aa r X i v : . [ m a t h - ph ] F e b Comment on the Bekenstein bound
Roberto Longo ∗ Dipartimento di Matematica, Universit`a di Roma Tor VergataVia della Ricerca Scientifica, 1 - 00133 Roma, ItalyEmail [email protected]
Feng Xu
Department of Mathematics, University of California at RiversideRiverside, CA 92521E-mail: [email protected]
Dedicated to Alain Connes on the occasion of his 70th birthday
Abstract
We propose a rigorous derivation of the Bekenstein upper limit for the entropy/infor-mation that can be contained by a physical system in a given finite region of space withgiven finite energy. The starting point is the observation that the derivation of such abound provided by Casini [6] is similar to the description of the black hole incrementalfree energy that had been given in [23]. The approach here is different but close in thespirit to [6]. Our bound is obtained by operator algebraic methods, in particular Connes’bimodules, Tomita-Takesaki modular theory and Jones’ index are essential ingredientsinasmuch as the von Neumann algebras in question are typically of type
III . We relyon the general mathematical framework, recently set up in [26], concerning quantuminformation of infinite systems. ∗ Supported in part by the ERC Advanced Grant 669240 QUEST “Quantum Algebraic Structures andModels”, MIUR FARE R16X5RB55W QUEST-NET, GNAMPA-INdAM and Alexander von HumboldtFoundation. Introduction
The Bekenstein bound is an universal limit on the entropy that can be contained in a physicalsystem of given size and total energy. If a system of total energy E , including rest mass, isenclosed in a sphere of radius R , then the entropy S of the system is bounded by S ≤ λRE , where λ > λ = 2 π is often proposed).In [6], H. Casini gave an interesting derivation for this bound, based on relative entropyconsiderations. It was observed, following [30], that, in order to get a finite measure for theentropy carried by the system in a region of the space, one should subtract from the bareentropy of the local state the entropy corresponding to the vacuum fluctuations, which isentirely due to the localisation. A similar subtraction can be done to define a localised formof energy.The argument in [6] is following. One considers a space region V and the von Neumannalgebra A ( O ) of the observables localised in the causal envelop O of V . The restriction ρ V of a global state ρ to A ( O ) has formally an entropy given by von Neumann’s entropy S ( ρ V ) = − Tr( ρ V log ρ V ) , that is known to be infinite. So one subtracts the vacuum state entropy S V = S ( ρ V ) − S ( ρ V )with ρ V the density matrix of the restriction of the vacuum state ρ to A ( O ).Similarly, if K is the Hamiltonian for V , the one considers the difference of the expec-tations of K in the given state and in the vacuum state K V = Tr( ρ V K ) − Tr( ρ V K ) . (1)The version of the Bekenstein bound in [6] is S V ≤ K V , namely S ( ρ V ) − S ( ρ V ) ≤ Tr( ρ V K ) − Tr( ρ V K ) (2)which is equivalent to S ( ρ V | ρ V ) ≡ Tr (cid:0) ρ V (log ρ V − log ρ V ) (cid:1) ≥ , namely to the positivity of the relative entropy. One is then left to estimate the right handside of (2). Here the (dimensionless) local Hamiltonian K is defined by ρ V = e − K / Tr( e K ),up to a scalar shifting that does not affect the definition of (1).The above argument, thought in terms