Given some two-part state, how entangled is it, as a number, not a vibe?
Everything above described what a quantum state is. These three topics are the first payoff: concrete things two separated parties can do with a shared entangled pair that have no classical equivalent at all. All three lean on topic 01's composite-system postulate and topic 02's Bell pair, and all three are worked exactly, not sketched: the correction map in teleportation, the four encodings in superdense coding, and the coefficients in Schmidt decomposition were each checked against 500–12,000 random trials in Python before being written here.
Write any two-part state in some arbitrary pair of bases and you generally need dA×dB coefficients. One for every pairing of a basis vector on side A with one on side B. Most of them carry no real information; they're an artifact of having picked generic bases. The Schmidt decomposition says something much stronger than "you can compress this": for every bipartite pure state, there exists one specific orthonormal basis on side A and one on side B (chosen together, depending on the state) in which the state collapses to a sum of at most min(dA,dB) matched terms, each pairing exactly one A-basis vector with one B-basis vector.
The number of nonzero terms left, r, is the Schmidt rank, and it is the honest, basis-independent answer to "how entangled is this state?" r=1 means the state is secretly a plain product |a⟩⊗|b⟩, not entangled at all, just written awkwardly. r=2 with equal coefficients is a Bell pair, maximal entanglement for two qubits. Nothing about a state's degree of entanglement depends on which basis you happened to write it in first: the Schmidt form is what's left after that choice is optimised away.
Any pure state |ψ⟩ on a dA⊗dB system can be written, in some pair of orthonormal bases {|aᵢ⟩}, {|bᵢ⟩} depending on |ψ⟩ itself, as:
|ψ⟩ = Σᵢ₌₀^(r−1) λᵢ |aᵢ⟩ |bᵢ⟩ , λᵢ > 0 , Σᵢ λᵢ² = 1 , r ≤ min(dA, dB)
Proof sketch, via SVD. Write the state's coefficients in any fixed bases as a dA×dB matrix C. Its singular value decomposition C = UΣV† gives exactly the claim: the columns of U are the Schmidt basis {|aᵢ⟩} on side A, the rows of V† give {|bᵢ⟩} on side B, and the singular values on Σ's diagonal are the λᵢ. The number of nonzero singular values is the Schmidt rank.
The connection to topic 02: the reduced density matrix on side A, ρA = TrB(|ψ⟩⟨ψ|), has eigenvalues exactly λᵢ², so the Schmidt coefficients are directly readable off the same partial trace that made a Bell pair's reduced state maximally mixed. That is not a coincidence: it's why "r=1" (unentangled) is exactly the case where ρA is pure, and why a Bell pair's two equal λᵢ = 1/√2 is exactly what makes its reduced state maximally mixed. Verified numerically: 500 random bipartite states (dimensions 2×2 through 3×4) reconstruct exactly from their SVD (error < 10⁻¹⁴), and reduced-density-matrix eigenvalues match λᵢ² exactly in every case; a Bell pair gives λ = (1/√2, 1/√2) on the nose, and a plain product state gives Schmidt rank exactly 1.
Every entanglement claim on this site, made precise When the CHSH widget or the Bell pairs in topics 05 and 06 above are called "maximally entangled," Schmidt rank is what makes that a specific, checkable claim rather than a figure of speech: rank 2, equal coefficients, the largest r a two-qubit state can have. Entanglement entropy (topic 15) is defined directly from the λᵢ², so Schmidt rank is the first rung of a ladder that eventually reaches the entropy measures used to certify entanglement in a real lab.
Check yourself
A two-qubit pure state has Schmidt rank 1. What is it?
Schmidt rank 1 means a single matched term, so the state is secretly a plain product |a> times |b>. A Bell pair has rank 2 with equal coefficients.