Uncollapsing the wavefunction by undoing quantum measurements

Abstract

We review and expand on recent advances in theory and experiments concerning the problem of wavefunction uncollapse: given an unknown state that has been disturbed by a generalised measurement, restore the state to its initial configuration. We describe how this is probabilistically possible with a subsequent measurement that involves erasing the information extracted about the state in the first measurement. The general theory of abstract measurements is discussed, focusing on quantum information aspects of the problem, in addition to investigating a variety of specific physical situations and explicit measurement strategies. Several systems are considered in detail: the quantum double dot charge qubit measured by a quantum point contact (with and without Hamiltonian dynamics), the superconducting phase qubit monitored by a SQUID detector, and an arbitrary number of entangled charge qubits. Furthermore, uncollapse strategies for the quantum dot electron spin qubit, and the optical polarisation qubit are also reviewed. For each of these systems the physics of the continuous measurement process, the strategy required to ideally uncollapse the wavefunction, as well as the statistical features associated with the measurement are discussed. We also summarise the recent experimental realisation of two of these systems, the phase qubit and the polarisation qubit.

Keywords:

• wavefunction uncollapse
• measurement reversal
• quantum Bayesianism

Acknowledgements

The work was supported by NSA and IARPA under ARO Grant W911NF-08-1-0336, the National Science Foundation under Grant No. DMR-0844899, and the University of Rochester.

Notes

1. One of the possible procedures is the following. If the post-measured state is not pure, we can apply more measurements to make it pure, and then probabilistically apply a unitary operation from an easily calculable set, which creates a mixture of pure states identical to the initial state.

2. An exception is when the initial state belongs to a very limited set, in which the measurement result corresponds to only one state.

3. The classical Bayes rule relates the posterior probability of a hypothesis k (after observing an event m) to the prior probability of this hypothesis and the conditional probability of the event m. The hypotheses k should form a complete and mutually exclusive set.

4. Strictly speaking, the QPT language may not be applicable to the uncollapsing experiment, because it requires linearity of the quantum operation, while after selection of particular realisations and state normalisation the quantum operation is not necessarily linear. However, the linearity is preserved when the selection probability does not depend on the initial state, which is exactly the case for a perfect uncollapsing – see Equation (32). It is also possible to show that even in the imperfect non-linear case the uncollapsing fidelity defined via the ‘naive’ QPT language practically coincides with the rigorous definition via the average state fidelity 23.

5. We use normalisation of the shot noise, in which , where is the QPC transparency.

6. In this example, Plato could change his uncollapsing strategy to simply apply a tailored unitary to shift the disturbed state |ψ_1,m ⟩ back to its original state |ψ₁⟩. Under this modified strategy, the success probability will be , which may or may not exceed , depending on the strength of the measurement. However, in this case, Plato himself will not know whether the strategy succeeded or not, and therefore cannot claim to have an uncollapse procedure.

7. It is easy to generalise Equations (44) –(47) to include the detector non-ideality and cross-correlation between the output and back-action noises (as in 13). Non-ideality leads to the extra dephasing term −γ _d σ₁₂ in Equation (47) for , while the cross-correlation brings the term iK[I(t) − I ₀]σ₁₂ into the same equation. The formulation of Equations (44) –(47) corresponds to the approach developed in 39.