Entanglement
Entanglement is a correlation between quantum systems so strong that the joint state cannot be written as a product of individual states.
Definition
Two qubits are entangled when their joint state cannot be factored into a state for qubit A times a state for qubit B. The canonical example is the Bell state (|00> + |11>)/sqrt(2). There is no way to write this as (a|0>+b|1>) tensor (c|0>+d|1>); the attempt forces a contradiction.
What entanglement does
In (|00> + |11>)/sqrt(2), neither qubit has a definite state on its own, yet measuring one instantly determines the other: measure the first as 0 and the second is certainly 0, measure the first as 1 and the second is certainly 1. The correlation holds regardless of the distance between them, and it holds across different measurement bases in ways no classical shared randomness can reproduce (see Bell inequalities).
No faster-than-light signalling
Entanglement does not transmit information faster than light. Each party sees random local outcomes; the correlation is only visible after they compare results over a classical channel. This is why entanglement cannot be used for signalling despite its non-local flavour.
A computational resource
Entanglement is a resource that quantum computing consumes. It underlies teleportation, superdense coding, and the correlations that give algorithms their reach. A computation using only product states offers no advantage over classical simulation; entanglement is a necessary ingredient of quantum speedup, though not by itself sufficient.
- Product (separable) state: |0> tensor |1> = |01>
- Entangled: (|00> + |11>)/sqrt(2)
- Partially entangled: 0.8|00> + 0.6|11>
Reduced states
Looking at one half of an entangled pair, you find a mixed state even though the whole is pure. This is captured by the reduced density matrix, and the degree of mixing measures how entangled the pair is.