Qubit, An Intuition #2 — Inner Product, Outer Product, and Tensor Product in Bra-ket Notation

TL;DR; 
2 qubits inner product, outer product, and tensor product in bra-ket notation, with examples.
Please refer to the previous article (published in July 12 2021), “Qubit, An Intuition #1 — First Baby Steps in Exploring the Quantum World” for a discussion on a single qubit as a computing unit for quantum computation.

Quantum Bit

Indeed, a single quantum bit (qubit) is exciting with its nature of being in superposition. Hence |Ψ> = α|0> + β|1>. The quantum state Psi (|Ψ>) is in the linear combination of |0> and |1> with the probability of being in state |0> is |α|², and the probability of being in state|1> is |β|².

Inner Product, Outer Product, and Tensor Product

However, we need more qubits to do meaningful quantum computation. Let’s say that we have the following two qubits, Psi|Ψ> and Phi |Φ>. Each has its respected quantum state.

Bra-ket notation for two-qubit operations: inner product, outer product, and tensor product.

Inner Product — <ΨΦ>

A product of two quantum states bra Psi <Ψ| and ket Phi |Φ> is called an inner product, producing a value. An inner product is also called an overlap, the overlap between quantum states.

Outer Product — |ΨΦ|

A product of two quantum states, ket Psi|Ψ> and bra Phi <Φ|, is called an outer product, producing a matrix. An outer product is also called a projection.

Tensor Product — |ΨΦ>

A product of two quantum states, ket Psi|Ψ> and ket Phi |Φ> is called a tensor product, producing a column vector with length 2ⁿ (where n is the number of qubits).

An example of quantum gates: NOT (flip gate), CNOT (Controlled-NOT), H (Hadamard), and Z (phase flip gate). Quantum gates are used to perform quantum computation in a quantum circuit.

Moving Forward

We can do a full or partial measurement of qubits. We will discuss this further in the next article, “Qubit, An Intuition #3 — Quantum Measurement, Full and Partial Qubits.”

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