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.

 

[Entropia]

The Bloch sphere model is a useful tool for understanding the behavior of a single spin state. However, when dealing with multiple spins, the model becomes more complex and difficult to visualize. This is because the Bloch sphere represents the state of a single spin in three-dimensional space, but when dealing with multiple spins, we need to consider the state of the entire system in a higher-dimensional space.

One approach to using a Bloch-coordinate system for multiple spins is to extend the model to include a Bloch sphere for each spin. This is known as the "multi-spin Bloch sphere" model. In this model, each Bloch sphere represents the state of a single spin, and the overall state of the system is described by the combination of all the individual Bloch spheres.

Another approach is to use a mathematical tool called the density matrix to represent the state of the system. The density matrix is a matrix that contains information about the probabilities of each possible state of the system. By manipulating this matrix, we can use the Bloch sphere model to understand the behavior of multiple spins.

Overall, the Bloch sphere model is a useful tool for understanding single spin states, but it becomes more complex when dealing with multiple spins. There are various approaches to extending the model for multiple spins, but they all require a deeper understanding of the mathematics involved.