# Measure or Amplify the least probable state

Qubits can be represented by a state vector $$\psi \in \mathbb{C}^{2^N}$$, where $$N$$ is the number of qubits.

The higher $$| \psi_i |$$ is, for any index $$i$$, the more likely it is to be measured. But is there any way to reverse this? The closer the value $$| \psi_i |$$ is to 0, the more probable the measurement of $$i$$ is?

In a rephrased way, is there a method to amplify probability amplitudes closer to 0? Such as Grover’s algorithm, which amplifies negative probability amplitudes. But instead for values closer to 0.

• did you mean to write "the close the value $|\psi_i|$ is to 0, the least probable measuring $i$ is"? I mean, you simply have $|\psi_i|=\sqrt{p_i}$ where $p_i$ is the probability of measuring the $i$-th outcome, so yes the two quantities are indeed tightly related in a rather straightforward way
– glS
Oct 19, 2022 at 23:43
• I'm pretty sure they're asking for a process to amplify low-probability states and minimize high-probability states. Maybe for example some process that takes $|\psi\rangle = \sum_{j} \sqrt{p_j} |j\rangle$ to some state like $|\psi'\rangle \propto \sum_{j} (1-\sqrt{p_j}) |j\rangle$ where you are most likely to measure the originally least likely states. Oct 19, 2022 at 23:50
• @ChrisE Yes, I am exactly asking that. That is also a process I considered, but I don’t see how it can be imported as a unitary (or even a matrix). Oct 20, 2022 at 15:30
• @glS I don’t understand how this is a textbook question— I think you maybe misunderstood. I will add Chris E’s rephrased version as well. Oct 20, 2022 at 15:32
• well, yes, the edit changes quite a bit what you were asking. So by "method" you're asking for a quantum circuit doing that, or something else? Also, are you asking about a coherent operation doing that, or a way to post-process data to do it? What are the circumstances? Surely you can't do it in general: imagine you have $\psi_0=1$ and $\psi_i=0$ for $i>0$. Then with such procedure you'd end up with a highly non-normalised state.
– glS
Oct 20, 2022 at 16:05