Quantum Stochastic Calculus and Quantum Gaussian Processes

In this lecture we present a brief outline of boson Fock space stochastic calculus based on the creation, conservation and annihilation operators of free field theory, as given in the 1984 paper of Hudson and Parthasarathy. We show how a part of this architecture yields Gaussian fields stationary under a group action. Then we introduce the notion of semigroups of quasifree completely positive maps on the algebra of all bounded operators in the boson Fock space $\Gamma (\mathbb{C}^n)$ over $\mathbb{C}^n.$ These semigroups are not strongly continuous but their preduals map Gaussian states to Gaussian states. They were first introduced and their generators were shown to be of the Lindblad type by Vanheuverzwijn. They were recently investigated in the context of quantum information theory by Heinosaari, Holevo and Wolf. Here we present the exact noisy Schr\"odinger equation which dilates such a semigroup to a quantum Gaussian Markov process

On the maximal dimension of a completely entangled subspace for finite level quantum systems

Let $\mathcal{H}_i$ be a finite dimensional complex Hilbert space of dimension $d_i$ associated with a finite level quantum system $A_i$ for $i = i, 1,2, ..., k$. A subspace $S \subset \mathcal{H} = \mathcal{H}_{A_{1} A_{2}... A_{k}} = \mathcal{H}_1 \otimes \mathcal{H}_2 \otimes ... \otimes \mathcal{H}_k$ is said to be {\it completely entangled} if it has no nonzero product vector of the form $u_1 \otimes u_2 \otimes ... \otimes u_k$ with $u_i$ in $\mathcal{H}_i$ for each $i$. Using the methods of elementary linear algebra and the intersection theorem for projective varieties in basic algebraic geometry we prove that $$\max_{S \in \mathcal{E}} \dim S = d_1 d_2... d_k - (d_1 + ... + d_k) + k - 1$$ where $\mathcal{E}$ is the collection of all completely entangled subspaces. When $\mathcal{H}_1 = \mathcal{H}_2$ and $k = 2$ 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 a {\it perfectly entangled} subspace for a multipartite quantum system, construct an example using the theory of stabilizer quantum codes and pose a problem