Exponential arcs in manifolds of quantum states

Abstract

The manifold under consideration consists of the faithful normal states on a sigma-finite von Neumann algebra in standard form. Tangent planes and approximate tangent planes are discussed. A relative entropy/divergence function is assumed to be given. It is used to generalize the notion of an exponential arc connecting one state to another. The generator of the exponential arc is shown to be unique up to an additive constant. In the case of Araki’s relative entropy, every self-adjoint element of the von Neumann algebra generates an exponential arc. The generators of the composed exponential arcs are shown to add up. The metric derived from Araki’s relative entropy is shown to reproduce the Kubo–Mori metric. The latter is the metric used in linear response theory. The e- and m-connections describe a dual pair of geometries. Any finite number of linearly independent generators determines a submanifold of states connected to a given reference state by an exponential arc. Such a submanifold is a quantum generalization of a dually flat statistical manifold.

1 Introduction

The goal of the present paper is to show that the theory of quantum statistical manifolds can be formulated without reference to density matrices. It is tradition to describe the statistical state of a quantum model by a density matrix. In many cases this suffices, in particular when the Hilbert space of wave functions is finite-dimensional. However, even simple models such as the quantum harmonic oscillator or the hydrogen atom require an infinite-dimensional Hilbert space. This involves handling of unbounded operators which cause considerable technical complications. These complications are avoided in the present work.

A one-to-one correspondence between density matrices and quantum states is usually accepted. The quantum states form the sample space of the statistical description. An alternative description emerged in the past century, which introduced the notion of a mathematical state on an algebra of observables which can be realized as an algebra of bounded operators on Hilbert space. See for instance [1–5].

Equilibrium states of quantum statistical mechanics are described by the quantum analogue of the probability distribution of Gibbs, which is a density matrix ρ of the formwith H a Hermitian matrix, β a parameter the inverse temperature, and Z a function of β used to normalize density matrix ρ so that its trace equals 1. Models described in this way can belong to a quantum exponential family. They possess an intriguing property called the Kubo–Martin–Schwinger (KMS) condition [6]. The KMS condition describes a symmetry property of the time evolution of quantum states. This symmetry coincides with the symmetry between left and right multiplication of operators, which is studied in the Tomita–Takesaki theory [7]. [5] can be used as a reference text for this theory.

The notion of a statistical manifold is studied in information geometry ([8–12]). It is a manifold of probability distributions. The quantum analogue is described in Chapter 7 of [11] as a manifold of k by k density matrices. The book of Petz [13] reviews several aspects of quantum statistics, including the basics of quantum information and quantum information geometry.

The generalization of Amari’s dually flat geometry from statistical models with a finite number of parameters to Banach manifolds of mutually equivalent probability measures started with the work of [14]. Non-commutative versions were formulated by [15–19].

The convex set of faithful normal states on a σ-finite von Neumann algebra is in general not a Banach manifold. The point of view taken in the present work is that the set should, by definition, be a quantum statistical manifold. This raises the question of how to transfer common notions of differential geometry and of Banach manifolds to this quantum setting. The present work contributes to this effort.

The relative entropy of Umegaki [20] is the starting point to implement Amari’s dually flat geometry on the quantum manifold. It should be noted that relative entropy is called a divergence function in mathematical literature. Araki [21–23] generalizes Umegaki’s relative entropy to the context of mathematical states on an algebra of bounded operators on a Hilbert space. The use of Araki’s relative entropy replacing that of Umegaki’s is the core of the present work.

Exponential arcs were introduced in [24, 25] and used in [26]. These arcs can be considered one-parameter exponential families embedded in the manifold. The maximal exponential model centered at a given probability distribution p equals the set of all probability distributions connected to p by an open exponential arc. Exponential arcs were studied in the quantum setting by [27]. Here, the definition is generalized. The exponential arcs are used to define quantum statistical manifolds as submanifolds of the manifold of all quantum states.

The Radon–Nikodym Theorem plays an important role in probability theory. For each measure absolutely continuous with respect to the reference measure, there exists an essentially unique probability distribution function. The problem that arises in the non-commutative context is the non-uniqueness of the Radon–Nikodym derivative. This leads to different definitions of the relative entropy and of the exponential arcs. First attempts to reformulate the theory of the quantum statistical manifold in terms of states on a C*-algebra are found in [28,29] and in [27]. These two approaches differ in the choice of the Radon–Nikodym derivative. In the present work, the definition of an exponential arc is generalized so that it depends explicitly on the choice of relative entropy and in that way on the choice of the Radon–Nikodym derivative.

The alternative approach of [30] relies on the Lie Theory for the group of bounded operators with bounded inverse. The state space is partitioned into the disjoint union of the orbits of an action of the Lie group. Under mild conditions, it is shown that the orbits are Banach manifolds. The restriction to bounded operators implies that the orbits do not connect quasi-equivalent states when the Radon–Nikodym derivatives are unbounded operators.

Sections 2–4 give a short introduction on KMS states, on the theory of the modular operator, and on positive cones. Section 5 gives a definition of the manifold under study as the convex set of faithful normal states on a sigma-finite von Neumann algebra. The tangent space consists of linear functionals on the algebra. Its extent depends on the chosen topology, and it is not obvious how to find a good compromise. Therefore, the notion of approximate tangent vectors is considered in Section 6.

A dense subset of the manifold consists of states majorized by a multiple of the reference state. This subset of states is mentioned in Section 7 because it is easier to handle.

Section 8 gives a new definition of exponential arcs. It generalizes existing concepts and is broad enough to cover different approaches. The definition depends on the choice of a relative entropy/divergence function. Such an exponential arc can be seen as a one-dimensional sub-manifold and as a straightforward example of a quantum statistical manifold. Duality properties well-known for models of information geometry are elaborated in Section 9.

