Markovianity of Quantum Random Fields

We present a notion of quantum Markov random field based on a concept of conditional independence replacing the (usual) requirement of conditional expectation onto a desired algebra by conditional expectation onto a subalgebra, adopted to the special case that all subalgebras of the filtration are type I factors. It is immediate to introduce the classical notions like pairwise, local, global and factorizing Markov properties [11]. These share the same relations as in the classical case except the Hammersley-Clifford theorem, which remains open in the quantum case. 1 Conditional Independence At the heart of the study of Markov random fields in classical probability there is the notion of conditional independence. One says that two random variables X ,Y are independent conditionally under a third random variable Z (notation X⊥⊥Y |Z) if P(X ∈ A,Y ∈ B|Z) = P(X ∈ A|Z)P(Y ∈ B|Z) (1) for any two Borel sets A,B. Equivalently it holds that P(X ∈ A|Y,Z) depends on Z only. Then one considers the following key properties [11] (C1) If X ⊥⊥Y |Z then Y ⊥⊥X |Z. ∗This work was partially supported by INTAS grants N 96-0698 and 99-00545 and DAADDST grant †GSF — National Research Centre for Environment and Health, Institute of Biomathematics and Biometry, Ingolstädter Landstr.1, D–85758 Neuherberg, Germany, email: [email protected]