Quantum logic gates are created from superposition of spin states. Should’t there be an infinite number of possible directions that the spin can point to – so an infinite number of states?

Quantum Logic Gates and Spin States

1. The Spin Hilbert Space is Finite-Dimensional

For a spin-1/2 particle (like an electron or a qubit in quantum computing), the quantum state is represented as a vector in a two-dimensional Hilbert space spanned by the basis states:

|0⟩ = [1, 0], |1⟩ = [0, 1]

These states are eigenstates of the Pauli Z-operator (σz), corresponding to spin “up” (↑) and spin “down” (↓) along the z-axis.

2. Superposition of Spin States

A general spin state is a superposition of these basis states:

|ψ⟩ = α |0⟩ + β |1⟩, where |α|² + |β|² = 1

This means that the state is always confined to a two-dimensional space, even though the spin vector may appear to point in any direction in real space.

3. Bloch Sphere Representation

A single qubit state can be visualized on the Bloch sphere:

|ψ⟩ = cos(θ/2) |0⟩ + eiφ sin(θ/2) |1⟩

Here, θ and φ define a point on the sphere. While there are an infinite number of possible directions, these do not correspond to distinct independent quantum states—they are just different superpositions of the same two basis states.

4. Quantum Measurement and Finite Outcomes

Even though spin can be in a superposition, measurement collapses the state into just one of two possible outcomes. For a spin-1/2 particle measured along any axis:

  • The result is always either spin-up or spin-down along that axis.
  • The probability depends on the state before measurement.

Thus, while the state space is continuous (because of the superposition coefficients), the measurement outcomes are always discrete.

Conclusion

Yes, spin can point in infinitely many directions, but the Hilbert space for a qubit is still two-dimensional because any state can be represented as a superposition of just two basis states. Quantum logic gates manipulate these superpositions, but they don’t require an infinite number of states—just a continuous set of possible transformations within this two-dimensional space.