A composite quantum state is entangled when it cannot be written as a product of independent subsystem states.

For example,

$$|\Phi^+\rangle=\frac{|00\rangle+|11\rangle}{\sqrt2}$$

is entangled. Measuring both subsystems in the $|0\rangle,|1\rangle$ basis gives perfectly correlated outcomes: either $00$ or $11$.

Correlation without predetermined local values

The joint state predicts correlations between measurements, but an individual subsystem need not possess a definite outcome for every possible measurement before measurement is performed.

For the state above, each local measurement in that basis is individually random:

$$P(0)=P(1)=\frac12.$$

The information lies in the joint statistics.

No faster-than-light signalling

Although entangled correlations can persist across large separations, one observer cannot choose the random local outcome. The local outcome distribution does not reveal which measurement was chosen at the distant subsystem.

Entanglement therefore does not by itself provide a channel for controllable faster-than-light communication.

Basis dependence

The strength and form of observed correlations depend on which observables are measured. Bell experiments exploit correlations across several alternative measurement settings to distinguish quantum predictions from a broad class of local hidden-variable models.