of a cutoff theory, breaks for general QuantumField Theory as the local von Neumann algebras A ( O ) are not of type I ; under generalassumptions, A ( O ) is a factor of type III so no trace Tr and no density matrix ρ is definable.Yet, as is well known, modular theory and Araki’s relative entropy S ( ϕ | ψ ) are definable ingeneral. We aim at a different argument, close in the spirit to the above discussion, thatmakes rigorous sense.The point is that the above argument is quite similar to the rigorous description of theblack hole incremental free energy and entropy given in [23] in the general Quantum Field2heory framework. Recently, this work led to a universal formula for the incremental freeenergy, that can be interpreted in several different contexts. This paper is an illustration ofthis fact.We take the point of view that relative entropy is a primary concept and other entropyquantities should be expressed in terms of relative entropies (cf. also [29]). This is the case,for example, for the von Neumann entropy. The von Neumann entropy S ( ϕ ) of a state ϕ ofa von Neumann algebra M may be expressed in terms of the relative entropy: S ( ϕ ) = sup ( ϕ i ) X i S ( ϕ | ϕ i )where the supremum is taken over all finite families of positive linear functionals ϕ i of M with P i ϕ i = ϕ (see [32]). Clearly S ( ϕ ) cannot be finite unless M is of type I .However, rather than tracing back the Bekenstein bound to the positivity of the rela-tive entropy, here we are going to rely on the positivity of the incremental free energy, orconditional entropy, which can be obtained in two possible ways: by the monotonicity ofthe relative entropy in relations to Connes-Størmer’s entropy [10], or by linking it to Jones’index [17]. In this respect, our argument is close to the derivation of the bound in [5], thatrelies on the monotonicity of the relative entropy. We now are going to compare two states of a physical system, ω in is a suitable referencestate, e.g. the vacuum in QFT, and ω out is a state that can be reached from ω in by somephysically realisable process (quantum channel). We relate the incremental energy and theentropy. With N , M be von Neumann algebras, an N − M bimodule is a Hilbert space H equippedwith a normal representation ℓ of N on H and a normal anti-representation r of M on H ,and ℓ ( n ) commutes with r ( m ), for all n ∈ N , m ∈ M . For simplicity, here we assume that N and M are factors. A vector ξ ∈ H is said to be cyclic for H if it is cyclic for the vonNeumann algebra ℓ ( N ) ∨ r ( M ) generated by ℓ ( N ) and r ( M ). Proposition 2.1.
Let α : N → M be a completely positive, normal, unital map and ω afaithful normal state of M . Then there exists an N − M bimodule H α , with a cyclic vector ξ α ∈ H and left and right actions ℓ α and r α , such that ( ξ α , ℓ α ( n ) ξ α ) = ω out ( n ) , ( ξ α , r α ( m ) ξ α ) = ω in ( m ) , (3) with ω in ≡ ω , ω out ≡ ω in · α . The pair ( H α , ξ α ) with this property is unique up to unitaryequivalence.Conversely, given an N − M bimodule H with a cyclic vector ξ ∈ H , with ω = ( ξ, r ( · ) ξ ) faithful state of M , there is a unique completely positive, unital, normal map α : N → M such that ( H , ξ ) = ( H α , ξ α ) , the cyclic bimodule associated with α by ω . roof. For the construction of ( H α , ξ α ), let M acts on a Hilbert space with cyclic and