In Section 2.4 we learned that a composite quantum system, composed of two subsystems A and B, is associated with a tensor product space HA? HB. Composite quantum systems have already played an important role in many instances in this text, especially in open systems and measurement models. In this chapter we dive deeper into tensor product spaces, discovering some strange implications of the quantum theory of composite quantum systems.

As a result of its tensor product structure, quantum theory has embedded into it the phenomenon of entanglement, which is often seen as the foremost quantum feature. As early as in 1935 it puzzled Erwin Schrödinger, who introduced the German term Verschränkung [127] and Albert Einstein, who, together with Boris Podolski and Nathan Rosen, used the specific example of an entangled state to argue that quantum theory is incomplete [58]. In the 1960s John Bell demonstrated that the consequences of entanglement are incompatible with the Einstein– Podolsky–Rosen model of local realism [11], [12]. In recent years entanglement has been recognized as the resource for quantum information processing.

Nowadays entanglement theory is a highly developed subject, and we can present only the mathematical basics. For the reader with a deeper interest in entanglement we recommend the recent reviews by Horodecki et al. [80] and Plenio et al. [118], which also cover the quantum information aspects of entanglement. To simplify our discussion we will assume that the Hilbert spaces are finite dimensional.