Measure how entangled a two-qubit pure state |ψ⟩ = a|00⟩ + b|11⟩ is. Enter the amplitude a (with b = √(1−a²)) to get the entanglement entropy in ebits and the concurrence. A Bell state (a = 0.707) is maximally entangled.
A two-qubit pure state written in Schmidt form is |ψ⟩ = a|00⟩ + b|11⟩ with a² + b² = 1. Its entanglement is quantified by the entanglement entropy — the von Neumann entropy of one qubit's reduced state — S = −p₀log₂p₀ − p₁log₂p₁, where p₀ = a² and p₁ = b². It ranges from 0 (separable) to 1 ebit (maximally entangled).
The concurrence C = 2|a|√(1−a²) is an equivalent measure that runs from 0 to 1. Both peak when a = 1/√2 ≈ 0.707, the balanced superposition of a Bell state, and vanish for a product state where a is 0 or 1.
For the underlying physics, read our explanation of nonlocality and entanglement.
It is the von Neumann entropy of one qubit after tracing out its partner, measuring how much information the two qubits share. It runs from 0 (no entanglement) to 1 ebit (maximal, a Bell state).
A Bell state such as (|00⟩ + |11⟩)/√2, where a = 1/√2 ≈ 0.707. It has 1 ebit of entanglement entropy and concurrence 1 — measuring one qubit instantly fixes the other.
Concurrence is an entanglement measure for two qubits ranging from 0 to 1. For the pure state a|00⟩ + b|11⟩ it equals 2|a|·|b| = 2|a|√(1−a²).
When the state is separable (a product state), i.e. a = 0 or a = 1. Then one qubit's outcome tells you nothing about the other, and both entropy and concurrence are zero.