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Summer Research Fellowship Programme of India's Science Academies

A study of quantum channels

Prem Nigam Kar

School of Mathematical Sciences, National Institute of Science Education and Research, Tehsildar Office, Khurda, Pipli, Near, Jatni, Odisha,752050.

Professor B.V. Rajarama Bhat

Professor, Statistics and Mathematics Unit, Indian Statistical Institute, Bangalore Centre, 8th Mile, Mysore Road, RVCE Post, Bengaluru, Karnataka, 560059.

Abstract

The main aim of the project is to explore quantum channels, especially entanglement breaking channels, and their properties in detail. The first half focuses on developing the mathematical prerequisites of the study of quantum channels like diagonalization, the spectral theorem, projections, spectral form of normal matrices, positive matrices, tensor products, and some classical probability. Later, we develop the theory of quantum probability by modifying the concepts of classical probability theory to suit a quantum mechanical set up. The event space of a quantum probability space is a Hilbert space and the events are the subspaces, or equivalently the projections into the subspaces of the Hilbert space. The events of a quantum probability space do not always commute, thus yielding a rich noncommutative algebraic structure to study. The probability measure in a quantum probability space is known as a state. The states are equivalently given by positive matrices of unit trace known as density matrices. This equivalence is established by Gleason's theorem. We then define the quantum mechanical equivalent of random variables, namely observables, which are the self adjoint operators on the Hilbert space and explore their probability distribution. We then define a quantum channel, which are completely positive maps on the set of all states or by linear extension on the algebra of matrices. We explore the properties of quantum channels in detail and give a complete characterization of completely positive maps by proving the ChoiKraus Theorem. After that, we study the simultaneous evolution of multiple quantum systems and define entangled and separable states. Then, we look into the properties of entanglement breaking quantum channels, the statement of the PPT square conjecture and the complete characterization of entanglement breaking quantum channels given by the Holevo forms We then move on to define some static distance measures for quantum information, namely the trace distance and fidelity.

Keywords: quantum probability, completely positive maps, entanglement breaking channels, entanglement fidelity

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