The important example of the algebra of n-by-n matrices is considered in Section 10.

Starting with Section 11 the paper specializes to the case of Araki’s relative entropy. It is shown in Section 13 that each self-adjoint element h of the von Neumann algebra defines an exponential arc defined relative to Araki’s relative entropy and starting at the reference state ω. The initial derivative of the arc exists as a Fréchet derivative and belongs to the tangent plane . The inner product between two such tangent vectors reproduces the metric which is used in the Kubo–Mori Theory of linear response. This is shown in Section 14. The exponential arcs are shown to be geodesics for the e-connection which is, by definition, the dual of the m-connection.

Section 16 applies the results obtained so far to show that manifolds generated by a finite number of exponential arcs have the properties one expects from a quantum statistical manifold.

A few points of concern are discussed in the final Section 17.

2 KMS states

Equilibrium states of quantum statistical mechanics satisfy the KMS condition. In the GNS representation, an equilibrium state becomes a faithful state on aσ-finite von Neumann algebra of operators on a complex Hilbert space. The state is defined by a normalized cyclic and separating vector in the Hilbert space.

The state of a model of statistical physics can be described by a mathematical state on a C*-algebra . It can be represented by a normalized vector Ω (a wave function) in a Hilbert space . This is known as the GNS (Gelfand–Naimark–Segal) representation theorem. Observable quantities are represented by self-adjoint operators on . The quantum expectation ⟨x⟩ of operator x is then given bywith in the right-hand side the scalar product of the two vectors xΩ and Ω. It should be noted that the mathematical convention is followed that the scalar product (inner product) is linear in its first argument and conjugate-linear in the second argument. In Dirac’s bra-ket notation, it reads

For convenience, one works with a von Neumann algebra of bounded operators on the Hilbert space . Observables of interest, when unbounded, are represented by operators affiliated with . The state ω on the C*-algebra extends to a vector state on again denoted ω. It is given by

The vector Ω is cyclic for , which means that the subspace is dense in the Hilbert space . It is also assumed in what follows that the state ω is faithful, i.e., ω(x*x) = 0 implies x = 0. This implies that Ω is a separating vector for , i.e., xΩ = 0 implies x = 0 for any x in , and hence it is a cyclic vector for the commutant of , the algebra of all operators commuting with all of .

Equilibrium states of statistical mechanics are characterized by the KMS (Kubo–Martin–Schwinger) condition [6]. Roughly speaking, this condition states that the quantum time evolution of the model has an analytic extension into the complex plane. This is made more precise in what follows.

The time evolution is described by a strongly continuous one-parameter group of unitary operators which leave the algebra unchanged, i.e., implies that belongs to for all t. The operators u_t are determined by a self-adjoint operator Hwhich is the generator of the time evolution in the GNS representation. The time derivative of x_t satisfies

This equation has the same form as Heisenberg’s equation of motion.

The KMS condition requires that for any pair x, y of operators in , there exists a complex function F(w), defined and continuous on the strip −β ≤ Im w ≤ 0 and analytical inside with boundary values

In the mathematics literature, the parameter β, which is the inverse temperature of the model, is usually taken equal to 1 or -1.

An immediate consequence of the KMS condition being satisfied is that the state ω is invariant. Indeed, take y equal to the identity operator. Then, one has F (t—iβ) = F(t) for all t in . If in addition, x is self-adjoint, then F(t) is a real function. From the Schwarz reflection principle, one then concludes that F(w) is a constant function. This implies ω(x_t) = ω(x) for all self-adjoint x and hence for all x. The GNS theorem then guarantees that the vector Ω can be taken to be invariant, i.e., u_tΩ = Ω for all t.

3 The modular operator

The quantum-mechanical time evolution coincides with the modular automorphism group of Tomita–Takesaki theory.

The KMS condition, when satisfied, expresses a symmetry which is present in the context of non-commuting operators. The symmetry is the inversion of the order of multiplication of operators. In non-commutative groups, the modular function links left and right Haar measures. The analogue in functional analysis is studied in the theory of the modular operator, also called the Tomita–Takesaki theory [7].

The operator e^−βH with H the generator of the quantum-time evolution is traditionally denoted as Δ_Ω. It is the modular operator of the Tomita–Takesaki theory. It is in general an unbounded operator such that is in the domain of the definition of the square root of Δ_Ω. Hence, the expressionis well-defined for 0 ≥ Im w ≥−β/2. The other half of the strip 0 ≥ Im w ≥−β is covered by the Schwarz reflection principle. Indeed, if x and y are self-adjoint, then one can show with the Tomita–Takesaki theory that the map t↦F (t − iβ/2) is a real function. Hence, the principle can be applied to obtain .

The unitary time evolution operator u_t can be written asThe time evolution of an operator x in the Heisenberg picture is then given by

The action of the group is called the modular automorphism group.

The modular conjugation operator J of the Tomita–Takesaki Theory represents the symmetry which is at the basis of the theory. It is a conjugate-linear operator satisfying J = J* and . Operator x belongs to the von Neumann algebra if and only if JxJ belongs to the commutant algebra . The latter is the space of operators commuting with all operators in . The product is denoted as S_Ω and has the property of

4 Dual cones

The natural positive coneis needed in subsequent sections. One reason for making use of it is that there exists a one-to-one correspondence between normal states onand normalized vectors in.

Section 4 of [22]introduces the cones , 0 ≤ α ≤ 1/2, of the vectors in . The self-dual cone is called the natural positive cone and is denoted as .

By definition, is the closure of the cone

The cone is used in [27] to introduce exponential arcs. It is equal to the closure of the setTo see this note thatwith y = Jx*J. The latter is an arbitrary element of the commutant .

