Bloch sphere model for many spins?

[Gerenuk]

The Bloch sphere helps understanding the mathematical results for a one-spin state. One could think of the state as a spin pointing in direction $\hat{n}$. Then the probability for measureing the spin in the direction $\hat{m}$ is simply
$$P=|<\hat{m}|\hat{n}>|^2=\frac{1+\hat{n}\cdot\hat{m}}{2}$$
and that's all one need to know for one spin.

But what about multiple spins? How can I use a similar "Bloch-coordinate system" for more spins? In a way one only needs to know the scalar product? I came up with an expression but it's still quite complex.

Does any know an easier treatment of multiple spins but in terms of these "Bloch-coordinates"?

 

[azntoon]

The easiest way to think about multiple spins is to use the tensor product of the single spin states. This is a generalization of the scalar product for two spins and allows us to calculate the probability of measuring a given state for an arbitrary number of spins. For example, if we have two spins in a state $\left| \psi_1 \right\rangle$ and $\left| \psi_2 \right\rangle$, then the probability of measuring a given state $\left| \phi \right\rangle$ is given byP = \left\langle \phi \middle| \left| \psi_1 \right\rangle \otimes \left| \psi_2 \right\rangle \right\rangle^2where the tensor product is defined as $\left| \psi_1 \right\rangle \otimes \left| \psi_2 \right\rangle = \left| \psi_1 \psi_2 \right\rangle$.This can be extended to an arbitrary number of spins by simply taking the tensor product of all the single spin states. This gives us a convenient way to calculate the probability of measuring a given state for multiple spins using the same Bloch-coordinate system.