+ a∗(ξ) the Clifford multiplication. H If T is a bonded linear operator on with T 1, we denote by Λ(T ), as in Sect. H || || ≤ 2.1, the bounded linear operator on Λ( ) such that Λ(T ) n = T T T , the n-fold H |Λ (H) ∧ ···∧ tensor product of T on Λk( ). Note that Λ(T ) 1. H || || ≤ The Klein transform k : Λ( ) Λ( ) is the linear operator such that k = 1 on the H → H even part and i on the odd part of Λ( ). H

14 Let (K) be the von Neumann algebra generated by c(ξ),ξ and J , ∆ the M { ∈ H} K K modular conjugation and operator of (K) with respect to the vacuum vector Ω. Then M −1 it it JK = k Λ(ijK ) , ∆K = Λ(δK ) , where jK and δK are the modular conjugation and operator of the standard subspace K, see [1] or [31] (in this section we use use small letters to denote modular operators and conjugations on the one-particle Hilbert space, and capital letter on the fermi Fock Hilbert space). We have (K)′ = J (K)J = k−1 (K⊥)k , M K M K M with K⊥ the orthogonal of K with respect to the real part of the scalar product; (K⊥) M is the graded commutant of (K). Note that the formulas for the orthogonal projections: M ′ ⊥ j K = K = (iK) , P ′ = 1 P =1+ iP i = iP ⊥ i . K K − iK K − K It follows that P = iP ⊥ i = i(1 P )i = P ⊥ . ijK K − (iK) − − iK K We shall denote ˆjK = ijK . Then P = P ⊥ . (26) ˆjK K K From now on we specialise to = L2(S1; C), with the complex structure on given by H H i(2P 1), with P is the projection onto the Hardy space. − The net of standard subspaces on S1 is given by I L2(I) L2(I; C) . 7→ ≡ 2 If K = L (I) with I the upper semicircle, then ˆjK is given by Theorem in Section 14 of [31]: 1 ˆj (f)= f(z−1) . (27) K z

3.2.1 Explicit formula in a special case On S1, we consider the following four “symmetric intervals” π π I = eiθ : 0 <θ< , I = eiθ : <θ< 0 , (28) 1 2 2 − 2  π  π I = e−iθ : 0 <θ< , I = e−iθ : <θ< 0 . (29) − 1 2 − 2 − 2   We note that I2 = ( I )2 = I = eiθ : 0 θ π , where I2 = z2 : z I . We denote by 1 − 1 { ≤ ≤ } 1 { ∈ 1} χ ,χ ,χ ,χ the characteristic functions of I ,I , I , I respectively. 1 2 −1 −2 1 2 − 1 − 2 Now let K = L2(I ) L2( I ). Our goal is to have an explicit formula for ˆj : this is 1 ⊕ − 1 K based on the formula for the modular operator for disjoint intervals in the free fermion case in [22], [29] and [7]. We will start with the following linear isomorphism from to which is inspired H⊕H H by Prop. 3 in [29]

2 2 β (φ1(z), φ2(z)) = ψ(z)= φ1(z )+ zφ2(z )  2 1
φ1(z )= 2 ψ(z)+ ψ( z) (30)  − φ (z2)= 1 ψ(z) ψ( z
) 2 2z − − 
15 where (φ , φ ) , ψ is in = L2(S1; C). In terms of basis one can check from the 1 2 ∈H⊕H H above definition that β(zn, 0) = z2n, β(0, zn)= z2n+1 for all integers n. Note that βP P = P β where P is the projection onto the Hardy space. ⊕ The inverse of β is given by

−1 β ψ(z) = φ1(z), φ2(z) . (31)

2 Now, for the free fermion net associated with φ1, if K = L (I) with I the upper semicircle, ˆ 1 −1 we have jK (f)= z f(z ) by (27). Using (30) and (31), we can transform the diagonal action of ˆjK on to an action −1 H⊕H of j on ψ(z) such that j = βˆjK β . We arrive at the following

