Can You Hear the Shape of a Drum?

If you had perfect pitch and listened to a recording of the sounds a drum made when struck, could you determine the shape of the drum? This question is an example of an inverse problem; inverse problems arise in medical imaging, oil prospecting, spectroscopy, and many other fields. We’ll first discuss the analogous question in the simpler setting of plucking a string. Then we’ll tackle the problem for drums and see that there are some surprises. Finally, I will give a brief indication of how this problem relates to some of my recent research. The emphasis will be on the ideas rather than on the technical details, so there will be pretty pictures instead of equations.

Strong Local Passivity in Finite Quantum Systems

Passive states of quantum systems are states from which no system energy can be extracted by any cyclic (unitary) process. Gibbs states of all temperatures are passive. Strong local (SL) passive states are defined to allow any general quantum operation, but the operation is required to be local, being applied only to a specific subsystem. Any mixture of eigenstates in a system-dependent neighborhood of a nondegenerate entangled ground state is found to be SL passive. In particular, Gibbs states are SL passive with respect to a subsystem only at or below a critical system-dependent temperature. SL passivity is associated in many-body systems with the presence of ground state entanglement in a way suggestive of collective quantum phenomena such as quantum phase transitions, superconductivity, and the quantum Hall effect. The presence of SL passivity is detailed for some simple spin systems where it is found that SL passivity is neither confined to systems of only a few particles nor limited to the near vicinity of the ground state.