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Unformatted text preview: qitd181 Quantum Information Types Robert B. Griffiths Version of 6 February 2012 References: R. B. Griffiths, Types of Quantum Information , Phys. Rev. A 76 (2007) 062320; arXiv:0707.3752 Contents 1 Introduction 1 2 Information Types 1 2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2.2 Compatible and incompatible types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3 Mutuallyunbiased types 3 4 Quantum Channels 4 5 Von Neumann Entropy 5 1 I n t r o d u c t i o n Since the world (so far as we know) is quantum mechanical, “classical” information must be some sort of “quantum information.” • The precise sense in which this is so remains a subject of investigation at the present time. Despite (or because of?) an enormous number of publications on the subject, the community has not arrived at any consensus. ◦ E.g., the term “classical information” is widely used in the literature by authors who would be embar rassed if asked to actually define what they are talking about. The purpose of these notes is to indicate a scheme based on types or species of quantum information which represents at least one way of seeing how quantum information theory is connected to classical infor mation theory as developed by Shannon and his successors, and in what way new phenomena arise when quantum effects are taken into account. • These new phenomena, which have no (direct, at least) classical analogs, include: teleportation, dense coding, quantum cryptography, quantum error correction, and, of course, quantum computing. 2 I n f o r m a t i o n T y p e s 2.1 Definition Let { P j } represent a decomposition of the identity on the Hilbert space H we are interested in. I.e., each P j = P j = P 2 j is a projector, and I = j P j , P j P k = δ jk P j . (1) 1 ◦ We assume that none of the P j in such a decomposition is equal to the zero operator. The zero operator is, on the other hand, part of the corresponding event algebra defined below. • Such a decomposition is the quantum counterpart of a sample space in ordinary probability theory: it represents a collection of mutuallyexclusive properties or “events”, one and only one of which is “true” or “occurs”. The corresponding event algebra , continuing the analogy with ordinary probability theory, is the collection of all projectors which can be written as sums of one or more of the P j ’s in the decomposition, with the zero projector 0 and the identity itself included for good measure. Probabilities { p j } can then be assigned to the elements of the sample space, and thereby to projectors in the event algebra in the usual way: Pr( P 1 + P 2 + P 7 ) = p 1 + p 2 + p 7 ....
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