1 1 1 1 (jf)(z)= + f(z−1) f( z−1) . (32) 2 2z2  − 2 − 2z2  −

Note that j maps L2(I ) to L2(I I ). We will denote by M the multiplication operator 1 2 ∪− 2 I by χI . 2 2 By definition, MI1 and jMI1 j are orthogonal projections whose ranges L (I1) and jL (I1) are also orthogonal. Hence P12 := MI1 + jMI1 j is a projection. Note that L2(I ) jL2(I ) is the canonical type I standard subspace which is interme- 1 ⊕ 1 diate between L2(I ) and L2(I I I ). 1 1 ∪ 2 ∪− 2 Theorem 3.1. 2 1 P12 = Mg + MhR on L (S ) , where 1 1 g(z)= (z2 + 1)2χ (z2 1)2χ + χ , (33) 4z2 2 − 4z2 − −2 1 1 h(z)= (z4 1)(χ χ ) , (34) 4z2 − 2 − −2 R(f)(z)= f( z) . (35) − Proof. By (32) we have

1 2 2 1 2 2 1 4 jMI1 jf = 2 (z + 1) χ2 2 (z 1) χ−2 f(z)+ 2 (z 1)χ2 χ−2 f( z) (36) 4z − 4z −  4z − −  − and the theorem follows. 

It is now useful to examine the Fourier modes of g and h.

n n Proposition 3.2. Consider the Fourier series g(z)= n gnz and h(z)= n hnz where −3 −2 g, h are as in Th. 3.1. Then gn = O(n ) and hn =PO(n ). P | | | |

16 n Proof. Consider the Fourier expansion χI1 (z)= n γnz . Then

π P 2 1 −inθ 1 i −inπ/2 γn = e dθ = (e 1) . 2π Z0 2π n − Thus 1 1 1 1 1 1 g(z)= γ + γ + γ γ γ + γ ( 1)p + γ zp , 4 2−p 2 −p 4 −2−p − 4 2−p − 2 −p 4 −2−p − p Xp 

 so g = 0 when n is even as γ = γ if n 0 is even. n n − −n ≥ When n is odd 1 g = (γ + γ )+ γ (37) n 2 2−n −2−n n i −iπn 1 1 1 1 i −iπn 4 = (e 2 1) + = (e 2 1) − = O(n−3) . (38) 2π − n − 2 n 2 n + 2 2π − n(n2 4)  −
 − As h(z)= ( γ + γ )zn, we have n − −2−n 2−n P h = γ + γ = O(n−2) . n − −2−n 2−n 

3.2.2 General symmetric interval case On S1, we consider the following general four “symmetric intervals”

I = eiθ : 0 <θ<φ , I = eiθ : φ π<θ< 0 , (39) 1 { } 2 { − } I = e−iθ : 0 <θ<φ , I = e−iθ : φ π<θ< 0 , 0 <φ<π. (40) − 1 { } − 2 { − } We note that all results in this section simplify in the previous section when φ = π/2. Denote by I := eiθ : 0 <θ< 2φ . We shall consider the action of SU(1, 1) on S1 0 { } which is given by z az+b with a 2 b 2 = 1. The M¨obius group Mob is the subgroup → ¯bz+¯a | | − | | ± of SU(1, 1) of elements with determinant a 2 b 2 = 1. The action z 1 is orientation | | − | | → z reversing. az+b 1 If m(z)= ¯bz+¯a , the unitary action of m on S is given by (See Section 4 of [31])

(U f)(z) = (α βz¯ )−1f(m−1z) . (41) m − Since (a ¯bz)−1 is holomorphic for z < 1 and a > b , U commutes with the Hardy − | | | | | | m space projection P . The flip map (F f)(z) = 1 f( 1 ) clearly satisfies P F P = 1 P . By 1 z z 1 − combining this we get an action of SU(1, 1) on which is of the form H −1 (Umf)(z)= αm(z)f(m z) , (42) where m(z)= az+b , α (z) = (α βz¯ )−1. ¯bz+¯a m −

