MATHEMATICAL TRIPOS Part III PAPER 57 QUANTUM ...

3
4
Alice and Bob (located at x A and x B respectively) share a maximally entangled
state |Φ + 〉 d1 d 2
= 1 √
2
(| ↑ z 〉 d1 | ↑ z 〉 d2 + | ↓ z 〉 d1 | ↓ z 〉 d2 ) of two spin- 1 2 particles d 1 and d 2 . In
addition, they hold spin- 1 2
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 = |x A −x B |.)
(a) Describe an explicit protocol which allows Alice and Bob to perform an instantaneous
non-demolition verification that |ψ〉 AB is a state of zero z-component of the spin.
(b) Write down the product eigenstates and eigenvalues of the operator
(σ A z ⊗I B +I A ⊗σ B z )mod 4 (1)
and use the result obtained in (a) to show how to perform an instantaneous nondemolition
measurement of (1).
(c) An operator on the tensor product H A ⊗H B has the following eigenstates
|Φ + θ 〉 = cosθ| ↑ z↑ z 〉 AB +sinθ| ↓ z ↓ z 〉 AB
|Φ − θ 〉 = sinθ| ↑ z↑ z 〉 AB −cosθ| ↓ z ↓ z 〉 AB
|Ψ + θ 〉 = cosθ| ↑ z↓ z 〉 AB +sinθ| ↓ z ↑ z 〉 AB
|Ψ − θ 〉 = sinθ| ↑ z↓ z 〉 AB −cosθ| ↓ z ↑ z 〉 AB ,
(2)
where 0 θ π/4. Consider a hypothetical instantaneous non-demolition measurement
of this operator.
(i) Show that the possibility of such operation would contradict relativistic causality
unless θ = 0,π/4.
(ii) Suggest and describe methods for realising such measurement in cases when
θ = 0 and θ = π/4. [For θ = π/4 you may use the results obtained in parts (a)
and (b). Assume that Alice and Bob share two maximally entangled states of
the type |Φ + 〉 d1 d 2
as a resource.]
Part III, Paper 57