How can we measure these Hermitian operators?

[Heidi]

TL;DR Summary

generalization of Pauli matrices

Hi Pf,
I am reading this article about generalization of Pauli matrices
https://en.wikipedia.org/wiki/Gener...es#Generalized_Gell-Mann_matrices_(Hermitian)
When i receive a qubit in a given density matrix , i can measure the mean values of the Pauli matrices by measuring the spin projections in the x, y, z directions.
I read that the generalized Gell-mann matrices are also hermitian, with null trace and like pauli matrices in a higher dimension.
suppose that the density matrix is no more 2*2 but 4*4
I wonder how one could measure the (1,4) element of the density matrix.
it is given by the trace of the product rho * gellmann(1,4)
but how to get it?

 

[Heidi]

My first idea was to make something like the EPR experiment: Bob and Alice share the pairs. They make measurements along different directions. At the end they meet and compare their results the correlations. With the hope to have both measured the density matrix.
It was a bad idea. Only one experimentalist must make measurements on each pair when it is produced. At the source.

I read that quantum logic gates can act on two qubits. and the way they act is described with a matrix. If a logic gate had for matrix the Gell-Mann form it would answer my question.

 

[Heidi]

It is easy to find how to measure the density matrix of one particle.
Why is it so hard to find the same things about pairs of particles?

 

[f95toli]

There are many ways to do this, but it is never trivial
A good search term would be "two qubit tomography".
Put that in Google scholar and you should get lots of hits.

 

[Heidi]

Good idea. I only used google. not google scholar.

 

[f95toli]

Good idea. I only used google. not google scholar.

The main keyword here is "tomography" because I think that is what you are actually asking about.

 

[Heidi]

If ρ is the bipartire density matrix, i know that it is the sum of 16 orthonormal matrices in the sense of the trace.
Tr(ΩiΩj=δij
with
ρ=ΣisiΩi
i have only to compute the 16 mean values s_i of the Omegas in the state rho si=Tr(ρΩi)

 

[Heidi]

I wonder how to relate this to the local measurements of Alice and Bob say with their Stern Gerlach.

 

[Heidi]

take the example of the matrix Omega =
0001
0000
0000
1000
how to get its mean value s from the mean value of the local 2*2 Pauli matrices (and coincidences labels)?

 

[Heidi]

We have 4 diagonal projectors
|dd><dd| , |du><du| , |ud><ud| , |uu><uu|
and 6 pairs of off diagonal Omegas. from
(|dd><du| + |du><dd|)/$\sqrt 2$ and (i|dd><du| - i |du><dd|)/$\sqrt 2$
to
(|ud><uu| + |uu><ud|)/$\sqrt 2$ and (i|ud><uu| - i |uu><ud|)/$\sqrt 2$

when the local observers measure the spin projection on z they get -1/2 or 1/2
for the pairs we will find -1 0 or 1.
Given a peculiat Omega in these 12 cases what is the recipe to get its mean value from the local measurements? what have we to throw away among the results before averaging?