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Matrix Analysis and Operator Theory with Applications to Quantum Information Theory

Shayne Plosker

Published Jul 1, 2013• 1 citation
OpenAlex

Abstract

We explore the connection between quantum error correction and quantum cryptography through the notion of conjugate (or complementary) channels. This connection is at the level of subspaces and operator subsystems; if we use a more general form of subsystem, the link between the two topics breaks down. We explore both the subspace and subsystem settings.\nError correction arises as a means of addressing the issue of the introduction of noise to a message being sent from one party to another. Noise also plays a role in quantum measurement theory: If one wishes to measure a system that is in a particular state via a measurement apparatus, one can first act upon the system by a quantum channel, which can be thought of as a noise source, and then measure the resulting system using a different measurement apparatus. Such a setup amounts\nto the introduction of noise to the measurement process, yet has the advantage of preserving the measurement statistics. Preprocessing by a quantum channel leads to the partial order "cleaner than" on quantum probability measures. Other meaningful partial orders on quantum probability measures exist, and we shall investigate that of cleanness as well as that of absolute continuity.\nLastly, we investigate partial orders on vectors corresponding to quantum states; such partial orders, namely majorization and trumping, have been linked to entanglement theory. We characterize trumping first by means of yet another partial order, power majorization, which gives rise to a family of examples. We then characterize trumping through the complete monotonicity of certain Dirichlet polynomials corresponding to the states in question. This not only generalizes a recent characterization of trumping, but the use of such mathematical objects simpli es the derivation of the result.

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Quantum computing exists at the intersection of mathematics, physics, chemistry, and engineering; the main goal of quantum computing is the creation of devices and algorithms which use the properties of quantum mechanics to store, manipulate and measure information. There exist many families of algorithms, which, using non-classical logical operations, can outperform traditional, classical algorithms in terms of memory and processing requirements. In addition, quantum computing devices are fundamentally smaller than classical processors and memory elements; since the physical models governing their performance are applicable on all scales, as opposed to classical logic elements, whose underlying principles rely on the macroscopic nature of the device in question. \n \nQuantum algorithms, for the most part, are predicated on a theory of resources. It is often assumed that quantum computers can be placed in a precise fiducial state prior to computation, and that logical operations are perfect, inducing no error on the system which they affect. These assumptions greatly simplify algorithmic design, but are fundamentally unrealistic. In order to justify their use, it is necessary to develop a framework for using a large number of imperfect devices to simulate the action of a perfect device, with some acceptable probability of failure. This is the study of fault-tolerant quantum computing. In order to pursue this study effectively, it is necessary to understand the fundamental nature of generic quantum states and operations, as well as the means by which one can correct quantum errors. Additionally, it is important to attempt to minimize the use of computational resources in achieving error reduction and fault-tolerant computing. \n \nThis thesis is concerned with three projects related to the use of error-prone quantum systems to transmit and manipulate information. The first of these is concerned with the use of imperfectly-prepared states in error-correction routines. Using optimal quantum error correction, we are able to deduce a method of partially protecting encoded quantum information against preparation errors prior to encoding, using no additional qubits. The second of these projects details the search for entangled states which can be used to transmit classical information over quantum channels at a rate superior to classical states. The third of these projects concerns the transcoding of data from one quantum code into another using few ancillary resources. The descriptions of these projects are preceded by a brief introduction to representations of quantum states and channels, for completeness. \n \nThree techniques of general interest are presented in appendices. The first is an introduction to, and a minor advance in the development of optimal error correction codes. The second is a more efficient means of calculating the action of a quantum channel on a given state, given that the channel acts non-trivially only on a subsystem, rather than the entire system. Finally, we include documentation on a software package developed to aid the search for quantum transcoding operations.

