Hilbert spaces from the mathematician’s perspective

Yesterday I came across these notes by Terence Tao on his blog for one of his Analysis courses at UCLA. They treat Hilbert spaces as part of mathematical analysis: a new perspective for those of us who first came across the topic while learning quantum mechanics. They’re embedded with exercises, and quite readable. Also, this is Tao, so they’re definitiely worth having a look at:

• 245B Notes 5: Hlibert spaces
• Higher order Hilbert spaces

Quantum metrology

According to Wikipedia, quantum metrology is the study of making high-resolution and highly sensitive measurements of physical parameters using quantum theory to describe the physical systems, in particular exploiting quantum entanglement.

If you are interested, you might want to take a look at this. It is a paper entitled Ensemble based quantum metrology.  Quoting the abstract:

We consider measurement of magnetic field strength using an ensemble of spins, and we identify a third essential resource: the initial system polarisation, i.e. the low entropy of the original state. We find that performance depends crucially on the form of decoherence present; for a plausible dephasing model, we describe a quantum strategy which can indeed beat the standard quantum limit.