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Bengtsson I., Zyczkowski K. Geometry of Quantum States: An Introduction to Quantum Entanglement

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Bengtsson I., Zyczkowski K. Geometry of Quantum States: An Introduction to Quantum Entanglement
Cambridge University Press, 2006. — 479 p. — ISBN 0521814510.
The geometry of quantum states is a highly interesting subject in itself, but it is also relevant in view of possible applications in the rapidly developing fields of quantum information and quantum computing.
But what is it? In physicswords like ‘states’ and ‘system’ are often used. Skipping lightly past the question of what these words mean – it will be made clear by practice – it is natural to ask for the properties of the space of all possible states of a given system. The simplest state space occurs in computer science: a ‘bit’ has a space of states that consists simply of two points, representing on and off. In probability theory the state space of a bit is really a line segment, since the bit may be ‘on’ with some probability between zero and one. In general the state spaces used in probability theory are ‘convex hulls’ of a discrete or continuous set of points. The geometry of these simple state spaces is surprisingly subtle – especially since different ways of distinguishing probability distributions give rise to different notions of distance, each with their own distinct operational meaning. There is an old idea saying that a geometry can be understood once it is understood what linear transformations are acting on it, and we will see that this is true here as well.
Convexity, colours and statistics
Geometry of probability distributions
Much ado about spheres
Complex projective spaces
Outline of quantum mechanics
Coherent states and group actions
The stellar representation
The space of density matrices
Purification of mixed quantum states
Quantum operations
Duality: maps versus states
Density matrices and entropies
Distinguishability measures
Monotone metrics and measures
Quantum entanglement
Basic notions of differential geometry
Basic notions of group theory
Geometry: do it yourself
Hints and answers to the exercises
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