L6 Unitary Matrix and Complete Basis

Overview

Hiu-Yung Wong develops core quantum linear algebra tools: spectral reconstruction from eigenpairs, the unitary condition U†U = I, and how unitary transformations preserve inner products. He then derives basis-change coefficients from overlaps and shows that a complete orthonormal basis satisfies the resolution of identity, ∑ᵢ |i⟩⟨i| = I.

Key takeaways

For an operator with a complete orthonormal eigenbasis, its spectral form is A = ∑ᵢ λᵢ|λᵢ⟩⟨λᵢ|; the outer product creates a matrix contribution for each eigenpair.
A unitary matrix satisfies U†U = UU† = I, which ensures that applying U preserves vector inner products and overlaps.
A square matrix is unitary exactly when its columns are orthonormal: each has norm one and every distinct pair has inner product zero.
To convert coordinates from an old basis {|j⟩} to a new basis {|i′⟩}, form Uᵢⱼ = ⟨i′|j⟩, with new-basis bras as rows and old-basis kets as columns.
In an orthonormal basis expansion |v⟩ = ∑ᵢ αᵢ|i⟩, the coefficient αⱼ is obtained directly as ⟨j|v⟩.
Every complete orthonormal basis resolves the identity as I = ∑ᵢ |i⟩⟨i|; for the qubit basis {|0⟩, |1⟩}, the two outer products sum to the 2 × 2 identity.

Chapters

0:00 Course Notes and Reconstructing an Operator from Eigenpairs

Hiu-Yung Wong reminds students that the next two lectures will be provided as videos and Assignment 1 is due in about two weeks.
In an operator's eigenvector basis, its matrix is diagonal, with eigenvalues along the diagonal.
For a complete orthonormal eigenbasis, the operator can be reconstructed as A = ∑ᵢ λᵢ|λᵢ⟩⟨λᵢ|.

8:00 Recovering the Pauli X Matrix from Its Eigenvectors

The Pauli X eigenvectors are |+⟩ = (|0⟩ + |1⟩)/√2 and |−⟩ = (|0⟩ − |1⟩)/√2, with eigenvalues +1 and −1.
Each eigenpair contributes an eigenvalue-weighted outer product, λ|λ⟩⟨λ|, rather than an inner product.
Adding the two contributions gives σₓ = [[0, 1], [1, 0]], demonstrating reconstruction in the computational basis.

15:00 Unitary Matrices: The Adjoint Is the Inverse

A matrix U is unitary when U† = U⁻¹, unlike a Hermitian matrix, which satisfies A† = A.
The definition is equivalent to U†U = UU† = I.
Hermitian operators are associated with measurements, while unitary matrices describe quantum gates.

18:00 Why Unitary Transformations Preserve Inner Products

For vectors |f⟩ and |g⟩ transformed to U|f⟩ and U|g⟩, their new inner product is ⟨f|U†U|g⟩.
Since U†U = I, the transformed inner product equals the original ⟨f|g⟩.
This invariance preserves overlaps and supports the physical consistency of quantum gate operations.

25:00 Reading U†U as Inner Products Between Matrix Columns

Writing U as an n × n matrix with entries aᵢⱼ makes each column interpretable as a vector.
Taking the adjoint transposes the matrix and complex-conjugates its entries, turning the columns into corresponding rows.
Each entry of U†U is an inner product between two columns of U, and the result must equal the identity matrix.

37:00 The Orthonormal-Column Test for Unitarity

The identity condition implies ⟨uᵢ|uⱼ⟩ = δᵢⱼ for columns uᵢ and uⱼ of U.
Each column must have unit norm, and distinct columns must be orthogonal.
Checking orthonormal columns can be more convenient than multiplying every entry of U†U, especially for large matrices.

39:00 A 2D Rotation as a Model for Unitary Transformation

A vector with coordinates (α, β) in the {|0⟩, |1⟩} basis provides a two-dimensional example of a transformation.
Rotating the vector counterclockwise can be described equivalently by rotating the coordinate axes clockwise and expressing the vector in the new basis.
The coordinate-system analogy helps build intuition for transformations in quantum state spaces, where direct geometric visualization is limited.

44:00 Deriving the 2D Rotation Matrix by Changing Coordinates

The old basis vectors are written in the rotated basis using sine and cosine: |0⟩ = cos θ|0′⟩ + sin θ|1′⟩ and |1⟩ = −sin θ|0′⟩ + cos θ|1′⟩.
Substituting these expressions into |v⟩ = α|0⟩ + β|1⟩ yields new coefficients α cos θ − β sin θ and α sin θ + β cos θ.
The resulting transformation is [[cos θ, −sin θ], [sin θ, cos θ]], acting on the original coefficient vector (α, β).

52:00 Quantum Gates as State Transformations

The rotation example illustrates the general idea that a quantum gate transforms a state's coordinate vector.
Transforming the vector in one direction corresponds to an inverse change of coordinates in the opposite direction.
The rotation matrix is an example of a unitary transformation because its columns are orthonormal and it preserves inner products.

55:00 Constructing a Basis-Change Matrix from Basis Overlaps

For old basis {|j⟩} and new basis {|i′⟩}, the transformation matrix has entries Uᵢⱼ = ⟨i′|j⟩.
Rows correspond to bras from the new basis and columns to kets from the old basis.
For a 2D rotation, the overlaps produce [[cos θ, −sin θ], [sin θ, cos θ]], providing a direct method when geometric drawings are impractical.

1:01:30 Completeness: Expressing Every Vector in a Basis

A basis is complete for a space if every vector in that space can be written as a linear combination of its basis vectors.
The vectors x̂ and ŷ span a real 2D plane but are not complete for 3D space because they cannot represent a z component.
An n-dimensional space requires n suitable basis vectors to express every vector in that space.

1:05:00 Finding a Basis Coefficient with an Inner Product

For |v⟩ = ∑ᵢ αᵢ|i⟩ in an orthonormal basis, taking ⟨j|v⟩ isolates the coefficient αⱼ.
The result follows from ⟨j|i⟩ = δᵢⱼ, which makes every term vanish except the one with i = j.
This projection method extracts amplitudes from a superposition and assumes orthonormal basis vectors unless stated otherwise.

1:09:00 From Basis Expansion to the Resolution of Identity

Replacing each coefficient αᵢ in |v⟩ = ∑ᵢ αᵢ|i⟩ with ⟨i|v⟩ gives |v⟩ = ∑ᵢ |i⟩⟨i|v⟩.
Because the equation holds for every |v⟩, the operator ∑ᵢ |i⟩⟨i| must be the identity.
A complete orthonormal basis therefore satisfies I = ∑ᵢ |i⟩⟨i|, an identity that can be inserted into operator expressions without changing them.

1:12:00 Verifying the Identity with the Qubit Basis

For the two-dimensional computational basis {|0⟩, |1⟩}, the completeness sum is |0⟩⟨0| + |1⟩⟨1|.
The outer products are [[1, 0], [0, 0]] and [[0, 0], [0, 1]].
Adding them gives the 2 × 2 identity matrix, confirming the resolution-of-identity formula.

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Quantum Computing, TCAD, Semicond by Hiu-Yung Wong.