We describe some instances in quantum information processing where numerical range techniques arise. We focus on two basic settings: higher-rank numerical ranges and their relevance in theoretical quantum error correction, and the classical numerical range and its use for comparing quantum information processing operations. We present the basic theory, discuss examples and formulate open problems.Keywords: numerical range; higher-rank numerical range; quantum information processing; quantum error correction; gate fidelity AMS Subject Classifications: 15A60; 15A90; 47N50; 81P68 IntroductionThe tools of matrix analysis and operator theory arise in a growing number of diverse scientific settings. Wherever matrices and operators are in use, it is also natural to expect that numerical range techniques will find application. The emerging disciplines of quantum information science [17] are no different. As two examples in quantum computing, for instance, higher-rank numerical ranges have been recently introduced in the context of quantum error correction [3,4], and numerical range techniques have recently been applied in quantum information processing and quantum optimal control [6,20,21]. There are certainly other instances of note, but for the sake of this article we shall focus on these two. We thus begin this article with a brief introduction to the basic mathematical setting for quantum computing. Then we describe in some detail these two scenarios, including examples and open problems.
Noise poses a challenge for any real-world implementation in quantum information science. The theory of quantum error correction deals with this problem via methods to encode and recover quantum information in a way that is resilient against that noise. Unitarily correctable codes are an error correction technique wherein a single unitary recovery operation is applied without the need for an ancilla Hilbert space. Here, we present the first optical implementation of a nontrivial unitarily correctable code for a noisy quantum channel with no decoherence-free subspaces or noiseless subsystems. We show that recovery of our initial states is achieved with high fidelity (≥ 0.97), quantitatively proving the efficacy of this unitarily correctable code.
We define and investigate the notion of entropy for quantum error correcting codes. The entropy of a code for a given quantum channel has a number of equivalent realisations, such as through the coefficients associated with the Knill-Laflamme conditions and the entropy exchange computed with respect to any initial state supported on the code. In general the entropy of a code can be viewed as a measure of how close it is to the minimal entropy case, which is given by unitarily correctable codes (including decoherence-free subspaces), or the maximal entropy case, which from dynamical Choi matrix considerations corresponds to non-degenerate codes. We consider several examples, including a detailed analysis of the case of binary unitary channels, and we discuss an extension of the entropy to operator quantum error correcting subsystem codes. Open Syst. Inf. Dyn. 2008.15:329-343. Downloaded from www.worldscientific.com by UNIVERSITY OF AUCKLAND LIBRARY -SERIALS UNIT on 03/15/15. For personal use only.
We give an introduction to the Bloch sphere interpretation of single qubit quantum channels and operations. We then present an algorithm that starts with an arbitrary mathematical description of such a channel, and derives the experimentally relevant geometric parameters that characterize the channel. Code is presented for an implementation of the algorithm via Maple.
We give a brief introduction to the algebraic formulation of error correction in quantum computing called operator algebra quantum error correction (OAQEC). Then we extend one of the basic results for subsystem codes in operator quantum error correction (OQEC) to the OAQEC setting: Every hybrid classical-quantum code is shown to be unitarily recoverable in an appropriate sense. The algebraic approach of the proof yields a new, less technical proof for the OQEC case.
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