17 Let m Mob be such that mI is the upper half circle. Let m = m−1F m. It is ∈ 0 1 1 straightforward to see that

z(1 + e2iφ)/2 e2iφ m (eiφ)= − . (43) 1 z (1 + e2iφ)/2 − Define −1 (F0f)(z)= αm1 (z)f(m1 z) . (44) Using (30) and (31), we can transform the diagonal action of F on to an action j 0 H⊕H on such that j = βF β−1. H 0 We then arrive at the following expression 1 z 1 z (jf)(z)= α (z2) + f(u)+ α (z2) f( u) , (45) m1 2 2u m1 2 − 2u −

2 2 where u = m1(z ). Note that j maps L2(I ) to L2(I I ). We will denote by M the multiplication 1 2 ∪− 2 I operator by χI , the characteristic function of interval I. 2 2 By definition, MI1 and jMI1 j are orthogonal projections whose ranges L (I1) and jL (I1) are also orthogonal. Hence P12 := MI1 + jMI1 j is a projection. Note that L2(I ) jL2(I ) is the canonical type I standard subspace which is interme- 1 ⊕ 1 diate between L2(I ) and L2(I I I ). 1 1 ∪ 2 ∪− 2 Theorem 3.3. 2 1 P12 = Mg + MhR on L (S ) , where (u + z)2 (u z)2 g(z)= χ (u) − χ ( u)+ χ (z) , (46) 4uz I1 − 4uz I1 − I1 z2 u2 h(z)= − (χ (u) χ ( u)) , (47) 4uz I1 − I1 − (R(f)(z)= f( z), u2 = m (z2) . (48) − 1 Proof. This follows directly from (45). 

To get a generalisation of Prop. 3.2 we need the following lemma: Lemma 3.4. Let f(z) be a continuous function, where z = eiθ and the support of f lies in θ [a, b] [0, 2π]. Suppose that f ′′ exists except for finitely many points in [a, b], and ∈ ⊂ f ′′(z) M < . Suppose that f = f zn, z = eiθ. Then f = O(n−2). | |≤ ∞ n n n P Proof. First suppose that f ′′ exists except at a, b. We have

1 2π 1 b−ǫ f = f(z)z−n−1dz = lim f(z)z−n−1dz . n + 2π Z0 ǫ→0 2π Za+ǫ By doing integration by parts once and use the continuity of f at a, b we get

1 1 b−ǫ f = lim f ′(z)z−n−2dz . n + n + 2 2π ǫ→0 Za+ǫ

18 −2 Integration by part one more time and use our assumption we conclude that fn = O(n ). ′′ In general if f does not exist for z1, ..., zk, we can divide the interval [a, b] into a union of finitely many intervals where f ′′ exists on each interval except at end points, and apply the argument above. 

We note that continuity assumption in the above Lemma is crucial. For example χI1 verifies the assumption of the above Lemma except continuity, and the conclusion of the Lemma does not hold for χI1 .

n n Proposition 3.5. Consider the Fourier series g(z)= n gnz and h(z)= n hnz where −2 −2 g, h are as in Th. 3.3. Then gn = O(n ) and hn =PO(n ). P | | | | Proof. It is sufficient to check that g and h verify Lemma 3.4. Let us check this for g. First let us check g(z) is continuous. It suffices to do this at the end points of intervals I ,I , I . 1 2 − 2 At these endpoints by construction u = z, and it is easy to see that g is continuous by ± using the fact that g is invariant under u u. For an example let us show that the right →− limit and left limit of g are equal to each other at eiφ. Let z = eiθ and suppose that θ φ−. → Note that u I or u I , hence g(z) = 1. When θ φ+, since g is invariant under ∈ 2 ∈ − 2 → u u, we can assume that u I and u eiφ. By the formula for g(z) it follows that → − ∈ 1 → limθ→φ+ g(z) = 1. Similarly one can show continuity of g at all other end points. Let us show that g′′ exists and is bounded except possibly at the end points of intervals I1,I2, I2. Fix such z which are not the end points of intervals I1,I2, I2. − 2 2 2 − Note that u = m1(z ), we can choose u = m1(z ) (cf. equation (43) for an explicit formula of m1(z)) that is smooth in a neighbourhoodp of z. g is independent of such choice. ′′ It is also clear that g exists and is bounded on such points, since m1(z) is a smooth rational function on the unit circle and m (z) = 1. Similarly h verify Lemma 3.4.  | 1 |

