ASSIGNMENT
PH534 Spring 2022-2023
(Submission deadline: 11 April 2023, 19:15 hours)
• Please include your NAME and LDAP ID in your solution sheet.
• Please answer all the questions.
1. Pure and mixed state entanglement
a) Given two vectors:
𝐚 = {𝑎! , 𝑎" , … , 𝑎# } and 𝐛 = {𝑏! , 𝑏" , … , 𝑏# },
we can define 𝐚 ⊗ 𝐛 = {𝑎! 𝑏! , 𝑎! 𝑏" , 𝑎! 𝑏$ , … , 𝑎# 𝑏# }, which is similar to forming diagonal
matrices with entries 𝐚 and 𝐛 and taking the tensor product of the two. Now, prove that
if 𝐚 ≺ 𝐜 and 𝐛 ≺ 𝐝 then 𝐚 ⊗ 𝐛 ≺ 𝐜 ⊗ 𝐝. Interpret this mathematical result physically in
terms of LOCC on pure bipartite quantum systems. (5 marks)
" !
b) Show that for the state |𝜙⟩ = 2% (|00⟩ + |11⟩) + (|22⟩ + |33⟩) we cannot have a
√!'
! !
deterministic transformation under LOCC to the state |𝜓⟩ = |00⟩ + (|11⟩ + |22⟩),
√" "
but the transformation, |𝜙⟩ ⊗ |𝜒⟩ ⟶ |𝜓⟩ ⊗ |𝜒⟩, is possible under LOCC, for the state
$ "
|𝜒⟩ = 2 |00⟩ + 2 |11⟩. (5 marks)
% %
2. Computation of entanglement
Use Mathematica or Python (or any other language) to do the computation. You are free
to use in-built functions and packages to perform partial trace/ transposition. Consider the
𝕀
Werner state for two qubits: 𝜌() (𝑝) = (1 − 𝑝) + + 𝑝|𝜓 , ⟩⟨𝜓 , |. Numerically estimate and
write down the quantum mutual information, negativity, logarithmic negativity,
entanglement of formation and concurrence for the Werner state, 𝜌() , for the values of
𝑝 ∈ [0, 1] increasing in steps of 0.1. (10 marks)
3. Quantum circuits and algorithms
a) Construct a CNOT gate using a controlled-Z gate and two Hadamard gates. Mention
the control and target qubits. (5 Marks)
PH534 Spring 2022-2023
(Submission deadline: 11 April 2023, 19:15 hours)
• Please include your NAME and LDAP ID in your solution sheet.
• Please answer all the questions.
1. Pure and mixed state entanglement
a) Given two vectors:
𝐚 = {𝑎! , 𝑎" , … , 𝑎# } and 𝐛 = {𝑏! , 𝑏" , … , 𝑏# },
we can define 𝐚 ⊗ 𝐛 = {𝑎! 𝑏! , 𝑎! 𝑏" , 𝑎! 𝑏$ , … , 𝑎# 𝑏# }, which is similar to forming diagonal
matrices with entries 𝐚 and 𝐛 and taking the tensor product of the two. Now, prove that
if 𝐚 ≺ 𝐜 and 𝐛 ≺ 𝐝 then 𝐚 ⊗ 𝐛 ≺ 𝐜 ⊗ 𝐝. Interpret this mathematical result physically in
terms of LOCC on pure bipartite quantum systems. (5 marks)
" !
b) Show that for the state |𝜙⟩ = 2% (|00⟩ + |11⟩) + (|22⟩ + |33⟩) we cannot have a
√!'
! !
deterministic transformation under LOCC to the state |𝜓⟩ = |00⟩ + (|11⟩ + |22⟩),
√" "
but the transformation, |𝜙⟩ ⊗ |𝜒⟩ ⟶ |𝜓⟩ ⊗ |𝜒⟩, is possible under LOCC, for the state
$ "
|𝜒⟩ = 2 |00⟩ + 2 |11⟩. (5 marks)
% %
2. Computation of entanglement
Use Mathematica or Python (or any other language) to do the computation. You are free
to use in-built functions and packages to perform partial trace/ transposition. Consider the
𝕀
Werner state for two qubits: 𝜌() (𝑝) = (1 − 𝑝) + + 𝑝|𝜓 , ⟩⟨𝜓 , |. Numerically estimate and
write down the quantum mutual information, negativity, logarithmic negativity,
entanglement of formation and concurrence for the Werner state, 𝜌() , for the values of
𝑝 ∈ [0, 1] increasing in steps of 0.1. (10 marks)
3. Quantum circuits and algorithms
a) Construct a CNOT gate using a controlled-Z gate and two Hadamard gates. Mention
the control and target qubits. (5 Marks)