The following characterization of the natural positive cone is found in Section 2.5 of [5].

Proposition 1: The cone equals the closure of the set of vectors

This result can be understood as follows. Take Φ in of the form (3), i.e., Φ = xJxΩ with x in . Let

This expression can be inverted toso that

Assume now that one could prove that the operator y defined by (4) belongs to ; then, the above calculation would show that Φ is of the form with a = yy* a positive element of . The actual proof of the proposition uses that belongs to .

The cone is independent [22] of the choice of the cyclic and separating vector Ω in , and the isometry J is the same for all these choices. For this reason, it is said to be universal.

From (3), it is easy to see that each vector in is an eigenvector with eigenvalue 1 of the modular conjugation operator J. Indeed, one hasHere, use is made of JΩ = Ω and the fact that the operators x and JxJ commute with each other.

5 A manifold of quantum states

A manifoldof vector states on the von Neumann algebrais defined. Tangent vector fields are Fréchet derivatives of paths in.

Introduce the notation ω_Φ for the vector state defined by the normalized vector Φ in . It is given by

A manifold of states on the von Neumann algebra is defined by

The equilibrium state ω = ω_Ω is taken as a reference point in . The subset of is the natural positive cone introduced in the previous section.

The topology on the manifold is that of the operator norm. One has

Several topologies can be defined on the algebra . Particularly relevant is the σ-weak topology. For what follows, it is important to know that in the present context, a state ω on is said to be normal if and only if it is σ-weakly continuous and if and only if it is a vector state. See for instance, Theorems 2.4.21 and 2.5.31 of [5].

Any tangent vector is a σ-weakly continuous linear functional on the von Neumann algebra . Let t↦γ_t be a Fréchet differentiable map defined on an open interval of with values in the manifold . The derivativeis required to exist as a Fréchet derivative, i.e., it satisfies

From the normalization, γ_t (1) = 1 for all t in the domain of the map, one obtains . From one obtains . Hence, the linear functional is Hermitian.

There are several ways to define the tangent space at the point ω in . Intuitively, a tangent vector is a derivative, defined in some sense, of a path t↦γ_t in passing through the point ω. The states of the manifold belong to the space of all σ-weakly continuous linear functionals on the algebra (see Proposition 2.4.18 of [5]). Hence, one expects that tangent vectors belong to as well.

In this section, the requirement is made that the path t↦γ_t is Fréchet-differentiable. This may be too restrictive. In what follows, we adopt the definition that the tangent space consists of all Hermitian χ in , satisfying χ(1) = 0. It should be noted that it is well-possible that for certain elements χ of this space, there is no smooth curve passing through ω with the property that the derivative at ω equals χ.

6 Approximate tangents

Approximate tangent vectors can be defined in an intrinsic manner.

An alternative definition of the tangent space starts from the following observation.

Proposition 2: The set defined byis a linear subspace of the tangent space .

Proof:

Let ϕ and ψ be two states in such that . Construct a Fréchet-differentiable path γ by

The state γ_t belongs to the manifold because the latter is a convex set. In particular, one has ω = γ_1/2 and is a tangent vector. This shows that ϕ − ψ and hence also λ(ϕ − ψ) belongs to . One concludes that .

Assume now that λ(ϕ − ψ) and λ′(ϕ′ − ψ′) both belong to . We have to show thatbelongs to . If λ = 0 or λ′ = 0, then the claim is clearly satisfied. Without restriction, assume λ = 1.

If λ′ > 0, then chooseBoth ϕ^″ and ψ^″ belong to because the latter is a convex set. One verifies that ϕ^″ + ψ^″ = 2ω and

This shows that the latter sum belongs to .

In the case that λ′ < 0, one choosesto reach the same conclusion. This finishes the proof that is a linear subspace of .

We introduce the notationsand

The construction of is analogous to the construction of the approximate tangent space in Chapter 3 of [31]. Clearly, . Further properties are derived below.

Proposition 3: If γ is a Fréchet-differentiable path in , then belongs to with ω = γ_t.

Proof:

Let γ be a Fréchet-differentiable path in . Without restriction of generality, assume that γ₀ = ω. For any ϵ > 0 and δ > 0, there exists t ≠ 0 such thatand

Then, ϕ defined bysatisfies ‖ϕ—ω‖ < ϵ, and (γ_t—γ_−t)/2t belongs to . Hence, (5) shows that the tangent vector belongs to the closure of . Because ϵ > 0 is arbitrary, it also belongs to the intersection, which is .

Lemma 1: is a linear subspace of .

Proof:

Take χ and ξ in . There exist ϕ and ψ in such that and with ‖ϕ—ω‖ < ϵ and ‖ψ—ω‖ < ϵ. Therefore, there exist real λ, μ and states ϕ₁, Φ₂, Ψ₁, Ψ₂ in such thatand

If λ = 0 or μ = 0, then χ + ξ belongs to without further argument. Assume, therefore, that λ ≠ 0 and μ ≠ 0. If λμ > 0, then χ + ξ belongs to , with π = (1 − α)ϕ + αψ and α given by

Indeed, let

Then both π₁ and π₂ belong to and satisfyand

In addition,

One concludes that in this case, χ + ξ belongs to .

The case that λμ < 0 is similar. That implies is straightforward. One can conclude that is a linear space. It clearly is a subspace of .

Proposition 4: is a closed linear subspace of .

Proof:

The lemma shows that is a linear subspace of , which is a space closed in norm. Hence, also the norm closure of is a subset of this space and therefore also of .

7 Majorized states

The subset of states majorized by a multiple of the reference stateωis considered.

