1. Dirac Notation and Spectral decomposition Michele Mosca

2. Dirac notation For any vector, we let denote, the complex conjugate of. We denote by the inner product between two vectors and defines a linear function that maps (I.e. … it maps any state to the coefficient of its component )

3. More Dirac notation defines a linear operator that maps (Aside: this projection operator also corresponds to the “density matrix” for ) More generally, we can also have operators like (I.e. projects a state to its component)

4. More Dirac notation For example, the one qubit NOT gate corresponds to the operator e.g. The NOT gate is a 1-qubit unitary operation.

5. Special unitaries: Pauli Matrices The NOT operation, is often called the X or σ X operation.

6. Special unitaries: Pauli Matrices

7. What is ?? It helps to start with the spectral decomposition theorem.

8. Spectral decomposition l Definition: an operator (or matrix) M is “normal” if MM t =M t M l E.g. Unitary matrices U satisfy UU t =U t U=I l E.g. Density matrices (since they satisfy  =  t ; i.e. “Hermitian”) are also normal

9. Spectral decomposition l Theorem: For any normal matrix M, there is a unitary matrix P so that M=P  P t where  is a diagonal matrix. l The diagonal entries of  are the eigenvalues. The columns of P encode the eigenvectors.

10. e.g. NOT gate

11. Spectral decomposition

15. Verifying eigenvectors and eigenvalues

17. Why is spectral decomposition useful? Note that So recall Consider e.g.

18. Why is spectral decomposition useful?

19. Same thing in matrix notation

22. “Von Neumann measurement in the computational basis” l Suppose we have a universal set of quantum gates, and the ability to measure each qubit in the basis l If we measure we get with probability

23. In section 2.2.5, this is described as follows l We have the projection operators and satisfying l We consider the projection operator or “observable” l Note that 0 and 1 are the eigenvalues l When we measure this observable M, the probability of getting the eigenvalue is and we are in that case left with the state