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For questions about the principle and application of quantum entanglement. It is a physical phenomenon which occurs when pairs or groups of particles are generated, interact, or share spatial proximity in ways such that the quantum state of each particle cannot be described independently of the state of the other(s), even when the particles are separated by a large distance—instead, a quantum state must be described for the system as a whole. (Wikipedia)

For a system with a Hilbert space $$H_A$$, the state of that system can be written as $$\left|\psi\right>_A = \sum_j\alpha_j\left|\phi_j\right>_A$$, for a basis $$\left\lbrace\left|\phi_j\right>_A\right\rbrace$$ with coefficients $$\alpha_j$$. Similarly, for a system with a Hilbert space $$H_B$$, the state of that system can be written as $$\left|\psi\right>_B = \sum_j\beta_j\left|\phi_j\right>_B$$, for a basis $$\left\lbrace\left|\phi_j\right>_B\right\rbrace$$ with coefficients $$\beta_j$$.

For two systems with Hilbert spaces $$H_A$$ and $$H_B$$, forming an overall Hilbert space $$H=H_A\otimes H_B$$, the state can then be written as $$\left|\psi\right> = \sum_{j, k}c_{jk}\left|\phi_j\right>_A\otimes\left|\phi_k\right>_B$$.

When there exists any states $$\left|\psi\right>_A$$ and $$\left|\psi\right>_B$$ such that $$\left|\psi\right>$$ can be written as $$\left|\psi\right> = \left|\psi\right>_A\otimes\left|\psi\right>_B$$, the state can be described as being a separable or product state. Otherwise, the state is said to be entangled.

An example of an entangled state is $$\left|\psi\right> = \frac{1}{\sqrt{2}}\left(\left|0\right>_A\otimes\left|0\right>_B \pm \left|1\right>_A\otimes\left|1\right>_B\right),$$ often denoted as $$\left|\psi\right> = \frac{1}{\sqrt{2}}\left(\left|00\right>\pm\left|11\right>\right).$$

When system A is measured (usually by someone known as Alice), this collapses system B into the appropriate state, so that when system B is measured (by someone called Bob), the outcomes are correlated. In this example, If Alice measures in the same basis as written here and the system is found to be in state $$\left|0\right>$$, Bob's system will also be in state $$\left|0\right>$$ immediately after Alice's measurement. Similarly if Alice's system is found to be in state $$\left|1\right>$$, Bob's system will also be found to be in state $$\left|1\right>$$.