As part of my studies and research, I have learned to work with three different measurement formalism which I define to avoid any ambiguity with the nomenclature:

  1. General measurements, which are described by a collection of self-adjoint operators $\{M_m\}_m$ satisfying $\sum_m M_m^\dagger M_m= I$.
  2. POVMs, which are described by a collection of positive semi-definite operators $\{E_m\}_m$ satisfying $\sum_m E_m = I$.
  3. Projective measurements, which are described by a collection of positive semi-definite operators $\{P_m\}_m$ satisfying $\sum_m P_m= I$ and $P_m P_{m'}= \delta_{m,m'}P_m$.

As I am from a theoretical computer science background, I have but a limited understanding of the physical implementation of quantum measurements. I know that all three formalism are equivalent by using ancilla states and unitary evolution, however this could be hard to realise on a physical implementation of a quantum system. My question therefore is:

Is there a measurement formalism that is (in general) easier to realise across all known physical implementation of quantum states?

My intuition leads me to believe that projective measurements are the simplest to implement as they are they are a special case of the other two (when no ancilla is available), but this intuition might be flawed. As mentionned in Nielsen and Chuang, projective measurements are non-destructive, but there are measurement devices that destroy the state in the process of measuring it.

A maybe easier question, but one to which the answer would still satisfy me would be: what kind of measurement operator (i.e. formalism 1, 2 or 3) are being realised by devices that measure qubit systems.


1 Answer 1


Could you elaborate on what you meant by "measurement formalism"? In particular, do you mean the type of measurement, or the mathematical representation thereof? I should also note that projective measurements (3) are not distinct from what you called "general measurements" (1).

There is no measurement type or representation that is best for all measurements across all physical systems. In fact, weak and projective measurements generally aren't interchangeable: While implemented very very similarly, they achieve very different effects.

However, it's perhaps relevant to note that the majority of measurements of various types all culminate in the same small set of projective or destructive measurements, perhaps of some secondary system (an ancilla, detector, auxiliary EM modes, etc.).

For example, fluorescence measurements of the state of atomic and ionic qubits, NV centers, and other systems all involve shining light to induce spontaneous emission (via fluorescence), and then use standard photodetection to detect that light. Photodetection (described by Glauber back in the day) is a destructive measurement of photon number, intensity, and any properties related to occupation number = energy.

As another example, dispersive measurement of superconducting qubits (and maybe also NV centers) culminates in homodyne detection of the outcoming EM modes. Homodyne detection is used for basically all interferometric measurements of light as well, such as evaluating quadatures (which generalize position and momentum). A straightforward generalization of homodyne detection is heterodyne detection, which uses another frequency of light. Some heterodyne measurements are necessarily nonprojective (e.g., weak). Most of these interferometric techniques require integrating signals over time, and thus are not particularly fast, though they may be very effective (especially when measuring the light that leaks out of cavities).

I would also say that all measurements use ancillas in some sense. Whether it's the detector itself or some auxiliary modes that couple to the system and then the detector, you need extra degrees of freedom to make any kind of measurement. In principle, you could measure $Z$ on an atomic qubit using a CNOT gate that flips the detector from 0 to 1 if the qubit is in the state 1, and does nothing otherwise. This is the "minimal representation" of the measurement (see below), but in practice, fluorescence measurement actually requires introducing an EM mode and a bunch of electrons in a detector --- i.e., two extra infinite-dimensional Hilbert spaces! The statement that is most generally true is that measurements involve entangling a system with other degrees of freedom, culminating in encoding the state of the system in a detector, which itself is classically readable and stable to decoherence. In practice, this tends to require a lot of extra degrees of freedom.

Your final question isn't answerable as far as I know, because these are not mutually exclusive. All projective measurements (3) are captured by the Kraus representation (1). The representation in terms of Kraus operators captures a subset of POVMs (2). So you could argue that POVMs are most general. And that's true, except that there are still other representations (see below).

Anyway, if you mean to ask which type of measurement (e.g., projective, destructive, nondemolition, weak, generalized, POVM, etc.) is easiest to implement, then the answer depends entirely on the physical system and the quantity being measured. So there isn't a general answer. However, all of these measurement types are achieved using similar techniques: A unitary interaction entangles something with the system, and eventually something is entangled to a classical system (the detector), whose state indicates the outcome. The only one that can do different things is POVM. POVMs and continuous measurements have the potential to take longer, like the interferometry measurements I mentioned above. But this depends on the system and observable being measured, so I don't see that there's anything useful to say here.

Instead, I'll assume that you're referring to measurement representations, where your (1) is equivalent to Kraus's first representation via "Kraus operators." This can be generalized to POVMs (2). There are POVMs (2) that cannot be captured by Kraus operators (1), and those should be expressed as POVMs. So POVMs are most general. Projective measurements can always be captured by Kraus operators (1) as far as I know.

However, there also exists a unitary representation of measurements on a dilated (enlarged) Hilbert space; this results from Kraus's second representation theorem. If the Kraus representation is known, the unitary version is easy to recover. But POVMs should also admit a unitary representation, because the derivation only assumes that the quantum operation or channel is completely positive, which is axiomatically guaranteed. Then Stinespring's dilation theorem and Choi's theorem get you pretty much there.

Crucially, these dilated unitaries correspond exactly to the unitary time evolution of the system and apparatus during the measurement process. The dilated Hilbert space is simply the product of the physical Hilbert space and that of the measurement apparatus. This holds for every projective and destructive measurement I know of, after binning distinct states of the apparatus corresponding to the same measurement outcome and tracing out auxiliary degrees of freedom that don't encode the outcome. This process recovers the "minimal Stinespring representation", which is unique up to the initial state of the apparatus. In practice, the actual physical unitary realizes a nonminimal representation.

I've also summarized this measurement formalism in my answers here, here, and here. This dilated "Stinespring" unitary is also equivalent to the unitary corresponding to evolution under von Neumann's "pointer Hamiltonian," described in these notes. This representation is easy to extend to weak, generalized, and other measurements. I haven't done it explicitly yet, but I'm confident it applies to POVMs, since it holds for all valid quantum operations.

Importantly, the unitary Stinespring representation gives the most information about how the measurement needs to be implemented in experiment. This is true for any type of measurement. It is also very useful in describing quantum protocols with outcome-dependent operations, and has been used to prove Lieb-Robinson-type bounds for arbitrary quantum protocols involving measurements and feedback. So it is analytically powerful, directly connected to experiment, and conceptually transparent. So if you're asking which "representation" of measurements is best, I'd say this one.

But again, which representation is relevant depends on the particulars of the experiment. For trapped ions, e.g., fluorescence measurements can be very close to projective. This also holds in some ultra-cold atomic gases, I believe. In cavity QED systems coupled to atoms, people use the von Neumann version fairly directly to describe the continuous measurement of photons that escape the cavity. This describes destructive measurements of photon number, homodyne and heterodyne detection, etc.


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