Abstract: There are limits to how much we can learn by a quantum measurement. To start
with, a measurement in general disturbs the measured quantum system. Also,
because of uncertainty relations, two or more non-commuting observables
cannot even simultaneously have well-defined values. Nevertheless, quantum
measurements can be optimised in clever ways. These lectures will give an
introduction to non-standard quantum measurements and quantum metrology.
When probe states are "classical", measurement accuracy is limited by the
so-called standard quantum limit. Non-classical states, including squeezed
states and entangled states, can be used to enhance the precision beyond the
standard quantum limit, reaching the Heisenberg limit. But how do we measure
whether a state is entangled in the first place? This is in itself another
non-trivial quantum measurement situation. In principle, we can try to
determine what the state of a quantum system is, and then from that try to
estimate whether it is entangled or not. But this may not be very reliable.
One other option is to test whether our quantum state violates a so-called
Bell inequality. This will be illustrated by a recent experiment to verify
that the orbital angular momentum of two photons, resulting from parametric
down-conversion, was entangled in at least 11 x 11 dimensions.
When it comes to distinguishing between different quantum signal states e.g.
in a communication situation, again, quantum measurements can be optimised
in various ways. The optimal measurements go beyond standard projective
measurements, in the eigenbasis of some observable. The resulting
"generalised quantum measurements" are not merely theoretical constructs,
but can be realised in a range of physical systems, including photons and
trapped ions. This will be illustrated for example by looking at the case of
distinguishing between two non-orthogonal states. Such a situation arises
for example whenever two signal states have passed though a lossy channel,
so that they are no longer perfectly distinguishable. Recent work on
measurements for distinguishing between non-orthogonal quantum states
includes optimising the measurements for when real and therefore imperfect
detectors are used.