Exponential arcs in the manifold of vector states on a sigma-finite von Neumann algebra
aa r X i v : . [ m a t h . OA ] J a n Exponential arcs in the manifold ofvector states on a σ -finitevon Neumann algebra Jan NaudtsPhysics Department, Universiteit Antwerpen,Universiteitsplein 1, 2610 Antwerpen Belgiume-mail: [email protected]: 0000-0002-4646-1190
Abstract
This paper introduces the notion of exponential arcs in Hilbert space andof exponential arcs connecting vector states on a sigma-finite von Neumannalgebra in its standard representation. Results from Tomita-Takesaki theoryform an essential ingredient. Starting point is a non-commutative Radon-Nikodym theorem that involves positive operators affiliated with the com-mutant algebra. It is shown that exponential arcs are differentiable and thatparts of an exponential arc are again exponential arcs. Special cases of prob-ability theory and of quantum probability are used to illustrate the approach.
Keywords
Exponential arc, Exponential family, Tomita-Takesaki theory, In-formation Geometry, Probability theory, Quantum Probability
Araki [1] introduces α -families of cones in Hilbert space H . They are character-ized by a σ -finite von Neumann algebra A of operators in H and a vector x in H which is cyclic and separating for A . The parameter α satisfies ≤ α ≤ / .The α -cone and the ( − α ) -cone are each others dual. Hence, the cone with α = 1 / is self-dual. It is called the natural positive cone and it plays a specialrole in Tomita-Takesaki theory [2]. For each value of α one can define generalizedRadon-Nikodym operators. In the case α = 0 they belong to the algebra A , in thecase α = 1 / they belong to the commutant algebra A ′ . The present paper is re-stricted to the α = 1 / -case in the belief that α = 1 / simplifies the introduction1n a non-commutative context of the notion of models belonging to an exponentialfamily.Amari [3, 4, 5] introduces an α -collection of geometries on manifolds of non-degenerate probability distributions. The α and − α geometries are each other dual.The self-dual case α = 0 involves the Levi-Civita connection. Models belongingto an exponential family have a flat geometry in the α = 1 connection.The analogy between Amari’s α -geometries and Araki’s α -cones is strikingbut may be superficial. Both theories have in common that the introduction of the α -dependence enriches the theory. In addition, the self-dual choice of α is techni-cally more demanding. The study of statistical models belonging to an exponentialfamily is more easy when using a flat geometry, this is, α = 1 . Goal of the presentpaper is to explore the use of Araki’s α -cones with α = 1 / in the context of on-going efforts to generalize Information Geometry to a non-commutative setting.A main concept in Information Geometry [4, 5, 6] is the notion of a modelbelonging to an exponential family. In Statistical Physics such a model is called aBoltzmann-Gibbs distribution. The generalization [7, 8, 9, 10, 11, 12] to quan-tum models is feasible, at least when the Hilbert space of the model is finite-dimensional and the states on the algebra can be described by density matrices.As soon as it is infinite dimensional technical problems appear on top of the prob-lems due to non-commutativity. Operators on Hilbert space need not anymore tobe continuous and their domain of definition is only a subspace. Selfadjoint op-erators play a crucial role. Non-uniqueness of selfadjoint extensions is a seriousproblem. Early attempts to tackle these problems include [13, 14, 15, 16, 17].Cones of positive-definite matrices are studied in [18, 19, 20, 21]. These conesare self-dual for the Hilbert-Schmidt inner product. With each such cone corre-sponds a Euclidean Jordan algebra. The product of this algebra is commutative.This simplifies the introduction of Amari’s dually flat geometry. It is not immedi-ately clear how to generalize this approach to infinite dimensions.The recent approach of [22, 23] relies on Lie theory for the group of boundedoperators with bounded inverse and does not yet attack the above mentioned prob-lems with unbounded operators.The present approach involving the commutant algebra A ′ has already beenexplored [24, 25] in the familiar context of density matrices. It is also studied in[26] where the exponential function is replaced by a function with linear growth at + ∞ . The latter idea is tried out in the context of commutative probability theorywith the intent to better control problems of unboundedness [27, 28, 29, 30]. It isnot used in the present paper because Tomita-Takesaki theory relies heavily on theproperties of the exponential function. 2 tructure of the paper The next section reviews elements of Tomita-Takesaki-theory and fixes some of the notations. Section 3 introduces exponential arcs inHilbert space. Section 4 introduces exponential arcs in the space of vector states.Section 5 considers special cases. The final section contains a short discussion.
Throughout the text A is a fixed σ -finite von Neumann algebra of operators ona Hilbert space H and x is a fixed normalized vector in H that is cyclic andseparating for A . Each vector y in H defines a linear functional ω y on A by ω y ( A ) = ( Ay, y ) , A ∈ A (1)If || y || = 1 then ω y is a vector state on A . The set of vector states of A is denoted V . A state ω on A is said to be faithful if ω ( A ∗ A ) = 0 implies A = 0 . The vectorstate ω x is faithful if x is separating for A . The subset of faithful vector states isdenoted ˚ V . The closure of the map Ax A ∗ x, A ∈ A is denoted S x . Let S x = J x | ∆ x | / be its polar decomposition. The modularautomorphisms τ t , t ∈ R , are defined by τ t ( A ) = ∆ itx A ∆ − itx , A ∈ A . The anti-linear isometry J x satisfies J ∗ x = J x and J x = I .A cone P x associated with the couple A , x is defined by taking the closure ofthe set of vectors { AJ x Ax : A ∈ A} . If x is cyclic and separating for A then P x is independent [1] of the choice of x .It is called the natural positive cone . It has the properties that P x = P y := P and J x = J y := J for all x, y ∈ P . For any x in P and A in A is ( AJ Ax, x ) ≥ with equality if and only if A = 0 .Introduce the notations C x = { y ∈ H : ( y, Ax ) ≥ holds for all A ≥ , A ∈ A} C ∗ x = { y ∈ H : ( y, A ′ x ) ≥ holds for all A ′ ≥ , A ′ ∈ A ′ } . The sets C x and C ∗ x are cones closed in the norm topology of H . A vector y belongs to C x if and only if J x y belongs to C ∗ x . The cones C x and C ∗ x form a dualpair in the sense that C x = { y ∈ H : ( y, z ) ≥ for all z ∈ C ∗ x } and C ∗ x = { y ∈ H : ( y, z ) ≥ for all z ∈ C x } . The notations C x and C ∗ x are used rather than P ♭x , respectively P ♯x , becausefurther indices will be added later on. In [1] the notation V α with α = 1 / , re-spectively α = 0 is used. The cone V / is then the natural positive cone.The next result is important for what follows. Proposition 2.1
A vector y ∈ H belongs to C x if and only if there exists a non-negative selfadjoint operator Y affiliated with A ′ such that y = Y x . For a proof, see for instance Proposition 2.5.27 of [31].
