1 Definitions and Examples 1.1 Metric and Normed Spaces Definition 1.1. A metric space is a pair (X, d), where X is a set and d is a function from X ×X to R such that the following conditions hold for every x, y, z ∈ X. 1. Non-negativity: d(x, y) ≥ 0. 2. Symmetry: d(x, y) = d(y, x). 3. Triangle inequality: d(x, y) + d(y, z) ≥ d(x, y) . 4. d(x, y) = 0 if and only if x = y. Elements of X are call...

In 2002 Coecke and Martin created a Bayesian order for the finite dimensional spaces of classical states in physics and used this to define a similar order, the spectral order on the finite dimensional quantum states. These orders gave the spaces a structure similar to that of a domain. This allows for measuring information content of states and for determining which partial states are approxim...

In quantum cryptography unconditional security proofs are derived under the assumption that Alice's and Bob's apparata (private spaces) are completely inaccessible by an eavesdropper who, therefore, can only attack the signal systems which are transmitted through the quantum communication channel connecting the two parties. Under this assumption, secret-key rates and security thresholds are der...

We propose a theory of learning aimed to formalize some ideas underlying Coquand’s game semantics and Krivine’s realizability of classical logic. We introduce a notion of knowledge state together with a new topology, capturing finite positive and negative information that guides a learning strategy. We use a leading example to illustrate how non-constructive proofs lead to continuous and effect...

Through the example of the quantum symplectic 4-sphere, we discuss how the notion of twisted spectral triple fits into the framework of quantum homogeneous spaces.