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Linear and Multilinear Algebra

Joint higher rank numerical range of Pauli group

Sayyed Ahmad Mousavi, Abbas Salemi

Published Feb 14, 2014• 2 citations
Publisher

Abstract

For a noisy quantum channel, a quantum error correction code of dimension k exists if and only if the joint rank- numerical range associated with the error operators of the channel is non-empty. In this paper, the joint rank- numerical range of an -tuple of elements in the -qubit Pauli group are discussed. Let be a Hermitian triple in the -qubit Pauli group. We state a constructive method to obtain the joint rank- numerical range of . Moreover, we show that .

DOI
10.1080/03081087.2013.872638

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For a noisy quantum channel, a quantum error correction code of dimension k exists if and only if the joint rank- numerical range associated with the error operators of the channel is non-empty. In this paper, the joint rank- numerical range of an -tuple of elements in the -qubit Pauli group are discussed. Let be a Hermitian triple in the -qubit Pauli group. We state a constructive method to obtain the joint rank- numerical range of . Moreover, we show that .
The rank-k numerical range has a close connection to the construction of quantum error correction code for a noisy quantum channel. For a noisy quantum channel, a quantum error correcting code of dimension k exists, if and only if the associated joint rank-k numerical range is non-empty. In this paper, the no- tion of joint rank-k numerical range is generalized, and some state- ments of (2011, Generalized numerical ranges and quantum error correction, J. Operator Theory, 66: 2, 335-351.) are extended.
Suppose noise is characterized by a quantum channel whose error operators are members of an arbitrary (not necessarily abelian) Pauli subgroup of -qubit Pauli group, where . In this paper, using a maximal abelian subgroup of , we obtained the largest , for which the joint rank- numerical range associated with members of is non-empty. It is proved that qubits codeword encodes data qubits. Moreover, we obtain recovery operators such that the decoded state automatically appears in the output with no syndrome measurements for such channels.
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 higher rank numerical ranges are obtained and their implications to quantum computing are discussed. It is shown that if the dimension of the underlying Hilbert space of the quantum states is sufficiently large, the joint higher rank numerical range of operators is always star-shaped and contains a non-empty convex subset. In case the operators are infinite dimensional, the joint infinite rank numerical range of the operators is a convex set closely related to the joint essential numerical ranges of the operators.
We solve the fundamental quantum error correction problem for bi-unitary channels on two-qubit Hilbert space. We construct qubit codes for such channels on arbitrary dimension Hilbert space, and identify correctable codes for Pauli-error models not detected by the stabilizer formalism. This is accomplished through an application of a new tool for error correction in quantum computing called the ``higher-rank numerical range''. We describe its basic properties and discuss possible further applications.
We introduce and initiate the study of a family of higher rank matricial ranges, taking motivation from hybrid classical and quantum error correction coding theory and its operator algebra framework. In particular, for a noisy quantum channel, a hybrid quantum error correcting code exists if and only if a distinguished special case of the joint higher rank matricial range of the error operators of the channel is non-empty. We establish bounds on Hilbert space dimension in terms of properties of a tuple of operators that guarantee a matricial range is non-empty and hence additionally guarantee the existence of hybrid codes for a given quantum channel. We also discuss when hybrid codes can have advantages over quantum codes and present a number of examples.
We introduce and initiate the study of a family of higher rank matricial ranges, taking motivation from hybrid classical and quantum error correction coding theory and its operator algebra framework. In particular, for a noisy quantum channel, a hybrid quantum error correcting code exists if and only if a distinguished special case of the joint higher rank matricial range of the error operators of the channel is non-empty. We establish bounds on Hilbert space dimension in terms of properties of a tuple of operators that guarantee a matricial range is non-empty, and hence additionally guarantee the existence of hybrid codes for a given quantum channel. We also discuss when hybrid codes can have advantages over quantum codes and present a number of examples.