Corollary 2.2
Any vector y of C x satisfies S ∗ x y = y . In particular, one has ( y, Ax ) = ( x, Ay ) for all A ∈ A . Proof
One has for all A in A that ( S x Ax, y ) = ( A ∗ x, Y x ) = ( Y x, Ax ) = ( y, Ax ) . Hence, y is in the domain of S ∗ x and satisfies S ∗ x y = y . One has ( y, Ax ) = ( y, S x A ∗ x ) = ( A ∗ x, S ∗ x y ) = ( A ∗ x, y ) = ( x, Ay ) . (cid:3) Definition 2.3
Let C x denote the subset of normalized elements of C x . Let ˘ C x = A ′ x and let ˘ C x denote the subset of normalized vectors in ˘ C x . Proposition 2.4
The set ˘ C x is dense in C x ; The set ˘ C x is dense in C x . Proof
Let y ∈ C x . By Proposition 2.1 there exists a non-negative operator Y affiliatedwith the commutant A ′ such that y = Y x . The operator Y can be approximatedby non-negative operators Y n belonging to the commutant A ′ in such a way thatthe vectors y n = Y n x are normalized and converge to y in norm. (cid:3) .2 Properties of C-cones The operator Y in Proposition 2.1 may be non-unique. Therefore the choice madein the proof of the proposition is made explicit. Definition 2.5
Given y in C x let X y,x denote the square of the Friedrichs extensionof the operator Ax Ay , A ∈ A . For further use note that the operator X y,x is affiliated with the commutant A ′ of A and that the vector X t/ y,x x belongs to the cone C x for any t in [0 , . Proposition 2.6
Take y in C x . The following statements are equivalent. (a) y is cyclic for A ; (b) The operator X / y,x is one-to-one; (c) X t/ y,x x is separating for A for all t ∈ [0 , ; (d) X t/ y,x x is cyclic for A for all t ∈ [0 , . Proof(a) implies (b)
Assume that X / y,x u = 0 . Then one has for all A in A X / y,x u, Ax ) = ( u, X / y,x Ax ) = ( u, Ay ) . Since y is cyclic by assumption this implies that u = 0 . This shows that the linearoperator X / y,x is one-to-one. (b) implies (c) Let v = X t/ y,x x . Assume Av = 0 with A in A . Then one has Av = AX t/ y,x x = X t/ y,x Ax.
Because X y,x is one-to-one it follows that Ax = 0 . Because x is separating for A one concludes that A = 0 . Hence, v is separating for A .5 c) implies (d) Let v = X t/ y,x x . Without loss of generality assume v is normal-ized. Then ω v is a faithful state. Hence there exists a vector z in the natural positivecone P such that ω y = ω z . See for instance Theorem 2.5.31 of [31]. The vector z is separating for A . It is well-known [1] that vectors in P are separating if andonly if they are cyclic. See for instance Proposition 2.5.30 of [31]. Hence z iscyclic. This implies that the isometry U in A ′ defined by U Av = Az A ∈ A , is actually a unitary operator. Assume now that ( u, Av ) = 0 for all A in A . Thenone has u, Av )= ( u, AU z )= ( U u, Az ) , A ∈ A . Because z is cyclic for A it follows that U u = 0 and hence u = 0 . This shows that v is cyclic for A . (d) implies (a) This follows by taking t = 1 . (cid:3) The notion of an open exponential arc was introduced in [32, 33] and is used forinstance in [34]. Below is given a definition of exponential arcs with values inHilbert space. A small difference in concept follows from referring to the arcs bytheir end points, or by a midpoint and one end point, rather than by two distinctpoints belonging to the interior. In addition, uni-directional arcs are introducedinstead of the equivalence relations of [32]. The advantage is that they can bedefined with any vector in C x as the end point. Definition 3.1
Let be given y in C x and let X y,x be the selfadjoint operator af-filiated with the commutant A ′ as defined in Definition 2.5. The exponential arc connecting y to x is the map t ∈ [0 , γ y,x ( t ) defined by γ y,x ( t ) = e − ζ y,x ( t ) X t/ y,x x, t ∈ [0 , , (2) where the normalization ζ y,x ( t ) is given by ζ y,x ( t ) = log || X t/ y,x x || .
6y the concavity of the function u u t for t in [0 , one has ζ y,x ( t ) = log || X t/ y,x x || ≤ t log || y || = 0 . Note that ζ y,x (0) = ζ y,x (1) = 0 and y = x and y = y .For each t in [0 , the vector γ y,x ( t ) belongs to C x . It is properly normalizedand one has for all positive A in A that ( γ y,x ( t ) , Ax ) = e − ζ y,x ( t ) (cid:0) X t/ y,x x, Ax (cid:1) = e − ζ y,x ( t ) (cid:0) X t/ y,x x, AX t/ y,x x (cid:1) ≥ . Proposition 3.2
The operator X t/ y,x is the Friedrichs extension of the symmetricoperator Ax AX t/ y,x x with A in A . Proof
Let v = X t/ y,x x and let Y denote the operator Y : Ax AX t/ y,x x, A ∈ A . Introduce the sesquilinear form s defined by s ( Ax, Bx ) = (
Av, Bx ) , A, B ∈ A . It is straightforward to verify that s is well-defined, symmetric and positive-semi-definite. The domain of its closure s consists of all u ∈ H for which there exist u n in A x converging to u such that s ( u n − u m , u n − u m ) tends to 0 as n, m tend to ∞ .It is straightforward to verify that the domain of X t/ y,x , and a fortiori that of X t/ y,x ,is a subset of domain of s . Hence, both Y F and X t/ y,x are selfadjoint extensionsof the operator Y . However, the Friedrichs extension Y F is the unique extensionwith domain included in dom s — see Theorem 2.11, p. 326 of [35]. Hence, bothoperators coincide. (cid:3) Corollary 3.3 If z = γ y,x ( t ) with < t ≤ then X / z,x = e − ζ y,x ( t ) X t/ y,x . Theorem 3.4
Let x be a normalized vector cyclic and separating for the von Neu-mann algebra A . For each y in C x and < t ≤ the map s γ y,x ( st ) is theexponential arc connecting γ y,x ( t ) to x . The normalization functions satisfy ζ y,x ( st ) = ζ z,x ( s ) + sζ y,x ( t ) with z = γ y,x ( t ) . (3)7 roof For convenience, let z = γ y,x ( t ) . We have to show that γ y,x ( st ) = v with v = γ z,x ( s ) . The above corollary implies that X / z,x = e − ζ y,x ( t ) X t/ y,x and X / v,x = e − ζ z,x ( s ) X s/ z,x = e − ζ z,x ( s ) e − sζ y,x ( t ) X st/ y,x and X / v,x = e − ζ y,x ( st ) X st/ y,x . Comparison of the latter two gives (3). Application of the latter on the vector x gives v = X / v,x x = e − ζ y,x ( st ) X st/ y,x x = γ y,x ( st ) . (cid:3) The following proposition mimics part of Proposition 6 of [26].
