Geometry of Quantum States:
  An Introduction to Quantum Entanglement

I. Bengtsson and K. Życzkowski

Cambridge University Press, Cambridge, 2006


After going through the chapter on entanglement you will have reached the end of our book. As the subtitle suggests, its aim was literally to present an introduction to the subject of quantum entanglement.

We have left untouched several important aspects of quantum entanglement, including multipartite systems, infinite dimensional systems, and continuous variables.  Moreover, we believe that some key ideas presented in the book might be extended  much further then we have managed to do. For instance the maps--states duality, illustrated in chapter 11, might be used to find relations between capacities of quantum channels and measures of entanglement of the corresponding states of an extended system.

In the book we  have consistently used a geometric approach to highlight similarities and differences between the classical and quantum spaces of states. What is the knowledge gained by studying the book good for? We hope it will contribute to a better understanding of quantum mechanics. We hope also that it will provide a solid foundation for a new, emerging field of science -- the theory of quantum information processing. Quantum entanglement plays a decisive role in all branches of the  field including quantum cryptography, quantum error correction, and quantum computing.

In trying to describe the intricate geometry of  the space of quantum states, we have deliberately restricted ourselves to discussing the statics of quantum theory. We have presented the arena, in which quantum information can be processed. We have not attempted to inject any concrete dynamics into our arena, but hope that readers equipped with some knowledge of its properties may introduce into it spectacular action.

In a sense we have characterised all the peculiarities of football fields of various sizes, without even specifying the rules of the game. Having at your disposal a huge flat grassy field, you can play soccer, cricket, American football, rugby, or Australian football, according to your mood and wishes.

In a similar way you can play different games in the multi--dimensional  arena of quantum states. It stays there right in the center of the beautiful Platonic world of quantum theory, accessible to all of us.

Its rich structure provides a real challenge especially for young researchers.
We wish you a good game in fine company!

Good luck !


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