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Introduction to quantum error correction

Andrew M. Steane

An introduction to quantum error correction (QEC) is given, and some recent developments are described. QEC consists of two parts: the physics of error processes and their reversal, and the construction of quantum error–correcting codes. Errors are caused both by imperfect quantum operations, and by coupling between the quantum system and its environment. Any such process can be analysed into a sum of ‘error operators’, which are tensor products of Pauli spin operators. These are the analogues of classical error vectors. A quantum error correcting code is a set of orthogonal states, ‘quantum codewords’, which behave in a certain useful way under the action of the most likely error operators. A computer or channel which only uses such states can be corrected by measurements which determine the error while yielding no information about which codeword or superposition of codewords is involved. Powerful codes can be found using a construction based on classical error–correcting codes. An analysis which allows even the corrective operations themselves to be imperfect leads to powerful and counter–intuitive results: the quantum coherence of a long quantum computation can be preserved even though every qubit in the computer relaxes spontaneously many times before the computation is complete.

Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences•Aug 15, 1998•38 citations
We explore the connection between quantum error correction and quantum cryptography through the notion of conjugate (or complementary) channels. This connection is at the level of subspaces and operator subsystems; if we use a more general form of subsystem, the link between the two topics breaks down. We explore both the subspace and subsystem settings.\nError correction arises as a means of addressing the issue of the introduction of noise to a message being sent from one party to another. Noise also plays a role in quantum measurement theory: If one wishes to measure a system that is in a particular state via a measurement apparatus, one can first act upon the system by a quantum channel, which can be thought of as a noise source, and then measure the resulting system using a different measurement apparatus. Such a setup amounts\nto the introduction of noise to the measurement process, yet has the advantage of preserving the measurement statistics. Preprocessing by a quantum channel leads to the partial order "cleaner than" on quantum probability measures. Other meaningful partial orders on quantum probability measures exist, and we shall investigate that of cleanness as well as that of absolute continuity.\nLastly, we investigate partial orders on vectors corresponding to quantum states; such partial orders, namely majorization and trumping, have been linked to entanglement theory. We characterize trumping first by means of yet another partial order, power majorization, which gives rise to a family of examples. We then characterize trumping through the complete monotonicity of certain Dirichlet polynomials corresponding to the states in question. This not only generalizes a recent characterization of trumping, but the use of such mathematical objects simpli es the derivation of the result.
We make an explicit connection between fundamental notions in quantum cryptography and quantum error correction. Error-correcting subsystems (and subspaces) for quantum channels are the key vehicles for contending with noise in physical implementations of quantum information processing. Private subsystems (and subspaces) for quantum channels play a central role in cryptographic schemes such as quantum secret sharing and private quantum communication. We show that a subsystem is private for a channel precisely when it is correctable for a complementary channel. This result is shown to hold even for approximate notions of private and correctable defined in terms of the diamond norm for superoperators.
This paper addresses and expands on the contents of the recent Letter [Phys. Rev. Lett. 111, 030502 (2013)] discussing private quantum subsystems. Here we prove several previously presented results, including a condition for a given random unitary channel to not have a private subspace (although this does not mean that private communication cannot occur, as was previously demonstrated via private subsystems) and algebraic conditions that characterize when a general quantum subsystem or subspace code is private for a quantum channel. These conditions can be regarded as the private analog of the Knill-Laflamme conditions for quantum error correction, and we explore how the conditions simplify in some special cases. The bridge between quantum cryptography and quantum error correction provided by complementary quantum channels motivates the study of a new, more general definition of quantum error-correcting code, and we initiate this study here. We also consider the concept of complementarity for the general notion of a private quantum subsystem.
The most general method for encoding quantum information is not to encode the information into a subspace of a Hilbert space, but to encode information into a subsystem of a Hilbert space. Recently this notion has led to a more general notion of quantum error correction known as operator quantum error correction. In standard quantum error-correcting codes, one requires the ability to apply a procedure which exactly reverses on the error-correcting subspace any correctable error. In contrast, for operator error-correcting subsystems, the correction procedure need not undo the error which has occurred, but instead one must perform corrections only modulo the subsystem structure. This does not lead to codes which differ from subspace codes, but does lead to recovery routines which explicitly make use of the subsystem structure. Here we present two examples of such operator error-correcting subsystems. These examples are motivated by simple spatially local Hamiltonians on square and cubic lattices. In three dimensions we provide evidence, in the form a simple mean field theory, that our Hamiltonian gives rise to a system which is self-correcting. Such a system will be a natural high-temperature quantum memory, robust to noise without external intervening quantum error-correction procedures.