Proposition 3.5
Take y in C x . One has (a) ≤ ζ y,x ( st ) ≤ sζ y,x ( t ) for all s, t in [0 , ; (b) The function t ζ y,x ( t ) is convex on the interval [0 , ; (c) The function t ζ y,x ( t ) is continuous at t ↓ .If x belongs to the domain of log X y,x then (d) The left derivative of t ζ y,x ( t ) at t = 0 exists and one has dd t (cid:12)(cid:12)(cid:12)(cid:12) t =0 ζ y,x = 12 ([log X y,x ] x, x ) . (4) Proof a) This follows from (3). (b)
Let X y,x = Z ∞ λ d E λ denote the spectral decomposition of the selfadjoint operator X y,x . Let r = (1 − s ) t + st with s in [0 , and ≤ t < t ≤ . Introduce the abbreviation d µ ( λ ) = λ t d( E λ x, x ) . Without restriction of generality assume that the measure µ is non-trivial. One has Z ∞ λ r d( E λ x, x ) = (cid:20)Z ∞ d µ ( λ ) (cid:21) R ∞ λ s ( t − t ) d µ ( λ ) R ∞ d µ ( λ ) ≤ (cid:20)Z ∞ d µ ( λ ) (cid:21) (cid:20) R ∞ λ t − t d µ ( λ ) R ∞ d µ ( λ ) (cid:21) s = (cid:20)Z ∞ d µ ( λ ) (cid:21) − s (cid:20)Z ∞ λ t − t d µ ( λ ) (cid:21) s . Take the logarithm to obtain ζ y,x ( r ) = 12 log Z ∞ λ r d( E λ x, x ) ≤ − s (cid:20)Z ∞ d µ ( λ ) (cid:21) + s (cid:20)Z ∞ λ t − t d µ ( λ ) (cid:21) = (1 − s ) ζ y,x ( t ) + sζ y,x ( t ) . This shows the convexity of t ζ y,x ( t ) . (c) This follows from (a) in combination with (b). (d)
Starting point are the inequalities ≤ e t − − t ≤ t , − ≤ t ≤ . Let < ǫ < . One has for ≤ t ≤ / ( − ǫ )0 ≤ Z /ǫǫ (cid:2) λ t − − t log λ (cid:3) d( E λ x, x ) ≤ t Z /ǫǫ [log λ ] d( E λ x, x ) . Z /ǫǫ [log λ ] d( E λ x, x ) ≤ Z /ǫǫ λ t − t d( E λ x, x ) ≤ Z /ǫǫ [log λ ] d( E λ x, x ) + t Z /ǫǫ [log λ ] d( E λ x, x ) . Let R ( ǫ ) given by R ( ǫ ) = (cid:20)Z ǫ + Z + ∞ /ǫ (cid:21) [log λ ] d( E λ x, x ) . One obtains − R ( ǫ ) + ([log X y,x ] x, x ) = Z /ǫǫ [log λ ] d( E λ x, x ) ≤ Z /ǫǫ λ t − t d( E λ x, x ) ≤ − R ( ǫ ) + ([log X y,x ] x, x ) + t || [log X y,x ] x || Note that R ( ǫ ) tends to zero as ǫ ↓ because by assumption x belongs to thedomain of log X y,x . Hence, one concludes that dd t (cid:12)(cid:12)(cid:12)(cid:12) t =0 || X t/ y,x x || = ([log X y,x ] x, x ) . This leads to (4). (cid:3)
An important property of exponential arcs is differentiability.
Proposition 3.6
Given y in C x one has dom X t/ y,x log X y,x ⊃ dom X / y,x , < t < . Proof
Let X y,x = Z ∞ λ d E λ X y,x . It follows fromthe spectral theorem that the domain of X t/ y,x log X y,x is given by dom X t/ y,x log X y,x = { u : Z + ∞ λ t (log λ ) d( E λ u, u ) < + ∞} . Note that for ≤ λ < the function λ t (log λ ) is bounded while for large enough λ one has λ t (log λ ) < λ . If u belongs to dom X / y,x then one has Z + ∞ λ d( E λ u, u ) < + ∞ so that u belongs to dom X t/ y,x . (cid:3) Theorem 3.7
Let x be a normalized vector cyclic and separating for the von Neu-mann algebra A . Take y in C x and consider the exponential arc t γ y,x ( t ) con-necting y to x . The Fr´echet derivative ˙ γ y,x ( t ) of t γ y,x ( t ) exists for < t < and is given by ˙ γ y,x ( t ) = (cid:20)
12 log X y,x − ζ ′ y,x ( t ) (cid:21) γ y,x ( t ) , < t < , (5) where ζ ′ y,x ( t ) is the derivative of the normalization function ζ y,x ( t ) . Proof
Let X y,x = Z ∞ λ d E λ denote the spectral decomposition of the selfadjoint operator X y,x . One has for < t < t + ǫ < || X ( t + ǫ ) / y,x x − X t/ y,x x − ǫ X t/ y,x log X y,x ] x || = Z + ∞ f t ( √ λ, ǫ ) d( E λ x, x ) with f t ( µ, ǫ ) = µ t + ǫ − µ t − ǫµ t log µ. Note that f t (0 , ǫ ) = f t (1 , ǫ ) = 0 . Hence, µ f t ( µ, ǫ ) attains its maximum in theinterval [0 , at some intermediate value µ of µ that is the solution of the equation ( t + ǫ ) [ µ ǫ −
1] = ǫt log µ.