Definition 1: A state ϕ on is said to be majorized by a multiple of the state ω if there exists a positive constant λ such that

Take a′ ≠ 0 in the commutant algebra and let

Then, the state ω_Φ is majorized by a multiple of the state ω. Indeed, one has for any positive x in

It is well-known that all states majorized by a multiple of the state ω are obtained in this way. This is the content of the following proposition.

Proposition 5: If the vector state ω_Φ is majorized by a multiple of the state ω, then there exists a unique element a′ of the commutant such that Φ = a′Ω.

Proof:

An operator a′ is densely defined by

It satisfies a′Ω = Φ. It is well-defined because xΩ = 0 impliesso that xΦ = 0.

The operator a′ is bounded because

The operator a′ commutes with any x in becauseand Ω is cyclic for .

The operator a′ is unique. Indeed, assuming b′ in satisfies Φ = b′Ω. Then, one has for all x in

Hence, a′ − b′ vanishes on which is dense in the Hilbert space because Ω is cyclic for . Because a′ − b′ is a bounded and hence continuous operator, it vanishes everywhere so that a′ = b′.

Item (8) of Theorem 3 of [22] implies the following.

Proposition 6: If a vector state ω_Φ, defined by a vector Φ in the natural positive cone , is dominated by a multiple of the state ω, then there exists a unique element a in the algebra such that Φ = aΩ and

Proof:

Proposition 5 shows that a′ in the commutant exists such that Φ = a′Ω. Because Φ and Ω both belong to , one has Φ = JΦ = Ja′JΩ.

Let a = Ja′J. From , it follows that a belongs to . This shows the existence.

The element a is unique because the correspondence between vector states on and vectors in is one-to-one and Ω is a separating vector for .

If is a commutative algebra, then a*a is the Radon–Nikodym derivative of the state ω_Φ with respect to the reference state ω.

The subset of states of majorized by a multiple of the state ω is dense in in the sense that for any state ϕ in , there exists a sequence of elements of with the property that a_nΩ is a Cauchy sequence and

See Propositions 1.5 and 2.5 of [32].

Proposition 7: A tangent vectorχbelongs to the subspace of the tangent space if and only if it is proportional to the difference of two states ϕ and ψ in , both majorized by a multiple of the state ω.

Proof:

If χ belongs to , then by definition, there exist states ϕ and ψ in such that χ = λ(ϕ − ψ) and ϕ + ψ = 2ω. The latter implies that both ϕ and ψ are majorized by 2ω.

Conversely, assume that ϕ and ψ in are both majorized by a multiple of the state ω and let χ = λ(ϕ − ψ). This implies the existence of μ ≥ 1 and ν ≥ 1 such that ϕ ≤ μω and ψ ≤ νω.

Without restriction, assume that λ > 0.

Introducewith ρ still to be chosen. By construction, it holds that ϕ′ + ψ′ = 2ω and ϕ′ − ψ′ = 2ρχ. Hence, if ϕ′ and ψ′ are states in and ρ ≠ 0, then one can conclude that χ belongs to .

Fromandone obtains

Let ρ be equal to the inverse of the maximum of λμ and λν to prove the positivity of the functionals ϕ′ and ψ′. Normalization ϕ′(1) = ψ′(1) = 1 follows from χ(1) = 0. The functions are σ-weakly continuous as well. Hence, they are states in . This ends the proof that χ belongs to .

8 Exponential arcs

[27]introduces the notion of an exponential arc in the Hilbert space, inspired by the notion of exponential arcs in probability space as introduced by[24, 25]. Here, a definition is given which depends on the choice of a relative entropy.

In the present context, a divergence function D (ϕ‖ψ) is a real function of two states ϕ and ψ in the manifold . It cannot be negative, and it vanishes if and only if the two arguments are equal. A value of + ∞ is allowed. An energy function is an affine function defined on a convex subset of the set of normal states on the algebra .

The following definition of an exponential arc in the manifold assumes that a divergence function D (ϕ‖ψ) is given.

The energy function is the generator of the exponential arc. The arc is said to connect the state γ₁ to the state γ₀.

A subclass of energy functions is formed by the functions for which there exists a self-adjoint operator h in the von Neumann algebra so that

In such a case, h is called the generator as well. The exponential arcs defined in [27] agree with the above definition with a generator defined by an unbounded operator affiliated with the commutant algebra .

Proposition 8: Expression (6) impliesand

It should be noted that with s = 0, expression (9) reduces to (6).

Proof:

Take ψ = γ_s in (6) to find

In particular, with s = t, this implies (8).

To prove (9), use (10) to write the right-hand side as

Next, eliminate D (γ₀‖γ_t) and D (ψ‖γ_s) with the help of (6). This gives

To obtain the last line, use (8).

Corollary 1: Ift↦γ_t is an exponential arc with generator that connects γ₁ to γ₀, then for any s, t in [0, 1], the map ϵ↦γ_{(1−ϵ)s+ϵt} is an exponential arc with generator that connects γ_t to γ_s.

Corollary 2: If t↦γ_t is an exponential arc with generator that connects γ₁ to γ₀, then t↦γ_1−t is an exponential arc with generator , connecting the state γ₀ to the state γ₁.

The following two propositions deal with the uniqueness of an exponential arc and of its generator.

Proposition 9: Let ω and ϕ be two states in . Fix an energy function . There is at most one exponential arc t↦γ_t with generator that connects ϕ to ω.

Proof:

Assume both t↦γ_t and t↦δ_t are exponential arcs connecting the state ϕ to the state ω. Subtract (6) from the same expression with γ_t replaced by δ_t and take s = 0. This gives

Take ψ equal to δ_t. Then, one obtains

On the other hand, with ψ = γ_t, one obtains

The two expressions together yield

This implies D (γ_t‖δ_t) = 0. By the basic property of a divergence, one concludes that γ_t = δ_t.

