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Pauli Operators
Pauli Operators
The Pauli operators (along with the identity) are fundamental in describing single qubit systems. σ=(X,Y,Z), where:
Hermitian Operators:
Hermitian Operators:
An operator is Hermitian if it is equal to its own conjugate transpose (Hermitian conjugate): The conjugate transpose is obtained by taking the transpose of the operator and then taking the complex conjugate of each element:
Properties of Hermitian Operators:
Properties of Hermitian Operators:
Real Eigenvalues: The eigenvalues of a Hermitian operator are always real numbers. This is consistent with physical observables having real measurement outcomes.
Orthogonal Eigenvectors: Eigenvectors corresponding to distinct eigenvalues of a Hermitian operator are orthogonal.
Spectral Theorem: Any Hermitian operator can be diagonalized by a unitary transformation, and its eigenvectors form a complete orthonormal basis for the Hilbert space. (This means that for any Hermitian matrix A, there exists a unitary matrix U (a matrix such that U*U=UU*=I, where I is the identity matrix) and a real diagonal matrix Λ such that: A = UΛ U*.
We Show That the Pauli Matrices are Hermitian:
We Show That the Pauli Matrices are Hermitian:
Eigenvalues and Eigenvectors of Pauli Matrices
Eigenvalues and Eigenvectors of Pauli Matrices
Commutator and Anticommutator
Commutator and Anticommutator
Commutator:
Commutator:
The commutator of two operators A and B is defined as: [A,B] = AB-BA.
Commuting Operators Share Eigenbasis:
If two Hermitian operators A and B commute (i.e. [A,B] =0, so AB=BA), then there exists a common set of eigenvectors that are eigenvectors of both A and B.
Brief Proof Sketch: Let ∣ψ⟩ be an eigenvector of A with eigenvalue a: A ∣ψ⟩ = a ∣ψ⟩. Consider AB ∣ψ⟩. Since AB=BA, we have
AB ∣ψ⟩ = BA ∣ψ⟩ = B(a ∣ψ⟩) = a(B ∣ψ⟩). This equation shows that B ∣ψ⟩ is also an eigenvector of A with the same eigenvalue a. If a is non-degenerate, B ∣ψ⟩ must be proportional to ∣ψ⟩, making ∣ψ⟩ an eigenvector of B (The eigenspace for a non-degenerate eigenvalue has
dimension 1. In this case, the eigenspace for eigenvalue a is spanned only by ∣ψ⟩ (and its scalar multiples). ). If a is degenerate, B acts within the eigenspace, and eigenvectors of B within that subspace are also eigenvectors of A.The eigenspace for a degenerate eigenvalue has a dimension greater than 1. Let E_a denote the eigenspace of A corresponding to the eigenvalue a. E_a={v∈V∣Av=av}. The linear operator B maps vectors that are in the eigenspace E_a to other vectors that are also in the eigenspace E_a. The eigenspace E_a is an invariant subspace under the action of B. Consider the matrix (or operator) B restricted to acting only on the vectors within the subspace Ea. Since E_a is a finite-dimensional vector space (assuming we are in a finite-dimensional setting), B, when restricted to E_a, will have its own eigenvectors within E_a. Let ∣ϕ⟩ be such an eigenvector of B that lies within E_a.
By definition of being an eigenvector of B within E_a, B∣ϕ⟩=λ∣ϕ⟩ for some scalar λ.
By definition of ∣ϕ⟩ being within the eigenspace E_a, ∣ϕ⟩ is an eigenvector of A with eigenvalue a: A∣ϕ⟩=a∣ϕ⟩.
Anticommutator
Anticommutator
An Important Pauli Vector Identity
An Important Pauli Vector Identity
Hilbert-Schmidt Inner Product
Hilbert-Schmidt Inner Product
Definition:
The Hilbert-Schmidt inner product is an inner product defined on the space of linear operators (matrices) acting on a Hilbert space. For two operators and acting on a -dimensional Hilbert space, their Hilbert-Schmidt inner product is:
where Tr denotes the trace of the matrix (sum of the diagonal elements) and A^{dagger} is the Hermitian conjugate of .
Properties:
Spectral Decomposition of a Single Qubit State
Spectral Decomposition of a Single Qubit State
Application in Quantum Computing (and beyond):
Application in Quantum Computing (and beyond):
Von Neumann Entropy
Von Neumann Entropy
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