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Conditional expectations associated with quantum states

An extension of the conditional expectations (those under a given subalgebra of events and not the simple ones under a single event) from the classical to the quantum case is presented. In the classical case, the conditional expectations always exist; in the quantum case, however, they exist only if a certain weak compatibility criterion is satisfied. This compatibility criterion was introduced among others in a recent paper by the author. Then, state-independent conditional expectations and quantum Markov processes are studied. A classical Markov process is a probability measure, together with a system of random variables, satisfying the Markov property and can equivalently be described by a system of Markovian kernels (often forming a semigroup). This equivalence is partly extended to quantum probabilities. It is shown that a dynamical (semi)group can be derived from a given system of quantum observables satisfying the Markov property, and the group generators are studied. The results are presented in the framework of Jordan operator algebras, and a very general type of observables (including the usual real-valued observables or self-adjoint operators) is considered.


I. INTRODUCTION
A classical Markov process is a probability measure together with a system of random variables, satisfying the Markov property (which characterizes a certain memorilessness of the process). A classical Markov process can equivalently be described by a system of Markovian kernels forming a semi-group when the process is stationary.
The quantum analogues of probability measures, random variables and Markovian kernels are the states, observables and unital positive linear maps. A quantum Markov process is usually modeled as a semi-group of unital positive linear maps on a W*-algebra (von Neumann algebra). The major result of the present paper is the derivation of such a semigroup from a stochastic process given as a quantum state together with a system of observables, satisfying a certain directed weak compatibility criterion, the Markov property and the stationarity criterion. This partly extends the classical equivalence between stationary Markov processes and semi-groups of Markovian kernels to the quantum case. As in the classical case, the major tools are the Radon-Nikodym theorem and conditional expectations, the extension of which to the quantum case is a major result of the present paper.
The quantum probabilities are considered in the general framework of Jordan operator algebras, particularly the so-called JBW algebras. A justification for this is given in Ref. 10. An observable then becomes a homomorphism between such algebras. A classical random variable can be considered an observable on a classical space with values in a classical space. A real-valued quantum observable (spectral measure of a self-adjoint operator) can be considered an observable on a non-classical quantum space with values in a classical space (the real numbers). This paper, however, uses the most general type, i.e., an observable on a non-classical quantum space with values in a non-classical quantum space; this is a homomorphism between two non-associative Jordan operator algebras.
A Jordan algebra 4   We shall later see that the condition 1 µ  →  F ensures the existence of a reasonable conditional expectation of F under 1 in the state µ. This condition is satisfied e.g., when 1 and F operator-commute, or when is the tensor product 11 of 1 and 2 with F∈ 2 , or when µ is a trace state. 13 A linear map (state, homomorphism, observable) on a JBW algebra is called normal if it is continuous with respect to the weak topology on generated by its predual. Normal linear maps are completely additive for any orthogonal family of events. The paper restricts to the study of normal states and normal observables although it would be desirable to include σadditive states and standard observables that are σ-additive and not necessarily normal, but then some methods needed from the theory of JBW algebras would not apply (see also Sec. VII).
The paper is organized as follows: In Sections II and III, some results that are well-known for C*-/W*-algebras are extended to the Jordan operator algebras for later use; these are Kadison's generalized Schwarz inequality and Sakai's Radon-Nikodym theorem. The extension of conditional expectations and Markov processes to the quantum case is introduced in Sections IV and V, respectively. It is shown that a semi-group of positive linear maps is associated with each such Markov process. Finally the generator of this semi-group is studied in Section VI.
Although the motivation for the work presented here primarily stems from a quantum probability model proposed in Refs. 9, 10, and 11, the paper is written in such a way that most of it can be understood without knowledge of that quantum probability model. However, some knowledge of Jordan operator algebras is required, and the monograph Ref. 4 is recommended as an excellent reference.

