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Linear and Multilinear Algebra

Higher rank matricial ranges and hybrid quantum error correction

Ningping Cao, David W. Kribs, Chi-Kwong Li, Mike I. Nelson, Yiu‐Tung Poon, Bei Zeng

Published Apr 7, 2020• 6 citations
Publisher

Abstract

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.

DOI
10.1080/03081087.2020.1748852

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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.
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.
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.
Clifford codes are a natural generalization of quantum stabilizer codes based primarily on representation theory. This class of codes has previously been extended to the setting of quantum subsystem codes. We formulate a two-fold generalization of Clifford codes, for both the hybrid classical and quantum information and projective representation theory settings. This leads to new classes of hybrid subspace and subsystem Clifford codes. We extend the fundamental representation theoretic quantum error correction theorem to include these codes, based on the operator algebra quantum error correction framework. We also discuss several examples throughout the presentation, of both stabilizer and non-stabilizer type.
In this dissertation, we consider quantum errors, represented by quantum channels defined on the space of n x n complex matrices, M(n). We begin with a quantum channel T defined on M(n). The Kraus operators of this channel can be used to generate an operator system S, which is guaranteed to have a basis of Hermitian matrices A_1,...,A_m in M(n). Given natural numbers k and m, we find a lower bound on n above which there is always an n x k matrix U with orthonormal columns such that U*A_1U,...,U*A_mU are diagonal k x k matrices. We then extend this to a lower bound on n above which there is always a k-dimensional quantum error correction code for any quantum channel T on M(n) with an associated operator system S with dim(S) < m+2. Such a bound is equivalent to a lower bound on n above which the joint rank-k numerical range of any m Hermitian matrices in M(n) is non-empty. So, we further extend our result to a lower bound on n above which the rank-k numerical range of any m Hermitian matrices in M(n) is star-shaped.