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.
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.
As mentioned previously, a classical Markov semigroup of transition operators { T t , t ≥ 0} can be generated from a classical Markov process { X t , t ≥ 0}. On the other hand, given a Markov semigroup, a Markov process { X t , t ≥ 0} can be constructed based on the Markov semigroup { T t , t ≥ 0} using the Kolmogorov consistency theorem. In the context of quantum probability space ( A , ρ ), a Markov dilation (i.e., construction of a quantum Markov process from a given quantum Markov semigroup of operators { I t , t ≥ 0}) turns out to be a nontrivial matter. This is partly due to the fact that, although the general concept of conditional expectation of an observable given a sub-von Neumann algebras B ⊂ A can be defined and required properties can be described, an explicit construction of such a conditional expectation is still unavailable in general. The main objective of this chapter is to introduce relevant concepts and to develop some properties of quantum Markov processes in the content of quantum probability explored in previous chapters. There are 2 major components in this chapter: (i) introduction of concepts and derivation of properties of a quantum Markov processes based on some assumed and/or derived properties of conditional expectation E ρ [· | A t ] ] given a filtration of sub-von Neumann algebras { A t ] , t ≥ 0} of A , and (ii) Markov dilation or construction of a weak quantum Markov flow (WQMF) from a given quantum Markov semigroup { I t , t ≥ 0} using the weak conditional expectation E [· | F t ], where { F t , t ≥ 0} is a filtration of orthogonal projection operators defined on the complex Hilbert space ℍ. The main objective of the first component is to introduce the relevant concept and develop some properties of quantum Markov processes in the content of quantum probability explored in previous chapters. Although the theory of (classical) Markov processes is well known in the probability community (see, for example, Dynkin [Dyn62], Ethier and Kurtz [EK85]), we outline some basic concept and results from the classical theory of homogeneous Markov processes below for comparison purpose.
Abstract Given the state of a quantum system, we define the conditional expectation of an observable with respect to a second observable, or commuting family of observables, as the real-valued function of the latter which best approximates the observable, in the sense of minimizing the quantum mechanical expectation of the difference squared. This corresponds to the notion of best predictor of a random variable by another random variable in non-parametric statistical regression. A basic example is the conditional expectation of the quantum mechanical momentum with respect to position, which can be identified with the Bohm momentum. We re-examine the de Broglie–Bohm theory from this perspective. As a second example we consider the simplest quantum system, that of a spin- 1 / 2 paticle in a magnetic field. We study the conditional expectations of the Pauli spin-operators S 1 , S 2 , S 3 with respect to S 3 , and determine their dynamics if the pure-state wave-function solves the relevant Schrödinger equation. Considering S 3 as a beable we can, following Bell, introduce a generalized guidance law in the form of a Markov jump process on the spectrum of S 3 which is compatible with the time evolution of the Born probabilities of the two eigenstates of S 3 . The conditional expectations of the remaining spin operators are then shown to follow a piece-wise deterministic Markov process, defined by an explicit SDE. We finally examine the relationship between the conditional expectations of this paper, and the von Neumann-algebraic conditional expectation onto the sub-algebra generated by the commuting family of conditioning operators when the joint spectrum of the latter is discrete, and derive a general expression for the conditional expectation dynamics in that context.
A concept of conditional expectations in quantum theory is established with interrelations to previously introduced concepts of the Cycon–Hellwig conditional expectations and a posteriori states, which are analogous to the existing interrelations in the classical probability theory among conditional expectations related to random variables, those related to σ subalgebras and conditional probability distributions. These three concepts are shown to have satisfactory statistical interpretation in the quantum measuring processes. For the above purpose, we introduce an integration with respect to functions with values in the states of operator algebras and positive operator valued measures such that the resulting indefinite integrals are completely positive map valued measures. Eventually, it is proved that in the von Neumann algebraic formulation, the Cycon–Hellwig conditional expectations always exist as completely positive map valued measures.
Quantum Sufficiency for Self-Adjoint Statistical Models via Likelihood-Type Operators on Real ∗-Subalgebras and Real Jordan Algebras
We develop a theory of quantum sufficiency on real *-subalgebras and real Jordan algebras for models consisting of general self-adjoint operators, including derivatives of states. Square-root likelihood ratios and symmetric logarithmic derivatives serve as self-adjoint likelihood-type objects, allowing ordinary quantum statistical models and local models to be treated within a unified framework that admits degenerate reference states.
