Proof of the orthogonal measurement conjecture for two states of a qubit

Abstract

In this thesis we prove the orthogonal measurement hypothesis for two states of a qubit. The accessible information is a key quantity in quantum information and communication. It is defined as the maximum of the mutual information over all positive operator valued measures. It has direct application in the theory of channel capacities and quantum cryptography. The mutual information measures the amount of classical information transmitted from Alice to Bob in the case that Alice either uses classical signals, or quantum states to encode her message and Bob uses detectors to receive the message. In the latter case, Bob can choose among different classes of measurements. If Alice does not send orthogonal pure states and Bobs measurement is fixed, this setup is equivalent to a classical communication channel with noise. A lot of research went into the question which measurement is optimal in the sense that it maximizes the mutual information. The orthogonal measurement hypothesis states that if the encoding alphabet consists of exactly two states, an orthogonal (von Neumann) measurement is sufficient to achieve the accessible information.

In this thesis we affirm this conjecture for two pure states of a qubit and give the first proof for two general states of a qubit.