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Basis State

A basis state is one of a set of reference states — mutually orthogonal and normalized — from which every other quantum state can be built as a superposition.

What it means

A basis is a set of orthonormal states that spans the state space: any state |ψ⟩ can be written uniquely as a linear combination |ψ⟩ = Σᵢ cᵢ|i⟩ of the basis states |i⟩, with ⟨i|j⟩ = δᵢⱼ.For a single qubit, the standard choice is the computational basis {|0⟩, |1⟩}, but it is not the only one: the Hadamard basis {|+⟩, |−⟩}, with |±⟩ = (|0⟩ ± |1⟩)/√2, is equally valid, and a state that is a superposition in one basis can be a basis state in another — |+⟩ is 'both 0 and 1' in the computational basis but perfectly definite in the Hadamard basis.For n qubits, the computational basis has 2ⁿ states, |00...0⟩ through |11...1⟩, one for each classical bit string.Basis states matter operationally: a measurement is always defined with respect to a basis, and its possible outcomes are exactly the basis states.Choosing the right basis is a recurring trick in quantum algorithms and quantum communication.

Everyday analogy

Basis states are like the plain building blocks in a box of LEGO: every fancy castle you can build is just those same few basic bricks combined in different amounts. In the quantum world, |0⟩ and |1⟩ are the basic bricks, and every qubit state is built by mixing them.
A basis is like the north-south and east-west streets of a city grid: any trip through the city can be described as 'so many blocks north plus so many blocks east'. And just as you could rotate the grid (use diagonal avenues instead), you can choose a different basis — the trip is the same, only the description changes.

Common misconceptions

  • The computational basis {|0⟩, |1⟩} is NOT the only basis — any pair of orthonormal qubit states forms a valid basis, and whether a state 'is in superposition' depends on which basis you describe it in.
  • A basis state is not the same as a 'ground state' — in quantum computing, 'basis state' refers to a member of a chosen reference set (usually the computational basis), not to the lowest-energy state of a physical system, even though hardware often encodes |0⟩ as the ground state.
  • Having 2ⁿ basis states does NOT mean an n-qubit register stores 2ⁿ readable values — a measurement returns only one n-bit outcome.

Key takeaways

  • Basis states are orthonormal (⟨i|j⟩ = δᵢⱼ) and span the state space: every state is a unique linear combination of them.
  • The computational basis {|0⟩, |1⟩} corresponds to classical bit values, but other bases such as {|+⟩, |−⟩} are equally valid.
  • Superposition is basis-relative: |+⟩ is a superposition in the computational basis yet a definite basis state in the Hadamard basis.
  • An n-qubit system has 2ⁿ computational basis states, one for each classical bit string.

Check your understanding

How many computational basis states does a 3-qubit system have?

  1. A.3
  2. B.6
  3. C.8
  4. D.9
Show the answer

Answer: C. 8

Why: An n-qubit system has 2ⁿ computational basis states. For n = 3 that is 2³ = 8 states: |000⟩, |001⟩, |010⟩, |011⟩, |100⟩, |101⟩, |110⟩, |111⟩.

Builds on

Qubit

Primary source: Nielsen & Chuang, Quantum Computation and Quantum Information (2010), doi:10.1017/CBO9780511976667

Graded 2026-07-10 (human sign-off): established — orthonormal bases and the computational basis per Nielsen & Chuang (2010) §1.2/§2.1.2 and Preskill Ph219. Pending human grading.

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