In accordance with the no-deleting theorem (https://en.wikipedia.org/wiki/No-deleting_theorem), there is no unitary operator that can take a qubit in an arbitrary state a|0> + b|1> and transform it into (e.g.) the |0> state (without knowing the amplitudes a and b to start with), yet this is a prerequisite for any quantum gate computation. How then does one do this in practice? I assume that it must be via some thermalization process where a qubit is allowed to drop to its lowest energy state via interaction with its environment. However, this is exactly the kind of environmental interaction that one needs to avoid in a computational process. No doubt it can be done (this is trivial in classical computing) but how is this achieved in the cryogenic conditions required for protecting qubits and with what fidelity?
Note that a projective measurement on the state a|0> + b|1> will only yield the |0> state with a probability |a|2, which does not solve the problem of reliably setting this state or even with a known probability (since one does not know the amplitude a to start with).
Hello,
You can look into the concept of psuedo-pure states and pulse sequences to create them.... these states act as approximations to pure states in qubit dynamics studies.
Stam Nicolis you cannot transform an arbitrary state a|0> + b|1> to |0> with a magnetic field since you would need to know the coefficients a and b beforehand in order to choose the correct field orientation, strength and time of application. One cannot know this unless the initial state has already been prepared in a known state, which just begs the question (since any state a|0> + b|1> can be taken to be the |0> state in some basis). Do you see the problem?
Debopam Ghosh Thanks for the suggestion - I shall take a look into this.
Of course it’s possible, since it’s possible to define a projector operator and act with it. The phase is global and can’t affect anything.
The projector projects upon a known direction and that direction is what matters.
It doesn’t matter what state the qubit was in before the projection; what matters is the state it will be found after the projection. And that's determined (up to a phase that can't be observed anyway) by acting on a qubit, in whatever state, by a device that acts as a polarization filter. That's why the initial state is irrelevant-the polarization filter selects the final state, which will be the initial state for whatever experiment one is interested in.
There isn't any such theorem as a ``no-deletion'' theorem; there's a ``no-cloning'' theorem. It's not possible to clone a qubit by a unitary transformation. So it's not possible to store ``perfectly'' the initial condition.
It is, of course, possible to transform any state a|0>+b|1> into any other state on the Bloch sphere by a unitary transformation, so the statement is wrong. It suffices to act on the qubit by Pauli matrices.
However, if the initial state isn't known, therefore the unitary transformation isn't known, it is possible to use a projection operator onto whatever state one wishes.
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