3.3 Hankel operator Let K = L2(I ), K = L2(I I I ), F = L2(I )+jL2(I ) as in the beginning of section 1 1 2 1 ∪ 2 ∪ −2 1 1 3.2.2. We shall prove that (1 P )P P q for 2

Proof. We note that

P P ′ P = P ((2P 1)(1 P )(2P 1)P F F F F − − F − F = 4[P , P ](1 P )[P, P ] F − F F

19 and [P , P ] = (1 P )P P P P (1 P ) = (1 P )P P (1 P )P P ∗. F − F − F − − F − − F q q
Hence, if (1 P )P P , then P P ′ P . − F ∈L F F F ∈L Let λn be the singular values of PF PF ′ PF ; then S(PF PF ′ PF )= n λn log λn. Since q { } q − λn < , and x log x

(1 P )P P = (1 P )P P = (1 P )M P + (1 P )M RP . − F − 12 − g − h Note that RP = PR = (1 P )M P + (1 P )M RP . − g − h In terms of the basis zn of L2(S1), we have

P M (1 P )(zn)= g zk , g − k−n Xk≥0 these are Hankel operators (cf. [28]). n −α 3 q Theorem 3.7. Suppose f = n fnz , fn = O(n ) with α > 2 . Then P Mf (1 P ) 1 − ∈L with 1 >q> 1 . P α− 2

k Proof. (First proof.) As in [18], let ξn = k≥0 fk+nz , n> 0, P P M (1 P )(zn)= f zk = ξ , n< 0 . f − k−n −n Xk≥0 It follows that P M (1 P )= ( , zn)ξ = ( , z−n)ξ . f − · −n · n nX0 q q q By [28, Appendix 1], we have T + R q T q + R q for 0

1 1 Note that ξ = f 2 2 = O(n−α+ 2 ), so the theorem follows. || n|| k≥0 | k+n|
q 1 (Second proof.)P According to Page 243 of [28] P Mf (1 P ) if the function f (z)= n − ∈L n≥0 fn+1z is in the Besoz space Bq, i.e. P (k) 1 z 2 kf 1 (z) Lp(D, dA/(1 z 2) , − | | ∈ − | |
for any integer pk > 1. Here dA is the Lebesgue measure on the disk D. Let us show that if f = O(n−α), then f 1 B . n ∈ q Let k = k1 + k2 with qk1 > 1, 0

(k) 1 (1 z 2)k1q (1 z 2)k2 f 1 (z) q dA ZZD − | | − | | | (1 z 2)2
− | | 1 1 2 k1q 2 (1 z ) (k) 2 2 − | | (1 z 2)k2 f 1 (z) dA (1 z 2)k1q−2dA . ≤ ZZD (1 z 2)2 − | | | |  ZZD − | |  − | |

20 It is sufficient to check that

(k) (1 z 2)k1q−2 (1 z 2)2k2 f 1 (z) 2dA < . ZZD − | | − | | | | ∞
We have

(k) (1 z 2)k1q−2 (1 z 2)2k2 f 1 (z) 2dA ZZD − | | − | | | |
2 k1q−2+2k2 2k−2α 2n c1 (1 z ) n z dA ≤ ZZD − | | | | nX≥1

2k−2α 2 k1q−2+2k2 2 k1q−2+2k2 c2 n (1 z ) dA (1 z ) dA . ≤ · ZZD − | | ZZD − | | nX≥1 Using polar coordinates, we have

1 (1 z 2)k1q−2+2k2 z 2ndA = 2π (1 r2)k1q−2+2k2 r2nrdr ZZD − | | | | Z0 − 2π 1 = (1 r2)k1q−2+2k2 rndr 2 Z0 − Γ(k q + 2k 1)Γ(n + 1) = π 1 2 − . Γ(k q + 2k 1+ n 1) 1 2 − − By using Sterling’s formula Γ(n + 1) √2πn n n, we have ∼ e
Γ(k q + 2k 1)Γ(n + 1) 1 2 − c n1−k1q−2k2 Γ(k q + 2k 1+ n 1) ≤ 3 1 2 − − when n . →∞ It follows that

