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A portion of the mathematical tripos part iii exam paper from the university of cambridge, held on june 11, 2001. It includes three questions related to quantum information physics, covering topics such as the elitzur-vaidman quantum bomb testing scheme, bell singlet states, and entangled werner states. The questions require the application of quantum mechanics principles and the derivation of certain results.
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Monday 11 June 2001 9 to 11
Attempt THREE questions. The questions are of equal weight.
1 The Elitzur-Vaidman quantum scheme for bomb testing can be represented ab- stractly as follows. A photon qubit is prepared in the state | 0 >p, and a unitary rotation U is then applied to it. The rotated state then undergoes a brief interaction with a bomb, which may be live or dud, by the following rules:
| 0 〉p + dud → | 0 〉p + dud , | 1 〉p + dud → | 1 〉p + dud , | 0 〉p + live → | 0 〉p + live , | 1 〉p + live → explosion.
If there is no explosion, a further unitary rotation U ′^ is applied to the photon qubit after this interaction. It is then measured by a projective measurement in the basis | 0 〉p, | 1 〉p. If the bomb did not explode, and if the measurement does not determine with certainty whether the bomb is dud or live, the test is repeated.
Show that, by choosing U, U ′^ appropriately and repeating the test appropriately, the probability of identifying a live bomb without detonating it can be increased to arbitrarily close to 1/2 within this scheme.
Show that any entangled bipartite pure state of two qubits can be transformed, with non-zero probability, to a Bell singlet state
by local measurements and unitary operations at A and/or B.
State the CHSH inequality. (You need not prove it.) Hence show that the correlations of outcomes of a general sequence of measurements on any entangled bipartite pure state cannot be reproduced by a local hidden variable theory.
3 A and B are separated and share an entangled Werner state
WF = F |Ψ−〉〈Ψ−| + 13 (1 − F )
where F > 12 and the Bell states are defined by
Show that it is impossible for them to create a more entangled Werner state WF ′ , with F ′^ > F , by local operations and classical communication alone.
Paper 61
