Quantum Entanglement Measure Calculator
Quantifies the degree of entanglement in a two-qubit pure quantum state using concurrence, von Neumann entropy, or negativity. Used by quantum information scientists to characterize and verify entangled qubit pairs.
About this calculator
A two-qubit pure state is written as |ψ⟩ = α|00⟩ + β|01⟩ + γ|10⟩ + δ|11⟩, where α, β, γ, δ are probability amplitudes satisfying |α|² + |β|² + |γ|² + |δ|² = 1. Concurrence C = |2(αδ − βγ)| ranges from 0 (separable) to 1 (maximally entangled). Von Neumann entropy S = −Σ pᵢ log₂(pᵢ) where pᵢ = |amplitude|² quantifies the mixedness of the reduced density matrix; S = 1 ebit for a maximally entangled Bell state. Negativity measures the negative eigenvalues of the partial transpose of the density matrix: N = max(0, |αδ − βγ| − √(α²+β²)·√(γ²+δ²)). All three measures equal zero for product (unentangled) states and reach their maximum for Bell states such as (|00⟩ + |11⟩)/√2.
How to use
Take the Bell state |Φ⁺⟩ = (|00⟩ + |11⟩)/√2, so α = 1/√2 ≈ 0.7071, β = 0, γ = 0, δ = 1/√2 ≈ 0.7071. For concurrence: C = |2(αδ − βγ)| = |2 × (0.7071 × 0.7071 − 0)| = |2 × 0.5| = 1.0 — maximally entangled. For entropy: p_α = 0.5, p_δ = 0.5, p_β = p_γ = 0; S = −0.5 log₂(0.5) − 0.5 log₂(0.5) = 1.0 ebit. Enter these amplitudes, select your measure, and verify the maximum entanglement of a Bell state.
Frequently asked questions
What does a concurrence value of 1 mean for a two-qubit state?
A concurrence of 1 indicates the two qubits are maximally entangled — measuring one qubit instantly determines the state of the other, regardless of the physical distance between them. This is the hallmark of a Bell state, such as (|00⟩ + |11⟩)/√2. Concurrence of 0 means the qubits are fully separable with no quantum correlations. Values between 0 and 1 represent partially entangled states, and the concurrence directly quantifies the amount of entanglement as a resource for quantum communication protocols.
How is von Neumann entropy different from classical Shannon entropy in quantum information?
Classical Shannon entropy measures uncertainty over a probability distribution of definite classical states. Von Neumann entropy S = −Tr(ρ log₂ ρ) generalizes this to quantum density matrices, capturing both classical uncertainty and quantum superposition. For a pure entangled state, the von Neumann entropy of the reduced density matrix (after tracing out one qubit) directly measures the entanglement — it is zero for a separable state and 1 ebit for a maximally entangled Bell state. Unlike classical entropy, it can be nonzero even when the global state is perfectly known (pure).
Why must the sum of squared amplitudes equal 1 when using this calculator?
The amplitudes α, β, γ, δ are quantum probability amplitudes, and the Born rule requires |α|² + |β|² + |γ|² + |δ|² = 1 so that probabilities of all outcomes sum to 100%. If this normalization condition is not satisfied, the state is unphysical and the entanglement measures lose their meaning. Before entering values, normalize your amplitudes — for example, if you have raw amplitudes, divide each by √(|α|²+|β|²+|γ|²+|δ|²). The calculator assumes normalized input for all three entanglement measures.