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More specific, short form of my question:

POVMs assign expectation values of positive operators (summing to identity) to probabilities of events, but not a particular measurement scheme, leading to no well-defined notion of canonical post-measurement state. Projective measurement assigns expectation values of particular projectors to probabilities of particular outcomes, however they give a particular canonical post-measurement state. This is often stated [1] as a key difference between the formalisms but is it truly the case that projective measurement forces us towards taking a special post measurement state without ambiguity?

[1] See second page, first column of this paper: https://arxiv.org/pdf/2011.11364.pdf

"For projective measurements, there is a canonical description of the post-measurement state corresponding to a given outcome (i) in terms of the corresponding projection operator, [...] There is no such canonical association for POVMs."


Long form:

In the context of generalised measurements and POVMS, one of the key distinguishing properties of the POVM formalism from that of Von Neumann measurement is that it does not force one into a particular realisation of the physical measurement process - and thus fails to force a particular 'canonical' definition for the post-measurement state as different measurement processes realising the same POVM can yield different post-measurement states.

This has lead me to consider turning this scenario on it's head and asking the same question of the projective formalism: Do ALL realisations of the measurement defined by some set of orthogonal projectors $\{P_i\}$ yield the same final state?: Given an initial state $\rho$ to be measured, the post-measurement state that follows an observation of outcome $i$ corresponding to the projection operator $P_i$ is canonically taken to be $\rho' = P_i\rho P_i/\text{Tr}[P_i\rho P_i].$ And if not, why do we canonically use this particular state as our 'specially' chosen post-measurement state?

Apriori, I note that since each $P_i$ is Hermitian and idempotent, we have the trivial decomposition of each projection operator $P_i$ into the Kraus form $P_i=P_i^{\dagger}P_i$ - that is, the RHS shows that the set consisting of just a single $P_i$ is a Kraus operator for each $P_i$ itself. However, this decomposition is never unique as for any unitary $U$, the set consisting of the single operator $\{M_i=UP_i\}$ is an alternative decomposition for $P_i$ i.e. $P_i = M_i^{\dagger}M_i$ so using this set of operators $\{M_i\}$ gives a different realisation of the same POVM defined by the projection operators $\{P_i\}$. Furthermore, in general, all Kraus decompositions must be unitarily equivalent to one another in this way so this, I believe, is the only form the Kraus decomposition for a projection operator can actually take(?).

However, following the usual prescription for defining the state update given a particular realisation of a measurement process, the realisation with $M_i=UP_i$ would give rise to the post-measurement state $UP_i\rho P_i U^{\dagger}/\text{Tr}[...]$ which is not generally equivalent to the canonical post-measurement state given by $M_i=P_i$ so it seems that even projective measurement doesn't force us into having a unique sensible choice of post measurement state. Is it therefore impossible to claim that the non-existence of a canonical post-measurement state is a contrasting feature of the Von Neumann Measurement and POVM measurement formalisms?

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    $\begingroup$ You should focus on one question, and streamline the question a bit -- it is really not easy to tell what you are really after. (Also, "Kraus decomposition" seems non-standard, and more akin to channels.) $\endgroup$ Jan 19 at 17:58
  • $\begingroup$ @NorbertSchuch Noted. POVMs assign expectation vals of +ive operators (summing to identity) to probabilities of events, but not a particular measurement scheme, leading to no well-defined notion of canonical post-measurement state. Projective measurement assigns expectation values of projectors to outcomes, however they give a particular canonical post-measurement state. This is often stated as a key difference between the formalisms but is it truly the case that projective measurement forces us towards taking a special post measurement state without ambiguity? $\endgroup$ Jan 19 at 19:13
  • $\begingroup$ (1) No weird abbreviations, please ("+ive"?! Why?). (2) Please edit the question instead/in addition of commenting. (3) This kind of distinction between POVM and projective is kind of arbitrary. Projective is really just a special case of POVM. Everything else is just a definition of how you use words. $\endgroup$ Jan 19 at 19:45
  • $\begingroup$ " This is often stated as a key difference " -- reference needed. $\endgroup$ Jan 19 at 20:04
  • $\begingroup$ @NorbertSchuch There were character limits to work around and I'm relatively new to using the site. I have edited the question. $\endgroup$ Jan 19 at 20:04

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I think it depends on what you mean by projective measurement. The canonical way that projective measurement is discussed does uniquely fix the post-measurement state (so long as the projectors considered are rank one). But there are generalised measurements that give rise to the same statistics as a projective measurement would, but completely different post-measurement states.