We introduce sufficient real unital positive maps and show that their Hermitian fixed points form a real Jordan algebra admitting a sufficient Jordan conditional expectation. The real and complex *-subalgebras generated by this algebra admit sufficient real and complex completely positive conditional expectations, respectively. We characterize the minimal sufficient real *-subalgebra by the likelihood-ratio set and ρ-modular invariance. The minimal sufficient real Jordan algebra is generated by the likelihood-ratio set together with the operator obtained by normalizing every irreducible diagonal block of the reference state, relative to the minimal sufficient real *-subalgebra, to trace one. We also obtain Koashi-Imoto type decompositions for real *-subalgebras and real Jordan algebras admitting sufficient conditional expectations, explicitly accounting for inequivalent irreducible Jordan representations. These results separate the likelihood-ratio aspect of sufficiency from its noncommutative modular aspect and identify real Jordan structure as a natural framework for quantum statistical sufficiency.
Given a normalized state-vector ψ, we define the conditional expectation $\mathbb{E }_{ψ} (A | B ) $ of a Hermitian operator $A $ with respect to a strongly commuting family of self-adjoint operators $B $ as the best approximation, in the operator mean square norm associated to ψ, of $A $ by a real-valued function of $B . $ A fundamental example is the conditional expectation of the momentum operator $P $ given the position operator $X $, which is found to be the Bohm momentum. After developing the Bohm theory from this point of view we treat conditional expectations with respect to general $B $, which we apply to non-relativistic spin 1/2-particles. We derive the dynamics of the conditional expectations of momentum and spin with respect to position and the third spin component. These dynamics can be interpreted in terms of classical continuum mechanics as a two-component fluid whose components carry intrinsic angular momentum. Interpreting the joint spectrum of the conditioning operators as a space of beables, we can introduce a classical-stochastic particle dynamics on this space which is compatible with the time-evolution of the Born probability, by combining the de Broglie-Bohm guidance condition with a Markov jump process, following an idea of J. Bell. This results in a new Bohm-type model for particles with spin. A basic problem is that such auxiliary particle dynamics are far from unique. We finally examine the relation of our conditional expectations with the conditional expectations of the theory of $C^* $-algebras and, as an application, derive a general evolution equation for conditional expectations for operators acting on finite dimensional Hilbert spaces. Two appendices re-interpret the classical Bohm model as an integrable constrained Hamiltonian system, and provide the details of the two-fluid interpretation.
In this chapter, we define and explore basic properties of a quantum Markov semigroup { I t , t ≥ 0} of linear maps on the C* -algebra or von Neumann algebra A that characterizes the quantum system. The quantum Markov semigroup (QMS) plays a key role in describing quantum Markov processes that are to be explored in the subsequent chapters. The concept of QMS extends the semigroup of probability transition operators { T t , t ≥ 0} for a classical Markov process to its noncommutative counterpart. Suppose the classical Markov process { X t , t ≥ 0} is defined on the complete filtered classical probability space (Ω, Ƒ, P ; {Ƒ t , t ≥ 0}) and with values in a measurable space (핏, B (X)). Recall that for each t ≥ 0, the probability transition operator T t : L ∞ (X, B (X)) → L ∞ (X, B (X)) by ( T t f )( X s ) = E x [ f ( X s + t ) | Ƒ s ] ∀ s , t ≥ 0. A semigroup of linear maps on the C* -algebra or von Neumann algebra A is said to be a quantum dynamical semigroup (QDS) if (i) I 0 = J (the identity operator on A ); (ii) I t I s = I t + s for all t , s ≥ 0; (iii) I t is completely positive for each t ≥ 0; and (iv) I t is σ weakly continuous on A , i.e., a ↦ tr ( ρI t ( a )) is continuous from A to ℂ for each ρ ∈ S ( A ) (the space of quantum states) and for each t ≥ 0. If in addition I t ( I ) = I , (respectively, I t ( I ) ≤ I ) for all t ≥ 0, then the QDS is said to be a quantum Makrov semigroup (QMS) (respectively, quantum sub-Markov semigroup).