quantum information and computing

1. Quantum Information HW 2: Tensors & Density Matrices PH534, January 31, 2018
1: Tensor Products & Partial Traces
1. Whats the tensor product of the 2 × 2 matrix ρ =
2
3 0.3
0.3 1
3
and
|ψ = cos(θ
2)|0 + sin(θ
2)|1 ?
2. Consider the quantum state |x = |ψ ⊗ |φ , where |ψ = a0|0 + a1|1 + a2|2 and
|φ = b0|0 + b1|1 . Compute the tensor product state explicitly. What is |x x| ?
3. Consider the matrices Oij = σi ⊗ σj, where i, j = 0, 1, 2, 3 and σj are the Pauli matrices with
σ0 = I. Explicitly write down all 16 Oij matrices.
4. The partial trace (over the subsystem E) is given by
TrE(|a b|S ⊗ |α β|E) = k k|E(|a b|S ⊗ |α β|E)|k E. Compute the partial trace of
X =




1 0 0 1
0 0 1 0
0 1 0 0
1 0 0 1



 , (1)
over the first and second two-dimensional subsystems.
5. Compute the partial trace of α|00 + β|11 , with |α|2 + |β|2 = 1 over the first qubit. Do the
same over the second qubit.
2: Density Matrices
6. Consider the matrix ρ =
1 −1/2
−1/2 0
. Check if this matrix is a density matrix.
7. Consider the matrix ρ =
1/3 0
0 2/3
. Check if this matrix is a density matrix.
8. Consider the quantum state |Ψ = α(|01 +|10 )
√
2
+ β|+x + x , where |+x = |0 +|1
√
2
. Bring this
quantum state into the Schmidt form. After this, compute the partial trace over each
subsystem.
9. Consider the quantum state |GHZ = 1√
2
(|0A0B0C + |1A1B1C ) (its named the
Greenberger-Horne-Zeilinger state). Compute the partial trace over the first, the second and
the third system. Compute the marginal state of the three subsystems as well.
10. Consider the quantum state |W = 1√
3
(|0A0B1C + |0A1B0C + |1A0B0C ). Compute the
partial trace over the first, the second and the third system. Compute the marginal state of
the three subsystems as well.
1