This largely self-contained book on the theory of quantum information focuses on precise mathematical formulations and proofs of fundamental facts that form the foundation of the subject. It is intended for graduate students and researchers in mathematics, computer science, and theoretical physics seeking to develop a thorough understanding of key results, proof techniques, and methodologies that are relevant to a wide range of research topics within the theory of quantum information and computation. The book is accessible to readers with an understanding of basic mathematics, including linear algebra, mathematical analysis, and probability theory. An introductory chapter summarizes these necessary mathematical prerequisites, and starting from this foundation, the book includes clear and complete proofs of all results it presents. Each subsequent chapter includes challenging exercises intended to help readers to develop their own skills for discovering proofs concerning the theory of quantum information.

The Theory of Quantum Information

Quantum theory of open systems

Fundamental theory: Holomorphic functions and domains of holomorphy Implicit functions and analytic sets The Poincare, Cousin, and Runge problems Pseudoconvex domains and pseudoconcave sets Holomorphic mappings Theory of analytic spaces: Ramified domains Analytic sets and holomorphic functions Analytic spaces Normal pseudoconvex spaces Bibliography Index.

Function theory in several complex variables

In this paper, we obtain a complete description of the invariant subspace structure of an interesting new class of algebras which we call free semigroup algebras. This enables us to prove that they are reflexive, and moreover to obtain a quantitative measure of the distance to these algebras in terms of the invariant subspaces. Such algebras are called hyper-reflexive. This property is very strong, but it has been established in only a very few cases. Moreover the prototypes of this class of algebras are the natural candidate for a non-commutative analytic Toeplitz algebra on n variables. The case we make for this analogy is very compelling. In particular, in this paper, the key to the invariant subspace analysis is a good analogue of the Beurling theorem for invariant subspaces of the unilateral shift. This leads to a notion of inner–outer factorization in these algebras. In a sequel to this paper [13], we add to this evidence by showing that there is a natural homomorphism of the automorphism group onto the group of conformal automorphisms of the ball in Cn. A free semigroup algebra is the weak operator topology closed algebra generated by a set S1, . . . , Sn of isometries with pairwise orthogonal ranges. These conditions are described algebraically by S∗ i Sj = δijI for 1 ≤ i, j ≤ n; (F)

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Invariant Subspaces and Hyper-Reflexivity for Free Semigroup Algebras

The non-commutative analytic Toeplitz algebra is the wot– closed algebra generated by the left regular representation of the free semigroup on n generators. We develop a detailed picture of the algebraic structure of this algebra. In particular, we show that there is a canonical homomorphism of the automorphism group onto the group of conformal automorphisms of the complex n-ball. The k-dimensional representations form a generalized maximal ideal space with a canonical surjection onto the ball of k × kn matrices which is a homeomorphism over the open ball analogous to the fibration of the maximal ideal space of H∞ over the unit disk. In [6, 17, 18, 20], a good case is made that the appropriate analogue for the analytic Toeplitz algebra in n non-commuting variables is the wotclosed algebra generated by the left regular representation of the free semigroup on n generators. The papers cited obtain a compelling analogue of Beurling’s theorem and inner–outer factorization. In this paper, we add further evidence. The main result is a short exact sequence determined by a canonical homomorphism of the automorphism group onto this algebra onto the group of conformal automorphisms of the unit ball of Cn. The kernel is the subgroup of quasi-inner automorphisms, which are trivial modulo the wot-closed commutator ideal. Additional evidence of analytic properties comes from the structure of k-dimensional (completely contractive) representations, which have a structure very similar to the fibration of the maximal ideal space of H∞ over the unit disk. An important tool in our analysis is a detailed structure theory for wot-closed right ideals. Curiously, left ideals remain more obscure. The non-commutative analytic Toeplitz algebra Ln is determined by the left regular representation of the free semigroup Fn on n generators z1, . . . , zn which acts on `2(Fn) by λ(w)ξv = ξwv for v, w in Fn. In particular, the algebra Ln is the unital, wot-closed algebra generated by the isometries Li = λ(zi) for 1 ≤ i ≤ n. This algebra and its norm-closed version (the noncommutative disk algebra) were introduced by Popescu [19] in an abstract sense in connection with a non-commutative von Neumann inequality and 1991 Mathematics Subject Classification. 47D25. March 9, 1997; October 9, 1997 final draft. First author partially supported by an NSERC grant and a Killam Research Fellowship. Second author partially supported by an NSF grant.

