The Schroedinger equation for a particle in a potential is introduced and the general properties of its solutions are discussed; the uncertainity relations are derived; the Gram--Schimdt procedure for orthonormalizing a set of independent wave functions is introduced; the time evolution of the expectation values of the position and momentum operatorsfor a particle in a potential and in an electromagnetic field are derived.

We show that for two classical Brownian particles there exists an analog ofcontinuous-variable quantum entanglement: The common probability distributionof the two coordinates and the corresponding coarse-grained velocitiescannot be prepared via mixing of any factorized distributions referring tothe two particles in separate. This is possible for particles which interactedin the past, but do not interact in the present. Three factors are crucial forthe effect: (1) separation of time-scales of coordinate and momentum whichmotivates the definition of coarse-grained velocities; (2) the resulting uncertaintyrelations between the coordinate of the Brownian particle and thechange of its coarse-grained velocity; (3) the fact that the coarse-grained velocity,though pertaining to a single Brownian particle, is defined on a commoncontext of two particles. The Brownian entanglement is a consequenceof a coarse-grained description and disappears for a finer resolution of theBrownian motion. We discuss possibilities of its experimental realizations inexamples of macroscopic Brownian motion.

This is a translation of an anonymous report published about Einstein’s seminar in Berlin in November of 1931 dicussed in detail in Chapter 1. The report describes Einstein discussing the meaning of Heisenberg’s uncertainty relations and describing his famous photon-box thought experiment.

This is a translation of notes by Schrödinger wherein he draws the implications of Einstein’s photon-box thought experiment.

In this chapter we present an alternative path to base security in challenging settings. We will discover that physical assumptions on the adversary, such that they have a bounded or a noisy quantum memory, can be leveraged to design secure protocols for tasks, such as 1-2 oblivious transfer, for which there cannot exist an unconditionally secure protocol. To prove security we make a fresh use of uncertainty relations introduced earlier in the context of quantum key distribution.

A quantum key distribution (QKD) protocol allows two honest users Alice and Bob to harness the advantages of quantum information processing to generate a shared secret key. The most well-known, and indeed the first QKD protocol that was discovered, is called BB’84, after its inventors Bennett and Brassard and the year in which their paper describing the protocol was published. In this chapter we describe the BB’84 protocol and we introduce the main ideas for showing that the protocol is secure.