How do three electrons entangle with each other? In a Quantum entity of two electrons entangled to each other, we know that when one electron's spin is known, then the other electron's spin is the opposite.
But, What's the case in a quantum entity that involves 3 electrons entangled to each other ?
I have two questions about this


*

*How does the information of spin is shared across three electrons since there are only two
states of spin that is up and down ?

*What do we know about other two electrons' spin when we know the spin of one electron ?


 A: Entanglement of a quantum state $|\Psi\rangle$ that has two subsystems 1 and 2 is defined as any state that cannot be decomposed in the form
$$|\Psi\rangle=|\psi\rangle_1\otimes|\phi\rangle_2$$ for some subsystem states $|\psi\rangle_1$ and $|\phi\rangle_2$.
Entanglement can manifest in many ways, one of which is a particular state
$$\frac{|\uparrow\rangle_1\otimes|\downarrow\rangle_2-|\downarrow\rangle_1\otimes|\uparrow\rangle_2}{\sqrt{2}},$$
in which any measurement of the spin of states 1 and 2 in the same basis will have exactly anticorrelated results, as mentioned in the question. However, not all entangled states have that property: another exemplary entangled state is $$\frac{|\uparrow\rangle_1\otimes|\uparrow\rangle_2-|\downarrow\rangle_1\otimes|\downarrow\rangle_2}{\sqrt{2}};$$ this state will have highly correlated spin-measurement results, not highly anticorrelated results.
Given that example, it is straightforward to extend to a "GHZ-type" entangled state of three spins:
$$\frac{|\uparrow\rangle_1\otimes|\uparrow\rangle_2\otimes|\uparrow\rangle_3-|\downarrow\rangle_1\otimes|\downarrow\rangle_2\otimes|\downarrow\rangle_3}{\sqrt{2}}.$$
Now, to answer your exact question in the title, how can one make this state that has three entangled electron spins? One route is by using an "entangling gate" such as the CNOT gate, which does a different operation on the spin of subsystem 2 depending on the spin of subsystem 1. For example, a CNOT gate acting on systems 1 and 2 will perform the operation
$$\mathrm{CNOT}_{12}\left(\frac{|\uparrow\rangle_1+|\downarrow\rangle_1}{\sqrt{2}}\otimes|\downarrow\rangle_2\right)=\frac{|\uparrow\rangle_1\otimes|\uparrow\rangle_2+|\downarrow\rangle_1\otimes|\downarrow\rangle_2}{\sqrt{2}};$$ an entangled state has been generated from a separable state. Performing the same operation after adding a third subsystem, we find
$$\mathrm{CNOT}_{13}\left(\frac{|\uparrow\rangle_1\otimes|\uparrow\rangle_2+|\downarrow\rangle_1\otimes|\downarrow\rangle_2}{\sqrt{2}}\otimes|\downarrow\rangle_3\right)=\frac{|\uparrow\rangle_1\otimes|\uparrow\rangle_2\otimes|\uparrow\rangle_3+|\downarrow\rangle_1\otimes|\downarrow\rangle_2\otimes|\downarrow\rangle_3}{\sqrt{2}}.$$ So this CNOT gate can be used to take separable states to entangled ones. The physical implementation of such entangling operations is one of the goals of quantum computing, and is possible with a variety of different physical systems.