2 k1q−2 2 2k2 1(k) 2 2k−2α+1−k1q−2k2 (1 z ) (1 z ) f (z) dA c4 n < ZZD − | | − | | | | ≤ ∞
nnX≥1 if k q + 2k 1 + 2α 2k > 1. Note that k q > 1. − 1 2 − − 1 These two inequalities imply that 1 2α 2 < k < − , q 1 2 q − 2α−2 1 1  which is possible iff 2−q q > 0 iff q > 1 . − α− 2

3.4 Finiteness of von Neumann entropy for free Fermi nets Let K = L2(I ), K = L2(I I I ), F = L2(I )+ jL2(I ) as in the beginning of 1 1 2 1 ∪ 2 ∪− 2 1 1 section 3.2.2. Note that we have the inclusion

(K ) (F ) (K ) . M 1 ⊂ M ⊂ M 2

21 Theorem 3.8. (1) (1 P )P P q for 2

Remark 3.9. (1) It is an interesting question to calculate the entropy in Th. 3.8. Note that 1 1 by Th. 3.18 of [23] that the entropy in Th. 3.8 is bounded from below by 6 ln cos(φ/2) . We expect that the theory of Hankel operators as in [28] will be useful in solving this question. For heuristic computations using replica methods in some holographic models, see [11]. (2) Since any open intervals I, I with I¯ I can be mapped to the symmetric intervals ⊂ in the above theorem by elements in Mob, it follows that the above theorem is also true for 2 2 e e K1 = L (I), K2 = L (I). e 3.4.1 Real fermion case

Define conjugate linear operator Q = Mz−1 C on where C is complex conjugation and −1 H−1 n M −1 is multiplication by z . We have (Qf)(z)= z f(z). On basis z of we have z H Q(zn)= z−n−1 . (49)

2 Lemma 3.10. (1) Q = I and Q commutes with Um as defined in equation (41) and flip operator F1 as defined after equation (41); (2) QPQ = 1 P,Qi(2P I)Q = i(2P I); − − − (3) QMf Q = Mf and Q commutes with P12 as in Th. 3.8. Proof. (1) follows from definitions. For (2), it is enough to check that QPQ = 1 P − since Q is conjugate linear. One checks easily that QPQ = 1 P on zn from equation − (49). To prove (3), note that g = g, h = h where g, h are as in Th. 3.8. (3) follows from − CM C = M , CM C = M ,CR = RC, RM −1 R = M −1 .  g g h − h z − z

Denote by Q := 1 (I Q) the projections from to the eigenspaces of Q with ±1 2 ± H H±1 eigenvalues 1. Note that = while by slightly abuse of notations in this ± H H+1 H−1 section only we use to mean orthogonalL with respect to the real part of inner product on . Note that M−i L+1 = −1 since Q is conjugate linear. H H H For an interval I we let L2(I) = L2(I) . The real free fermion net associates to ±1 ∩H±1 interval I the von neumann algebra (L2(I) ) generated by a(f)+ a(f)∗, f L2(I) . M +1 ∀ ∈ +1 We note that (L2(I)) is the graded tensor product of (L2(I) ) and (L2(I) ). M M +1 M −1 Let K = L2(I ), K = L2(I I I ), F = L2(I )+ jL2(I ) as in the beginning 1 1 2 1 ∪ 2 ∪− 2 1 1 of section 3.2.2. Note by Lemma 3.10 K , K , F are Q-invariant. Let (K ) := K 1 2 1 ±1 1 ∩ , (K ) := K , F := F . Then we have ((K ) ) (F ) H±1 2 ±1 2 ∩ H±1 ±1 ∩ H±1 M 1 ±1 ⊂ M ±1 ⊂ ((K ) ), where (F ) is the canonical type I factor. Note that we have S(F ) = M 2 ±1 M ±1 S(F+1)+ S(F−1). Since (F−1) = Λ( i) (F−1)Λ(i) and the vacuum is preserved by M −1 M Λ(i), it follows that S(F+1)= S(F−1)= 2 S(F ). We have therefore proved the following: Theorem 3.11. We have ((K ) ) (F ) ((K ) ), and S(F )= S(F )= M 1 ±1 ⊂ M ±1 ⊂ M 2 ±1 +1 −1 1 S(F ) < where S(F ) is the von Neumann entropy in Th. 3.8. 2 ∞

