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Quantum systems, channels, information. A mathematical introduction. (English) Zbl 1332.81003

de Gruyter Studies in Mathematical Physics 16. Berlin: de Gruyter (ISBN 978-3-11-027325-0/hbk; 978-3-11-027340-3/ebook). xiii, 349 p. (2012).
The concept “channel” is central in information theory from the very beginning of the classical up to the quantum information theory. The fundamentals, introduced by C. Shannon in the forties of the latter century, have been developed and extended remarkably, especially with respect to the quantum theory because the non-classical features entanglement, state operations, and uncertainty relations have to be recognized. All these topics, from the classical coding theorem through their quantal generalizations, are explained and represented with mathematical rigor. They are arranged in five parts and twelve chapters. Notes and references are added at the end each chapter. The book is suited for researchers, and lecturers who find the respective matter worked out and well ordered, as well as for students who may check their grasp, since any step is accompanied by task-oriented exercises.
Part I introduces to the quantum theory of finite level systems, statistical state discrimination, entanglement, dense coding, and teleportation are included. Part II considers first the classical channel and then the classical-quantum channel. The latter consists in encoding a classical input letter \(x\) into a quantum state \(S_x\) followed by a measurement of an observable defined by the POV measure \(M_{(\cdot)}\) to get a classical output letter \(y\) with the conditional probability \(p(y|x) = \mathrm{Tr}(S_xM_y)\). For both kinds of channels, the coding theorems are formulated, proven, and compared. Part III introduces general state operations as completely positive maps. The Stinespring dilation theorem is explained and proven in the finite dimensional case. Quantum measurement processes, entanglement breaking, PPT, and covariant channels are considered. The Bloch ball representations for the qubit case is given. Then, the v. Neumann entropy and related information quantities are introduced, and their properties are derived. These results are used in Part IV to treat the general channel where \(p(y|x) = \mathrm{Tr}(\Phi[S_x]M_y)\), i.e., the quantum information is transmitted through a quantum channel described by the state operation \(\Phi\). The respective coding theorem is proven and specialized for the covariant, and the depolarizing channel. Then, the additivity problem is extensively considered, several propositions and theorems are proven for special cases. Entropy inequalities and the non-additivity of entropy quantities are represented and discussed. Entanglement-assisted channels using dense coding to transmit classical information are considered and respective coding theorems are proven. The transmission of quantum information is more involved. Quantum error correcting codes are needed. The theorem of E. Knill and R. Lalamme (1997) stating necessary and sufficient conditions for a code to correct errors of a class \(\mathscr{E}\) is explained and proven. The possibility of perfect error correction by transmitting a purification of the original state (coherent transmission) is considered, too. Estimates for metric distances on the set of quantum states are derived and motivate the introduction of fidelity measures. Relations between them are derived. The quantum capacity of quantum information transmitting channels is defined. It vanishes for entanglement breaking channels. Fidelities are discussed. The results are specialized to coherent transmission and so called degradable channels. Another special treatment concerns the classical private capacity and the quantum capacity. Part IV closes with the formulation and proof of the direct coding theorem for the quantum capacity. Part V is devoted to infinite systems, i.e., the dimension of the Hilbert space is infinite. In this case the channel inputs need to be constrained to be treatable. After some analytic definitions and statements on states, operators, and functionals the constrained classical-quantum (c-q) channels for discrete infinite and continuous alphabets as well as the constrained quantum channel are considered including entanglement assisted and entanglement braking channels. The final chapter twelve of the book concerns the problem of Gaussian channels. After explaining the multidimensional version of Stone’s theorem and a related treatment of the harmonic oscillator, its coherent states are introduced. The application to quantum optics leads to a model of noise disturbed classical signals, a mixture of coherent states \(S_{\mu}\) belonging to the class of so called ‘classical states’ in quantum optics, \(\mu\) being a complex number. Methods developed treating the c-q channel are applied to put constraints and to get the capacity for a c-q Gaussian channel. A thorough recapitulation of the canonical quantization of mechanical systems of finite degrees of freedom based on Weyl operators, symplectic vector spaces, associated complex structures and gauge transformations prepares the introduction of Gaussian states. For a quantum channel Hilbert spaces \({\mathcal H}_A \otimes {\mathcal H}_D\) for the input side and \({\mathcal H}_B \otimes {\mathcal H}_E\) for the output side, where \({\mathcal H}_A\) and \({\mathcal H}_B\) concern the the sent and received informations, \({\mathcal H}_D\) and \({\mathcal H}_E\) concern the environments, respectively, are considered. These Hilbert spaces are representation spaces of Weyl quantizations of the symplectic vector spaces \(Z_A,Z_B,Z_D,Z_E\), respectively. Then, Gausssian states are defined and their density operators, entropies, separability and purifications are considered. The properties of an open channel to be ‘linear bosonic’ and ‘Gaussian’ are defined and a necessary and sufficient condition for the latter property to hold true is proven. Also, a criterion for complete positivity and further properties of the channel operation are given. Gaussian observables defined by POV-measures on the Borel sets of \(Z_B\) acting on \({\mathcal H}_A\) are introduced, and the existence of suitable bosonic systems is proven. A criterion for a Gaussian channel to be entanglement breaking is proven. Gauge covariant channels are considered, a propostion on the coherent information is obtained, and a conjecture on the classical capacity is formulated. Eventually, the simpler case of Gaussian one mode channels, i.e. \(\dim Z =2\), is treated.
This book mediates the origins and the development up to the present state of art in a very instructive manner. It can be best recommended.

MSC:

81-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to quantum theory
81-02 Research exposition (monographs, survey articles) pertaining to quantum theory
81P45 Quantum information, communication, networks (quantum-theoretic aspects)
94A40 Channel models (including quantum) in information and communication theory
81P70 Quantum coding (general)
94A24 Coding theorems (Shannon theory)
94A17 Measures of information, entropy
94B60 Other types of codes
81P94 Quantum cryptography (quantum-theoretic aspects)
94A60 Cryptography
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