Quantum computation and quantum information. A mathematical perspective (Q6555070)

This book is for a one-semester course in quantum computing and quantum information theory, emphasizing the mathematical aspects and the historical continuity of both algorithms and information theory.\N\NThe synopsis of the book goes as follows.\N\N\begin{itemize}\N\item[Chapter 1] presents classical and probabilistic computation via linear algebra in order to pave the way for quantum computation.\N\N\item[Chapter 2] covers aspects of quantum mechanics needed for quantum computation.\N\N\item[Chapter 3] covers the basics of quantum computing and the standard quantum algorithms. It presents the algorithms of Grover and Simon, respectively in \S 3.1 and \S 3.2. Elementary number theory needed for the algorithms of Miller-Rabin and Shor is presented in \S 3.3. \S 3.4 presents the Miller-Rabin primality test. \S 3.5 discusses admissible quantum gates. \S 3.6 is concerned with Shor's algorithm.\N\N\item[Chapter 4] is concerned with classical information theory. \S 4.1 deals with toy examples leading up to the notion of channel capacity. \S 4.2 gives a preliminary version of Shannon's noiseless channel theorem. \S 4.3 studies entropy as a function on probability distributions with many interesting properties. \S 4.4 gives the statement and proof of Shannon's noiseless channel theorem, along with several variants. \S 4.5 establishes the noisy channel theorem.\N\N\item[Chapter 5] serves as preparation for the quantum information theory presented in Chapter 6. \S 5.1 presents a convenient reformulation of the postulates of quantum mechanics in terms of density operators, which gives rise to several new concepts and ways of viewing composite systems as discussed in \S 5.2. \S 5.3 discusses distances between density operators. \S 5.4 discusses briefly singular value and polar decompositions. \S 5.5 defines quantum channels, describing several presentations of them.\N\N\item[Chapter 6] is concerned with quantum information. \S 6.1 presents a quantum version of Shannon's noiseless channel theorem. \S 6.2 discusses properties of von Neumann entropy. \S 6.3 introduces conditional von Neumann entropy, establishing the strong subadditivity of von Neumann entropy. \S 6.4 discusses information-theoretic consequences of strong additivity, including a bound on classical teleportation. \S 6.5 introduces a generalization of quantum teleportation, called LOCC, presenting classical results of Hardy-Littlewood-Polya and Birkhoff as well as Nielson's theorem on LOCC. \S 6.6 gives an overview of further measures of entanglement.\N\N\item[Chapter 7] is concernes with representation theory and quantum information. \S 7.1 is a crash course in representation theory focusing on Schur-Weyl duality. \S 7.2 makes use of Schur-Weyl duality to obtain a representation-theoretic interpretation of the quantum \(\varepsilon\)-typical subspaces. \S 7.3 discusses the quantum marginal problem addressed by Klayatchko and independtly by Christandl and Mitchison. \S 7.4 presents a generalization of the quantum marginal results to arbitrary density operators on tripartite states from [\textit{M. Christandl} et al., Ann. Henri Poincaré 19, No. 2, 385--410 (2018; Zbl 1383.81367)]. \S 7.5 establishes the finite generation of the semigroup Kron\(_{m,n,k}\)\ consisting of nonvanishing Kronecker coefficients.\N\N\item[Appendix A] is concerned with algebra and linear algebra.\N\N\item[Appendix B] is concerned with probability.\N\end{itemize}