sep-arating vector ξ such that ω ( m ) = ( ξ, mξ ). The GNS representation of the algebraic tensorproduct N ⊙ M o ( M o the opposite algebra of M ), associated with the state determined by n ⊙ m o ( ξ, α ( n ) J M m ∗ J M ξ ) (4)gives ( H α , ξ α ), see [26].Conversely, let H be a N − M bimodule with left/right actions ℓ/r be given with cyclicvector ξ and ω = ( ξ, r ( · ) ξ ) faithful. Then define α : N → M by r ( α ( n )) p = J M p ℓ ( n ∗ ) p J M (5)with p ∈ r ( M ) ′ the projection onto r ( M ) ξ and J M the modular conjugation of r ( M ) p, ξ .In order to show that ( H , ξ ) = ( H α , ξ α ), by eq. (4) we have to show that( ξ, ℓ ( n ) r ( m ) ξ ) = ( ξ α , ℓ α ( n ) r α ( m ) ξ α ) , n ∈ N , m ∈ M . Indeed M acts on the right on p H , r p ( m ) ≡ r ( m ) | p H , with cyclic and separating vector ξ and we may identify p H with the identity M − M bimodule with left action ℓ p ( m ) ≡ J M r p ( m ∗ ) J M . We have( ξ α , ℓ α ( n ) r α ( m ) ξ α ) = ( ξ, ℓ p ( α ( n ∗ )) r p ( m ) ξ ) = ( ξ, J M r p ( α ( n )) J M r p ( m ) ξ )= ( ξ, J M r ( α ( n )) pJ M r ( m ) ξ ) = ( ξ, pℓ ( n ) p r ( m ) ξ ) = ( ξ, ℓ ( n ) r ( m ) ξ ) . (cid:3) Different normal, faithful initial states ω give unitary equivalent bimodules [26].Let α : N → M be a normal, unital completely positive map, ω in a faithful normal stateof M as above and H α , ξ α as in Prop. 2.1. Suppose that α is faithful, so ω out = ω in · α isfaithful. The converse of Prop. 2.1, interchanging left and right actions, gives a completelypositive, normal, unital map α ′ : M → N such that ω in ( α ( n ) m ) = ω out ( nα ′ ( m )) , called the transpose of α w.r.t. ω in . Similarly as eq. (5), α ′ is given by ℓ α ( α ′ ( m )) q = J N q r α ( m ∗ ) q J N , (6)with q ∈ ℓ α ( N ) ′ the orthogonal projection onto ℓ α ( N ) ξ α .We shall say that α is left invertible w.r.t. ω in is α ′ α = id N . We shall say that a state ω in is full for α if ξ α is cyclic for r α ( M ) and α is left invertible w.r.t. ω in . Lemma 2.2.
Suppose that ξ α is cyclic for r α ( M ) . Then ω in is full for α iff there existsa conditional expectation ε : r α ( M ) ′ → ℓ α ( N ) preserving the state ( ξ α , · ξ α ) . In this case α ′ ( m ) = ε ( J M r α ( m ) J M ) . Proof.
By equations (5) and (6), with ℓ ≡ ℓ α and r ≡ r α we have α ( n ) = r − (cid:0) J M ℓ ( n ∗ ) J M (cid:1) , (7)thus ℓ (cid:0) α ′ · α ( n ) (cid:1) q = J N q (cid:0) r − r (cid:0) J M ℓ ( n ) J M (cid:1)(cid:1) qJ N = J N qJ M ℓ ( n ) J M qJ N . It follows that α ′ ( α ( n )) = n , for all n ∈ N , iff J N qJ M = q , namely J N = J M q = qJ M .This is equivalent to say that the map ε is a conditional expectation by Prop. 3.1. (cid:3) ϕ, ψ be normal, faithful, positive linear functionals of a von Neumann algebra M and ξ ψ be a vector on the underlying Hilbert space such that ψ = ω ξ ψ | M , where ω ξ ψ = ( ξ ψ , · ξ ψ ).We denote Araki’s relative entropy by S ( ϕ | ψ ) [2], see [32]: S ( ϕ | ψ ) = − ( ξ ψ , log( dϕ/dψ ′ ) ξ ψ ) ;where ψ ′ is the positive linear functional on M ′ given by ψ ′ = ω ξ ψ | M ′ and dϕ/dψ ′ is Connes’spatial derivative [7] w.r.t. ϕ and ψ ′ .Essentially all properties of the relative entropy follow at once from Kosaki’s variationalexpression [19]. In particular, the relative entropy is non-negative and monotone: S ( ϕ | ψ ) ≥ , S ( ϕ · α | ψ · α ) ≤ S ( ϕ | ψ ) , with α : N → M a completely