entanglement entropy

Entanglement entropy is a number that measures how strongly two parts of a quantum system are entangled. Take a system in a definite, pure overall state, mentally split it into two pieces, and ask how much you can know about one piece on its own. The answer is given by the von Neumann entropy of that piece's reduced description: zero when the two parts are independent, and growing larger the more entangled they are.

The logic is subtle but beautiful. The whole system can be in a perfectly definite state while each half, looked at alone, appears genuinely uncertain — a mixed bag of possibilities. That local uncertainty is not ignorance about hidden details; it is the price of having traded definite local properties for shared, non-local ones. The more the halves are entangled, the more uncertain each half looks by itself, and the larger the entanglement entropy.

For a pair of maximally entangled qubits, like a Bell state, this entropy reaches its maximum value of one bit, meaning each qubit on its own is as undetermined as a fair coin. The same measure has become a central tool well beyond simple pairs: it characterises quantum phases of matter, guides the design of efficient simulations, and even appears in the physics of black holes, where the entropy of the horizon is read as an entanglement between inside and outside.