Mary Beth Ruskai

Proof of the strong subadditivity of quantum-mechanical entropy

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

Entanglement Breaking Channels

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)=∑kRkTrFkρ where each Rk is a density matrix and Fk>0. If, in addition, Φ is trace-preserving, the {Fk} 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.

Minimal entropy of states emerging from noisy quantum channels

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 nonunital 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. For the class of unital channels on C/sup 2/, we show that additivity of minimal entropy implies that the Holevo (see IEEE Trans. Inform. Theory, vol.44, p.269-73, 1998 and Russian Math. Surv., p.1295-1331, 1999) 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.

An analysis of completely-positive trace-preserving maps on M2

Abstract 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.

Inequalities for quantum entropy: A review with conditions for equality

This article presents self-contained proofs of the strong subadditivity inequality for von Neumann’s quantum entropy, S(ρ), and some related inequalities for the quantum relative entropy, most notably its convexity and its monotonicity under stochastic maps. Moreover, the approach presented here, which is based on Klein’s inequality and Lieb’s theorem that the function A→Tr eK+log A is concave, allows one to obtain conditions for equality. In the case of strong subadditivity, which states that S(ρ123)+S(ρ2)⩽S(ρ12)+S(ρ23) where the subscripts denote subsystems of a composite system, equality holds if and only if log ρ123=log ρ12−log ρ2+log ρ23. Using the fact that the Holevo bound on the accessible information in a quantum ensemble can be obtained as a consequence of the monotonicity of relative entropy, we show that equality can be attained for that bound only when the states in the ensemble commute. The article concludes with an Appendix giving a short description of Epstein’s elegant proof of Lieb’s the...

Bounds for the adiabatic approximation with applications to quantum computation

We present straightforward proofs of estimates used in the adiabatic approximation. The gap dependence is analyzed explicitly. We apply the result to interpolating Hamiltonians of interest in quantum computing.

Monotone Riemannian metrics and relative entropy on noncommutative probability spaces

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 (parametrized 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 metrics, and geodesic distances obtained by our procedure all contract under completely positive, trace-preserving maps. We then define and study the maximal contraction associated with these quantities.

Multiplicativity of Completely Bounded p-Norms Implies a New Additivity Result

AbstractWe prove additivity of the minimal conditional entropy associated with a quantum channel Φ, represented by a completely positive (CP), trace-preserving map, when the infimum of S(γ12) − S(γ1) is restricted to states of the form

Qubit Entanglement Breaking Channels

(\mathcal{I} \otimes \Phi)\left( | \psi \rangle \langle \psi | \right)