Hi I have a question about POVMsim confused as to how they

[mtak0114]

Hi I have a question about POVMs

im confused as to how they correspond to actual measurements we do in a lab, and how non-orthogonal measurements fit into this context. For example we have some POVM with elements $E_i$. I read in Nielsen and Chuang that we do a measurement and get outcome m say. Now what does this correspond to? if the POVMs were projectors then for the different elements $E_i$ we would have different experimental setups is this the same for POVMs or is it one setup with multiple outcomes?

And I guess for non-orthogonal measurements I am just confused as to whether they can be represented by POVMs

thanks

Mark

 

[genneth]

A useful theorem to keep in mind about POVM's is Naimark's dilation theorem, which basically states that you can always imagine a POVM to be a straight projection measurement on an enlarged system of your system + some ancilla. Physically, this corresponds better to what we really mean when we make measurements: we couple the system to a measurement apparatus (the ancilla in this case), the latter is *very* large compared to the former, and after some stabilisation we measure a small set of commuting (i.e. classical looking) observables on the ancilla. Naimark's theorem is a bit more restrictive than this physical situation, but it should be clear that in the limit of a very large ancilla then the physical situation can be very closely approximated.

POVM's then state that these generalised measurements can be thought of as operators on the system alone (rather like when we go from pure states of system + ancilla to just a density matrix of the system). The assumption is that the ancilla is generic/universal in some sense, so that one can safely ignore/coarse grain over it. Many "paradoxes" that people hang themselves over are results of coarse graining over the ancilla when it is actually important to the measurement.

 

[mtak0114]

Thanks for the reply that's quite helpful.

I have not heard of the Dilation theorem. Do you have any good sources detailing some theory and examples perhaps of this in regards to measurement. I had a look at Nielsen and Chuang which had a tiny bit but was a bit of detail. Specifically what did you mean by

\ldots and after some stabilisation we measure a small set of commuting (i.e. classical looking) observables on the ancilla. Naimark's theorem is a bit more restrictive than this physical situation, but it should be clear that in the limit of a very large ancilla then the physical situation can be very closely approximated.

thanks

M

 

[genneth]

What I mean is that the dilation theorem strictly says that (POVM on system) == (projective measurement on system + ancilla), but really we usually do (projection measurement on ancilla only). We get away with just projection measurements because the degrees of freedom of the ancilla is *vast* (it's an actual classical object, like the pointer on an ammeter). In addition, we only pick a very small set of commuting/almost commuting variables of the ancilla to measure (position of macroscopic pointer, for example).

It intuitively feels true that in the limit of ancilla going to infinite size, a set of projective measurements on the ancilla alone should approximate arbitrarily well a set of POVMs on the system. However, I don't know of any theorems regarding this...

 

[mtak0114]

Thanks for the help

I think an example might help me gain some intuition, for a qubit would the following two projectors constitute a POVM ?
$P_0 = |0 \rangle \langle0 |$ and $P_1 = |1 \rangle \langle 1 |$

If so if I choose to measure along different directions on the bloch sphere, say $x$ and $y$ are these added to the POVM or do they constitute a different POVM?

I had another questions about POVM's specifically informationally complete POVM's.
I understand that an informationally complete POVM is a POVM which is sufficient to reconstruct a state $\rho$ is that correct? In relation to projective measurements what would an informationally complete POVM look like for qubits

thanks again and sorry for the confused questions