positive, unital, normal map.Let now ω be a faithful normal state of M and α : N → M is a completely positive,unital, normal map as above. We setH ω ( α ) ≡ sup ( ω i ) X i S ( ω | ω i ) − S ( ω · α | ω i · α ) , where the supremum is taken over all finite families of positive linear functionals ω i of M with P i ω i = ω .The conditional entropy H ( α ) of α is defined byH( α ) = inf ω H ω ( α ) , where the infimum is taken over all full states ω for α . If no full state exists, we putH( α ) = ∞ . Clearly H( α ) ≥ ω ( α ) ≥ α is a quantum channel if its conditional entropy H( α ) is finite.Let α : N → M be a quantum channel, that we assume to be faithful for simplicity.Let H α the bimodule with cyclic vector ξ and left/right action ℓ/r associated with α andthe faithful normal state ω ≡ ω in of M by Prop. 2.1. As α is faithful, the output state ω out = ω in · α is a faithful normal state of N .With ε : r ( M ) ′ → ℓ ( N ) the minimal conditional expectation (see refs. in [12]), the left modular operator ∆ α,ω in of α with respect to the initial state ω in is the spatial derivativebetween the states ω out · ℓ − · ε of r ( M ) ′ and ω in · r − of r ( M )∆ α,ω in = d ( ω out · ℓ − · ε ) (cid:14) d ( ω in · r − ) , (8)thus ∆ α,ω in = d ( ω ξ · ε ) (cid:14) d ( ω ξ | r ( M ) ), with ω ξ = ( ξ, · ξ ). Then ∆ ≡ ∆ α,ω in is a positive,non-singular selfadjoint operator on H α and we have∆ it ℓ ( n )∆ − it = ℓ ( σ out t ( n )) , ∆ it r ( m )∆ − it = r ( σ in t ( m )) , (9)with σ in / out the modular group of ω in / out .The right modular operator is∆ ′ α,ω in = d ( ω out · ℓ − ) (cid:14) d ( ω in · r − · ε ′ ) , (10)5ith ε ′ : ℓ ( N ) ′ → r ( M ) the minimal expectation.log ∆ and log ∆ ′ are called the left and right modular Hamiltonian of α w.r.t. the initialstate ω in . We have log ∆ ′ = log ∆ + H( α ) . The physical Hamiltonian K at inverse temperature β = T >
0, associated with α and theinitial state ω in , is a shifting the modular Hamiltonian with natural functoriality properties,and also rescaled in order to get the β -KMS property. By [26] and Prop. 3.2, K is given by K = − β − log ∆ − β − H( α ) = − β − log ∆ ′ + 12 β − H( α ) . (11) K may be considered as a local Hamiltonian associated with α and the state transfer withinput state ω in .The entropy S ≡ S α,ω in of α is here defined as S = − ( ˆ ξ, log ∆ ′ ˆ ξ ) , where ˆ ξ is a vector representative of the state ω in · r − · ε ′ in H α . S is thus Araki’s relativeentropy S ≡ S ( ω ξ | ℓ ( N ) | ω ξ · ε ′ ) w.r.t. the states ω ξ | ℓ ( N ) of ℓ ( N ) and ω ξ · ε ′ of ℓ ( N ) ′ , with ξ ≡ ξ α . Thus S ≥ E = ( ˆ ξ, K ˆ ξ )is the relative energy w.r.t. the states ω in and ω out .The free energy is now defined by the relative partition function F = − β − log( ˆ ξ, e − βK ˆ ξ ) . Proposition 2.3. F satisfies the thermodynamical relation F = E − T S . (12)
Proof.
Similarly as in [26], we have F = β − H( α ), indeed by (11) we have βF = − log( ˆ ξ, e − βK ˆ ξ ) = − log( ˆ ξ, ∆ ′ ˆ ξ ) + 12 H( α ) = − log || ∆ / ξ, ˆ ξ ˆ ξ || + 12 H( α )= − log || J ∆ / ξ, ˆ ξ ˆ ξ || + 12 H( α ) = − log || ξ || + 12 H( α ) = 12 H( α ) , with J the modular conjugation w.r.t. ξ, ˆ ξ .Thus eq. (12) follows by evaluating the linear relation between K and log ∆ ′ in (11) onthe vector state given by ˆ ξ . (cid:3) We then have the following general version of the Bekenstein bound.