Proposition 10: If the exponential arc t↦γ_t has two generators and , then these generators differ by a constant on their common domain of definition.

Proof:

It follows from (6) thatfor all states ψ in the intersection of the domains of and . This implies that a constant c exists so thatfor all ψ in the common domain.

The requirement (6) is a stability condition. The generator is a perturbation which shifts the state γ₀ to the state ψ. This interpretation will become clear further on. The effect on the relative entropy of the shift along the arc t↦γ_t is linear. In the standard case, the relative entropy is based on the logarithmic function. This justifies calling the path t↦γ_t an exponential arc.

It should be noted that the Pythagorean relation [33, 34]is satisfied for all ψ with the same energy as the state γ_s, i.e., with

If the divergence function is interpreted as the square of a pseudo-distance, then the aforementioned relation states that for an arbitrary state ψ, the point γ_s of the arc which has the same energy is the point with minimal distance.

9 The scalar potential

The exponential arc has a dual structure similar to that found in information geometry[10, 11].

Given an exponential arc t↦γ_t with generator , introduce the potential Φ_γ defined by

Its Legendre transform is given by

Because divergences cannot be negative, this implies that is non-decreasing. Assume now that . Then, it follows that

On the other hand, one can use (b) to obtain

The optimal choice t = s yields the lower bound .

A dual parameter η of the exponential arc γ, dual to the parameter t, is the value of the generator . By item (a) of the proposition, it is a strictly increasing function of t. It is almost equal everywhere to the derivative of the value of the potential along the path.

10 The matrix case

Ifρandσare two density matrices, then the obvious definition of an exponential arc connectingσtoρiswith normalizationζ(t)given byIt is shown below that the corresponding states given byform an exponential arc for the relative entropy of Umegaki[20]in the GNS-representation of the stateσ₀.

Fix a non-degenerate density matrix ρ of size n-by-n. It is a positive-definite matrix with trace Tr ρ equal to 1.

Umegaki’s relative entropy for the pair of density matrices σ, τ is given by

Assume now a mapwith normalization ζ(t) and with h given byThis is the obvious definition of an exponential arc in terms of density matrices. The corresponding potential iswith

The map (14) is also an exponential arc in the sense of Definition 2. To see this, consider any density matrix τ and calculate

This is of the form (6) except that the relative entropy is expressed in terms of density matrices in instead of vector states in the GNS representation of the state defined by the density matrix ρ.

An explicit construction of the GNS representation is possible. See for instance, the appendix of [28]. Let ω = σ₀ denote the state determined by the density matrix ρfor any n-by-n matrix A with entries in . Such a matrix A is represented on the Hilbert space by the operator , where is the n-by-n identity matrix. The von Neumann algebra is the space of operators .

The matrix ρ can be diagonalized. This gives the spectral representationwhere (e_i)_i is an orthonormal basis in . Let

It is a normalized vector in . One readily verifies thatfor any n-by-n matrix A. In this way, any density matrix ρ defines a vector Ω in . The vector Ω is cyclic and separating for if ρ is non-degenerate. Hence, there is a one-to-one correspondence between non-degenerate density matrices and states in the manifold . It is then straightforward to replace the density matrices by states in the expressions obtained in the first part of this section.

11 The relative modular operator

Araki [35]introduces the relative modular operatorΔ_Φ,Ψfor any pair of vectorsΦ and Ψin the natural positive cone.

Assume that Φ and Ψ are vectors in which are separating for the algebra . Then, a conjugate–linear operator is defined by

It is well-defined because by assumption, xΨ = 0 implies that x = 0 so that also x*Φ = 0. It is a closable operator. Indeed, assume the sequence x_nΨ converges to 0. Then, one has for any y in the commutant thatconverges to 0. By assumption, Ψ is separating for so that it is cyclic for the commutant . Hence, if the sequence converges, then it converges to 0. This shows the closability of the operator.

Let S_Φ,Ψ denote the closure of this operator. It satisfies

Its inverse equals S_Ψ,Φ.

The relative modular operator Δ_Φ,Ψ is defined by

Important properties of the relative modular operator arewhere J is the modular conjugation operator for the vector Φ.

12 Araki’s relative entropy

Araki [22, 23] uses the relative modular operator Δ_Φ,Ψ to define the relative entropy/divergence D (ϕ‖ψ) of the corresponding states ϕ = ω_Φ and ψ = ω_Ψ by

Proposition 12: The divergence D (ϕ‖ψ) satisfies D (ϕ‖ψ) ≥ 0 with equality if and only if ϕ = ψ.

Proof:

Letdenote the spectral decomposition of the operator Δ_Φ,Ψ. From the concavity of the logarithmic function, it follows that

This shows that the divergence cannot be negative.

If ϕ = ψ, then one hasbecause Δ_ϕΦ = Φ.

Finally, D (ϕ‖ψ) = 0 implies that Φ is in the domain of log  Δ_Φ,Ψ and that log  Δ_Φ,ΨΦ = 0. The latter implies thatThis shows that D (ϕ‖ψ) = 0 vanishes only when Φ = Ψ.

Theorem 2.4 of [35] shows that

Because Φ belongs, by assumption, to the natural positive cone , it satisfies Φ = JΦ. Hence, one has also

13 A theorem

Each self-adjoint elementhof the von Neumann algebradefines an exponential arc with a generator equal to the energy function defined byh.

