Expectation value of an operator

[suma]

When we say expectation value of an operator like the pauli Z=[1 0; 0 -1], like when <Z> = 0.6 what does it mean?

What is difference between calculating expectation value of Z and its POVM elements{E1,E2}?

thanks

 

[bhobba]

Hermitian operators are uniquely decomposable into O = ∑ Yi |bi><bi|. The Yi are the values associated with each possible outcome. If the observation is performed a conceptual large number of times then the average of the outcomes will be it's expected value. QM says that expected value is Trace (PO) where P is the systems state. To some extent this is implied by Gleasons theorem.

Operators are related to resolutions of the identity. A POVM is a generalisation of a resolution of the identity that removes the requirement for them to be disjoint. Resolutions of the identity, describe what are called Von Neumann measurements and can be described by Hermition operators. POVMs describe what are called generalised measurements and are not related to observables but by considering a system to be measured and a probe can be related to them.

See:
http://www.quantum.umb.edu/Jacobs/QMT/QMT_Chapter1.pdf

Thanks
Bill