Problem set 1
Problem 1: Warming Up Exercises
(a) Verify that |wi ≡ (1, 1) and |vi ≡ (1, −1) are orthogonal. What are the normalized
forms of these vectors?
(b) The Pauli matrices X, Y and Z can be considered as operators with respect to an
orthonormal basis |0i, |1i for a two-dimensional Hilbert space. Express each of the
Pauli operators in the outer product notation.
(c) Find the eigenvectors, eigenvalues, and diagonal representations of the Pauli matrices
X, Y and Z.
(d) Show that the eigenvalues of a projector P are all either 0 or 1.
(e) Show that the transpose, complex conjugation, and adjoint operations distribute over
the tensor product:
(A ⊗ B)∗ = A∗ ⊗ B ∗
(A ⊗ B)T = AT ⊗ B T
(1)
(A ⊗ B)† = A† ⊗ B †
(f) If A and B are two linear operators, show that
Tr(AB) = Tr(BA).
(2)
(g) The Hadamard operator on one qubit may be written as
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H = √ [(|0i + |1i)h0| + (|0i − |1i)h1|]
2
(3)
Show explicitly that the Hadamard transform on n qubits, H ⊗n , may be written as
P
H ⊗n = √12n x,y (−1)x·y |xihy|. Write out an explicit matrix representation for H ⊗2 .
Problem 2: Bloch Sphere for Qubits
(a) Show that an arbitrary density matrix for a mixed state qubit may be written as
ρ=
I + ~r · ~σ
,
2
(4)
where ~r is a real three-dimensional vector such that ||~r|| ≤ 1. This vector is known as
the Bloch vector for the state ρ.
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(b) What is the Bloch vector representation for the state ρ = I/2.
(c) Show that a state ρ is pure if and only if ||~r|| = 1.
Problem 3: Tensorproduct
(a) Write down A ⊗ B in matrix form where A and B are 2 × 2 matrices. With V and W
vector spaces and dV = dim(V ), dW = dim(W ), what is the dimension of the vectorspace
V
W
V ⊗ W ? If |iidi=1
and |jidj=1
are bases for V and W resp. what is a basis for V ⊗ W ?
(b) Show that Tr(A ⊗ B) = Tr(A) · Tr(B).
(c) Show that exp(A ⊗ I) = exp(A) ⊗ I and exp(A ⊗ I + I ⊗ B) = exp(A) ⊗ exp(B).
(d) The direct sum of two vector spaces V and W with V ∩ W = {~0} is denoted as V ⊕ W .
All vectors ~x in V ⊕ W can be written as ~x = ~v + w,
~ where ~v ∈ V and w
~ ∈ W . What
is the dimension of V ⊕ W ?
Problem 4: Density Matrices
(a) What is the density matrix of a qubit which equals 0 with probability p = 1/2 and 1 with
probability 1/2? What is the density matrix of a qubit which equals |+i =
with p = 1/2 and |−i =
√1 (|0i
2
√1 (|0i + |1i)
2
− |1i) with p = 1/2. Would a measurement be able to
distinguish between these two physical states?
(b) Alice and Bob hold a state
√1 (|00i + |11i)
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where Alice has the first qubit and Bob holds
the last qubit. What is the density matrix for Alice when Bob measures his qubit in the
|+i, |−i basis, but does not tell Alice the outcome of his measurement?
(c) Let ρA =
P
i
λi |ψi ihψi | where |ψi i are orthonormal vectors (the eigenvectors of ρ) in
the vector space CdA and λi ≥ 0 are the associated eigenvalues. Can you write down a
bipartite pure state |ψi on system A and some other system B (of arbitrary dimension)
such that TrB |ψihψ| = ρA ? Can you characterize all pure states |ψi which have ρA as
its reduced density matrix?
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