[21] constructs for each self-adjoint operator h in a vector Φ_h in the natural positive cone and calls h the relative Hamiltonian. Inspection of the explicit expression used in [21] shows thatwith operator X given by

The vector Φ_h defines a state ϕ_h by

Here, ξ(h) is the normalization

Theorem 3.10 of [35] implies that the state ϕ_h obtained in this way satisfies for all ψ in

Take ψ = ϕ_h and ψ = ω to find that the normalization ξ(h) is given by

Consider now the path γ defined by γ_t = ϕ_th. Then, (16) becomeswith

From this last expression, one obtains

From (15), we infer that γ_t converges to ω as t ↓ 0. Hence, D (γ_t‖ω) and D (ω‖γ_t) converge to 0 faster than t. This implies that the derivative exists and equals ω(h). This also implies that

Elimination of ζ(t) from (17) yieldsThis shows that γ is an exponential arc connecting γ₁ to γ₀ = ω.

Proposition 13: One haswith the operator T_Ω given by

It should be noted that this operator T_Ω was introduced in [36].

Proof:

From (15), one obtains

Writeand

The two contributions to (21) can now be taken together. One obtainsTake x = 1 to see thatso that it follows (19).

In summary, one can infer

Theorem 1: Let ω in be a vector state with cyclic and separating vector Ω. Choose the divergence function equal to the relative entropy of Araki as defined by (15). For each self-adjoint element h in , an energy function is defined by and there exists an exponential arc γ with generator connecting some state γ₁ of to the state γ₀ = ω. For any state ψ in , the derivative of D (ψ‖γ_t) at t = 0 exists and is given by ω(h) − ψ(h). The derivative of the exponential arc at t = 0 satisfies (19).

Further properties hold for the exponential arc of the above theorem.

Proposition 14: For any exponential arc γ constructed in Theorem 1, the derivative is a Fréchet derivative.

Proof:

Let Ξ(h) denote the remainder of order h² in (15), i.e.,

Then one can use (19) for

This yields

Each of the terms in the right-hand side of this expression is of order less than t as t tends to 0. Hence, is a Fréchet derivative.

Proposition 15 (Additivity of generators): If the state ϕ is connected to the state ω by the exponential arc with generator h and ψ is connected to ϕ by the exponential arc with generator k, then ψ is connected to ω by the exponential arc with generator h + k and ω is connected to ψ by the exponential arc with generator −h.

For the proof, see Proposition 4.5 of [21].

14 The metric

Eguchi [37]introduced the technique of deriving the metric of the tangent space by taking two derivatives of the divergence. Application here yields the metric which is used in the Kubo–Mori theory of linear response[38, 39].

Consider two exponential arcs t↦γ_t and s↦η_s with respective generators h and k. They connect the states γ₁ and η₁ to the reference state ω. The tangent vectors at s = t = 0 are and . They belong to the tangent space . The scalar product between them is by definition given by

Assume now that these exponential arcs are those constructed in Theorem 1. Then, one haswith the operator T_Ω defined by (20). It should be noted that in most applications, one assumes that the expectations ω(h) of the generator h and ω(k) of the generator k vanish. Then, the result obtained here coincides with that used in [36]. In what follows, a non-vanishing expectation of the generators is taken into account.

Let us now discuss some technical issues. The scalar product is well-defined by (22). This follows from

Lemma 2: If two exponential arcs with initial point ω with generators h, respectively k, both in , have the same initial tangent vector, then one has

Proof:

Let γ and η be two exponential arcs with generators h, respectively k in , such that γ₀ = η₀ = ω. Without restriction, assume that ω(h) = ω(k) = 0 and . Then, (19) implies that

Take x = h − k. Then, it follows that T_Ω(h − k)Ω = 0.

This lemma shows that the mapis one-to-one and identifies the tangent vector with the vector T_ΩhΩ in the Hilbert space .

Expression (22) defines a bilinear form. This follows from.

Lemma 3: The map (23) is linear.

Proof:

Let γ be an exponential arc with generator h in . Then, t↦γ_ϵt is an exponential arc with generator ϵh for any ϵ in [−1, 1] and the tangent vector is . Hence, (23) maps onto ϵT_ΩhΩ.

Next, consider a pair of exponential arcs γ and η with generators k, and h, respectively, in and with γ₀ = η₀ = ω. Let θ denote the exponential arc with generator h + k. It exists by Theorem 1. The state θ_t can then be written aswith Φ_th+tk being the unique element in the natural positive cone representing the state θ_t. Now, use (15) to write

This implies

Both observations together prove the linearity of map (23).

Proposition 16: Expression (22) defines a non-degenerate scalar product on the space of tangent vectors of the form with γ an exponential arc as constructed in Theorem 1.

Proof:

The two lemmas show that (22) is a well-defined bilinear form. Positivity of the form is clear. The symmetry follows from (22). It remains to be shown that it is non-degenerate.

Assume that . This implieswith h the generator of γ. The operator T_Ω is invertible—see the proof of Lemma II.2 of [36]. Hence, it follows thatBecause O is separating for , it follows that h is a multiple of the identity. The latter implies that .

15 Dual geometries

The geodesics of the e-connection are the exponential arcs. In the m-connection, the geodesics are made up by convex combinations of a pair of states. The m- and e-connections are each others’ dual with respect to the metric ofSection 14.

Consider two states ω and ϕ in the manifold . The tangent vectoris independent of t. Hence, it is a geodesic for the connection in which all parallel transport operators are taken equal to the identity operator. It should be noted that the tangent space coincides with the space of σ-weakly continuous linear functionals χ, satisfying χ(1) = 0 and hence it is the same everywhere . This connection is by definition the m-connection.

For t in (0, 1), the tangent vector belongs to the subspace of the tangent space which is introduced in Section 6. Conversely, every vector χ in is the tangent vector of an m-geodesic passing through the point γ_t. However, this does not imply that through parallel transport Π(γ_t↦γ_s), the space maps onto the space .

The transport operators Π* of the dual geometry are defined by

In this expression, V and W are vector fields and (⋅,⋅)_ω is the scalar product defined in the previous section and evaluated at the point ω of the manifold .

