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Dirac notation

Dirac notation

February 2, 2025

Dirac notation is a compact language for quantum states and operators.

Tensor products

Multi-qubit kets use the tensor product symbol (often omitted in writing):

01=01. \lvert 0 \rangle \otimes \lvert 1 \rangle = \lvert 01 \rangle.

Operators on separate qubits combine as Kronecker products ABA \otimes B.

Projectors and measurement

A projective measurement can be described using projectors onto orthogonal subspaces. For example, the projector onto 0\lvert 0 \rangle on a single qubit is P0=00P_0 = \lvert 0 \rangle \langle 0 \rvert.

When you read circuit diagrams, translate each time slice into the corresponding unitary acting on the tensor-product Hilbert space.

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Matrices as gates