Dirac Notation and Spectral decomposition

1. Dirac Notation and Spectral decomposition Michele Mosca

2. review”: 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 This is a scalar so I can move it to front (I.e. projects a state to its component Recall: this projection operator also corresponds to the “density matrix” for )

4. More Dirac notation More generally, we can also have operators like

5. Example of this Dirac notation For example, the one qubit NOT gate corresponds to the operator e.g. (sum of matrices applied to ket vector) This is one more notation to calculate state from state and operator The NOT gate is a 1-qubit unitary operation.

6. Special unitaries: Pauli Matrices in new notation The NOT operation, is often called the X or σX operation.

7. Recall: Special unitaries: Pauli Matrices in new representation Representation of unitary operator

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

9. Spectral decomposition • Definition: an operator (or matrix) M is “normal” if MMt=MtM • E.g. Unitary matrices U satisfy UUt=UtU=I • E.g. Density matrices (since they satisfy =t; i.e. “Hermitian”) are also normal Remember: Unitary matrix operators and density matrices are normal so can be decomposed

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

11. Example: Spectral decomposition of the NOT gate This is the middle matrix in above decomposition

12. Spectral decomposition: matrix from column vectors Column vectors

13. Spectral decomposition: eigenvalues on diagonal Eigenvalues on the diagonal

14. Spectral decomposition: matrix as row vectors Adjont matrix = row vectors

15. Spectral decomposition: using row and column vectors From theorem

16. Verifying eigenvectors and eigenvalues Multiply on right by state vector Psi-2

17. Verifying eigenvectors and eigenvalues useful

18. Why is spectral decomposition useful? Because we can calculate f(A) m-th power Note that recall So Consider e.g.

19. Why is spectral decomposition useful? Continue last slide M = f(i)

20. Now f(M) will be in matrix notation

21. Same thing in matrix notation

22. Same thing in matrix notation

23. Important formula in matrix notation

24. “Von Neumann measurement in the computational basis” • Suppose we have a universal set of quantum gates, and the ability to measure each qubit in the basis • If we measure we get with probability We knew it from beginning but now we can generalize

25. Using new notation this can be described like this: • We have the projection operators and satisfying • We consider the projection operator or “observable” • Note that 0 and 1 are the eigenvalues • When we measure this observable M, the probability of getting the eigenvalue b is and we are in that case left with the state

26. Polar Decomposition Left polar decomposition Right polar decomposition This is for square matrices

27. Gram-Schmidt Orthogonalization Hilbert Space: Orthogonality: Norm: Orthonormal basis: