Multiplicative properties of quantum channels

Abstract

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

Eventually entanglement breaking maps

“…Consider the channels E 1 , · · · , E k in Theorem 4.1. Each channel has trivial multiplicative domain, so by [16], its peripheral spectrum is trivial. Thus, we can apply Theorem 4.3 and conclude that lim n E n i (x) = Tr(x) 1 d i .…”

“…For any unital channel E and any k ∈ N, M E k+1 ⊆ M E k (See [16]), and hence there is some N ∈ N such that for any n ≥ N , M E n = M E N . Following [16], we denote this algebra M E ∞ and refer to it as the stabilized multiplicative domain of E. 16]). The multiplicative index of a unital quantum channel E is the minimum n ∈ N such that M E n = M E ∞ .…”

A New Bound on Quantum Wielandt Inequality

“…So M E ∞ is non-trivial. Following Theorem 2.5 in [15] we get M d = M E ∞ ⊕M ⊥ E ∞ and E is an automorphism on M E ∞ with inverse being E * . It follows that E n also is an automorphism on M E ∞ of every n with the inverse E * n .…”

“…This is because if we take a projection p ∈ M E such that E(p) ∈ M E , then E(p) is again a projection. By the definition of M E 2 (see [15]), p ∈ M E 2 . If M E 2 is still not C1, we get E(p) is not positive definite.…”

Asymptotics of quantum channels: conserved quantities, an adiabatic limit, and matrix product states

“…Rotating points and conserved quantities of the form from the previous subsection can be constructed for each block to form a canonical basis (i.e., a basis respecting the block structure). It is well-known among experts (see, e.g., [78]) that the conserved quantities {J } form a matrix algebra -a vector space (where the vectors are matrices) that is closed under multiplication and the conjugate transpose operation. It is important to keep in mind that all of this extra structure in does not put any constraints on the remaining parts {A , A } of A, the extension of E (as long as eqs.…”