22 3.5 Passing to free bosons Note first that

2 2 −1 (F )= L (I ) kJ c L (I ) J c k , (50) M M 1 ∨ M(K1) ∩M(K2)M 1 M(K1) ∩M(K2)

where (K )c denotes the graded commutant of (K ), and k is the Klein transformation. M 1 M 1 Let g be an automorphism of the net (L2(I)) implemented by a unitary U(g) on Λ( ) M H which commutes with grading operator. This means U(g)Ω = Ωand U(g) (L2(I))U(g)∗ = M (L2(I)) and U(g) commutes with Γ = k2 with k the Klein transformation. It is clear that M such adjoint action of U(g) preserves (F ) as a set. M Example of such automorphisms are Γ = k2 with k the Klein transformation and more generally the automorphisms implemented by the multiplication by λ, λ = 1, on L2(I). | | This is the action of U(1) on free fermion net. In fact one can also check from Th. 3.3 that the automorphisms implemented by this U(1) action preserves (F ) as a set. M In the notation of [33], the free fermion net is written as . The free bosonic net AU(1)1 with central charge c = 1 is the U(1) the fixed point net of under the action of U(1) as AU(1)1 described above (cf. Page 186 of [33]). Denote by G := U(1). Then U(g) U(g)∗ = (F ), MF M g G. ∈ We have (K )G (F )G (K )G , M 1 ⊂ M ⊂ M 2 where (F )G is the fixed point of (F ) under the action of G. As (F ) is a type I M M M factor and G is compact, (F )G is a type I von Neumann algebra, but (F )G is not a M M factor, rather (F )G is the direct sum of countably many type I factors. M Since the vacuum state ω = (Ω, Ω) is preserved by the action of G, by Proposition 6.7 · of [24] we have S(ω G ) S(ω ) < , |M(F ) ≤ |M(F ) ∞ where last inequality follows by Theorem 3.8. By the remark (2) after Theorem 3.8 we have therefore proved the following: Theorem 3.12. Let I, I be open intervals with I¯ I. For the free bosonic net G we ⊂ AU(1)1 have that e e G (I) (F )G G (I) , AU(1)1 ⊂ M ⊂AU(1)1 where (F )G is a type I discrete, von Neumann algebrae and the vacuum von Neumann M entropy S ω G < . |M(F ) ∞
Remark 3.13. By taking the graded tensor product of r-copies of the net G , Theorems AU(1)1 3.8 and 3.12 immediately generalise to the case of the conformal net of r free fermions.

4 Conclusion

Let OR denote the double cone in the Minkowski spacetime that is the causal envelop of a time zero ball of radius R> 0 centered at the origin. We have shown a natural way to define rigorously the entanglement entropy SA(OR′ , OR) of a QFT net of von Neumann algebras ′ associated with an inclusion O ′ O , R < R. We have shown that S (O ′ , O ) is A R ⊂ R A R R finite, in particular in low dimensional free models.

23 It will be natural to establish that this entropy is finite, also for higher dimensional models, in a model independent ground. We expect that SA(OR′ , OR) is finite if the modular nuclearity condition holds, see [15]. ′ As R R, the entropy S (O ′ , O ) should diverge. However, the divergence leading → A R R term should grow proportionally to the area of the sphere of radius R, according to a seemingly universal feature (area theorems). Results in this direction would put several fundamental issues in theoretical physics on a rigorous mathematical ground.

Acknowledgements. Our collaboration took place, in particular, during the CERN work- shop on “Advances in Quantum Field Theory” in March-April 2019 and the program “Op- erator Algebras and Quantum Physics” at the Simons Center for Geometry and Physics at Stony Brook in June 2019. We are grateful to both institutions for the invitations. R.L. acknowledges the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006.

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