This proposal concerns several related problems in quantum information theory, all arising from the study of noisy quantum channels. The first problem is to prove additivity of minimal entropy and multiplicativity of the maximal trace norms for a unital qubit channel. This result would imply additivity of the Holevo capacity for such channels, thereby settling one particular case of a longstanding conjecture. The second problem is to investigate a conjectured equality between the Shannon capacity of a noisy quantum channel and a new quantity which was introduced in earlier work of the P.I.. The third problem involves a study of one special class of non-product measurements for product channels, with the goal of providing an interesting laboratory for testing ideas about entangled measurements.
Advances in nanotechnology have provided the opportunity for scientists to study and manipulate the quantum properties of microscopic systems, including single-atom systems and single-photon states. Recent theoretical discoveries indicate that such systems may have extraordinary properties. One example is the quantum computer, which is a theoretical device capable of outperforming any standard computer. Another example is a protocol for unconditionally secure encryption, which would be achieved by encoding messages as quantum states. Mathematics played a key role in the development of both of these ideas. The current proposal concerns a similar mathematical investigation of quantum devices which would be used to transmit and store information. A fundamental problem is to determine the information capacity of such a system, and thereby find the quantum analog of Shannon's famous expression for the capacity of a noisy channel. The P.I. describes a strategy for determining this capacity in an important special case, namely a memoryless channel where "qubits" are used to encode the information. The method of solution would involve some new mathematical ideas which may have applications to other areas of quantum information theory.