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Part of Qiskit: Intro to Linear Algebra

Hilbert Spaces, Orthonormality, Inner Products

One of the premiere mathematical structures of Quantum Mechanics is the Hilbert Space. Here, we can think of Hilbert spaces as a state space in which all vectors representing quantum states exist. The difference between these spaces, and typical vector spaces, is that Hilbert space comes equipped with an inner product, an operation that can be performed on two vectors, that returns a scalar.

The scalar quantity returned from this operation represents the degree to which the first vector lies along the second vector. From this value, the probabilities of measurement in different quantum states can be calculated.

For two vectors a and b, in a Hilbert space, the inner product is described as ⟨a∣b⟩\langle a |b\rangle, where a and b are conjugate transposes of their original values, giving us:

⟨a∣b⟩ = (a1∗a2∗...an∗)(b1 b2 . . . bn) = a1∗b1 + a2∗b2 + ... + an∗bn\footnotesize \langle a | b \rangle \ = \ \begin{pmatrix} a_1^{*} & a_2^{*} & ... & a_n^{*} \end{pmatrix} \begin{pmatrix} b_1 \ b_2 \ . \ . \ . \ b_n \end{pmatrix} \ = \ a_1^{*} b_1 \ + \ a_2^{*} b_2 \ + \ ... \ + \ a_n^{*} b_n, where * denotes the complex conjugate of some vector.

One of the most important conditions for these spaces, is that the inner product of some vector with itself is equal to one: ⟨ψ∣ψ⟩=1\langle \psi | \psi \rangle = 1. This is the normalization condition expressed mathematically, which states the length of the vector squared (the components squared and summed), is equal to one. The physical significance is that the length of a vector in a particular direction is representative of the probability amplitude of the quantum system, with regard to the measurement of some quantum state.

Returning to the Bloch sphere:

The surface of this sphere is a valid Hilbert Space, along with the inner product between qubit states. A final note on Hilbert spaces is that they relate to Unitary matrices, which preserve inner product, meaning you can transform a vector under a series of unitary matrices without breaking the normalization condition that the inner product of two vectors must equate to one:

⟨ψ∣ψ⟩ = 1 ⇒ ∣ψ⟩ → U∣ψ⟩ = ∣ψ′⟩ ⇒ ⟨ψ′∣ψ′⟩ = (U∣ψ⟩)†U∣ψ⟩ = ⟨ψ∣U†U∣ψ⟩ = ⟨ψ∣ψ⟩ = 1\footnotesize \langle \psi | \psi \rangle \ = \ 1 \ \Rightarrow \ |\psi\rangle \ \rightarrow \ U |\psi\rangle \ = \ |\psi'\rangle \ \Rightarrow \ \langle \psi' | \psi' \rangle \ = \ (U |\psi\rangle)^{\dagger} U|\psi\rangle \ = \ \langle \psi | U^{\dagger} U |\psi\rangle \ = \ \langle \psi | \psi \rangle \ = \ 1

This means Unitary evolution of quantum systems sends quantum states to other valid quantum states- for a single qubit Hilbert space, represented by the Bloch sphere, unitary transformations correspond to vector rotations to different points on the sphere, without altering the length of the vector.

Referenced in

Qiskit: Intro to Linear Algebra