11n the limit of vanishing ǫ is log µ = − t − ǫ t + O ( ǫ ) . Use this to show that f t ( µ, ǫ ) = O ( ǫ ) . One obtains Z f t ( √ λ, ǫ ) d( E λ x, x ) ≤ f t ( µ, ǫ ) Z d( E λ x, x ) ≤ f t ( µ, ǫ )= O ( ǫ ) . (6)Next introduce the function g t ( ν, ǫ ) defined by f t ( µ, ǫ ) = √ λ g t ( ν, ǫ ) with ν = 1 /λ. One finds g t ( ν, ǫ ) = f − t ( ν, − ǫ ) . Hence, the maximum with ν in the interval [0 , is reached for a value ν satisfying g t ( µ, ǫ ) = O ( ǫ ) . One has now Z + ∞ f ( √ λ, ǫ ) d( E λ x, x ) ≤ g ( ν, ǫ ) Z + ∞ λ d( E λ x, x ) ≤ g ( ν, ǫ ) || X / y,x y || = g ( ν, ǫ )= O ( ǫ ) . (7)Both estimates (6) and (7) together show that (1 / X t/ y,x log X y,x ] x is the Fr´echetderivative of the map t X t/ y,x x . The existence of the Fr´echet derivative impliesthat also the normalization function ζ y,x ( t ) is differentiable. The result is (5). (cid:3) For convenience, introduce the following definition.
Definition 3.8
Given a vector y in C x the set ˚ γ y,x is the image of [0 , by theexponential arc t ∈ [0 , γ y,x ( t ) minus the endpoints x and y . ˚ γ y,x = { γ y,x ( t ) : t ∈ [0 , } \ { x, y } ⊂ H . roposition 3.9 If y in C x is cyclic for A then all vectors γ y,x ( t ) are cyclic andseparating for A . Proof
It follows from Proposition 2.6 that γ y,x ( t ) is a cyclic vector. It is separatingby Proposition 2.6. (cid:3) If y in C x is cyclic for A then the role of the vectors x and y may be interchangedand the arc can be inverted. Proposition 3.10 If y in C x is cyclic for A then x belongs to C y and t γ y,x (1 − t ) is the exponential arc connecting x to y . The normalization functions satisfy ζ x,y ( t ) = ζ y,x (1 − t ) . The sets ˚ γ y,x and ˚ γ x,y coincide. Proof If y is cyclic then the operator X y,x has a densely defined inverse. The vector x belongs to C y because it can be written as x = X − / y,x y with X − / y,x a selfadjointoperator affiliated with the commutant A ′ . The inverse of the Friedrichs extensionof the map Ax Ay is the Friedrichs extension of the map Ay Ax . Henceone has X / x,y = X − / y,x .The remainder of the proof is then straightforward and follows from X (1 − t ) / y,x x = X − t/ y,x y = X t/ x,y y. (cid:3) The following result extends that of Theorem 3.4 in the case that all vectorsof the exponential arc are cyclic for the von Neumann algebra A . To do so it usesProposition 3.10. Theorem 3.11
Let x be a normalized vector cyclic and separating for the vonNeumann algebra A . Assume y in C x is cyclic for A . Then it holds that for any s and t in [0 , the map r γ y,x ((1 − r ) s + rt ) is the exponential arc connecting γ y,x ( t ) to γ y,x ( s ) . Proof
Proposition 3.10 allows us without loss of generality to make the assumptionthat < s < t ≤ . 13or convenience, let z = γ y,x ( s ) and u = γ y,x ( t ) . We have to show that γ u,z ( r ) = γ y,x ((1 − r ) s + rt ) .Let q = r ( t − s ) / (1 − s ) . It follows from Theorem 3.4 that γ u,z ( r ) = γ y,z ( q ) .Proposition 3.10 yields γ y,z ( q ) = γ z,y (1 − q ) . Finally, apply Theorem 3.4 again toobtain γ z,y (1 − q ) = γ x,y ( rs + (1 − r ) t ) and Proposition 3.10 to obtain γ x,y ( rs +(1 − r ) t ) = γ y,x ((1 − r ) s + rt ) . (cid:3) Theorem 3.7 asserts that any exponential arc t γ y,x ( t ) is Fr´echet differentiableon the open interval (0 , . The end points t = 0 and t = 1 are not included. Thissuggests to consider exponential arcs that have x as a midpoint. Definition 3.12
Let ˚ γ x denote the union of all sets ˚ γ z,y where y and z are vectorsin C x that are cyclic and separating for A with the property that x = γ z,y (1 / . It follows from Theorem 3.4 that any vector in ˚ γ z,y with y and z in C x and withmidpoint x also belongs to C x . Hence one has the inclusion ˚ γ x ⊂ C x . Vectors y in C x are excluded from ˚ γ x if they are not cyclic for A or the exponential arc t γ y,x ( t ) cannot be extended beyond the initial vector x or the final vector y . Proposition 3.13
For any u in ˚ γ x , u = x , the exponential arc t ∈ [0 , γ u,x ( t ) can be extended to a map t ∈ [ − , γ u,x ( t ) with the following properties. (a) There exist s in (1 / , and normalized vectors y and z that are cyclicand separating for A such that γ u,x ( t ) = γ z,y (( s − / t + 1 / , − ≤ t ≤ (8) (b) The Fr´echet derivative ˙ γ u,x ( t ) of γ u,x ( t ) exists for all t in the open inter-val ( − , ; it is given by ˙ γ u,x ( t ) = ( s − /
2) ˙ γ z,y ( r ) (9) with r = ( s − / t + 1 / ; (c) The Fr´echet derivative ˙ γ u,x ( t ) satisfies the equation ˙ γ u,x ( t ) = (cid:20)
12 log X u,x − ζ ′ u,x ( t ) (cid:21) γ u,x ( t ) , − < t < , (10) with ζ u,x ( t ) = log || X t/ u,x x || . Proof a) From the definition of ˚ γ x it follows that there exist normalized vectors y and z that are cyclic and separating for A with the properties that x = γ z,y (1 / and u = γ z,y ( s ) for some s in (1 / , . It then follows from Theorem 3.4 that γ u,x ( t ) = γ z,y ( r ) holds for t in [0 , . Hence, the definition (8) is an extension ofthe exponential arc γ u,x from [0 , to [ − , . (b) The existence of the Fr´echet derivative is proved in Theorem 3.7. It satisfies ˙ γ z,y ( r ) = (cid:20)
12 log X z,y − ζ ′ z,y ( r ) (cid:21) γ z,y ( r ) . (11)This gives (9). (c) Apply Corollary 3.3 with u = γ z,y ( s ) and x = γ z,y (1 / to obtain X s/ z,y = e ζ z,y ( s ) X / u,y ,X / z,y = e ζ z,y (1 / X / x,y . (12)Together these expressions yield
