Armin Uhlmann

The “transition probability” in the state space of a ∗-algebra

Let ω,ϱ be two states of a ∗-algebra and let us consider representations of this algebra R for which ω and ϱ are realized as vector states by vectors x and y. The transition probability P(ω,ϱ) is the spectrum of all the numbers |(x,y)|2 taken over all such realizations. We derive properties of this straightforward generalization of the quantum mechanical transition probability and give, in some important cases, an explicit expression for this quantity.

Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in an interpolation theory

We show that the Wigner-Yanase-Dyson-Lieb concavity is a general property of an interpolation theory which works between pairs of (hilbertian) seminorms. As an application, the theory extends the relevant work of Lieb and Araki to positive linear forms of arbitrary *-algebras. In this context a “relative entropy” is defined for every pair of positive linear forms of a *-algebra with identity. For this generalized relative entropy its joint convexity and its decreasing under identity-preserving completely positive maps is proved.

Parallel transport and “quantum holonomy” along density operators

“Quantum holonomy” as defined by Berry and Simon, and based on the parallel transport of Bott and Chern, can be considerably extended. There is a natural “parallelity” W∗dW=(dW)∗W within the Hilbert-Schmidt operators W. This defines parallelity and holonomy along curves of density operators ϱ=WW∗. There is an intrinsic non-linearity in the parallel transport which dissolves for curves of projection operators. In the latter case one comes back to the Bott-Chern parallel transport.

A gauge field governing parallel transport along mixed states

At first, a short account is given of some basic notations and results on parallel transport along mixed states. A new connection form (gauge field) is introduced to give a geometric meaning to the concept of parallelity in the theory of density operators.

Entropy and Optimal Decompositions of States Relative to a Maximal Commutative Subalgebra

To calculate the entropy of a subalgebra or of a channel with respect to a state, one has to solve an intriguing optimalization problem. The latter is also the key part in the entanglement of formation concept, in which case the subalgebra is a subfactor.I consider some general properties, valid for these definitions in finite dimensions, and apply them to a maximal commutative subalgebra of a full matrix algebra. The main method is an interplay between convexity and symmetry. A collection of helpful tools from convex analysis is collected in an appendix.

The Metric of Bures and the Geometric Phase.

After the appearance of the papers of Berry [1], Simon [2], and of Wilczek and Zee [3], I tried to understand [4], whether there is a reasonable extension of the geometric phase - or, more accurately, of the accompanying phase factor - for general (mixed) states. A known recipe for such exercises is to use purifications: One looks for larger, possibly fictitious, quantum systems from which the original mixed states are seen as reductions of pure states. For density operators there is a standard way to do so by the use of Hilbert Schmidt operators (or by Hilbert Schmidt maps from an auxiliary Hilbert space into the original one).

Density operators as an arena for differential geometry

What I am going to describe may be called an interplay between concepts of differential geometry and the superposition principle of quantum physics. In particular, it concerns a metrical distance introduced by Bures [14] as a non-commutative version of a construction of Kakutani [24] on the one hand, and on the other hand the purifications of mixed states in physically larger systems, including the problem of geometric phases associated with a distinguished class of such extensions. The Bures distance and the general transition probability [15], [27] are discussed in [lo], [ll], [30]: and further papers. For the sake of clarity, and to avoid technicalities, I will be concerned with finitedimensional objects. Let ‘Ft denote a Hilbert space with complex dimension n,. The set of density operators defined on it is

Geometric phases and related structures

Abstract The parallel transport responsible for the geometric phase is reviewed emphasizing the role of transition probabilities and of the metric of Bures.

An energy dispersion estimate

Abstract Given the density operator ϱ 1 as an initial value of a Hamiltonian motion that evolves in a time interval Δ t to ϱ 2 . Then Δ t Δ E , Δ E being the energy dispersion (or energy uncertainty) of the motion, can be estimated from below by comparing the length of the Hamiltonian curve with a geodesic joining the initial and the final density operator. The lengths are calculated in the Bures metric.

On the Shannon entropy and related functionals on convex sets

We give the definition of functionals r(K,x) and r(K,S,x) defined on convex sets K without or with respect to locally convex topology with the help of a strongly convex function r(p) on a unit interval. If r = —p ln p we refer r(K,x) to be the Shannon entropy of x relative to the convex set K. In the case of the convex set Zn of density matrices this definition gives the usual Shannon–Gibbs entropy and yields a new defining inequality for the entropy which is independent of the representation of the algebra of n×n-matrices.