It can be shown that any exponential arc γ is a geodesic for this dual geometry. To do so, we have to show thatThe tangent vector at t = 0 is given by (19). Its value for arbitrary t is given by the following proposition.

Proposition 17: Let γ denote an exponential arc γ with generator h belonging to . Let Φ_t be the normalized vector in the natural positive cone representing the state γ_t. The derivative is given by

Proof:

The state γ₁ is connected to ω by the exponential arc with generator h and γ_t is connected to ω by the exponential arc with generator th. Let

It follows from Proposition 8 that s↦ψ_s is an exponential arc with generator (1 − t) h connecting γ_t to γ₁. Application of (19) to the latter arc giveswith Ψ = Φ_t. This implies (24) because .

Theorem 2: Any exponential arc γ with generator h in is a geodesic for the dual of the m-connection with respect to the metric introduced in Section 14.

Proof:

Let t↦ϕ_t be an exponential arc with generator k in such that ϕ₀ = γ_t. Fix t in [0, 1] and let Φ_t denote the normalized element of the natural positive cone representing the state γ_t. Let η be an exponential arc with generator k starting at γ_t, i.e., η₀ = γ_t. Because Π(γ_s↦γ_t) is the identity, the definition of the dual transport operator yieldswith l the generator of the arc s↦γ_(1−s)t+s. It equals l = (1 − t)h. This last expression equals

By proposition 16, the scalar product is non-degenerate. Therefore, one can conclude that

This shows that the exponential arc γ is a geodesic for the dual of the m-connection.

16 Finite-dimensional submanifolds

A finite set of linearly independent generators is shown to define a finite-dimensional submanifold in which all states are connected to the reference state by an exponential arc. The submanifold defined in this way is a dually flat quantum statistical manifold.

Let ω be the reference state of . It is a vector state with a cyclic and separating vector Ω. Choose an independent set of self-adjoint operators h₁, … , h_n in . By Theorem 1, there exists an exponential arc γ with generator h = θⁱh_i connecting some state γ₁ in to the state γ₀ = ω. A parameterized family of states ω_θ, is now defined by putting ω_θ = γ₁. These states form a submanifold of .

From the definition of an exponential arc, one obtains immediately that for any ψ in

Take ψ = ω_θ in this expression to find

Hence, the quantity θⁱω(h_i) is maximal if and only if ω_θ equals the reference state ω.

Proposition 18: Dual coordinates η_i are defined by

They satisfywith (⋅,⋅)_θ equal to the scalar product introduced in Section 14 and with basis vectors ∂_i equal to ∂ω_θ/∂θⁱ.

Proof:

Introduce the path γ⁽ⁱ⁾ defined by

It satisfies

By definition, is the end point of the exponential arc with generator . From Proposition 15, it then follows that γ⁽ⁱ⁾ is an exponential arc with generator h_i connecting to ω_θ. These arcs γ⁽ⁱ⁾ are used in the calculation that follows.

The definition of the scalar product at the beginning of Section 14 gives

Corollary 3: There exists a potential Φ(θ) such that

This follows because the scalar product is symmetric so that

This symmetry is a sufficient condition for the potential Φ(θ) to exist.

Consider the following generalization of the potential introduced in Section 9.

Apply (18) to the exponential arc γ⁽ⁱ⁾ which connects to ω_θ to find

This implies that Φ(θ) satisfies (28).

One can conclude that the selection of an independent set of self-adjoint operators h₁, … , h_n in defines a parameterized statistical model θ↦ω_θ of states on the von Neumann algebra . An obvious basis in the tangent plane is formed by the derivative operators ∂_i. The scalar product introduced in Section 14 starting from the relative entropy of Araki defines a Hessian metric on the tangent planes. Exponential arcs are geodesics for the e-connection which is the dual of the m-connection.

17 Discussion

References

• 1.

  DixmierJ. Les C*-algèbres et leurs représentations. Paris: Gauthier-Villars (1964).

  • Google Scholar

• 2.

  DixmierJ. Les algèbres d’operateurs dans l’espace Hilbertien. Paris: Gauthier-Villars (1969).

  • Google Scholar

• 3.

  RuelleD. Statistical mechanics, Rigorous results. New York: W.A. Benjamin, Inc. (1969).

  • Google Scholar

• 4.

  EmchG. Algebraic methods in statistical mechanics and quantum field theory. New Jersey, United States: John Wiley & Sons (1972).

  • Google Scholar

• 5.

  BratteliORobinsonD. Operator algebras and quantum statistical mechanics. New York, Berlin: Springer (1979).

  • Google Scholar

• 6.

  HaagRHugenholtzNMWinninkM. On the equilibrium states in quantum statistical mechanics. Commun Math Phys (1967) 5:215–36. 10.1007/bf01646342

  • CrossRef
  • Google Scholar

• 7.

  TakesakiM. Tomita’s theory of modular Hilbert algebras and its applications. In: Lecture notes in mathematics. Berlin: Springer-Verlag (1970).

  • Google Scholar

• 8.

  ChentsovNN. Statistical decision rules and optimal inference. In: Transl. Math. Monographs. Rhode Island, United States: American Mathematical Society (1972).

  • Google Scholar

• 9.

  EfronB. Defining the curvature of a statistical problem. Ann Stat (1975) 3:1189–242.

  • Google Scholar

• 10.

  AmariS. Differential-geometrical methods in statistics. In: Lecture notes in statistics. New York, Berlin: Springer (1985).

  • Google Scholar

• 11.

  AmariSNagaokaH. Methods of information geometry. In: Translations of mathematical monographs. Oxford, United Kingdom: Oxford University Press (2000).

  • Google Scholar

• 12.

