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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 [itex]\left\langle\frac{1+\sigma_{1}^{z}}{2} \right\rangle [/itex] 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 [itex]\left\langle\frac{1+\sigma_{1}^{z}}{2}\otimes\frac{1+\sigma_{2}^{z}}{2}\right\rangle[/itex] which is not the product of expectation values calculated earlier because we now have a term of the form [itex]\left\langle\sigma_{1}^{z}\otimes\sigma_{2}^{z} \right\rangle [/itex]

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 [itex]\left\langle1\otimes1\right\rangle [/itex], 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 [itex]\left\langle\sigma_{1}^{z}\otimes\sigma_{2}^{z} \right\rangle [/itex] 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 :)

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# Understanding correlations in a pair of two level systems

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