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arXiv:1610.00930 · Quantum Information Processing

Nuclear numerical range and quantum error correction codes for non-unitary noise models

Patryk Lipka-Bartosik, Karol Życzkowski

Published Dec 10, 2016• 6 citations
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Abstract

We introduce a notion of nuclear numerical range defined as the set of expectation values of a given operator A among normalized pure states, which belong to the nucleus of an auxiliary operator Z . This notion proves to be applicable to investigate models of quantum noise with block-diagonal structure of the corresponding Kraus operators. The problem of constructing a suitable quantum error correction code for this model can be restated as a geometric problem of finding intersection points of certain sets in the complex plane. This technique, worked out in the case of two-qubit systems, can be generalized for larger dimensions.

DOI
10.1007/s11128-016-1484-8

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We introduce a notion of nuclear numerical range defined as the set of expectation values of a given operator A among normalized pure states, which belong to the nucleus of an auxiliary operator Z . This notion proves to be applicable to investigate models of quantum noise with block-diagonal structure of the corresponding Kraus operators. The problem of constructing a suitable quantum error correction code for this model can be restated as a geometric problem of finding intersection points of certain sets in the complex plane. This technique, worked out in the case of two-qubit systems, can be generalized for larger dimensions.
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.
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.
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 among a certain subset of the set of all quantum states. One considers, for instance, the set of real states, the set of product states, separable states, or the set of maximally entangled states. We show exemplary applications of these algebraic tools in the theory of quantum information: analysis of k-positive maps and entanglement witnesses, as well as study of the minimal output entropy of a quantum channel. Product numerical range of a unitary operator is used to solve the problem of local distinguishability of a family of two unitary gates.
In the present paper we initiate the study of the product higher rank numerical range. The latter, being a variant of the higher rank numerical range [M.--D. Choi {\it et al.}, Rep. Math. Phys. {\bf 58}, 77 (2006); Lin. Alg. Appl. {\bf 418}, 828 (2006)], is a natural tool for studying a construction of quantum error correction codes for multiple access channels. We review properties of this set and relate it to other numerical ranges, which were recently introduced in the literature. Further, the concept is applied to the construction of codes for bi--unitary two--access channels with a hermitian noise model. Analytical techniques for both outerbounding the product higher rank numerical range and determining its exact shape are developed for this case. Finally, the reverse problem of constructing a noise model for a given product range is considered.