12 ( s − /
2) log X z,y = ζ z,y ( s ) + 12 log X u,y − ζ z,y (1 / −
12 log X x,y . Hence one has ˙ γ u,x ( t ) = (cid:18) s − (cid:19) ˙ γ z,y ( r )= (cid:18) s − (cid:19) (cid:20)
12 [log X z,y ] − ζ ′ z,y ( r ) (cid:21) γ u,x ( t )= (cid:20)
12 log X u,y −
12 log X x,y + ζ z,y ( s ) − ζ z,y (1 / − ζ ′ z,y ( r ) (cid:21) γ u,x ( t ) . Comparison with (5) gives [log X u,x ] γ u,x ( t ) = [log X u,y − log X x,y + f ( t )] γ u,x ( t ) , < t < , for some real function f ( t ) . (cid:3) The exponential arcs of vectors in Hilbert space define exponential arcs of vectorstates on the von Neumann algebra A . 15 efinition 4.1 Let V x denote the subset of V consisting of the vector states ω y where y belongs to C x . Proposition 4.2
Let x and y be normalized vectors in H that are cyclic and sep-arating for A . If ω x = ω y then there exists a unitary operator U in the commutantalgebra A ′ which maps C x onto C y and exponential arcs onto exponential arcs insuch a way that γ Uz,y ( t ) = U γ z,x ( t ) , z ∈ C x and t ∈ [0 , . (13) In particular, one has V x = V y . Proof
A unitary operator U is defined by U Ax = Ay for all A in A . It belongs to thecommutant algebra A ′ . Take now z in C x . Then one has ( U z, Ay ) = (
U z, U Ax ) = ( z, Ax ) ≥ , A ∈ A , A ≥ . This shows that
U z belongs to C y . Now one has ω Uz ( A ) = ( AU z, U z ) = (
U Az, U z ) = (
Az, z ) = ω z ( A ) , A ∈ A . This shows that ω z belongs to V y and hence that V x ⊂ V y . Equality follows byinterchanging x and y .The proof of (13) is straightforward. (cid:3) The above proposition justifies the following definition.
Definition 4.3
Given y in C x a map t ∈ [0 , ω t is an exponential arc con-necting the vector state ω y to the vector state ω x if there exists an exponential arc t ∈ [0 , γ y,x ( t ) connecting y to x such that ω t = ω y t with y t = γ y,x ( t ) for all t in [0 , . Because the map y ∈ C x ω y is one-to-one there is only one such exponentialarc for each y in C x . In addition, it follows immediately from Proposition 3.4 thatfor each t satisfying < t ≤ the map s ω y st is the unique exponentialarc connecting ω y t to ω x . Proposition 4.2 shows that the exponential arc does notdepend on the choice of vector x representing the state ω x .In summary, one has the following. Proposition 4.4
Any state ω of V x is connected to the state ω x by a unique expo-nential arc t ω t such that ω = ω x and ω = ω y . t ω t is an exponential arc connecting ω y to ω x and the normalizationfunction ζ y,x ( t ) is strictly convex then the set of vector states { ω y t : 0 < t < } isan exponential model. The primary chart is the map ω y t t . The tangent vectors ˙ ω y t belong to the predual A ∗ . This is a subspace of the dual A ∗ of A and is aBanach space for the norm of A ∗ . See for instance Section 2.4.3 of [31]. Thesetangent vectors are given by ˙ ω y t ( A ) = ( A ˙ γ y,x ( t ) , γ y,x ( t )) + ( Aγ y,x ( t ) , ˙ γ y,x ( t ))= e − ζ y,x ( t ) ( AX t/ y,x x, [log X y,x ] X t/ y,x x ) − ζ ′ y,x ( t ) ω y t ( A ) , A ∈ A . The Legendre transformation of ζ y,x ( t ) is given by ζ ∗ y,x ( s ) ≡ sup { st − ζ y,x ( t ) : 0 ≤ t ≤ } . If ζ y,x ( t ) is strictly convex then the supremum is reached at a unique value of s .The latter is denoted t ∗ and is given by t ∗ = ζ ′ y,x ( t ) . The following identity holds ζ y,x ( t ) + ζ ∗ y,x ( t ∗ ) = tt ∗ . The map ω t t ∗ = ζ ′ y,x ( t ) is the dual chart of the model. Definition 4.5
Fix a normalized vector x in H that is cyclic and separating for A . Let ˘ V x denote the subset of states of V that are majorized by a multiple of thestate ω x . Additional properties can be proved for vector states belonging to ˘ V x . SeeSection 9 of [1]. The following is an adaptation of some of these results to thepresent context. Proposition 4.6
Let ω ∈ V . Are equivalent: (a) ω belongs to ˘ V x ; (b) There exists y in C x such that ω = ω y and the operator X y,x is bounded. Proof a) implies (b) By assumption there exists z such that ω = ω z and there exists areal constant λ such that ω z ( A ) ≤ λω x ( A ) for all positive A ∈ A . The linear operator Z defined by Z : Ax Az, A ∈ A , is well-defined because x is cyclic and separating for A . It extends by continuityto a bounded operator in A ′ because || ZAx || = || Az || = ω z ( A ∗ A ) ≤ λω x ( A ∗ A ) = λ || Ax || , A ∈ A . Let Z = J | Z | be the polar decomposition of Z and let Y = | Z | and y = Y x .Then one has ω y = ω z and ( y, A ∗ Ax ) = || A | Y | / x || ≥ , A ∈ A . This shows that y belongs to C x and hence that ω belongs to V x . Finally note that X y,x = Y = Z ∗ Z . (b) implies (a) One has for all A in A ω y ( A ∗ A ) = ( AX / y,x x, AX / y,x x ) = ( X / y,x Ax, X / y,x Ax ) ≤ || X y,x || ( Ax, Ax ) = || X y,x || ω x ( A ∗ A ) . This shows that ω y is majorized by || X y,x || ω x . (cid:3) Corollary 4.7 ˘ V x ⊂ V x . If the operator X y,x is bounded then one does not need to take care about do-main problems. From Definition 3.1 one obtains for t ∈ [0 , γ y,x ( t ) = e − ζ y,x ( t ) exp (cid:18) − t H y,x (cid:19) x, with H y,x = − log X y,x . The normalization ζ y,x ( t ) can be written as ζ y,x ( t ) = 12 log (exp ( − tH y,x ) x, x ) . The following result follows immediately from Proposition 2.4.18 roposition 4.8 ˘ V x is norm-dense in V x . Proposition 4.9
