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I know that we can measure a state in standard basis (Z) where the post measurement state is either $|0\rangle$ or $|1\rangle$. But can you do a 'half-Z' basis measurement? I mean, you only measure with $|0\rangle \langle0|$ or $|1\rangle \langle1|$. So that when you measure $|+\rangle$, you would get $|0\rangle$ or $|1\rangle$ with equal probability. But if you measure $|0\rangle$ in $|1\rangle \langle1|$, then you would not get a detection with certainty. Is there any implementation of this sort of a 'half-basis'?

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    $\begingroup$ I’m confused as to how what you’re asking differs from normal Z basis measurement. Could you please clarify? $\endgroup$ – DaftWullie Mar 18 '20 at 4:51
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What you're talking about is called a "projection". The projection onto basis state $|\psi\rangle$ is given by:

$P_{\psi} = |\psi\rangle\langle\psi|$.

Any measurement operator $M$, such as $Z$ can be written as the sum over projectors, weighted by their respective eigenvalues, such that

$M = \sum_i \lambda_i P_i$

where $\lambda_i$ is the real-valued eigenvalue associated with outcome $i$, and $\{i\}$ is a set of orthogonal states such that $P_iP_j = \delta_{ij}$. A measurement basis is said to be complete if it has the property

$\sum_i P_i = I$

and hence $\{i\}$ forms an orthonormal basis. What you are describing is an "incomplete" measurement basis. Because such an operator does not span a complete basis of the space, one property of such a measurement operator is that the sum of probabilities associated with its measurement is less than one (when taken over all input states).

So, that's some maths, but what does this mean physically? If you imagine a photon in one of two waveguides, such that the $|0\rangle$ state is associated with it being in one waveguide and $|1\rangle$ in the other, then the measurement of the $Z$ operator is as simple as putting a detector at the end of each waveguide. In the case where your measurement operator is only $|0\rangle\langle 0|$, this equates to a system where you only have a a single detector at the end of the first waveguide.

What happens then? Well, what you would expect, if you put in a $|0\rangle$ state, you detect it with probability 1. On the other hand, you never detect the $|1\rangle$ state. Similarly, in the case of $|+\rangle$, you detect a photon in the first waveguide half of the time.

I guess the question that leaves us is how to think of this. Personally, I wouldn't think of it like a half-measurement, but just that you have measurement apparatus that has spans a limited basis and so the information it gathers over other basis states is simply incomplete. In the real world, this is always the case anyway. In the aforementioned photon experiment, it is often the case that no photon arrives at either detector due to a scattering event, and at the end of the day all this is telling us is that the measurement apparatus simply doesn't span the set of all possible states the system can take.

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