## Nielsen and Chuang Exercise 2.76

Exercise 2.76Extend the proof of the Schmidt decomposition to the case where and may have spaces of different dimensions.

We start with the state

Here the matrix of values can be non-square rectangular. We can again do a SVD, yielding

The dimensions of and allow that the products and still make sense, and the diagonal index .

## Nielsen and Chuang Exercise 2.75

Exercise 2.75For each of the four Bell states, find the reduced density operator for each qubit.

## Nielsen and Chuang Exercise 2.74

Exercise 2.74Suppose a composite of systems and is in the state , where is a pure state of system , and is a pure state of system . Show that the reduced density operator of system alone is a pure state.

and is thus a pure state.

## Help Needed with Nielsen and Chuang Exercise 2.73

Exercise 2.73Given density matrix , the minimal ensemble contains a number of elements equal to the rank of . Let be any state in the support of , where the support is defined as being spanned by the eigenvectors of with nonzero eigenvalues. Show there exists a minimal ensemble for that contains , and that appears with probability

where is the inverse that acts only on the support of .

The density matrix has the eigendecomposition . We are given state in the support of , meaning that it is spanned by the eigenvectors with nonzero eigenvalues. This means that we can expand

where the primed summation is only over those vectors in the support of . Among other things, this means that the rank of is no greater than . Since we can expand in the eigenvectors of , this also means that we can create a minimal ensemble for than contains .

Given with rank , consider the matrix . We can write the projection of onto via

I need help here, if anyone has any suggestions. Just can’t push the equations home.

## Nielsen and Chuang Exercise 2.72

Exercise 2.72Bloch sphere for mixed states. The Bloch sphere picture for pure states of a single qubit was introduced in Section 1.2. This description has an important generalization to mixed states as follows.

- Show that an arbitrary density matrix for a mixed state qubit may be written as
where is a real three-dimensional vector such that . This vector is known as the

Bloch vectorfor the state .- What is the Bloch vctor representation for the state ?
- Show that a state is pure iff .
- Show that for pure states the description of the Bloch vector we have given coincides with that of Section 1.2.

Given that

we can write, for a pure state,

Massaging this a little bit

We want this to have the form

For this to work,

Looking at the norm of ,

Thus, we have shown that a pure state corresponds to the proper form, with , which completes goal 3 and goal 4.

Suppose we have a mixed state density matrix with . Such a state corresponds to

We now have a linear combination of unit vectors , again with . This sum will correspond to some other vector , with . This completes goal 1 of the exercise.

The Bloch vector for the state (goal 2) corresponds to .

## Nielsen and Chuang Exercise 2.71

Exercise 2.71Let be a density operator. Show that , with equality iff is a pure state.

A density matrix is defined by , where are nonnegative and . Its square is given by . Given the above constraints on , . If is a pure state, , and thus for all , meaning that .

## Nielsen and Chuang Exercise 2.70

Exercise 2.70Suppose is any positive operator acting on Alice’s qubit. Show that takes the same value when is any of the four Bell states. Suppose a malevolent third part (‘Eve’) intercepts Alice’s qubit on the way to Bob in the superdense coding protocol. Can Eve infer anything about which of the four possible bit strings 00, 01, 10, 11 Alice is trying to send? If so, how so, or if not, why not?

For the first part,

If Eve eavesdrops, she can only measure one qubit, and, as we show above, this yields the same result for all four states. However, if Bob has both states, he can do a measurement on both qubits

which will not be the same for all four qubits.

## Nielsen and Chuang Exercise 2.69

Exercise 2.69Verify that the Bell basis forms an orthonormal basis for the two qubit state space.

The Bell basis is given by

We note that, for the two electron states

from which it is easy to show that for and among the Bell states, and thus that the states form an orthonormal basis.

## Nielsen and Chuang Exercise 2.68

Exercise 2.68Prove that for all single qubit states and .

Suppose that and . Then

For this to be equal to the desired state, , which means either or . However, if , then , and if , then . Which makes it impossible for the product of two states to be equal to .

## Nielsen and Chuang Exercise 2.67

Exercise 2.67Suppose is a Hilbert space with a subspace . Suppose is a linear operator which preserves inner products, that is, for any and in ,

Prove that there exists a unitary operator whichextends. That is, for all in , but is defined on the entire space . Usually we omit the prime symbol and just write to denote the extension.

This is yet another of these exercised whose point I suspect that I am missing. However, I note that the composite operator , where the identity is defined only on , preserves inner products and is unitary provided that is unitary.

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