Minimal math for quantum foundations

Context

A language older than its use. Quantum mechanics speaks Hilbert space — complex vector spaces with inner products — but the physicists who built the theory didn't know it. When Heisenberg worked out his strange arrays of numbers on Helgoland in 1925, he had never heard of a matrix; it was Max Born who stared at Heisenberg's multiplication rule and recognized, from lectures decades earlier, that this was matrix algebra. Schrödinger's waves looked like a completely different theory until the equivalence was proved. It took John von Neumann's 1932 Mathematical Foundations of Quantum Mechanics to name the common structure — "Hilbert space" — and legend has it Hilbert himself asked, after a seminar, what exactly a Hilbert space was. The math was sitting in the library the whole time, waiting for physics to need it.

How much you actually need. Not the working physicist's toolkit. You need enough fluency to read: what a two-qubit state is, what a unitary operator does, why observables are Hermitian, what an inner product measures. That's linear algebra with complex numbers — weeks of honest work, not years — and it pays compound interest, because Dirac's bra-ket notation (his one great act of typography, 1939) makes the rest of the quantum path legible instead of ornamental. This is the floor beneath the whole Quantum path: the difference between watching the Bell-inequality argument happen and being told about it.

How to read it

Anchor. Gilbert Strang, Introduction to Linear Algebra + the MIT OCW 18.06 lectures. Pick ~8 lectures covering vector spaces, basis, eigenvalues/eigenvectors, inner products, orthogonality, the spectral theorem. Don't do the whole book. Don't do every problem set. Concepts, not gradebook. ~20 hours.

Companion (probability). Pick one: Seeing Theory is free, fast, visual; Naked Statistics is the shortest serviceable book. Skip the other ten probability books for now.

Companion (complex numbers). One sitting: Needham's Visual Complex Analysis chapter 1, or a 3Blue1Brown video, or the Wikipedia article plus a few Euler's-formula exercises. You need enough to not flinch at e^iθ.

One-session calibration exercise. Write out by hand the two-qubit Bell state (|00⟩ + |11⟩)/√2. Compute the probabilities of each measurement outcome; compute them again after the first qubit is measured. Twenty minutes of algebra — it demystifies entanglement more than any amount of reading.