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arXiv:1204.2752 · Journal of Mathematical Physics

Extremality conditions for generalized channels

Anna Jenčová

Published Nov 28, 2012• 5 citations
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Abstract

A generalized channel is a completely positive map that preserves trace on a given subspace. We find conditions under which a generalized channel with respect to a positively generated subspace J is an extreme point in the set of all such generalized channels. As a special case, this yields extremality conditions for quantum protocols. In particular, we obtain new extremality conditions for quantum 1-testers with 2 outcomes, which correspond to yes/no measurements on the set of quantum channels.

DOI
10.1063/1.4764885

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A generalized channel is a completely positive map that preserves trace on a given subspace. We find conditions under which a generalized channel with respect to a positively generated subspace J is an extreme point in the set of all such generalized channels. As a special case, this yields extremality conditions for quantum protocols. In particular, we obtain new extremality conditions for quantum 1-testers with 2 outcomes, which correspond to yes/no measurements on the set of quantum channels.
Let K be a convex subset of the state space of a finite-dimensional C*-algebra. We study the properties of channels on K, which are defined as affine maps from K into the state space of another algebra, extending to completely positive maps on the subspace generated by K. We show that each such map is the restriction of a completely positive map on the whole algebra, called a generalized channel. We characterize the set of generalized channels and also the equivalence classes of generalized channels having the same value on K. Moreover, if K contains the tracial state, the set of generalized channels forms again a convex subset of a multipartite state space, this leads to a definition of a generalized supermap, which is a generalized channel with respect to this subset. We prove a decomposition theorem for generalized supermaps and describe the equivalence classes. The set of generalized supermaps having the same value on equivalent generalized channels is also characterized. Special cases include quantum combs and process positive operator valued measures (POVMs).
Abstract In this note we study completely positive, trace-preserving (CPT) maps M 4 ( C ) ↦ M 4 ( C ) , in particular the class of examples given by quantum Markov chains (QMCs) describing the evolution of one qubit on a 2-vertex graph. For this class of channels, and regarding its convex structure, we examine the problem of finding the extreme points both with respect to CPT maps and with respect to the set of QMCs. This is done by considering the class of maps whose image is contained in the set of block-diagonal matrices. Motivated by previous results in the literature, a practical criterion for extremality is obtained via the Choi–Jamiołkowski isomorphism and appropriate direct-sum subspaces. The study of extreme maps is natural from a structural point of view, as these play the role of indecomposable operations and provide fundamental information about the geometry of the set of quantum channels.
Completely positive and trace preserving (CPT) maps are important for Quantum Information Theory, because they describe a broad class of of transformations of quantum states. There are also two other related classes of maps, the unital completely positive (UCP) maps and the unital completely positive and trace preserving (UCPT) maps. For these three classes, the maps from a finite dimensional Hilbert space XXX to another one YYY is a compact convex set and, as such, it is the convex hull of its extreme points. The extreme points of these convex sets are yet not well understood. In this article we investigate the preservation of extremality under the tensor product. We prove that extremality is preserved for CPT or UCP maps, but for UCPT it is not always preserved.
This paper studies the class of stochastic maps, or channels, for which (I⊗Φ)(Γ) is always separable (even for entangled Γ). Such maps are called entanglement breaking, and can always be written in the form Φ(ρ)=∑ k R k Tr F k ρ where each R k is a density matrix and F k >0. If, in addition, Φ is trace-preserving, the {F k } must form a positive operator valued measure (POVM). Some special classes of these maps are considered and other characterizations given. Since the set of entanglement-breaking trace-preserving maps is convex, it can be characterized by its extreme points. The only extreme points of the set of completely positive trace preserving maps which are also entanglement breaking are those known as classical-quantum or CQ. However, for d≥3, the set of entanglement breaking maps has additional extreme points which are not extreme CQ maps.
Generalized quantum instruments correspond to measurements where the input and output are either states or more generally quantum circuits. These measurements describe any quantum protocol including games, communications, and algorithms. The set of generalized quantum instruments with a given input and output structure is a convex set. Here, we investigate the extremal points of this set for the case of finite dimensional quantum systems and generalized instruments with finitely many outcomes. We derive algebraic necessary and sufficient conditions for extremality.