The average of the three Pauli Matrices

[eviegirl]

Homework Statement

By using the general density matrix rho find the average of the three Pauli matrices. You can then tell how many independent experiments you must make in order to determine rho.

Homework Equations

The Attempt at a Solution

I know the Pauli matrices and their basic properties, but I don't know how to start finding the average of these three matrices.

 

[diazona]

Surely you know how to take an average?

But anyway: the problem asks you to use a general density matrix. What do you know about that concept?

 

[eviegirl]

Yeah ;) but I have the feeling I'm blanking out. The most general density matrix I can think of is:

rho = 1/2*(I + a*sigma)

Where I is the identity matrix, sigma the three Pauli matrices and a a three-dimensional vector. The trace of Pauli matrices are zero, and the trace of an identity matrix is 2 (in this case where I look at 2x2 matrices, so by taking the half of that - the trace of the density matrix is 1. Which I think is a property of the density matrix? Hermitian matrices which have a trace equal to 1. When the length of that vector is equal to or less than 1, we have a pure state. I think.

 

[diazona]

It's been quite a while since I did anything with density matrices, but I think what you're saying is correct. Unfortunately it doesn't really clarify what the question is asking for me, so I'm not sure what to tell you. Perhaps someone else who has better knowledge of the subject can come along and explain it.

 

[eviegirl]

Thank you for your thoughts ;) I just started taking a Quantum Optics course, and these first exercises are supposed to refresh whatever QM knowledge we had. Very interesting subject with difficult (or new) exercises.

 

[weejee]

I think the vector 'a' should always have a length shorter than or equal to 1, to have only non-negative eigenvalues. When the length is equal to 1, it will correspond to a pure state (having eigenvalues 0 and 1).

So, it seems that what you want to do is to calculate the trace of rho*sigma_i, where i=x,y,z ?

I guess it should be straightforward once you notice the multiplication properties of Pauli matrices, such as sigma_x *sigma_y = i*sigma_z.

 

[vela]

Homework Statement

By using the general density matrix rho find the average of the three Pauli matrices. You can then tell how many independent experiments you must make in order to determine rho.

Homework Equations

The Attempt at a Solution

I know the Pauli matrices and their basic properties, but I don't know how to start finding the average of these three matrices.

In terms of the density matrix ρ, the ensemble average [A] of an observable A is given by [A]=tr(ρA). I think the problem is asking you to find the averages for S_x, S_y, and S_z and show you can determine ρ completely from these three numbers.