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The authors, distinguished mathematical physicists, have written a markedly distinctive, dedicatedly pedagogical, suitably rigorous text, designed, in part, for advanced undergraduates familiar with the principles of quantum mechanics. The book, pleasing in character and enthusiastic in tone, has many stimulating diagrams and tables, as well as problem sets (with hints and answers supplied at the end). The diverse topics covered -conveniently all assembled here -reflect the geometrically-oriented, fundamental quantum-information-theoretic interests and expertise of the two authors. (Several of the areas surveyed are among the many discussed in another recent copiously illustrated book, Roger Penrose's The road to reality MR2116746 (2005k:83002), which has a broader, less specialized audience as its overall target.)In the first chapter ("Convexity, colours and statistics"), convexity is discussed using, most notably, the three-dimensional example of color-mixing -making use of chromaticity diagrams and MacAdam ellipses. The second chapter ("Geometry of probability distributions") deals with the geometry of probability distributions, setting the stage for the later quantum extension of these concepts. (The reviewer has examined a certain area of overlap between the classical and quantum treatments, in this regard,

Mutually unbiased bases for quantum degrees of freedom are central to all theoretical investigations and practical exploitations of complementary properties. Much is known about mutually unbiased bases, but there are also a fair number of important questions that have not been answered in full as yet. In particular, one can find maximal sets of N + 1 mutually unbiased bases in Hilbert spaces of prime-power dimension N = p m , with p prime and m a positive integer, and there is a continuum of mutually unbiased bases for a continuous degree of freedom, such as motion along a line. But not a single example of a maximal set is known if the dimension is another composite number (N = 6, 10, 12, . . . ).In this review, we present a unified approach in which the basis states are labeled by numbers 0, 1, 2, . . . , N − 1 that are both elements of a Galois field and ordinary integers. This dual nature permits a compact systematic construction of maximal sets of mutually unbiased bases when they are known to exist but throws no light on the open existence problem in other cases. We show how to use the thus constructed mutually unbiased bases in quantum-informatics applications, including dense coding, teleportation, entanglement swapping, covariant cloning, and state tomography, all of which rely on an explicit set of maximally entangled states (generalizations of the familiar two-q-bit Bell states) that are related to the mutually unbiased bases.There is a link to the mathematics of finite affine planes. We also exploit the one-toone correspondence between unbiased bases and the complex Hadamard matrices that turn the bases into each other. The Hadamard-matrix approach is instrumental in the very recent advance, surveyed here, of our understanding of the N = 6 situation. All evidence indicates that a maximal set of seven mutually unbiased bases does not exist -one can find no more than three pairwise unbiased bases -although there is currently no clear-cut demonstration of the case.

We analyze several product measures in the space of mixed quantum states. In particular we study measures induced by the operation of partial tracing. The natural, rotationally invariant measure on the set of all pure states of a N × K composite system, induces a unique measure in the space of N ×N mixed states (or in the space of K ×K mixed states, if the reduction takes place with respect to the first subsystem). For K = N the induced measure is equal to the Hilbert-Schmidt measure, which is shown to coincide with the measure induced by singular values of non-Hermitian random Gaussian matrices pertaining to the Ginibre ensemble. We compute several averages with respect to this measure and show that the mean entanglement of N × N pure states behaves as lnN − 1/2. 03.65.Ca, 03.65.Ud