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.

We prove several theorems about quantum-mechanical entropy, in particular, that it is strongly subadditive.

We give a useful new characterization of the set of all completely positive, trace-preserving maps Φ : M 2 → M 2 from which one can easily check any trace-preserving map for complete positivity. We also determine explicitly all extreme points of this set, and give a useful parameterization after reduction to a certain canonical form. This allows a detailed examination of an important class of non-unital extreme points that can be characterized as having exactly two images on the Bloch sphere.We also discuss a number of related issues about the images and the geometry of the set of stochastic maps, and show that any stochastic map on M 2 can be written as a convex combination of two "generalized" extreme points.

In this paper, we consider the minimal entropy of qubit states transmitted through two uses of a noisy quantum channel, which is modeled by the action of a completely positive trace-preserving (or stochastic) map. We provide strong support for the conjecture that this minimal entropy is additive, namely that the minimum entropy can be achieved when product states are transmitted. Explicitly, we prove that for a tensor product of two unital stochastic maps on qubit states, using an entanglement that involves only states which emerge with minimal entropy cannot decrease the entropy below the minimum achievable using product states. We give a separate argument, based on the geometry of the image of the set of density matrices under stochastic maps, which suggests that the minimal entropy conjecture holds for non-unital as well as for unital maps. We also show that the maximal norm of the output states is multiplicative for most product maps on n-qubit states, including all those for which at least one map is unital. * Partially supported by National Science Foundation Grant DMS-97-05779 † Partially supported by National Science Foundation Grant DMS-97-06981 and Army Research Office Grant DAAG55-98-1-0374 1 For the class of unital channels on C 2 , we show that additivity of minimal entropy implies that the Holevo capacity of the channel is additive over two inputs, achievable with orthogonal states, and equal to the Shannon capacity. This implies that superadditivity of the capacity is possible only for nonunital channels.

We prove several theorems about quantum-mechanical entropy, in particular, that it is strongly subadditive.with A = P123 and B = exp(-1np2 + Inp12 + InP23)' One finds F(P123) '" 8 123 + 8 2 -8 12 -8 23 "" Tr123 [exp(lnpI2 -lnP2 + Inp23) -P123]'

We use the relative modular operator to define a generalized relative entropy for any convex operator function g on (0, ∞) satisfying g(1) = 0. We show that these convex operator functions can be partitioned into convex subsets each of which defines a unique symmetrized relative entropy, a unique family (parameterized by density matrices) of continuous monotone Riemannian metrics, a unique geodesic distance on the space of density matrices, and a unique monotone operator function satisfying certain symmetry and normalization conditions. We describe these objects explicitly in several important special cases, including g(w) = − log w which yields the familiar logarithmic relative entropy. The relative entropies, Riemannian

Degradable quantum channels are among the only channels whose quantum and private classical capacities are known. As such, determining the structure of these channels is a pressing open question in quantum information theory. We give a comprehensive review of what is currently known about the structure of degradable quantum channels, including a number of new results as well as alternate proofs of some known results. In the case of qubits, we provide a complete characterization of all degradable channels with two dimensional output, give a new proof that a qubit channel with two Kraus operators is either degradable or anti-degradable and present a complete description of anti-degradable unital qubit channels with a new proof.For higher output dimensions we explore the relationship between the output and environment dimensions (d B and d E respectively) of degradable channels. For several broad classes of channels we show that they can be modeled with a environment that is "small" in the sense d E ≤ d B . Such channels include all those with qubit or qutrit output, those that map some pure state to an output with full rank, and all those which can be represented using simultaneously diagonal Kraus operators, even in a non-orthogonal basis. Perhaps surprisingly, we also present examples of degradable channels with "large" environments, in the sense that the minimal dimensionThese examples can also be used to give a negative answer to the question of whether additivity of the coherent information is helpful for establishing additivity for the Holevo capacity of a pair of channels.In the case of channels with diagonal Kraus operators, we describe the subclass which are complements of entanglement breaking channels. We also obtain a number of results for channels in the convex hull of conjugations with generalized Pauli matrices. However, a number of open questions remain about these channels and the more general case of random unitary channels.

New bounds are given on the contraction of certain generalized forms of the relative entropy of two positive semi-definite operators under completely positive mappings. In addition, several conjectures are presented, one of which would give a strengthening of strong subadditivity. As an application of these bounds in the classical discrete case, a new proof of 2-point logarithmic Sobolev inequalities is presented in an Appendix.