II. THE GENERALIZED SCHWARZ INEQUALITY
The following lemma will be needed to prove the generalized Schwarz inequality for positive linear maps between JB algebras. Lemma 2.1: Let X 1 ,...,X n be n positive elements in a JB algebra with unit 1I such that Σ Proof: Let ϕ be a state on . On the direct sum of n copies of we consider the positive-semi-definite inner product Then the Cauchy-Schwarz inequality holds [note that {Y k ,X k ,Y k }≥0 for Y k ,X k ∈ with X k ≥0] and, selecting Y k :=s k 1I for k=1,...,n and and, since this holds for every state ϕ, the lemma is proved.
The following proposition now provides the extension of Kadison's generalized Schwarz inequality for positive maps between C*-algebras 5 to the more general case of positive maps between JB algebras.

Proposition 2.2: Let V be a positive linear map from a unital JB algebra
to a unital JB algebra ! with V(1I )=1I . Then (V(X)) 2 ≤V(X 2 ) for every X∈ .
Proof: First, we assume that is a JBW algebra 4 and that X has the shape X= Σ with pairwise orthogonal idempotent elements E 1 ,...,E n ∈ and s 1 ,...,s n ∈ IR . Then 0≤V(E k ) for k=1,...,n and ΣV(E k )≤ 1I such that we can apply Lemma 2.1 getting: Since, due to the spectral theorem, every X∈ can be approximated in the norm topology by a sequence of elements having the above special shape and since a positive linear map is automatically norm-continuous, (V(X)) 2 ≤V(X 2 ) holds for every X∈ .
We now assume that is a JB algebra. Then the double dual spaces ** and !** are JBW algebras containing and !, respectively, as subalgebras 4 , such that Proposition 2.2 follows by applying the above to the map V**: **→!**.

Corollary 2.3: Let V be a bijective positive linear map from a unital JB algebra
to a unital JB algebra ! with V(1I )=1I such that V −1 is positive as well. Then V is a muliplicative homomorphism.

III. THE RADON-NIKODYM THEOREM
We shall now extend one of Sakai's Radon-Nikodym theorems 14 from W*-algebras to JBW algebras. Since all methods needed for the proof of the W*-case are available in the JBW case as well, the proofs are quite similar in these two cases. Proof: For X∈ we define µ X via µ X (Z):=µ(X Z) for Z∈ and consider the set K:={µ X |X∈, 0≤X≤1I } which is a non-empty convex subset of the predual * of . Moreover, K is compact with respect to the weak topology on * that is generated by . Note that {X∈, 0≤X≤1I } is compact and that the multiplication operation with one element fixed is continuous with respect to the weak topology on generated by * .
All we have to show is that ν∈K. We assume that ν∉K. From the Hahn-Banach theorem it then follows that there is an element A∈ and a real number r such that ν(A)>r and µ X (A)≤r for X∈ with 0≤X≤1I . Now let F be the support of the positive part A + of A; these are defined

IV. CONDITIONAL EXPECTATIONS
Let be a JBW algebra and o a JBW sub-algebra of with 1I ∈ o . Let µ be a normal state on and X∈ with 0≤X≤1I .  Let the restriction of µ to o be faithful, and let 0 µ  →  X ,X 1 ,X 2 hold with X,X 1 ,X 2 ∈, The faithfulness of µ on o is not really a strong restriction; moving over from to the JBW algebra {D,,D} with D being the support of µ, one could even assume that µ is faithful on . Note that the support is the smallest event E with µ(E)=1, which exists for normal states.
We If R is injective, there is one and only one element V R,µ (X)∈ with µ(X|R)=RV R,µ (X). Then 0≤V R,µ (X)≤1I . We now assume that 1 is a further sub-algebra of such that where the Markov property has been applied several times. With ( ) is satisfied for s<s']. The approach to Markov processes, presented here, is very similar to the one in classical mathematical probability theory (e.g., Ref. 8). The adaptation to the quantum case becomes possible due to the compatibility criterion (i) in Definition 5.1. The faithfulness of the underlying state µ and the injectivity of the R s are technical assumptions to avoid the difficulties involved with the µalmost-everywhere equivalence classes that are used in mathematical probability theory. Some further discussion of the assumptions will follow in the concluding remarks. We

