Quantum Information 2009/10 - Worksheet 1
October 8, 2009
Wherever possible try to use mathematica or maple or matlab to check your
answers. Think about writing specialized routines/functions in those programs
to do some of these calculations automatically.
Question 1 - single qubit pure states
Normalize the states
j 1i = 3 j0i + 1 j1i
j 2i = j0i j1i
j 3i = j0i + j+i
j 4i = j0i j+i
j 5i = j1i ei =3 j i
j 6i = ei =4 j0i + e i =4
Question 2 - representing pure states in the Bloch sphere
Find the bloch vectors and plot on a Bloch sphere the following pure states:
(i) The eigenstates of the Pauli X,Y,Z matrices
(ii) The states j 1 i ; ::: j 6 i from Question 1.
(iii) What are the quantum states (in the j0i ; j1i basis) which correspond
to the Bloch vectors:
~1 = [0; 1; 0]
~2 = [ ; 0; ]
~3 = [sin cos ; sin sin ; cos ]
How can we tell directly from the Bloch vector that these correspond to pure
Question 3 - unitary evolution of single pure qubits
The Pauli matrices are both Unitary and Hermitian. This means they can
play 3 roles - a unitary evolution operator in their own right, a Hamiltonian
which generates a unitary evolution and …nally an observable!
Firstly, for each of the states j 1 i ; ::: j 6 i from Q1 and the 6 eigenstates of
the Pauli matrices, make sure you understand geometrically how they evolve
under the action of each of the 3 Pauli matrices. (Essentially it is by re‡ ections
in the various planes). For example, where would the state X j 5 i be? How
do the Pauli operators evolve their own eigenstates? How about the eigenstates
of the other Pauli operators? You do not need to supply answers to all these
possibilities but you should ensure you have a good grasp on this.
(i) Show by expanding the exponentials that:
i2X cos 2 i sin 2
Rx ( ) e = ;
i sin 2 cos 2
i2Y cos 2 sin 2
Ry ( ) e = ;
sin 2 cos 2
i2Z e i2Z 0
Rz ( ) e =
0 ei 2 Z
These matrices act as rotations around the x; y; z axes. Note that if = h!t
then these matrices are the unitary evolution induced by a Hamiltonian e iHt
where the Hamiltonian is proportional to the particular Pauli matrix.
(ii) Does Rx;y;z (2 ) = Rx;y;z (0)?
(ii) In terms of the Bloch vector ~ of a state j i ; does Ry ( ) rotate or 2
radians around the y axis?
(iii) Construct a matrix which would rotate the single qubit state around
the axis de…ned by the vector ~ = [ p3 ; p3 ; p1 ]:
(iii) The following single qubit unitaries crop up often in quantum comput-
1 1 1 1 0 1 0
H=p ;S = ;T =
2 1 1 0 i 0 ei 4
In the context of quantum computing they are called the Hadamard, Phase
and =8 “gates” respectively.
Are these gates expressible as a rotation of the form e i(nx X+ny Y +nz Z) ?
Make sure you gain a geometric picture of how these gates act on generic Bloch
vectors, in particular the 6 Pauli eigenstates.
(iv) Do exercises (4.4) (4.5) and (4.7) in Nielsen and Chuang.
When doing calculations by hand in quantum information one often has to
choose whether to do it algebraically or as a matrix. An example of the former
is as follows: We can write the Hadamard gate as:
H = p (j0i h0j + j0i h1j + j1i h0j j1i h1j) :
So if j i = a j0i + b j1i then
Hj i = p (j0i h0j + j0i h1j + j1i h0j j1i h1j) (a j0i + b j1i)
a b a b
= p j0i + p j0i + p j1i p j1i = a j+i + b j i
2 2 2 2
since h0j 1i = 0:
To get some practise at acting operators on a state algebraically try the
(v) First …nd the unitary matrix U = ei 3 (X+Z) : Then by expanding the
expression algebraicially as above …nd U j+i : Then by using the fact that
e i 2 (^ ~ ) = cos 2 I i sin 2 (^ ~ ) …nd the answer (algebraically) an even
simpler way. (Be careful - n is a unit vector...)
(vi) In optics a common encoding of a qubit is by using the polarization
degree of freedom of a single photon. Explain carefully how an arbitrary single
qubit unitary rotation on such a qubit can be built from suitable combinations
of quater wave plates and half wave plates.
Question 4 - mixed states of single qubits
(i) Write down the denisty matrices of the 6 states in question 1.