Proposition 2.4. S ≤ βE . (13) Proof. As F = β − H( α ), we have F ≥ α ) ≥
0. So the inequality followsfrom the thermodynamical relation (12). (cid:3) As explained in the footnote in [26], the sign of F depends on the left or right modular Hamiltonianchoice. Here we consider the right modular Hamiltonian, so F ≥ .1.1 Fixing the temperature Above we have defined the modular Hamiltonian log ∆ and the physical Hamiltonian K associated with a quantum channel α , given a faithful normal initial state ω . K is obtainedby the modular Hamltonian by a shifting and a scaling − log ∆ shifting −−−−→ − log ∆ − log d ( α ) scaling −−−−→ β − (cid:0) − log ∆ − log d ( α ) (cid:1) , (left modular Hamiltonian case). Here log d ( α ) = H( α ), half of the conditional entropy of α , and is independent of ω . Indeed H( α ) is equal to the logarithm of the Jones index of H α (Prop. 3.2) and the dimension d ( α ) is a tensor categorical notion [25].Both the unitary one-parameter group generated by − log ∆ and − log ∆ − log d ( α )implement the modular flow (9), a property that is not affected by a scalar shifting of theHamiltonian. The shift is determined by functoriality properties. So both the modularHamiltonian and the physical Hamiltonian are intrinsic objects.Now, the scaling is not intrinsic; it corresponds to a choice of the parametrisation ofthe modular group to be matched with some physical dynamics (or evolution). The one-parameter group generated by K = β − (cid:0) − log ∆ − log d ( α ) (cid:1) : it is the unique rescaling of − log ∆ − log d ( α ) that satisfies the KMS condition w.r.t. the modular group at inversetemperature β . In other words, the rescaling is determined once is given by temperature ofthe system, which is the case of an equilibrium state for a thermodynamical system.In Quantum Field Theory, we may have a spacetime region O such that the modulargroup σ ωt of the local von Neumann algebra A ( O ) associated with O has a geometric mean-ing. Namely there is a geometric flow θ s : O → O and a re-parametrisation of σ ωt that actscovariantly w.r.t θ . Here ω may be the vacuum state or any other state. In other words,the world lines of θ in O have a modular origin, yet the modular flow t σ ωt is to bere-parametrised in order to correspond with the geometric flow θ .The main and well known illustration of the above situation concerns a Rindler wedgeregion O of the Minkowski spacetime. The vacuum modular group ∆ − it of A ( O ) w.r.t.the vacuum state is here equal to U ( βt ), with U the boost unitary one-parameter grouppreserving O with acceleration a and β the Unruh inverse temperature [4]. In this casethe re-parametrisation of the geometric flow is the rescaling by the inverse temperature β = 2 π/a .In general, the re-parametrisation is not just a scaling. As discussed in [9], it is howevernatural to define locally the inverse temperature by β s = (cid:13)(cid:13)(cid:13)(cid:13) dθ s ds (cid:13)(cid:13)(cid:13)(cid:13) , the Minkowskian length of the tangent vector to the modular orbit. Namely dτ = β s ds with τ proper time along modular trajectories.In the following, we shall use this point of view to fix β . This seems to be a good wayto compare locally the inverse temperatures of different modular flows. It also gives hereindications about the KMS temperatures, yet the right choice is a matter of discussion andis to be determined on physical grounds. We now read the above structure in a couple of situations in Quantum Field Theory. OurQFT is given by a net of von Neumann algebras A ( O ), on a Hilbert space H , associated7ith spacetime regions O (see [14]). By locality, the von Neumann algebra A ( O ′ ) associatedwith the causal complement O ′ of O commute with A ( O ). For the moment, our spacetime isgeneral, but we shall consider settings where there is a vector Ω ∈ H with the Reeh-Schliederproperty, namely Ω is cyclic and separating for A ( O ) if O and O ′ have non-empty interiors,and regions with geometric modular actions, namely the modular group associated with A ( O ) , Ω acts covariantly w.r.t. a geometric flow of O . For example, for a Wightman QFT,this is the case for a wedge region on the Minkowski space time, with Ω the vacuum vector[4]. In [23, 24], the relative entropy between the vacuum state ω and a localised chargedstate (state obtained by ω and a DHR charge [11]) is expressed by the thermodynamicalrelation (12) with the relative energy and the incremental free energy. DHR charges on acurved spatime are discussed in [13]. Here, we can deal with the output state obtained byany quantum channel, following the general formula in [26]. Let O be the Schwarzschild black hole region in the Schwarzschild-Kruskal spacetime ofmass M >
0, namely the region inside the event horizon (see [37]), and
N ≡ A ( O ) the localvon Neumann algebra associated with O on the underlying Hilbert space H . We considerthe Hartle-Hawking vacuum state ω (the global vacuum) ω ( X ) = (Ω , X Ω)and vacuum vector Ω. H is a N − N bimodule, indeed the identity
N − N bimodule L ( N )associated with Ω.The modular group of A ( O ) associated with ω is geometric and indeed corresponds tothe geodesic flow. The KMS Hawking-Unruh temperature is T = 1 / πM = 1 / πR , with R = 2 M the Schwarzschild radius (cf. [36, 34]). Thus our general formula (13) giveshere Proposition 2.5.
In the Schwarzschild black hole case as above, we have S ≤ πRE , with S the entropy associated with the Hartle-Hawking state and the output state transferredby a quantum channel, and E the corresponding relative energy. We now consider a Conformal Quantum Field Theory on the Minkowski spacetime of anyspacetime dimension. Let O R be the double cone with basis a radius R > A ( O R ) the associated local von Neumann algebra. The modular group of A ( O R ) w.r.t. the vacuum state ω has a geometrical meaning:∆ − isO R = U (cid:0) Λ O R (2 πs ) (cid:1) . U is the covariance unitary representation of the conformal group and Λ O R is a one-parameter group of conformal transformation leaving O R globally invariant and conjugateto the boost one-parameter group of pure Lorentz transformations [16]. Clearly we haveΛ O R ( s ) = δ R · Λ O ( s ) · δ /R , with δ R the dilation by R . We may compare the proper time at a point x with parameterof the flows dτ = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dds Λ O R ( s ) x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ds = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dds δ R · Λ O ( s ) · δ /R x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ds = R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dds Λ O ( s ) x R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ds (Minkowskian norm); in particular, in the center of the sphere, the proper time τ R of theflow Λ O R is R times the proper time of the flow Λ O (cf. [31]).Now, the inverse temperature β R = (cid:12)(cid:12)(cid:12)(cid:12) dds Λ O R ( s ) x (cid:12)(cid:12)(cid:12)(cid:12) s =0 in O R is maximal on the time-zerobasis of O R , in fact at the origin x = . Thus the maximal inverse temperatures β R in O R and β in O are related by β R = Rβ .This leads us to fix the KMS inverse temperature for Λ O R as β R = Rβ . One indeedcomputes that β = π , half of the Unruh value, and β R = πR .Our general formula (13) now gives: Proposition 2.6.
Let O R be a radius R > double cone in the Minkowski spacetime of anydimension as above, and A ( O R ) the local von Neumann algebra in a conformal QFT. Then S ≤ πRE . with S and E the entropy and energy associated with any quantum channel by the vacuumstate. Proof. S ≤ β R E = β RE ≤ πRE . (cid:3) The analysis in this section is rather interlocutory, less complete than the previous ones.Yet it shows up new aspects as the temperature depends on the distance from the boundary.We consider now a 1+1 dimensional Boundary CFT on the right Minkowski half-plane x >
0. The net A + of von Neumann algebras on the half-plane is associated with a localconformal net A of von Neumann algebras on the real line (time axis) by A + ( O ) = A ( I − ) ∨ A ( I + ) ;Here I − , I + are intervals of the time axis at positive distance with I + > I − ( I + is on thefuture of I − ) and O is the associated double cone O = I − × I + ≡ { ( t, x ) : t ± x ∈ I ± } .More generally, in the rational case, we have to consider a (necessarily finite-index)extension of A . However the following discussion remains the same.There is a natural state with geometric modular action [27], that corresponds to thechiral “2-interval state” and geometric action of the double covering of the M¨obius group[28]. We refer to this state as the “geometric state”.9ith I − = ( a , b ), I + = ( a , b ), in chiral coordinates u = x + t , v = x − t , the flow θ Os ( u, v ) = ( u s , v s ) has velocity field ( ∂u s , ∂v s ) given by ∂ s u s = 2 π ( u s − a )( u s − b )( u s − a )( u s − b ) Lu s − M u s + N , (14)with L = b − a + b − a , M = b b − a a , N = b a ( b − a ) + b a ( b − a ), and similarlyfor v s [27].Let us fix a double cone O with basis of unit length (say O is Lorentz conjugate to adouble cone with basis on the space half-axis with length one).With R >
0, let O R be the double cone associated with the intervals RI − = ( Ra , Rb ), RI + = ( Ra , Rb ), namely O R = δ R O , with δ R the dilation by R on the half-plane. Then θ O R = δ R · θ O · δ R − .As in Section 2.2.2, the maximal inverse temperatures are related by β O R = R β O . By choosing the KMS inverse temperatures equal to the maximal temperature we thus have:
Proposition 2.7.
In the above setting, with S and E the entropy and energy in O R withrespect to the geometric state and a quantum channel, we have S ≤ λ O RE where the constant λ O is equal to β O . It depends on the distance of O from the boundaryvia the above flow (14) . We collect here a couple of mathematical results.
Let
N ⊂ M be von Neumann algebras on a Hilbert space H and ξ ∈ H a cyclic andseparating unit vector for M . Let q ∈ N ′ be the projection onto N ξ .We may consider the associated modular conjugation J M of ( M , ξ ) on H and J N of( N q, ξ ) on q H . We may view J N as a anti-linear partial isometry on H by replacing J N with J N q . The map γ : M → N defined by γ ( m ) q = J N qJ M mJ M qJ N is normal, completely positive, unital and preserves the state ϕ ≡ ( ξ, · ξ ) on M , cf. [1, 20]. Proposition 3.1.
The following are equivalent: ( i ) There exists a conditional expectation ε of M onto N preserving ϕ ; ( ii ) J N = J M q = qJ M ; ( iii ) γ | N = id .In this case ε = γ . roof. Assuming ( i ), Takesaki’s theorem [35] implies that the modular operator of ( N , ξ )on q H is the restriction of the modular operator of ( M , ξ ), and this easily entails ( ii ). Theimplication ( ii ) ⇒ ( iii ) is immediate.Concerning ( iii ) ⇒ ( i ), notice that γ | N = id implies γ = γ , thus γ is an expectation of M onto N preserving ϕ . The rest is clear. (cid:3) Let N , M be factors. The index Ind( H ) of an N − M bimodule H is the Jones index[ r ( M ) ′ : ℓ ( N )]. The index Ind( α ) of a normal, unital, completely positive map α : N → M is the index of the
N − M bimodule H α , namely Ind( α ) ≡ Ind( H α ) (see [17, 18, 21, 22] and[12] for the non-factor case). The index is related to the conditional entropy. Proposition 3.2. H( α ) = log Ind( α ) . Proof.
By Lemma 2.2 and notations there, ε = α ′ · j with j an anti-isomorphism thatpreserves the state ω = ( ξ, · ξ ). This easily implies that H ω ( α ′ ) = H ω ( ε ). On the otherhand Connes-Størmer’s entropy H ω ( ε ) is equal to the logarithm of the index of ε [33, 15],so minimising H ω ( α ′ ) over all full states for α gives the logarithm of the minimal indexInd( H α ) of [ r α ( M ) ′ : ℓ α ( N )], namely H( α ′ ) = log Ind( H α ). Since Ind( H α ′ ) = Ind( H α ), wehave H( α ′ ) = log Ind( H α ′ ), hence the proposition follows by replacing α ′ with α . (cid:3) Acknowledgements.
We thanks Y. Kawahigashi for the invitation at the Seasonal Insti-tute of the Mathematical Society of Japan “Operator Algebras and Mathematical Physics”Tohoku University, Sendai, August 2016, where our collaboration started. We are gratefulto H. Casini and K.H. Rehren for helpful comments.
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