VI. THE DYNAMICAL GROUP AND ITS GENERATOR
We now assume a Markov process such that each R s (s∈S) is injective and s = for all We shall now briefly consider the generators of such groups, but will not go into the technical details of the different kinds of convergence, since most of it is well-known -if not for JBW algebras, then at least for the C*-/W*-algebras. If exists (convergence in some sense assumed) for a stationary Markov process, L is a linear operator → (or possibly defined only on a dense subset of ) and the following differential equation holds: Then ( ) . L is called the generator of the dynamical (semi-)group V t . From the generalized Schwarz inequality we get for Y∈ : and therefore (note that V o is the identity): In the reversible case, each "≥" can be replaced by "=" (Corollary 2.3). Then for Y∈ . Linear maps L satisfying this equation are called derivations. We call a linear map L satisfying ( ) for Y∈ a dissipation. Thus, we have shown that the generator of the dynamical group associated with a stationary Markov process is a dissipation, and is a derivation if the Markov process is stationary and reversible. With being the self-adjoint part of a W*-algebra and with the inner derivation L(X):= i[H,X] for X∈ [with some H∈ ], we get This provides the Schroedinger equation and its solution as a very special case of a more general approach. Derivations on C*-algebras are studied in Refs. 1 and 2. The above definition of a dissipation differs from the dissipations studied in Refs. 2 and 3. In Ref. 2, a linear map δ on a *-subalgebra o of a C*-algebra, satisfying δ(X*X)≤δ(X*)X+X*δ(X) for X∈ o , is called a dissipation. This implies, but is not equivalent to δ(X 2 )≤δ(X)X+Xδ(X)= 2X δ(X) for all selfadjoint X in o . Other authors use the reversed inequality δ(X*X)≥δ(X*)X+X*δ(X) for the definition of a dissipation. Note that our definition is based on this reversed inequality which, moreover, is required to hold for the self-adjoint elements only (which form the JB algebra).
Note that other authors immediately define a quantum Markov process as a pair consisting of a W*-algebra and a dynamical semi-group of (completely) positive normal maps on this W*-algebra, without starting from a stochastic process given as a family of observables and implicitly assuming the stationarity.

VII. CONCLUDING REMARKS
The concept of the classical conditional expectations has been extended to the quantum case, using the framework of Jordan operator algebras. An important condition for the existence of the conditional expectations is given by a certain weak compatibility criterion that was introduced earlier in Ref. 11. With these concepts, it has been possible to partly extend the classical equivalence between two different ways of describing a Markov process to the quantum case. Starting from a Markov process given as a family of observables, we have derived the positive maps V s s , ′ . In classical probability theory, the reverse is also possible; a Markov process consisting of a family of random variables can be reconstructed from a system of Markovian kernels by using the concept of product σ-algebras. The problem of finding a satisfying analogue of these product σ-algebras for the quantum case has been addressed, but only partly been solved in Ref. 11.
The appropriate framework for this approach to quantum Markov processes are Jordan operator algebras, but neither the only norm-complete JB algebras nor the weakly complete JBW algebras are really satisfying. The JB algebras do not contain sufficiently many idempotent elements (quantum events). If the theory is based upon JBW algebras (as here in this paper), an important example, the algebra consisting of the measurable real-valued functions on a measurable space (the real-valued random variables of classical probability theory) is ruled out. This algebra can be embedded in a JBW algebra [even in a W*-algebra 7 ], however, one is rather reluctant to work with this abstractly constructed JBW algebra instead of the well-understood algebra of measurable real functions. What is needed is a theory of monotone-sequentially complete JB algebras, similar to the one of the monotone-sequentially complete C*-algebras studied by Kadison, 6 Kehlet 7 and Pedersen. 12 This theory must include a Gleason-type theorem and a Radon-Nikodym theorem for σ-additive states. The Gleasontype theorem is needed for proving that unique conditional probabilities exist for σ-additive states defined on the system of events [as done for the JBW case in Ref. 9], and the Radon-Nikodym theorem is required for showing that the conditional expectations exist for the σadditive states.