Nielsen and Chuang Exercise 2.76
Exercise 2.76 Extend 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 2.75 For each of the four Bell states, find the reduced density operator for each qubit.
Nielsen and Chuang Exercise 2.74
Exercise 2.74 Suppose 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 2.73 Given 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 2.72 Bloch 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 vector for 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 2.71 Let
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 2.70 Suppose
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 2.69 Verify 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 2.68 Prove 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 2.67 Suppose
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
which extends
. 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.
2 comments