In this paper, we study the multiplicative behaviour of quantum channels, mathematically described by trace preserving, completely positive maps on matrix algebras. It turns out that the multiplicative domain of a unital quantum channel has a close connection to its spectral properties. A structure theorem (Theorem 2.5), which reveals the automorphic property of an arbitrary unital quantum channel on a subalgebra, is presented. Various classes of quantum channels (irreducible, primitive etc.) are then analysed in terms of this stabilising subalgebra. The notion of the multiplicative index of a unital quantum channel is introduced, which measures the number of times a unital channel needs to be composed with itself for the multiplicative algebra to stabilise. We show that the maps that have trivial multiplicative domains are dense in completely bounded norm topology in the set of all unital completely positive maps. Some applications in quantum information theory are discussed. * 2010 Mathematics Subject Classification: Primary 46L05; Secondary 46L60, 81R15 2. Given a quantum channel E : M d → M d , we can identify its dual map or adjoint map E * via the relation Tr(E(a)b) = Tr(aE * (b)) ∀ a, b ∈ M d where a, b = Tr(ab * ) for all a, b ∈ M d , defines an inner product on M d which makes M d a Hilbert space. This is known as Hilbert-Schmidt inner product. We will frequently denote the norm of an element x ∈ M d , arising from this inner product as x 2 H.S := x, x = Tr(xx * ). We are now ready to state the following theorem. The result was known before (see [14], [28]) but we present a different proof here. The technique used in this proof will be used significantly throughout the rest of the paper. Theorem 1.3. Let E : M d → M d be a unital quantum channel. Then

We analyze linear maps on matrix algebras that become entanglement breaking after composing a finite or infinite number of times with themselves. This means that the Choi matrix of the iterated linear map becomes separable in the tensor product space. If a linear map becomes entanglement breaking after finitely many iterations, we say the map has a finite index of separability. In particular we show that every unital PPT-channel has a finite index of separability and that the class of unital channels that have finite index of separability is a dense subset of the unital channels. We construct concrete examples of maps which are not PPT but have finite index of separability. We prove that there is a large class of unital channels that are asymptotically entanglement breaking. This analysis is motivated by the PPT-squared conjecture made by M. Christandl that says every PPT channel, when composed with itself, becomes entanglement breaking.

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We introduce and study the entanglement breaking rank of an entanglement breaking channel. We show that the entanglement breaking rank of the channel Z:Md→Md defined by Z(X)=1d+1(X+Tr(X)Id) is d2 if and only if there exists a symmetric informationally complete positive operator-valued measure in dimension d.

We investigate spectral properties of the tensor products of two completely positive and trace preserving linear maps (also known as quantum channels) acting on matrix algebras. This leads to an important question of when an arbitrary subalgebra can split into the tensor product of two subalgebras. We show that for two unital quantum channels the multiplicative domain of their tensor product splits into the tensor product of the individual multiplicative domains. Consequently, we fully describe the fixed points and peripheral eigen operators of the tensor product of channels. Through a structure theorem of maximal unital proper *-subalgebras (MUPSA) of a matrix algebra we provide a non-trivial upper bound of the recentlyintroduced multiplicative index of a unital channel. This bound gives a criteria on when a channel cannot be factored into a product of two different channels. We construct examples of channels which cannot be realized as a tensor product of two channels in any way. With these techniques and results, we found some applications in quantum information theory.