Quantum computing exists at the intersection of mathematics, physics, chemistry, and engineering; the main goal of quantum computing is the creation of devices and algorithms which use the properties of quantum mechanics to store, manipulate and measure information. There exist many families of algorithms, which, using non-classical logical operations, can outperform traditional, classical algorithms in terms of memory and processing requirements. In addition, quantum computing devices are fundamentally smaller than classical processors and memory elements; since the physical models governing their performance are applicable on all scales, as opposed to classical logic elements, whose underlying principles rely on the macroscopic nature of the device in question. \n \nQuantum algorithms, for the most part, are predicated on a theory of resources. It is often assumed that quantum computers can be placed in a precise fiducial state prior to computation, and that logical operations are perfect, inducing no error on the system which they affect. These assumptions greatly simplify algorithmic design, but are fundamentally unrealistic. In order to justify their use, it is necessary to develop a framework for using a large number of imperfect devices to simulate the action of a perfect device, with some acceptable probability of failure. This is the study of fault-tolerant quantum computing. In order to pursue this study effectively, it is necessary to understand the fundamental nature of generic quantum states and operations, as well as the means by which one can correct quantum errors. Additionally, it is important to attempt to minimize the use of computational resources in achieving error reduction and fault-tolerant computing. \n \nThis thesis is concerned with three projects related to the use of error-prone quantum systems to transmit and manipulate information. The first of these is concerned with the use of imperfectly-prepared states in error-correction routines. Using optimal quantum error correction, we are able to deduce a method of partially protecting encoded quantum information against preparation errors prior to encoding, using no additional qubits. The second of these projects details the search for entangled states which can be used to transmit classical information over quantum channels at a rate superior to classical states. The third of these projects concerns the transcoding of data from one quantum code into another using few ancillary resources. The descriptions of these projects are preceded by a brief introduction to representations of quantum states and channels, for completeness. \n \nThree techniques of general interest are presented in appendices. The first is an introduction to, and a minor advance in the development of optimal error correction codes. The second is a more efficient means of calculating the action of a quantum channel on a given state, given that the channel acts non-trivially only on a subsystem, rather than the entire system. Finally, we include documentation on a software package developed to aid the search for quantum transcoding operations.
Operator quantum error correction is a technique for robustly storing quantum information in the presence of noise. It generalizes the standard theory of quantum error correction, and provides a unified framework for topics such as quantum error correction, decoherence-free subspaces, and noiseless subsystems. This paper develops (a) easily applied algebraic and information-theoretic conditions that characterize when operator quantum error correction is feasible; (b) a representation theorem for a class of noise processes that can be corrected using operator quantum error correction; and (c) generalizations of the coherent information and quantum data processing inequality to the setting of operator quantum error correction.
An introduction to quantum error correction (QEC) is given, and some recent developments are described. QEC consists of two parts: the physics of error processes and their reversal, and the construction of quantum error–correcting codes. Errors are caused both by imperfect quantum operations, and by coupling between the quantum system and its environment. Any such process can be analysed into a sum of ‘error operators’, which are tensor products of Pauli spin operators. These are the analogues of classical error vectors. A quantum error correcting code is a set of orthogonal states, ‘quantum codewords’, which behave in a certain useful way under the action of the most likely error operators. A computer or channel which only uses such states can be corrected by measurements which determine the error while yielding no information about which codeword or superposition of codewords is involved. Powerful codes can be found using a construction based on classical error–correcting codes. An analysis which allows even the corrective operations themselves to be imperfect leads to powerful and counter–intuitive results: the quantum coherence of a long quantum computation can be preserved even though every qubit in the computer relaxes spontaneously many times before the computation is complete.