Understanding correlations in a pair of two level systems

1. Feb 10, 2013

McLaren Rulez

Hi,

I have a rather silly question so forgive me for that. Suppose I have two two-level atoms. Now, I have a state of the system and I can find the excited state population of the first atom by taking the expectation value $\left\langle\frac{1+\sigma_{1}^{z}}{2} \right\rangle$ and similarly for the second atom.

However, if I want to find the probability that both atoms are excited, we need to look at the expectation value of the operator $\left\langle\frac{1+\sigma_{1}^{z}}{2}\otimes\frac{1+\sigma_{2}^{z}}{2}\right\rangle$ which is not the product of expectation values calculated earlier because we now have a term of the form $\left\langle\sigma_{1}^{z}\otimes\sigma_{2}^{z} \right\rangle$

Thus, if we have the identity and the three Pauli matrices as a basis for the operators on each atom, I get a total of 16 operator combinations. I need to therefore have 16 numbers to fully characterize my system. Discounting the trivial $\left\langle1\otimes1\right\rangle$, that leaves 15 numbers.

My question is, are these fifteen numbers fully independent of each other? For instance, if I have both my atoms in the excited state, then isn't $\left\langle\sigma_{1}^{z}\otimes\sigma_{2}^{z} \right\rangle$ also determined? Or more generally, can I choose any fifteen numbers between -1 and 1 and still have it correspond to a physical state?

Thank you :)