Related papers: Nuclear Numerical Range and Quantum Error Correcti…
Quantum error correction protects quantum information against environmental noise. When using qubits, a measure of quality of a code is the maximum number of errors that it is able to correct. We show that a suitable notion of ``number of…
The main ideas of quantum error correction are introduced. These are encoding, extraction of syndromes, error operators, and code construction. It is shown that general noise and relaxation of a set of 2-state quantum systems can always be…
We work out a theory of approximate quantum error correction that allows us to derive a general lower bound for the entanglement fidelity of a quantum code. The lower bound is given in terms of Kraus operators of the quantum noise. This…
Quantum error correcting codes have been shown to have the ability of making quantum information resilient against noise. Here we show that we can use quantum error correcting codes as diagnostics to characterise noise. The experiment is…
For a noisy quantum channel, a quantum error correcting code exists if and only if the joint higher rank numerical ranges associated with the error operators of the channel is non-empty. In this paper, geometric properties of the joint…
Quantum error correction is a set of methods to protect quantum information--that is, quantum states--from unwanted environmental interactions (decoherence) and other forms of noise. The information is stored in a quantum error-correcting…
Numerical range of a Hermitian operator X is defined as the set of all possible expectation values of this observable among a normalized quantum state. We analyze a modification of this definition in which the expectation value is taken…
A quantum error-correcting code is defined to be a unitary mapping (encoding) of k qubits (2-state quantum systems) into a subspace of the quantum state space of n qubits such that if any t of the qubits undergo arbitrary decoherence, not…
This is an expository article aiming to introduce the reader to the underlying mathematics and geometry of quantum error correction. Information stored on quantum particles is subject to noise and interference from the environment. Quantum…
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…
Operator quantum error correction provides a unified framework for the known techniques of quantum error correction such as the standard error correction model, the method of decoherence-free subspaces, and the noiseless subsystem method.…
Quantum error correction allows to actively correct errors occurring in a quantum computation when the noise is weak enough. To make this error correction competitive information about the specific noise is required. Traditionally, this…
Some problems of the quantum error-correcting codes theory can be reduced to the investigation of the higher-rank numerical ranges of the operators related to the error operators. We constructively verify a conjecture on the structure of…
Known quantum error correction schemes are typically able to take advantage of only a limited class of classical error-correcting codes. Entanglement-assisted quantum error correction is a partial solution which made it possible to exploit…
Due to the fragility of quantum mechanical effects, real quantum computers are plagued by frequent noise effects that cause errors during computations. Quantum error-correcting codes address this problem by providing means to identify and…
In this introduction we motivate and explain the ``decoding'' and ``subsystems'' view of quantum error correction. We explain how quantum noise in QIP can be described and classified, and summarize the requirements that need to be satisfied…
While quantum weight enumerators establish some of the best upper bounds on the minimum distance of quantum error-correcting codes, these bounds are not optimized to quantify the performance of quantum codes under the effect of arbitrary…
This paper is an expanded and more detailed version of our recent work 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…
We verify a conjecture on the structure of higher-rank numerical ranges for a wide class of unitary and normal matrices. Using analytic and geometric techniques, we show precisely how the higher-rank numerical ranges for a generic unitary…
A quantum error correcting code is a subspace $\mathcal{C}$ such that allowed errors acting on any state in $\mathcal{C}$ can be corrected. A quantum code for which state recovery is only required up to a logical rotation within…