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The algebraic structure of non-commutative analytic Toeplitz algebras

General results of interpolation (e.g., Nevanlinna-Pick) by elements in the noncommutative analytic Toeplitz algebraF
 ∞ (resp., noncommutative disc algebraA
 n) with consequences to the interpolation by bounded operator-valued analytic functions in the unit ball of ℂn are obtained. Noncommutative Poisson transforms are used to provide new von Neumann type inequalities. Completely isometric representations of the quotient algebraF
 ∞/J on Hilbert spaces whereJ is anyw
 *-closed, 2-sided ideal ofF
 ∞, are obtained and used to construct aw
 *-continuous,F
 ∞/J-functional calculus associated to row contractionsT=[T
 1,…,T
 n] whenf(T1, …, Tn)=0 for anyf∈J. Other properties of the dual algebraF
 ∞/J are considered.

Noncommutative interpolation and Poisson transforms

Quantum operations frequently occur in quantum measurement theory, quantum probability, quantum computation, and quantum information theory. If an operator A is invariant under a quantum operation φ, we call A a φ-fixed point. Physically, the φ-fixed points are the operators that are not disturbed by the action of φ. Our main purpose is to answer the following question. If A is a φ-fixed point, is A compatible with the operation elements of φ? We shall show in general that the answer is no and we shall give some sufficient conditions under which the answer is yes. Our results will follow from some general theorems concerning completely positive maps and injectivity of operator systems and von Neumann algebras.

Fixed points of quantum operations

General results of interpolation (eg. Nevanlinna-Pick) by elements in the noncommutative analytic Toeplitz algebra $F^\infty$ (resp. noncommutative disc algebra $A_n$) with consequences to the interpolation by bounded operator-valued analytic functions in the unit ball of ${\bf C}^n$ are obtained. 
Non-commutative Poisson transforms are used to provide new von Neumann type inequalities. Completely isometric representations of the quotient algebra $F^\infty/J$ on Hilbert spaces, where $J$ is any $w^*$-closed, 2-sided ideal of $F^\infty$, are obtained and used to construct a $w^*$-continuous, $F^\infty/J$--functional calculus associated to row contractions $T=[T_1,\dots, T_n]$ when $f(T_1,\dots,T_n)=0$ for any $f\in J$. Other properties of the dual algebra $F^\infty/J$ are considered.

Noncommutative Interpolation and Poisson transforms

The framework of the paper is that of the full Fock space$\mathcal{F}^2 (\mathcal{H}_n )$ and the Banach algebraF∞ which can be viewed as non-commutative analogues of the Hardy spacesH2 andH∞ respectively.

Factorization and reflexivity on Fock spaces

A Banach space X is prime if every infinite-dimensional complemented subspace contains a further subspace which is isomorphic to X. A Banach space X is said to be primary if whenever X — Y © Z, X is isomorphic to either Y or Z. The classical examples of prime spaces are the spaces ipj 1 < p < oc. Many spaces derived from the £p-spaces in various ways are primary (see for example [AEO] and [CL]). The primarity of B(H) was shown by Blower [B] in 1990, and Arias [A] has recently developed further techniques which are used to prove the primarity of Cι, the space of trace class operators (this was first shown by Arazy [Arl, Ar2]). It has become clear that these techniques are not naturally confined to a Hubert space context; in the present paper we wish to extend the results to a variety of tensor products and operator spaces of ^-spaces (and in some cases £p-spaces). We also include some related results. Some of the intermediate propositions (on factoring operators through the identity) may actually be true for a wider class of Banach spaces (those with unconditional bases which have nontrivial lower and upper estimates). In fact, the combinatorial aspects of the factorization can be applied quite generally, and may have other applications. The proofs of primarity, however, rely on Pelczyήski's decomposition method which is not so readily extended. We have thus kept mainly to the case of injective and projective tensor products of tv spaces throughout. The results we obtain apply to the growing study of polynomials on Banach spaces since polynomials may be considered as symmetric multilinear operators with an equivalent norm (see [FJ], [M],

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Alvaro Arias

Papers

Noncommutative interpolation and Poisson transforms

Fixed points of quantum operations

Noncommutative Interpolation and Poisson transforms

Factorization and reflexivity on Fock spaces

On the structure of tensor products of $l_p$-spaces.