It is was recently proved that there are situations in which the quantum phenomenon called entanglement can increase the capacity of a quantum channel used to transmit classical information. Such channels are called non-additive; however, no explicit examples are known. Even more recently, it was observed that the anti-symmetry associated with the Pauli exclusion principle can be used to give an example of a channel which is non-additive for a related quantity, called the minimal output Renyi entropy, when p > 2. This grant is intended to construct more examples of non-additive channels and clarify our understanding of this phenomenon. In connection with this, the PI will closely examine the role of permutational symmetry in quantum information theory. The PI will also study entropy inequalities and cones of entropy vectors for composite systems.
This highly interdisciplinary work will clarify the role of the Pauli exclusion principle, which is often suppressed in quantum information theory. It will also be of interest to quantum chemists and condensed matter physicists, particularly those who used reduced density matrices. The work should help identify situations in which quantum systems have advantages over classical ones for communication, as well as computation.
This project studied several mathematical questions related to the ability of quantum particles (known as qubits) to transmit information, particularly in the presence of noise. One of the main issues studied is the question of under what circumstances a quantum phenomenon known as entanglement can be used to transmit more information when the channel is used repeatedly than in the classical case. This is known as "superadditivity" of the channel. Progress was made on identifying particular examples of channels which have the property of superadditivity. The ratio of the number of bits of information transmitted to the number of qubits needed to do this is known as the capacity of a channel. This is measured using quantum entropy. Some mathematical generalizations of quantum entropy and relative entropy were studied and classified. These generalizations are useful in studying a variety of mathematical problems associated with quantum channels. The P.I. also collaborated with students and postdocs at MIT to show that quantum error correcting codes can be used to obtain a counter-example to a question that was raised by a quantum chemist almost 40 years ago. We showed that although real physical particles have only 1-body and 2-body interactions, the 2-body reduced density matrix need not correspond to a unique the N-body state, even when it is known to come from the ground state of some N-body Hamiltonian. Quantum information theory is very interdisciplinary. Physicists, mathematicians, computer scientists, electrical engineers and quantum chemists are all involved in various ways. The P.I. organized an interdisciplinary program on quantum information theory at the Mittag-Leffler Institut in Stockholm in Fall, 2010; she facilitated the participation of graduate students, postdoctoral fellows, junior faculty and faculty at undergraduate institutions in the U.S. through a separate NSF block travel grant. She has received approval from the Banff International Research Station (partially funded by NSF) to hold a one week workshop in February, 2015 to bring together quantum information scientists and mathematicians with expertise in what is known as "hypercontracitivty and log Sobolev inequalities". Tools from this area of mathematics have recently been shown to be very useful in quantum information theory. This will be the first workshop in this "hot" new area.