  AyNJostJVân LêHSchwachhöferL. Information geometry. Berlin, Germany: Springer (2017).

  • Google Scholar

• 13.

  PetzD. Quantum information theory and quantum statistics. (Berlin: Springer-Verlag) (2008).

  • Google Scholar

• 14.

  PistoneGSempiC. An infinite-dimensional structure on the space of all the probability measures equivalent to a given one. Ann Stat (1995) 23:1543–61.

  • Google Scholar

• 15.

  GrasselliMRStreaterRF. On the uniqueness of the chentsov metric in quantum information geometry. Infin Dim Anal Quan Prob. Rel. Top. (2001) 4:173–82. 10.1142/s0219025701000462

  • CrossRef
  • Google Scholar

• 16.

  StreaterRF. Duality in quantum information geometry. Open Syst Inf Dyn (2004) 11:71–7. 10.1023/b:opsy.0000024757.25401.db

  • CrossRef
  • Google Scholar

• 17.

  StreaterRF. Quantum orlicz spaces in information geometry. Open Syst Inf Dyn (2004) 11:359–75. 10.1007/s11080-004-6626-2

  • CrossRef
  • Google Scholar

• 18.

  JenčováA. A construction of a nonparametric quantum information manifold. J Funct Anal (2006) 239:1–20. 10.1016/j.jfa.2006.02.007

  • CrossRef
  • Google Scholar

• 19.

  GrasselliMR. Dual connections in nonparametric classical information geometry. Ann Inst Stat Math (2010) 62:873–96. 10.1007/s10463-008-0191-3

  • CrossRef
  • Google Scholar

• 20.

  UmegakiH. Conditional expectation in an operator algebra. IV. Entropy and information. Kodai Math Sem Rep (1962) 14:59–85. 10.2996/kmj/1138844604

  • CrossRef
  • Google Scholar

• 21.

  ArakiH. Relative Hamiltonian for faithful normal states of a von Neumann algebra. RIMS (1973) 9:165–209.

  • Google Scholar

• 22.

  ArakiH. Some properties of modular conjugation operator of von Neumann algebras and a non-commutative Radon–Nikodym theorem with a chain rule. Pac J Math (1974) 50:309–54. 10.2140/pjm.1974.50.309

  • CrossRef
  • Google Scholar

• 23.

  ArakiH. Relative entropy of states of von Neumann algebras. Publ RIMS Kyoto Univ (1976) 11:809–33.

  • Google Scholar

• 24.

  PistoneGCenaA. Exponential statistical manifold. AISM (2007) 59:27–56. 10.1007/s10463-006-0096-y

  • CrossRef
  • Google Scholar

• 25.

  PistoneG. Nonparametric information geometry. In: NielsenFBarbarescoF, editors. Geometric science of information. Berlin, Germany: Springer (2013). p. 5–36.

  • Google Scholar

• 26.

  SantacroceMSiriPTrivellatoB. On mixture and exponential connection by open arcs. In: NielsenFBarbarescoF, editors. Geometric science of information. Berlin, Germany: Springer (2017). p. 577–84.

  • Google Scholar

• 27.

  NaudtsJ. Exponential arcs in the manifold of vector states on a σ-finite von Neumann algebra. Inf Geom (2022) 5:1–30. 10.1007/s41884-021-00064-4

  • CrossRef
  • Google Scholar

• 28.

  NaudtsJ. Quantum statistical manifolds. Entropy (2018) 20:472. 10.3390/e20060472

  • CrossRef
  • Google Scholar

• 29.

  NaudtsJ. Correction: Naudts, J. Quantum statistical manifolds. Entropy 2018, 20, 472. Entropy (2018) 20:796. 10.3390/e20100796

  • CrossRef
  • Google Scholar

• 30.

  CiagliaFMIbortAJostJMarmoG. Manifolds of classical probability distributions and quantum density operators in infinite dimensions. Inf Geom (2019) 2:231–71. 10.1007/s41884-019-00022-1

  • CrossRef
  • Google Scholar

• 31.

  SimonL. Lectures on geometric measure theory. In: Proceedings of the centre for mathematical Analysis. Australia: Australian National University (1983).

  • Google Scholar

• 32.

  NiesteggeG. Absolute continuity for linear forms on b*-algebras and a Radon-Nikodym type theorem (quadratic version). Rend Circ Mat Palermo (1983) 32:358–76. 10.1007/bf02848539

  • CrossRef
  • Google Scholar

• 33.

  CsiszárI. I-divergence geometry of probability distributions and minimization problems. Ann Probab (1975) 3:146–58.

  • Google Scholar

• 34.

  CziszárIMatús̆F. Generalized minimizers of convex integral functionals, Bregman distance, Pythagorean identities. Kybernetika (2012) 48:637–89.

  • Google Scholar

• 35.

  ArakiH. Relative entropy for states of von Neumann algebras II. Publ Rims, Kyoto Univ (1977) 13:173–92.

  • Google Scholar

• 36.

  NaudtsJVerbeureAWederR. Linear response theory and the KMS condition. Comm Math Phys (1975) 44:87–99. 10.1007/bf01609060

  • CrossRef
  • Google Scholar

• 37.

  EguchiS. Information geometry and statistical pattern recognition. Sugaku Expositions (2006) 19:197–216.

  • Google Scholar

• 38.

  KuboR. Statistical-Mechanical theory of irreversible processes. I General theory and simple applications to magnetic and conduction problems. J Phys Soc Jpn (1957) 12:570–86. 10.1143/jpsj.12.570

  • CrossRef
  • Google Scholar

• 39.

  MoriH. Transport, collective motion, and Brownian motion. Progr Theor Phys (1965) 33:423–55. 10.1143/ptp.33.423

  • CrossRef
  • Google Scholar