Matrix Differential Calculus with Applications in Statistics and Econometrics

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

It may seem inevitable that highly entangled quantum states are susceptible to disturbance through interaction with a decohering environment. However, certain multiqubit entangled states are well protected from common forms of decoherence as the quantum information is hidden in inherently nonlocal degrees of freedom. This review shows that this robustness is enabled by specific measurements on subsets of qubits, implementing a quantum version of an error correction process. Beginning with the basics, the latest understanding of the relation between this form of error correction and the concept of two-dimensional topological order in many-body physics is reviewed.

Quantum error correction for quantum memories

Journal of Physics A - Mathematical and General, Special Issue. SI Aug 11 2006 ?? Preface

We propose a family of exactly solvable toy models for the AdS/CFT correspondence based on a novel construction of quantum error-correcting codes with a tensor network structure. Our building block is a special type of tensor with maximal entanglement along any bipartition, which gives rise to an isometry from the bulk Hilbert space to the boundary Hilbert space. The entire tensor network is an encoder for a quantum error-correcting code, where the bulk and boundary degrees of freedom may be identified as logical and physical degrees of freedom respectively. These models capture key features of entanglement in the AdS/CFT correspondence; in particular, the Ryu-Takayanagi formula and the negativity of tripartite information are obeyed exactly in many cases. That bulk logical operators can be represented on multiple boundary regions mimics the Rindlerwedge reconstruction of boundary operators from bulk operators, realizing explicitly the quantum error-correcting features of AdS/CFT recently proposed in [1].

/pdf/holographic-quantum-error-correcting-codes-toy-models-for-1rssua6xhb.pdf

Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence

We present a unified approach to quantum error correction, called operator quantum error correction. Our scheme relies on a generalized notion of a noiseless subsystem that is investigated here. By combining the active error correction with this generalized noiseless subsystems method, we arrive at a unified approach which incorporates the known techniques--i.e., the standard error correction model, the method of decoherence-free subspaces, and the noiseless subsystem method--as special cases. Moreover, we demonstrate that the quantum error correction condition from the standard model is a necessary condition for all known methods of quantum error correction.

https://arxiv.org/pdf/quant-ph/0412076.pdf

Unified and generalized approach to quantum error correction

This paper is an expanded and more detailed version of the work [1] in which the Operator Quantum Error Correction formalism was introduced. This is a new scheme for the error correction of quantum operations that incorporates the known techniques -- i.e. the standard error correction model, the method of decoherence-free subspaces, and the noiseless subsystem method -- as special cases, and relies on a generalized mathematical framework for noiseless subsystems that applies to arbitrary quantum operations. We also discuss a number of examples and introduce the notion of "unitarily noiseless subsystems".

https://epiq.physique.usherbrooke.ca/data/files/publications/KLPL05a.pdf

Operator quantum error correction

Every countable directed graph generates a Fock space Hilbert space and a family of partial isometries These operators also arise from the left regular representations of free semigroupoids derived from directed graphs We develop a structure theory for the weak operator topology closed algebras generated by these representations, which we call free semigroupoid algebras We characterize semisimplicity in terms of the graph and show explicitly in the case of finite graphs how the Jacobson radical is determined We provide a diverse collection of examples including; algebras with free behaviour, and examples which can be represented as matrix function algebras We show how these algebras can be presented and decomposed in terms of amalgamated free products We determine the commutant, consider invariant subspaces, obtain a Beurling theorem for them, conduct an eigenvalue analysis, give an elementary proof of reflexivity, and discuss hyper-reflexivity Our main theorem shows the graph to be a complete unitary invariant for the algebra This classification theorem makes use of an analysis of unitarily implemented automorphisms We give a graph-theoretic description of when these algebras are partly free, in the sense that they contain a copy of a free semigroup algebra

Free Semigroupoid Algebras

https://chaos.if.uj.edu.pl/~karol/pdf/CKZ06al.pdf

Higher-rank numerical ranges and compression problems

We show that the theory of operator quantum error correction can be naturally generalized by allowing constraints not only on states but also on observables. The resulting theory describes the correction of algebras of observables (and may therefore suitably be called ``operator algebra quantum error correction''). In particular, the approach provides a framework for the correction of hybrid quantum-classical information and it does not require the state to be entirely in one of the corresponding subspaces or subsystems. We discuss applications to quantum teleportation and to the study of information flows in quantum interactions.

https://arxiv.org/pdf/quant-ph/0608071.pdf

David W. Kribs

Papers

Unified and generalized approach to quantum error correction

Operator quantum error correction

Free Semigroupoid Algebras

Higher-rank numerical ranges and compression problems

Generalization of quantum error correction via the Heisenberg picture.