Download Problem Set 2 - Massachusetts Institute of Technology

MASSACHUSETTS INSTITUE OF TECHNOLOGY
MIT 2.111/8.411/6.898/18.435
Quantum Information Science I
September 16, 2010
Problem Set #2
(due in class, 23-Sep-10)
1. Density matrices. A density matrix (also sometimes known as a density operator) is a representation
of statistical mixtures of quantum states. This exercise introduces some examples of density matrices,
and explores some of their properties.
(a) Let |ψi = a|0i + b|1i be a qubit state. Give the matrix ρ = |ψihψ|, which you may compute using
linear algebra using the vector representations of |ψi and hψ|. What are the eigenvectors and
eigenvalues of ρ?
(b) Let ρ0 = |0ih0| and ρ1 = |1ih1|. Give the matrix σ =
eigenvalues of σ?
ρ0 +ρ1
2 .
What are the eigenvectors and
(c) Compute tr(ρ2 ) and tr(σ 2 ). In general, tr(M 2 ) ≤ 1, with equality if and only if M is a pure state.
2. Exponential of the Pauli matrices. Let ~v be any real, three-dimensional unit vector and θ a real
number. Prove that
exp(iθ~v · ~σ ) = cos(θ)I + i sin(θ)~v · ~σ ,
where ~v · ~σ ≡
P3
i=1
(1)
vi σi , and σi are the Pauli matrices, σ1 = X, σ2 = Y , σ3 = Z.
3. Hadamard operator on n qubits. The Hadamard operator on one qubit may be written as
i
1 h
H = √ (|0i + |1i)h0| + (|0i − |1i)h1| .
2
(2)
Show explicitly that the Hadamard transform on n qubits, H ⊗n , may be written as
1 X
(−1)x·y |xihy|.
H ⊗n = √
2n x,y
(3)
Write out an explicit matrix representation for H ⊗2 .
4. Single qubit rotations. Define the rotation operator
θ
θ
I − i sin
(nx X + ny Y + nz Z) ,
Rn̂ (θ) ≡ exp(−iθ n̂ · ~σ /2) = cos
2
2
(4)
where n̂ is a real three-dimensional unit vector.
1. Prove that an arbitrary single qubit unitary operator can be written in the form U =
exp(iα)Rn̂ (θ), for some real numbers α and θ.
2. Find values for α, θ, and n̂ giving the Hadamard gate H.
3. Find values for α, θ, and n̂ giving the phase gate
1 0
S=
.
0 i
(5)