The bra-ket primitives

daitai-algebra v1.4 includes a small set of bra-ket primitives that let you write quantum and classical state in the same vocabulary. This page is the conceptual introduction; for the full operator catalogue, see Step semantics.

Why bra-ket in a general-purpose algebra?

Bra-ket notation is not exotic. It is the cleanest way to talk about:

• a state vector that may be a superposition of possibilities;
• the dual of that vector;
• the inner product that asks "how aligned are these two states?";
• the projection that asks "what is the probability of observing this outcome?".

Every one of these questions also appears in classical machine learning, statistics, and information theory — just disguised. Naming them once, with one notation, removes a class of bugs that comes from translating between notations.

The primitives

| Symbol | Sort | Meaning | |--------|------|---------| | |ψ⟩ | Ket | state vector | | ⟨ψ| | Bra | dual of a ket | | ⟨φ|ψ⟩ | Scalar | inner product | | |ψ⟩⟨φ| | Operator | outer product (rank-1 operator) | | A ⊗ B | Operator × Operator → Operator | tensor product |

A worked example: the Bell state

type Register = Tensor<Qubit, Qubit>

fn bell() -> Register {
  let q0 = |0⟩
  let q1 = |0⟩
  let h  = H(q0)              # superposition
  let r  = CNOT(h ⊗ q1)       # entangle
  r
}

# Classical observation collapses the state.
let step : Register × Measure -> Distribution<Bit × Bit> = observe

The observe function returns a Distribution, not a value. This is the algebra forcing you to be honest: measurement is non-deterministic, and the type system insists you handle the resulting distribution rather than silently sampling.

Reading next

• Step semantics — the canonical State × Event → State relation.
• The principles — why these primitives are constitutional and not "syntax sugar".