Quantum Information

Vlatko Vedral
Imperial College London

I will present a basic introduction to quantum information theory with the emphasis on the theory of communication [1]. First, I plan to introduce the classical information theory of Shannon and explain the physical importance behind the notion of entropy for communication. Other relevant measures of information will also be discussed (Lectures 1&2). I will then show how to generalise this notion to the quantum domain and obtain the von Neumann entropy from the information-theoretic perspective. I will generalise this result and indicate how to generalise any classical measure of information to obtain the corresponding quantum measure. To do so, I will need to introduce quantum mechanics from the more general perspective of mixed states and evolutions based on the super-operator formalism (L 3&4). This will allow me to derive one of the central results of quantum information theory - the Holevo bound, which limits the amount of information that can be encoded into one quantum bit to one classical bit of information (L 5). In spite of this severe limitation, I will argue that quantum information processing is still more efficient than classical. Examples will be given, such as the super dense coding and quantum teleportation, which illustrate protocols that are impossible based on classical physics, yet fully comply with the Holevo bound (L 6). The key ingredient of these two protocols is quantum entanglement, the correlations that exist between quantum systems and cannot be explained in any classical manner. I give a brief overview of our current understanding of entanglement based on quantum information theory (L 7). Also, I will review the theory of positive maps and completely positive maps that feature strongly in describing entanglement (as well as general quantum evolutions) (L8). A connection between thermodynamical cycles and quantum local operations on entangled states will be emphasised (L 9). I will talk about how to distil entanglement in order to achieve its maximal utility (L 10). Finally, I will briefly discuss some quantum algorithms [2] - Grover's search in particular - and the question of the role of entanglement in quantum computing (L 11). If time permits, I will talk about different ways of applying quantum algorithms, especially in the domain of quantum cryptography (L 12).

Ref:
[1] V. Vedral, Rev. Mod. Phys. 74, 197 (2002);
[2] M. Nielsen & I. Chuang, "Quantum Computing" (Cambridge University Press, 2000);