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Dirac Notation and Spectral decomposition

Michele Mosca

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)

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 )

More Dirac notation

More generally, we can also have operators like

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.

Special unitaries Pauli Matrices in new notation

The NOT operation, is often called the X or sX

operation.

Recall Special unitaries Pauli Matrices in new

representation

Representation of unitary operator

What is ??

It helps to start with the spectral decomposition

theorem.

Spectral decomposition

- Definition an operator (or matrix) M is normal

if MMtMtM - E.g. Unitary matrices U satisfy UUtUtUI
- 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

Spectral decomposition Theorem

- Theorem For any normal matrix M, there is a

unitary matrix P so that - MP?Pt where ? is a diagonal matrix.
- The diagonal entries of ? are the eigenvalues.
- The columns of P encode the eigenvectors.

Example Spectral decomposition of the NOT gate

This is the middle matrix in above decomposition

Spectral decomposition matrix from column vectors

Column vectors

Spectral decomposition eigenvalues on diagonal

Eigenvalues on the diagonal

Spectral decomposition matrix as row vectors

Adjont matrix row vectors

Spectral decomposition using row and column

vectors

From theorem

Verifying eigenvectors and eigenvalues

Multiply on right by state vector Psi-2

Verifying eigenvectors and eigenvalues

useful

Why is spectral decomposition useful? Because we

can calculate f(A)

m-th power

Note that

recall

So

Consider

e.g.

Why is spectral decomposition useful? Continue

last

M

f(? i)

Now f(M) will be in matrix notation

Same thing in matrix notation

Same thing in matrix notation

Important formula in matrix notation

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

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

Polar Decomposition

Left polar decomposition

Right polar decomposition

This is for square matrices

Gram-Schmidt Orthogonalization

Hilbert Space

Orthogonality

Norm

Orthonormal basis

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