One has the following. (a)
The set of states ˘ V x is convex; (b) The norm closure V x of the set of states V x is convex. Proof(a)
Take y and z in C x and λ in [0 , . If ω y and ω z belong to ˘ V x then they aredominated by multiples of ω x and so is λω y + (1 − λ ) ω z . It then follows fromProposition 4.6 that the convex combination belongs to ˘ V x . (b) Take ω (1) and ω (2) in V x and let λ ∈ [0 , . There exist states ω (1) n and ω (2) n in ˘ V x converging to ω (1) , respectively ω (2) . The states λω (1) n + (1 − λ ) ω (2) n belong to ˘ V x by item (a) of the proposition and converge to λω (1) + (1 − λ ) ω (2) . Hence, thelatter state belongs to the closure of ˘ V x which coincides with the closure of V x . (cid:3) Proposition 4.10 If y belongs to C x and is cyclic for A then one has ˘ V x = ˘ V y and V x = V y . Proof If y belongs to C x and is cyclic for A then Proposition 3.10 asserts that x belongs to C y as well. Hence there exist positive constants λ y,x and λ x,y such that ω y ≤ λ y,x ω x and ω x ≤ λ x,y ω y Take now z in ˘ V x . Then there exists λ z,x such that ω z ≤ λ z,x ω x ≤ λ z,x λ x,y ω y . Hence, ω z belongs to ˘ V y . This shows that ˘ V x ⊂ ˘ V y . By symmetry one concludesthat ˘ V x = ˘ V y . Because they are dense in V x respectively V y one concludes thatalso V x = V y . (cid:3) .3 A neighborhood of faithful states Definition 4.11
The set of vector states ω y with y in ˚ γ x is denoted ˚ V x . Proposition 4.12
Take y in C x . If X y,x is bounded with bounded inverse then allvector states ω u with u = γ y,x ( t ) and − < t < belong to ˚ V x . Proof
Let z = X − / y,x . Then x is the midpoint of the exponential arcs γ z,y ( t ) and γ y,z ( t ) . By Proposition 3.9 all vectors of these arcs are cyclic and separating for A . Hence, one has ˚ γ y,x ⊂ ˚ γ x . From the above definition it then follows that for all z in ˚ γ y,x the vector state ω z belongs to ˚ V x . (cid:3) The bounded operators with bounded inverse and belonging to the commutant A ′ form a real Banach-Lie group in the topology induced by the norm operatortopology. This observation is the starting point of [22, 23]. Theorem 4.13
Let x be a normalized vector cyclic and separating for the vonNeumann algebra A . Choose a vector state ω in ˚ V x . Let t ∈ [0 , ω t be theexponential arc connecting ω to ω x . One has the following. (a) The exponential arc t ∈ [0 , ω t can be extended to the interval [ − , in such a way that t ω t − is the exponential arc connecting ω to ω − ; (b) The Fr´echet derivative ˙ ω t of t ω t exists for t in the open interval ( − , and is given by ˙ ω t ( A ) = ( Ay t , [log X y,x ] y t ) − ( Ay t , y t )( y t , [log X y,x ] y t ) , where y t is the unique vector in C x such that ω t = ω y t . The operator X y ,x isthe positive operator affiliated with the commutant A ′ as given by Definition2.5. Proof(a)
From the definition of the exponential arc t ∈ [0 , ω t it follows that thereexist y in C x such that ω t = ω y t with y t = γ y,x ( t ) . Because ω = ω = ω y belongsto ˚ V x there exists u in ˚ γ x such that ω = ω u . From the definition of ˚ γ x it followsthat there exist y, z in C x such that u ∈ ˚ γ y,z , x is the midpoint of γ y,z and y and z A . By Theorem 3.11 x is also the midpoint of γ u,v forsome v belonging to the exponential arc γ y,z . By Proposition 3.13 the exponentialarc connecting u to x can now be extended in such a way that γ u,x ( −
1) = v . Now ω t can be extended to all of [ − , by putting ω t = ω y t with y t = γ u,x ( t ) . Byconstruction it is the exponential arc connecting u to v . (b) One calculates with the help of ˙ ω t ( A ) = dd t ( Ay t , y t )= ( A ˙ y t , y t ) + ( Ay t , ˙ y t ) with ˙ y t = (cid:20)
12 log X y,x − ζ ′ y,x ( t ) (cid:21) y t . It satisfies | ω t + ǫ ( A ) − ω t ( A ) − ǫ ˙ ω t ( A ) | = | ( Ay t + ǫ , y t + ǫ ) − ( Ay t , y t ) − ǫ ( Ay t , ˙ y t ) − ǫ ( A ˙ y t , y t ) |≤ | ( Ay t + ǫ , y t + ǫ ) − ( Ay t + ǫ , y t ) − ǫ ( Ay t , ˙ y t ) | + | ( Ay t + ǫ , y t ) − ( Ay t , y t ) − ǫ ( A ˙ y t , y t ) |≤ || A || || y t + ǫ − y t − ǫ ˙ y t ) || + || A || || y t + ǫ , y t − Ay t − ǫ ˙ y t || = o ( ǫ ) . This shows that ˙ ω t is the Fr´echet derivative of t ω t . (cid:3) Let A be the algebra of diagonal n -by- n matrices with complex entries. They acton the Hilbert space C n . Let x be a normalized element of C n with non-vanishingcomponents. The corresponding probabilities are denoted p i and are given by p i = | x i | . The state ω x on A is defined by ω x ( A ) = ( Ax, x ) = n X i =1 p i A i,i . C x consists of element y of C n satisfying y i x i ≥ for i = 1 , , · · · n .Hence, for each y in C x there exists a non-negative number ˜ y i such that y i = ˜ y i x i . In particular, the operator X y,x is the diagonal matrix with entries ( X y,x ) i,i = (˜ y i ) .One verifies that for any A in A one has ( AX / y,x x, X / y,x x ) = X i A i,i ( X y,x ) i,i | x i | = X i A i,i (˜ y i ) | x i | = X i A i,i | y i | = ω y ( A ) . Note that ˘ C x = C x because all operators on C n are bounded. The algebra A coincides with its commutant A ′ . Hence, cone and dual cone coincide. In thiscommutative example all cones C x are self-dual. The natural positive cone consistsof all vectors x with non-negative components x i ≥ . In Hilbert space