(ii) Find the density matrix of the mixed state formed when I prepare j 3 i
with probability 1/2 and j 4 i with probability 1/2. What is the Bloch vector
of this mixed state. Sketch the Bloch vectors of and j 3 i and j 4 i ; and show
that the Bloch vector of lies halfway along the line joining the Bloch vectors
of j 3 i and j 4 i :
(iii) Repeat (ii) but now for the case that I prepare j 3 i with probability 1/3
and j 4 i with probability 2/3. What fraction of the way along the line joining
the two pure state Bloch vectors does the mixed state Bloch vector now lie.
(iv) I now prepare a mixed state by choosing j0i with probability 1/2, j+i
with probability 1/4 and j i with probability 1=4: Sketch the Bloch vectors of
the mixed states and the pure states forming this particular convex decomposi-
(v)* If I wanted to prepare the same mixed state as in (iv) but in such a
way that the probability of choosing the j+i state is as large as possible, what
is the convex decomposition I should prepare?
(vi) Is there any way to prepare the mixed state of (iv) by choosing the
three “+1” eigenvalue eigenstates of X; Y; Z with some suitable probabilities?
Explain your answer geometrically.
(vii) Show that a mixed state can only be decomposed by some set of
(pure or mixed) states which have Bloch vectors ~i if the Bloch vector ~ of r
lies within the polyhedron formed by the set of vectors ~i :s
(vi)* By noting that a density matrix has only one eigendecomposition
(which is necessarily into orthogonal states since the density matrix is Her-
mitian), show that the eigen-vectors of a mixed state with Bloch vector ~ haver
^ r r
Bloch vectors r = ~= j~j :
(vii)** A spin-1/2 system is in a strong magnetic …eld pointed in the x-
direction, so that the Hamiltonian is H = gX for some coupling constant g
(basically g is the Bohr magneton times the strength of the magnetic …eld).
The energy eigenstates are j i : If the spin is also in thermal equilibrium with
a reservoir at temperature T; then …nd the density matrices (and plot them in
the Bloch sphere) of the qubit for temperatures T = 0; T = 2g; and T = 1
(taking kB = 1): For what temperature is the spin in a state equivalent to being
with probability 1/2 at T = 0 and with probability 1=2 at T =p1? For what
temperature is it equivalent to a mixture of a the two pure states 23 j+i 1 j i?
For what temperature is the state of the qubit 1 j+i h+j + 2 j i h j?
Question 5 - Projective measurements on qubits
Let i = j i i h i j be projectors onto the states from Question 1.
(i) Compute the orthogonal projectors i = i i .
(ii) For each of the complete (ie von-Neumann or “PVM” measurements
f i ; i g compute the probability of each outcome for a measurement performed
on a qubit in the mixed state of Question 4(ii). (You should be able to do this
via matrix multiplication or algebraically).
(iii) Note that a projector for a measurement, if it is rank 1, looks the same
as a density matrix for a system prepared in that state, and so can also be
represented in Bloch vector form. If m is such a Bloch vector for a measure-
ment projector show that the probability of obtaining this outcome when the
measurement is performed on a system in a mixed state with Bloch vector ~ r
T r ( ) = (1 + m ~) :
Question 6 - The generalized Bloch sphere
** The Bloch sphere is a particularly nice geometric representation of qubit
states. In fact for a d-dimensional quantum system there are always d2 traceless
Hermiation operators i ; i = 1; ::; d2 1; satisfying the same algebraic relations
as the Pauli matrices (we always take 0 = I so that there are d2 1 nontrivial
such matrices). So an arbitrary mixed state of a qu-d-it can be represented
= I +~ ~
for a Bloch vector r which lives in d2 1 dimensions. Do some research on
generalized Bloch sphere representations (ie search the quant-ph arxiv, read the
seminal paper by Joe Eberly etc), and write a list of the properties which the
generalized Bloch vector does and does not share with the 2d case. (One For
instance, do pure states still have unit length Bloch vectors? Is every vector on
the surface of the generalized Bloch ball a valid pure state? Do convex decom-
positions of a generalized mixed state still have the same geometric descriptions
they have in the qubit case. Why is the generalized Bloch sphere not used as
much as the qubit version?
Supplementary problems which you may …nd helpful to do:
Problems 3,4,8,9,10,11 in Steeb and Hardy, Problems and Solutions in Quan-
tum Computing and Quantum Information.