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The instructions and questions for part iii, paper 66 of the mathematical tripos exam, focusing on quantum foundations. Topics include instantaneous non-demolition verification of quantum states, quantum teleportation, and the chsh-bell inequality. Students are required to answer questions related to entangled states, bell-operators, and hidden variable models.
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Tuesday, 12 June, 2012 9:00 am to 11:00 am
Attempt no more than THREE questions. There are FOUR questions in total. The questions carry equal weight.
Cover sheet None Treasury Tag Script paper
You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
Alice and Bob (located at xA and xB respectively) share a maximally entangled state |Φ+〉d 1 d 2 = √^12 (| ↑z 〉d 1 | ↑z 〉d 2 + | ↓z 〉d 1 | ↓z 〉d 2 ) of two spin- 12 particles d 1 and d 2. In
addition they hold spin- 12 particles A and B prepared in advance by a third party in an unknown state |ψ〉AB. (In the following it is assumed that Alice and Bob each complete their local operations and measurements during time ∆t ≪ L/c, where L = |xA − xB|.)
(a) Describe an explicit protocol which will allow Alice and Bob to perform an instanta- neous non-demolition verification that |ψ〉AB is a state of zero total spin.
(b) Write down the product eigenstates and eigenvalues of the operator
(σAz ⊗ IB^ + IA^ ⊗ σBz )mod 4 (1) and use the result obtained in (a) to show how to perform an instantaneous non- demolition measurement of (1).
(c) Hence suggest a procedure to perform an instantaneous non-demolition measurement of the Bell-operator, i.e. show how the four Bell-states |Φ±〉AB , |Ψ±〉AB can be distinguished with certainty without being disturbed. [You may assume that Alice and Bob share two maximally entangled states of the type |Φ+〉d 1 d 2 as a resource.]
(d) Consider an operator on the tensor product HA ⊗ HB with the following eigenstates
|ψ 1 〉AB = | ↑z 〉A ⊗ | ↑z 〉B |ψ 2 〉AB = | ↑z 〉A ⊗ | ↓z 〉B |ψ 3 〉AB = | ↓z 〉A ⊗ (cos θ| ↑z 〉B + sin θ| ↓z 〉B ) |ψ 4 〉AB = | ↓z 〉A ⊗ (sin θ| ↑z 〉B − cos θ| ↓z 〉B ),
where 0 6 θ 6 π/4. Show that the possibility of an instantaneous non-demolition measurement of (2) would contradict relativistic causality unless θ = 0. Is there a simple way to perform such a measurement in the latter case? Comment on your result.
Part III, Paper 66
(a) State EPR’s criterion for identifying an element of physical reality.
(b) Consider an experiment in which two space-like separated parties, Alice and Bob, perform measurements on their local 2-level systems, which interacted in the past. We assume that each party may choose between two measurement settings a, a′ and b, b′^ respectively. Each measurement yields an outcome ±1. The experiment is characterised by the expectation values E(a, b), E(a, b′), E(a′, b), and E(a′, b′) of the product of the outcomes of local measurements. Explain how the criterion described in (a) leads to the assumption of a local hidden variables (LHV) model for such an experiment and explain its meaning. Under this assumption derive the CHSH-Bell inequality
| E(a, b) − E(a, b′) | + | E(a′, b) + E(a′, b′) | 6 2. (1)
(c) Now assume that the systems held by Alice and Bob are spin- 12 particles in the joint state ρ(x)AB =
− x
where |Φ±〉 are corresponding Bell-states and 0 6 x 6 12. Show that the correlations in ρ(0)AB can be simulated by a trivial LHV model. Hence, without performing calculations, deduce that ρ(0)AB will not violate CHSH inequality. Explain.
(d) Find the range of values of x, for which the correlations in ρ(x)AB do not admit LHV description. Comment on your result. [You may use the formula for the expectation value of an operator Aˆ for a mixed state ρ: 〈 Aˆ〉 = T r( Aρˆ ).]
Part III, Paper 66
In the standard von Neumann model of a projective measurement of a 2-level quantum system the initial product state of the system and the measurement device |Ψin〉SD =
1 i=0 ci|i〉S
⊗ | 0 〉D is transformed into |Ψf 〉SD =
i=0 ci|i〉S^ |i〉D^ , where 〈 i | j 〉 = δij. This transformation is governed by the Hamiltonian of interaction
Hint = g(t) π 4
IS^ ⊗ ID^ − σSz ⊗ ID
IS^ ⊗ ID^ − IS^ ⊗ σDx
where g(t) is a time-dependent coupling coefficient, which is non-zero only between times t 1 and t 2 , such that (^) ∫ t 2
t 1
g(t)dt = 1.
Here | 0 〉, | 1 〉 are eigenstates of σz in the corresponding Hilbert spaces.
(a) Show, via explicit calculation, that the action of Hint results in a CNOT-transformation
U (^) SDCN OT = | 0 〉〈 0 |S ⊗ ID^ + | 1 〉〈 1 |S ⊗ σDx (2)
being implemented on S and D, with S as a control and D as a target (assume ℏ = 1).
(b) By changing the basis to the complementary basis |±〉 ≡ √^12 (| 0 〉 ± | 1 〉) for both S and D show that (2) is equivalent to the CNOT with the roles of the control and the target swapped
U (^) SDCN OT = |+〉〈+|D ⊗ IS^ + |−〉〈−|D ⊗ σ˜Sx , where ˜σi correspond to Pauli matrices defined in the new basis.
(c) Use the result obtained in (b) to rewrite the interaction Hamiltonian (1) in the complementary bases.
(d) Hence deduce that (1) also governs the measurement, in which D acts as a measured system and S as a measurement device. Explain. What will the observable of D measured in this way be?
Part III, Paper 66