The exponential arc γ y,x ( t ) is given by (2). This gives ( γ y,x ( t )) i = e − ζ y,x ( t ) ( X t/ y,x ) i,i x i = e − ζ y,x ( t ) (˜ y i ) t x i . Note that y i = 0 implies that y ti = 0 for any t > . However, if t = 0 then byconvention (˜ y i ) t = 1 whatever the value is of ˜ y i . Hence, discontinuities can occurat t = 0 .The normalization function is given by ζ y,x ( t ) = log || X t/ y,x x || = 12 log X i (˜ y i ) t | x i | . If one or more of the y i vanish then the function ζ y,x ( t ) is discontinuous at t = 0 .If t > then the function is differentiable and one has ζ ′ y,x ( t ) = e − ζ y,x ( t ) X i ′ (log ˜ y i )(˜ y i ) t | x i | , where the sum Σ ′ is restricted to those terms for which ˜ y i = 0 . The second deriva-tive then becomes ζ ′′ y,x ( t ) = 2 e − ζ y,x ( t ) X i ′ (log ˜ y i ) (˜ y i ) t | x i | − (cid:2) ζ ′ y,x ( t ) (cid:3) e − ζ y,x ( t ) X i ′ (cid:2) log ˜ y i − ζ ′ y,x ( t ) (cid:3) (˜ y i ) t | x i | . The vector x is a midpoint of an arc ending at a vector y of C x if and only if y i = 0 for all i . If this is the case then by Proposition 3.13 the path t γ y,x ( t ) canbe extended to the interval [ − , . It is actually clear from the explicit expressiongiven above that it can be extended to all of the real line. The set ˚ γ x consists of all y in C x such that y i = 0 for all i . In particular, it does not depend on the choice of x . In state space
Starting from the assumption that x i = 0 for all i one can eas-ily show that any state on the algebra A of diagonal matrices is majorized by amultiple of the state ω x . Hence, by Proposition 4.6 all states on A belong to theneighborhood ˘ V x and a fortiori to V x .Take y in C x . Let p i = | x i | and q i = | y i | . The exponential arc connecting ω y to ω x satisfies the equation ˙ ω t ( A ) = ω t ( AH ) − ω t ( A ) ω t ( H ) with the generator H given by H i,i = [log X y,x ] i,i = log q i p i . In particular the tangent vector at t = 0 is given by ˙ ω t ( A ) (cid:12)(cid:12)(cid:12)(cid:12) t =0 = ω x ( AH ) + D ( ω x || ω y ) ω x ( A ) with the divergence D ( ω x || ω y ) given by D ( ω x || ω y ) = X i p i log p i q i . What follows here is partly based on [24, 25].Consider the von Neumann algebra of n -by- n matrices. In its standard formit acts on the Hilbert space H = C n ⊗ C n (up to a ∗ -isomorphism). Any vector x of H can be written as a linear combination of vectors of the form x (1) ⊗ x (2) .Any operator A of A is of the form A = A (1) ⊗ I , where A (1) is an n -by- n matrixand I is the identity matrix. Any element A ′ of the commutant A ′ is of the form A ′ = I ⊗ A (2) . 23or each state ω on A there exists a unique density matrix ρ ω such that ω ( A ) = Tr ρ ω A (1) . Let ( e i ) i be a basis of eigenvectors of ρ ω with corresponding eigenvalues p i . Thena vector x is given by x = X i √ p i e i ⊗ e i . (14)It satisfies ω x ( A ) = X i,j √ p i p j (cid:0) A (1) e i ⊗ e i , e j ⊗ e j (cid:1) = X i p i (cid:0) A (1) e i , e i (cid:1) = Tr ρ ω A (1) . Assume that p i > for all i . Then the vector x , as given by (14), is cyclicand separating for A . A straightforward calculation shows that the correspondingmodular operator ∆ x equals ρ ω ⊗ ρ − ω .Let ω y be an arbitrary vector state with y in C x . Then there exists a uniquedensity matrix ρ y such that ω y ( A ) = Tr ρ y A (1) , A ∈ A . It is shown in [25] that the operator X y,x is given by X y,x = S x (cid:2) ρ − x ρ y ⊗ I (cid:3) S x = J x (cid:0) ρ − / x ρ y ρ − / x ⊗ I (cid:1) J x and satisfies X y,x x = (cid:2) ρ y ρ − x ⊗ I (cid:3) x. The vector y is then given by y = X / y,x x . Exponential arcs
Take y in C x . One has, making use of J x = I and J x x = x , X t/ y,x x = J x (cid:0) [ ρ − / x ρ y ρ − / x ] t/ ⊗ I (cid:1) J x x = X i √ p i J x (cid:0) [ ρ − / x ρ y ρ − / x ] t/ e i ⊗ e i (cid:1) . Hence one finds ζ y,x ( t ) = log || X t/ y,x x || log || X i √ p i [ ρ − / x ρ y ρ − / x ] t/ e i ⊗ e i || = 12 log X i p i (cid:0) [ ρ − / x ρ y ρ − / x ] t e i , e i (cid:1) = 12 log Tr ρ x [ ρ − / x ρ y ρ − / x ] t . Take y in C x . The exponential arc t ω t connecting ω y to ω x is given by ω t ( A ) = e − ζ y,x ( t ) ( AX t/ y,x x, X t/ y,x x ))= e − ζ y,x ( t ) X i,j √ p i p j (cid:18) AJ x (cid:0) [ ρ − / x ρ y ρ − / x ] t/ e i ⊗ e i (cid:1) ,J x (cid:0) [ ρ − / x ρ y ρ − / x ] t/ e j ⊗ e j (cid:1) (cid:19) = e − ζ y,x ( t ) X i,j √ p i p j (cid:18) [ ρ − / x ρ y ρ − / x ] t/ e j ⊗ e j ,A ′ (cid:0) [ ρ − / x ρ y ρ − / x ] t/ e i ⊗ e i (cid:1) (cid:19) with A ′ = J x AJ x . Use that A ′ belongs to the commutant A ′ to write A ′ = I ⊗ R .One obtains ω t ( A ) = e − ζ y,x ( t ) X i,j √ p i p j (cid:18) [ ρ − / x ρ y ρ − / x ] t/ e j ⊗ e j , (cid:0) [ ρ − / x ρ y ρ − / x ] t/ e i ⊗ Re i (cid:1) (cid:19) = e − ζ y,x ( t ) X i,j √ p i p j (cid:0) [ ρ − / x ρ y ρ − / x ] t/ e j , [ ρ − / x ρ y ρ − / x ] t/ e i (cid:1) × ( e j , Re i ) . Now use that (see the appendix of [24]) ( e j , Re i ) = ( e i , A ∗ e j ) to obtain ω t ( A ) = e − ζ y,x ( t ) X i,j √ p j (cid:0) [ ρ − / x ρ y ρ − / x ] t/ e j , [ ρ − / x ρ y ρ − / x ] t/ e i (cid:1) × ( e i , ρ / x A ∗ e j )= e − ζ y,x ( t ) X j (cid:0) [ ρ − / x ρ y ρ − / x ] t/ ρ / x e j , [ ρ − / x ρ y ρ − / x ] t/ ρ / x A ∗ e j (cid:1) = Tr ρ t A ρ t given by ρ t = e − ζ y,x ( t ) ρ / x [ ρ − / x ρ y ρ − / x ] t ρ / x . (15)This exponential arc differs from the geodesic studied in [24, 25]. The latter isgiven by ρ t ∼ exp((1 − t ) log ρ x + t log ρ y ) . (16)The problem when generalizing (16) to infinite dimensions is that the operators log ρ x and log ρ y need not to commute and even need not to have a common domainso that the sum appearing in the exponential function of (16) may be ill-defined.Note that if ρ x and ρ y commute then (15) and (16) coincide. In this section the von Neumann algebra A is (isomorphic with) the space L ∞ ( R n , C ) of all essentially bounded complex functions on R n with its Lebesgue measure.The Hilbert space H coincides with the space of square-integrable complex func-tions L ( R n ) . The symbol x denotes an element of R n instead of a vector in H . Afunction A ( x ) , element of A , acts on a square-integrable function f ( x ) by point-wise multiplication ( Af )( x ) = A ( x ) f ( x ) . The commutant A ′ of A coincides with A . See [36], Part I, Ch. 7, Thm. 2.Any normalized element f of L ( R n ) determines a state ω f on A by ω f ( A ) = Z R n A ( x ) | f ( x ) | d x. The correspondence between probability measures | f ( x ) | d x and vector states ω f is one-to-one.The function f is cyclic and separating for A if it is strictly positive almosteverywhere. The cone C f consists of all square integrable functions g for whichthe ratio ˜ g = g/f is a non-negative function a.e.. The operator X / g,f equals thefunction ˜ g . Clearly is X / g,f f = g . The state ω g satisfies ω g ( A ) = ω f ( X g,f A ) . Theoperator X g,f is the Radon–Nikodym derivative of the measure | g ( x ) | d x w.r.t. themeasure | f ( x ) | d x .The square integrable function g belongs to ˘ C f if ˜ g is a bounded function. Exponential arcs
Take g in C f . The exponential arc γ g,f connecting g to f isgiven by γ g,f ( t )( x ) = e − ζ g,f ( t ) [˜ g ( x )] t f ( x ) , ζ g,f given by ζ g,f ( t ) = 12 log Z R n [˜ g ( x )] t | f ( x ) | d x. The Fr´echet derivative satisfies the equation ˙ γ g,f ( t )( x ) = (cid:2) − ζ ′ g,f ( t ) + log ˜ g ( x ) (cid:3) γ g,f ( t )( x ) with ζ ′ g,f ( t ) the derivative of ζ g,f ( t ) . It is well-defined when ˜ g ( x ) = 0 implies γ g,f ( t )( x ) = 0 . In particular, the tangent vector at t = 0 is given by ˙ γ g,f (0)( x ) = [log ˜ g ( x ) − ω f (log ˜ g ( x ))] f ( x ) . The exponential arc γ g,f ( t ) is discontinuous at t = 0 when ˜ g ( x ) is not strictlypositive a.e.. Hence, the set ˚ γ f consist of all vectors g in C f for which ˜ g ( x ) > a.e.. The corresponding probability measures | f ( x ) | d x and | g ( x ) | d x are thenequivalent measures. The set V x contains all states ω g such that the correspondingprobability measures are absolutely continuous w.r.t. the measure | f ( x ) | d x .The exponential arc connecting the state ω g to the state ω f is given by ω t ( A ) = Z R n A ( x ) | ˜ g ( x ) | t | f ( x ) | d x = Z R n A ( x ) | f ( x ) | − t ) | g ( x ) | t d x. This expression for the exponential arc is found for instance in [33].
The main result of the present paper is the definition of exponential arcs both inHilbert space H and in the space V of vector states on a σ -finite von Neumannalgebra A in its standard form [37]. A proof is given that these definitions arelogically consistent. See Theorems 3.4 and 3.11. Theorems 3.7 and 4.13 showthat the arcs are Fr´echet differentiable. The special cases treated in the previoussection show that the formulation in terms of σ -additive von Neumann algebrascovers both quantum models and models of classical probability.The present work makes use of powerful results of Tomita-Takesaki theory [2].The theory describes the symmetry that may exist between a von Neumann algebraand its commutant. The underlying structure is a symmetry between a pair of realsubspaces of the Hilbert space [38]. This raises the hope that eventually one canformulate a theory in which unbounded operators can be avoided.27he generalization of the notion of an exponential arc to a non-commutativecontext can be done in more than one way. The exponential arcs introduced in thepresent paper differ from the geodesics discussed in [24, 25]. Araki [1] introducesan α -family of cones in Hilbert space, with α running from to / . The presentpaper focuses on the α = 1 / -cone while [24, 25] is intended to correspond withthe self-dual case α = 1 / . Some of the technical problems one encounters in thelatter case, when trying to generalize to an infinite-dimensional Hilbert space, areavoided in the present paper. In the commutative case all α -cones coincide and thepresent approach reproduces known results.The definition of exponential arcs is only a first step in the study of the manifoldof vector states from the view point of Information Geometry. The next step is thestudy of tangent planes, the search for Banach or Hilbert charts and the introductionof a Riemannian metric. Acknowledgment
I am indebted to an anonymous referee carefully reading a previous version of thepaper and suggesting to work out the link with the natural positive cone.This leadto a complete overhaul of the paper.
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