Section "2.2.3 Quantum measurement" in Nielsen&Chuang uses very general measurement axioms:

Postulate 3: 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 $m$ refers 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 $$p(m) = ⟨ψ|M_m^\dagger M_m|ψ⟩ ,\qquad(2.92)$$ and the state of the system after the measurement is ...

In fact, I am not aware of any introductory quantum mechanics textbook using comparably general axioms. So I wondered why they did this, and how this might be related to the fact that it is a book on quantum computers:

Indeed, interaction is central. But what would happen if it were an interaction between a quantum sensor of a quantum computer instead of an interaction with a classical measurement device? The quantum computer might do some processing of the sensor input on the quantum level, and in the end, could output a combination of classical measurement results and special quantum output states. ...

Since here the output of the quantum computer was a combination of classical and quantum information, it makes sense to assume that the input too can be a combination of classical and quantum information. The difference is that quantum information cannot be cloned, while classical information can be copied at will and even be permanently recorded.

Quantum Computation and Quantum Information by Michael Nielsen and Isaac Chuang present axioms for quantum mechanics (and especially measurements) which fit well to the possible outputs of such a combination of quantum sensors with a quantum computer described above. ...

I really liked the idea of a quantum computer directly reading out a quantum sensor without forcing a classical measurement first. But then I started to wonder whether it is really that simple, even on a conceptual level. A quantum computer is still a "digital" device, in a certain sense (also with respect to its time evolution). On the other hand, the raw quantum output of a quantum sensor probably still lives in continuous time, and also might still be "analog". Discretizing the "analog" part should be no problem on a conceptual level, but I struggle to find the right abstraction level for the conversion from continuous time (because I want to avoid forcing a classical measurement). And I also wonder whether there might be even more "non-digital" quantum properties that must be taken into account for conversion before the raw quantum output can be fed into a quantum computer.

Edit A preprint from Dec 1 uses the imagery of this question already in its abstract:

An experimental setup that transduces data from a physical system to a stable quantum memory, and processes that data using a quantum computer, could have significant advantages over conventional experiments in which the physical system is measured and the outcomes are processed using a classical computer.

(The most important difference to the imagery in this question is the stable quantum memory and how it is used, which is central to that preprint and its predecessors.) The preprint itself doesn't answer my question, but some of its references look promising, especially the survey paper "Perspectives on quantum transduction". It reviews quantum transduction between microwave and optical photons. This is the sort of thing I struggle with "for the conversion from continuous time", because the (continuous) time modulation of the phase is so different between microwave and optical frequencies. (In my imagery, a classical computer uses a low pass filter in its analog to digital converter, but I am neither sure whether there is a purely quantum analog to a low pass filter, nor whether it would even help.)

  • $\begingroup$ Is your main question why the textbook starts from an axiomatic approach? This is actually a trend in more modern books on quantum theory, even introductory ones (but the latter often only speak of projective measurements) $\endgroup$ Oct 28, 2021 at 13:30
  • $\begingroup$ No, my main question is given by the title, and elaborated in the text below the quotes. The measurement axioms from Nielsen&Chuang are more general than POVM. Believe me, this is rare. $\endgroup$ Oct 28, 2021 at 13:45
  • $\begingroup$ Ok gotcha. The axioms are the same as defining Kraus operators, or quantum channels/maps/operations, as N&C do in 8.2.4, and they're even on wikipedia $\endgroup$ Oct 28, 2021 at 21:39
  • $\begingroup$ That wikipedia link (from 2020) is nice, especially for context like quantum instrument: "It combines the concepts of measurement and quantum operation." That is basically the intuition I "guessed" for those measurement axioms. But my "quantum computer" image suggests "too exact" (digital) control compared to the "quantum operation" concept. And my question asks whether this can be fixed, by having some intermediate quantum abstraction level for the raw output of a quantum sensor. $\endgroup$ Oct 29, 2021 at 10:01
  • $\begingroup$ that measurement formalism isn't really that much more general than POVMs. It essentially amounts to POVMs where you consider the possibility of post-measurement states. $\endgroup$
    – glS
    Oct 29, 2021 at 10:28


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