Checkpoint: Measure Entanglement

This checkpoint ties together the computable measures of the module. You will take a pure two-qubit state, extract its concurrence, and push it through Wootters' formula to get the entanglement of formation — the asymptotic cost, in Bell pairs, of building the state.

The chain of formulas

For a pure state $|\psi\rangle = a|00\rangle + b|01\rangle + c|10\rangle + d|11\rangle$:

The first step measures how entangled the state is (a number in $[0,1]$); the second converts that into the operational cost in Bell pairs (also in $[0,1]$ bit for two qubits). For a maximally entangled state $C=1$ and $E_F = 1$; for a product state $C = 0$ and $E_F = 0$.

The instance

We use $|\psi\rangle = \cos\theta\,|00\rangle + \sin\theta\,|11\rangle$ with $\theta = \pi/8$. Only $a = \cos\theta$ and $d = \sin\theta$ are nonzero, so

Then $C^2 = \tfrac12$, so

The state is more than halfway to maximal entanglement by cost, even though its concurrence is only about $0.71$ — a reminder that $E_F$ and $C$ carry the same ordering but not the same scale.

Try it

Implement the full chain — concurrence, then the binary-entropy formula — for $|\psi\rangle = \cos(\pi/8)|00\rangle + \sin(\pi/8)|11\rangle$, and return the entanglement of formation $E_F$.

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