## Nielsen and Chuang Exercise 2.47

Exercise 2.47Suppose that and are Hermitian. Show that is Hermitian.

## Nielsen and Chuang Exercise 2.44

Exercise 2.44Suppose that and and is invertible. Show that must be 0.

If , then and if , then . The only way for is for either or to be zero, and if is invertible it isn’t zero, and thus is zero.

## Nielsen and Chuang Exercise 2.43

Exercise 2.43Show that for

If , , and the terms are zero. Otherwise, the terms are give by the matrix products from exercise 2.40.

## Nielsen and Chuang Exercise 2.41

Exercise 2.41Anti-commutation relations for the Pauli matrices. Verify the anti-commutation relations , where are both chosen from the set 1,2,3. Also verify that () .

The anti-commutation relations follow from the Pauli products worked out in exercise 2.40. The Pauli squares were worked out in exercise 2.19.

## Nielsen and Chuang Exercise 2.40

Exercise 2.40Commutation relations for the Pauli matrices. Verify the commutation relations , , . There is an elegant way of writing this using , the antisymmetric tensor on three indices, for which , except for and :

From these it follows that:

## Nielsen and Chuang Exercise 2.39

Exercise 2.39The Hilbert-Schmidt inner product on operators: the set of linear operators on a Hilbert space is obviously a vector space—the sum of two linear operators is a linear operator, is a linear operator is is a linear operator and is a complex number, and there is a zero element 0. An important additional result is that the vector space can be given a natural inner product structure, turning it into a Hilbert space.

- Show that the function (,) on defined by is an inner product function. This inner product is known as the Hilbert-Schmidt or trace inner product.
- If has dimensions, show that has dimension .
- Find an orthonormal basis of Hermitian matrices for the Hilbert space .

- Show that the function (,) on defined by is an inner product function. This inner product is known as the Hilbert-Schmidt or trace inner product.
An inner product must satisfy:

- is linear in the second argument:
We can show this via:

- , with equality iff .
This is a sum of strictly positive values, so that the sum is , and is equal to zero iff all entries are zero.

- is linear in the second argument:
- If has dimensions, show that has dimension : For to fully span , it must be able to map any vector into itself and into any other vector . Thus there must be different transformation.
- Find an orthonormal basis of Hermitian matrices for the Hilbert space . Here we can do something along the lines of the Gramm-Schmidt orthogonalization. Taking the linear operators, we can Hermitize each one via , and then orthogonalize each matrix to all of the for by subtracting the inner product representation given by , above.

I think my answers to 2 and 3 are still incomplete. Please comment if you have suggestions.

## Nielsen and Chuang Exercise 2.38

Exercise 2.38Linearity of the trace: if and are two linear operators, show that , and, if is an arbitrary complex number, show that .

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