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

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
= 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: