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I have heard various talks at my institution from experimentalists (who all happened to be working on superconducting qubits) that the textbook idea of true "Projective" measurement is not what happens in real-life experiments. Each time I asked them to elaborate, and they say that "weak" measurements are what happen in reality.

I assume that by "projective" measurements they mean a measurement on a quantum state like the following:

$$P\vert\psi\rangle=P(a\vert\uparrow\rangle+ b\vert\downarrow\rangle)=\vert\uparrow\rangle \,\mathrm{or}\, \vert\downarrow\rangle$$

In other words, a measurement which fully collapses the qubit.

However, if I take the experimentalist's statement that real measurements are more like strong "weak"-measurements, then I run into Busch's theorem, which says roughly that you only get as much information as how strongly you measure. In other words, I can't get around not doing a full projective measurement, I need to do so to get the state information

So, I have two main questions:

  1. Why is it thought that projective measurements cannot be performed experimentally? What happens instead?

  2. What is the appropriate framework to think about experimental measurement in quantum computing systems that is actually realistic? Both a qualitative and quantitative picture would be appreciated.

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    $\begingroup$ To clarify the scope of the question: you're using superconducting qubits just to give some background, but your question is general, right? (As opposed to the more particular question 'Are true projective measurements possible experimentally using superconducting qubits?'). $\endgroup$ – agaitaarino Apr 30 '18 at 5:18
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    $\begingroup$ Good point, yes I referred to superconducting qubits but I am interested in the general question. Although, I've only heard this point of view from those who study superconducting qubits, but that may be my limited experience. $\endgroup$ – user157879 Apr 30 '18 at 5:50
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Let's step back from QC for a moment and think about a textbook example: the projector onto position, $|x\rangle$. This projective measurement is obviously unphysical, as the eigenstates of $|x\rangle$ are themselves unphysical due to the uncertainty principle. The real measurement of position, then, is one with some uncertainty. One can treat this either as a weak measurement of position, or as a projective measurement onto a non-orthonormal basis (a strong POVM), where the various basis elements have some support on multiple values of $x$: say pixels on a detector.

Going back into QC, most systems' measurements are pretty close to projective, or are 'strong' measurements at the least. In some systems, like ion traps, the readout can be thought of as a series of weak measurements that collectively form a strong one. A photon counter, on the other hand, is very close to a projective measurement with some odd projectors due to finite efficiency--no click corresponds to a projector onto $|0\rangle + (1-e)^n|n\rangle$, for instance.

On the other hand, that projector doesn't leave behind the state listed above, because the apparatus also absorbs the photon.

To sum up, thinking of things as POVMs (Positive operator-valued measures) is probably the most-right intuition, where you can think of the outcomes of the POVM mostly as non-orthonormal projectors. Non-projective POVMs also exist, but are less common in practice in systems I've thought about.

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    $\begingroup$ Thanks for the answer! I do have some concerns though. While the eigenstate of the position operator is unphysical for very fundamental reasons (special relativity, QFT etc.), the states of the harmonic oscillator are not unphysical. So I don't totally follow the logic here. Is it accurate to say that measurements in current implementations have too large uncertainties to be seen as projective? $\endgroup$ – user157879 Apr 30 '18 at 17:05
  • $\begingroup$ Also, could you go into a bit more detail about POVMs and how that formalism works? That's a concept I'm not familiar with. Thanks again! $\endgroup$ – user157879 Apr 30 '18 at 17:07
  • $\begingroup$ Yes--and measurements of harmonic-oscillator-like things tend to be more like textbook projective measurements than measurements of continuous variables. Photon number, for instance, is a harmonic oscillator almost precisely, and you can think of a perfect number-counting detector as pretty close to a projective measurement. Similarly, measuring the state of an electron's energy level, if done strongly, is very close to a projective measurement. It does take time to get signal, and so can be done 'weakly' as well, though not particularly usefully. $\endgroup$ – D. H. Smith Apr 30 '18 at 17:12
  • $\begingroup$ POVMs are to density matrices as projective measurements are to kets, roughly speaking. As long as 1. All input states output some measurement outcome and 2. some probability conservation requirements hold, it turns out that you don't need orthogonal projectors to make your measurement work. The simplest example is a 4-outcome qubit measurement: we choose randomly between measuring {$|0\rangle,|1\rangle$} and {$\0±1\rangle$}, and then measure in one of those bases. This whole operation can be treated as either a conditional gate and a projective meassurement, or as a 4-outcome POVM. $\endgroup$ – D. H. Smith Apr 30 '18 at 17:17
  • $\begingroup$ Could you incorporate the example you mentioned in your answer with a bit more detail? I'll accept your answer afterwards, thanks for the help! $\endgroup$ – user157879 Apr 30 '18 at 22:02
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An assumption in general measurements: The measuring device itself has no degrees of freedom and it does not couple with the qudit in any form of interaction, which is not true.

1) A projective measurement is ideal and non-realistic because it is always assumed that there is no extension of this Projector to a bigger Hilbert space or more degrees of freedom than the Qudit degrees of freedom. But actually what happens experimentally is the fact that, to measure on a qubit we always have to assign a classical operation called a "Pointer" that is a link between your classical outcome by the measurement and the quantum measurement. By doing this the system is always exposed to a non-unitary and open environment where the measurement becomes non-deal and the information is leaked in outer degrees of freedom when the system coupled with the measuring device. This in principle itself is a nature's inherent property that forbids an ideal Quantum Measurement.

2) To go about this, as you pointed out, the true realistic method is a weak measurement method. To minimize the coupling with the environement and be close to a true quantum measurment.

However, there are certain cases which are special, certain states called "Pointer states" allow true ideal measurement w.r.t particular Measurement operators (Because they retain their quantum properties like Coherence, entanglement, etc) in the smaller Hilbert space and do not couple with higher degrees of freedom of the measuring device.

Some literature about this which I read in detail is from this article by W.H. Zurek: https://arxiv.org/abs/quant-ph/0105127

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