Reverse Quantum Computing: How to unmeasure a qunit

Question

After taking some measure, how can a qunit be "unmeasured"? Is unmeasurement (ie reverse quantum computing) possible?

Answer 1

I am not really sure about what you mean by "unmeasuring" a qubit, but if you mean to recover the qubit that was measured by manipulating the post-measurement state then I am afraid that the answer is no. When a quantum state is measured, the supoerposition state of such is collpased to one of the possible outcomes of the measurement, and so the qubit is lost.

The third postulate of quantum mechanics explains measurments in the quantum world, and such postulate says the following:

Quantum measurements are described by a collection $\{M_m\}$ of measurement operators. These are operators acting on the state space of the system being measured. The index $m$ referes to the measurement outcomes that may occur in the experiment. If the state of the quantum system is $|\psi\rangle$ immediately before the measurement, then the probability that result $m$ occurs is given by \begin{equation} p(m)=\langle\psi|M_m^\dagger M_m|\psi\rangle, \end{equation} and the state of the sytem after the measurement is \begin{equation} \frac{M_m|\psi\rangle}{\sqrt{\langle\psi|M_m^\dagger M_m|\psi\rangle}}. \end{equation}

So the post-measurement state collapses into another state defined by the postulate 3, and the previous quantum state is lost irreversibly. See also this wikipedia entry for wave function collapse, where it explains the collapse of quantum states after measurement.

Consequently, if the same measurment wants to be done, the quantum state must be prepared again before the measurement and so the xperiment can be repeated.

Answer 2

You can compute by measuring - see cluster-based quantum computation - but the whole thing that makes measurement different in quantum mechanics is that it destroys the superposition. It can't be undone. Once you measure, the qudit isn't in a state $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle + ... +\gamma|n\rangle$ but in a state $|\psi\rangle = |0\rangle$ or $|\psi\rangle = |1\rangle$ or what have you based upon probability. When you measure the qubit again soon after, it stays as either $|0\rangle$ or $|1\rangle$. The superposition is gone. We can't get it back (except by doing the same operations that lead our qubit to that point, in which case it'll be very similar) because we can't clone a qubit, so we can't figure out what $\alpha$ and $\beta$ are.

Tl;dr: No.

Answer 3

Unitary operation is revesible, but measurement is a projection operation, which is not reveaible. Think about matrix inverse, projection matrix has lower rank and does not have inverse