CS 190/264: Quantum Computation Homework 3
Note: graduate students are not required to do problem 1, but are required to probelm 6. Undergraduates are required to do problems 1-5.
1. Consider the game played by Alice and Bob in Bell’s protocol. Suppose that the random bit that Alice receives XA = 1 and the random bit that Bob receives is XB = 0. Give the probabilities for each of combination for Alice and Bob’s output bits. That is, give the probability that a = 0 and b = 0. Then do the same for the other three possible values for a and b. What’s the probability that Alice and Bob win the game?
2. Work out a version of the quantum teleportation protocol if Bob and Alice are given the entangled pair 1/ √ 2(|01i − |10i) instead of 1/ √ 2(|00i + |11i).
3. Normally, we consider two quantum states that differ by a multiple of a ”global phase” e iθ to be equivalent, (|φi ≡ e iθ|φi), because any measurement performed on |φi will have the same likelihoods and outcomes as a measurement on e iθ|φi. Thus, the factor of e iθ is undetectable by any measurement.
(a) Prove that it is possible to multiply any normalized 1-qubit state by a phase e iθ so that the state is in the form a|0i + e iγb|1i, where a and b are non-negative real numbers that satisfy a 2 + b 2 = 1.
(b) Let |vi = a|0i+e iγb|1i be a normalized 1-qubit state. Define |v ⊥i to be the normalized state that is perpendicular to |vi. The state |v ⊥i will be unique up to a global phase. Express |v ⊥i in the standard basis.
(c) In class, we showed that the state |Φi = 1/ √ 2(|00i + |11i) can be expressed as 1/ √ 2(|ψi|ψi+|ψ ⊥i|ψ ⊥i), for any |ψi such that |ψi has real amplitudes in the standard basis. That is, |ψi = a|0i + b|1i, where a and b are real. This is in general not true if the state |ψi has a phase: |ψi = a|0i + e iγb|1i. Prove that the Bell state |Ψ−i = 1/ √ 2(|01i − |10i) can be expressed as |Ψ−i = √ 1 2 (|vv⊥i − |v ⊥vi) for any 1-qubit state |vi, up to a global phase. That is, it is OK to show that e iθ|Ψ−i = √ 1 2 (|vv⊥i − |v ⊥vi), for some θ.
4. (a) Describe the action of a CNOT gate if the target bit is |−i.
(b) Describe the action of a CNOT gate if the target bit is |+i.
(c) Now show that the following circuit is effectively a CNOT gate with the control and target qubits swapped (i.e. b is the control and a is the target).
5. The single qubit gate Uθ computes a rotation between the |0i and |1i states: Uθ = cos θ − sin θ sin θ cos θ . The circuit below uses a CNOT gate as well as a Uθ gate.
(a) What is the output on the circuit when the input is |0i ⊗ |φi, where |φi is an arbitrary 1-qubit state?
(b) What is the output on the circuit when the input is |1i ⊗ |φi, where |φi is an arbitrary 1-qubit state? (Hint: you will need the double-angle formulas from trigonometry.)
(c) Describe in words what the circuit does for a general input state. For graduate students: skip problem 1, and do the following problem:
6. Suppose that a 2-qubit state is shared by Alice and Bob. Suppose that Alice performs a unitary opertion U on her qubit and then Bob measures his qubit in some basis {|φi, |φ ⊥i}. Show that the probabilities of the outcomes from Bob’s measurement do not depend on the unitary operation chosen by Alice.