Mixed-State Entanglement and Quantum Communication

Abstract

Quantum entanglement is one of the most striking features of the quantum formalism [26]. It can be expressed as follows: If two systems interacted in the past it is, in general, not possible to assign a single state vector to either of the two subsystems [257]. This is what is sometimes called the principle of nonseparability. A common example of an entangled state is the singlet state [258],

$$ \psi _ - = \frac{1} {{\sqrt 2 }}(\left| {01} \right\rangle - \left| {10} \right\rangle ). $$

(5.1)

One can see that it cannot be represented as a product of individual vectors describing states of subsystems. Historically, entanglement was first recognized by Einstein, Podolsky and Rosen (EPR) [24] and by Schrödinger [5]. In their famous paper, EPR suggested a description of the world (called “local realism”) which assigns an independent and objective reality to the physical properties of the well-separated subsystems of a compound system. Then EPR applied the criterion of local realism to predictions associated with an entangled state to conclude that quantum mechanics was incomplete. The EPR criticism was the source of many discussions concerning fundamental differences between the quantum and classical descriptions of nature.