Given a family of transition probability functions between measure spaces and an initial distribution Kolmogorov’s existence theorem associates a unique Markov process on the product space. Here a canonical non-commutative analogue of this result is established for families of completely positive maps betweenC* algebras satisfying the Chapman-Kolmogorov equations. This could be the starting point for a theory of quantum Markov processes.

LetH_i, 1 ≤ i ≤n be complex finite-dimensional Hilbert spaces of dimension d_i,1 ≤ i ≤n respectively withd_i ≥ 2 for everyi. By using the method of quantum circuits in the theory of quantum computing as outlined in Nielsen and Chuang [2] and using a key lemma of Jaikumar [1] we show that every unitary operator on the tensor productH =H₁ ⊗H₂ ⊗... ⊗H_n can be expressed as a composition of a finite number of unitary operators living on pair productsH_i ⊗H_j,1 ≤i,j ≤n. An estimate of the number of operators appearing in such a composition is obtained.

LetH_ibe a finite dimensional complex Hilbert space of dimensiond_i associated with a finite level quantum system A_i for i = 1, 2, ...,k. A subspaceS ⊂$${\mathcal{H}} = {\mathcal{H}}_{A_1 A_2 ...A_k } = {\mathcal{H}}_1 \otimes {\mathcal{H}}_2 \otimes \cdots \otimes {\mathcal{H}}_k $$ is said to becompletely entangled if it has no non-zero product vector of the formu₁⊗u₂ ⊗ ... ⊗u_k with u_i inH_i for each i. Using the methods of elementary linear algebra and the intersection theorem for projective varieties in basic algebraic geometry we prove that$$\mathop {max}\limits_{S \in \varepsilon } dim S = d_1 d_2 ...d_k - (d_1 + \cdots + d_k ) + k - 1$$ where ε is the collection of all completely entangled subspaces.

When$${\mathcal{H}} = {\mathcal{H}}_2 $$ andk = 2 an explicit orthonormal basis of a maximal completely entangled subspace of$${\mathcal{H}}_1 \otimes {\mathcal{H}}_2 $$ is given.

We also introduce a more delicate notion of aperfectly entangled subspace for a multipartite quantum system, construct an example using the theory of stabilizer quantum codes and pose a problem.

By using a quantum probabilistic approach we obtain a description of the extreme points of the convex set of all joint probability distributions on the product of two standard Borel spaces with fixed marginal distributions.

Given any finite set $\mathcal{F}$ of $(n-1)$-dimensional subspaces of $\mathbb{R}^n$ we give examples of nonGaussian probability measures in $\mathbb{R}^n$ whose marginal distribution in each subspace from $\mathcal{F}$ is Gaussian. However, if $\mathcal{F}$ is an infinite family of such $(n-1)$-dimensional subspaces then such a nonGaussian probability measure in $\mathbb{R}^n$ does not exist.

Let 𝑇 be a bijective map on $\mathbb{R}^n$ such that both 𝑇 and $T^{-1}$ are Borel measurable. For any $\theta\in\mathbb{R}^n$ and any real $n\times n$ positive definite matrix 𝛴 , let $N(\theta,\Sigma)$ denote the 𝑛-variate normal (Gaussian) probability measure on $\mathbb{R}^n$ with mean vector 𝜃 and covariance matrix 𝛴 . Here we prove the following two results: (1) Suppose $N(\theta_j, I)T^{-1}$ is gaussian for $0\leq j\leq n$, where 𝐼 is the identity matrix and $\{\theta_j-\theta_0,1\leq j\leq n\}$ is a basis for $\mathbb{R}^n$. Then 𝑇 is an affine linear transformation; (2) Let $\Sigma_j=I+\varepsilon_ju_j{u'}_j, 1\leq j\leq n$ where $\varepsilon_j>-1$ for every 𝑗 and $\{u_j,1\leq j\leq n\}$ is a basis of unit vectors in $\mathbb{R}^n$ with ${u'}_j$ denoting the transpose of the column vector $u_j$. Suppose $N(0,I)T^{-1}$ and $N(0,\Sigma_j)T^{-1}, 1\leq j\leq n$ are gaussian. Then $T(x)=\Sigma_s 1_{E_s}(x)VsUx a.e.x$, where 𝑠 runs over the set of $2^n$ diagonal matrices of order 𝑛 with diagonal entries $\pm 1,U,V$ are $n\times n$ orthogonal matrices and $\{E_s\}$ is a collection of $2^n$ Borel subsets of $\mathbb{R}^n$ such that $\{E_s\}$ and $\{VsU(E_s)\}$ are partitions of $\mathbb{R}^n$ modulo Lebesgue-null sets and for every $j,VsU\Sigma_j(VsU)^{-1}$ is independent of all 𝑠 for which the Lebesgue measure of $E_s$ is positive. The converse of this result also holds.

Our results constitute a sharpening of the results of Nabeya and Kariya (J. Multivariate Anal. 20 (1986) 251–264) and part of Khatri (Sankhyā Ser. A 49 (1987) 395–404).