For example, consider a set of orthonormal states $\{|n\rangle\}$ for $n=0,1,2,\ldots$ and the corresponding projectors $P_n = |n\rangle\langle n|$, such that $\sum_nP_n=\hat{1}$. It is natural to associate these projectors to a non-degenerate observable $N = \sum_n n P_n$. I would say that a projective measurement of $N$ (or a projective measurement in the basis $\{|n\rangle\}$) is one that yields the outcome $n$ with probability $$p_n = {\rm tr}(P_n \rho),$$ (the Born rule) such that the conditional post-measurement state is the pure state $$\rho'_{n} = P_n.$$ This final equation is an assumption known as the Lüders rule and it is conventionally (albeit perhaps not universally) taught as a part of the process of projective measurement.

Another measurement yielding the same statistics is specified by an instrument with Kraus operators $K_n=|0\rangle \langle n|$, satisfying $\sum_n K_n^\dagger K_n = \hat{1}$. This measurement yields outcome $n$ with probability $${\rm tr}(K_n\rho K_n^\dagger) = {\rm tr}(P_n\rho) \equiv p_n, $$ but leaves the system in the conditional post-measurement state $$\rho'_n = |0\rangle \langle 0|,$$ irrespective of the measurement outcome. Thus, the measurement statistics are the same. Loosely speaking, both measurements give the same information on "which of the states $|n\rangle$ the system was in". But the post-measurement state is completely different.

Of course, we could interpret $|n\rangle$ as an $n$-photon Fock state, so that the observable $N$ is the photon number. The second measurement is then nothing but (an idealised model of) number-resolved photodetection, where the photons are absorbed by the detector. This is the only way one can actually measure the photon number in most experimental setups, because non-destructive photodetection is extremely challenging. Nevertheless, I would consider it bizarre to call this measurement "projective": it does not project the system onto the eigenstate pertaining to the observed measurement outcome.

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  • $\begingroup$ I think your latter example is very interesting. So is it fair to say that the conventional definition of "projective measurement" should not only enforce that the POVM elements $P_i$ are projectors, but also the Kraus operators $K_i$ are also projection operators? In that case, we are forced towards the specific decomposition into Kraus operators $K_i=P_i$? $\endgroup$ Jan 19 at 19:52
  • $\begingroup$ That is what I would call a projective measurement, yes. But as Norbert Schuch's answer and comments demonstrate, this is really a matter of terminology/convention. I think there exists a fairly large constituency of quantum physicists who would associate projective measurement with the Lüders postulate. However, strictly speaking, the basic concern of measurement theory is how to associate outcomes and probabilities with a given quantum state & observable/POVM. The post-measurement state is actually a question of dynamics and it is only relevant in a scenario with sequential measurements. $\endgroup$ Jan 21 at 15:03
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Projective measurements are just a special case of POVMs. Thus, they have precisely the same ambiguity POVMs have (assuming that all you fix is the probability $p_i(\rho)$ for different outcomes $i$, and you are asking for different ways to realize such a process with some post-measurement state). There is nothing special about projective measurements here.

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  • $\begingroup$ I see. This logic tracks with me. But then, why is ambiguity of post measurement state sometimes spoken of as a key difference between POVMs and the Von Neumann measurement scheme? $\endgroup$ Jan 19 at 19:23
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    $\begingroup$ @Theoreticalhelp Don't ask me. Ask those who claim that difference. -- In any case, it is not clear why "typical" measurement should have a post-measurement state. It is certainly not true for photons which hit a detector, which is certainly a very canonical measurement, nor for anything which hits a screen. It is really more a theoretical construct, and somewhat arbitrary. $\endgroup$ Jan 19 at 19:45

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