# What is the description of measurement in the Heisenberg picture?

In all the books I've read this picture is presented only briefly, by essentially saying that in the HP the whole time dependence is assigned to the operators (representing observables), whereas the state vectors do not depend on time, and remain unchanged no matter what. Then, the derivation of Heisenberg eq of motion is presented. This is pretty much all that can be found in books. I would dare say that this presentation is utterly incomplete and somehow misleading. The very important problem of how the measurement is described in HP is left out. Upon measurement the state vector DOES CHANGE even in the HP, so that immediately after a measurement of an observable $A_{H}(t)$, at time $t$ (the subscript H on the operator $A(t)$ standing for the Heisenberg picture), the state vector becomes $|a,t\rangle$, no matter what the state vector was before the measurement, where $|a,t\rangle$ is the eigenvector of $A_{H}(t)$ corresponding to the measured eigenvalue, say, $a$. This new state vector $|a,t\rangle$ will remain unchanged in time and will represent the system in the HP, for times later than $t$, until a new measurement is performed on the system. Please let me know if my thoughts are correct so far. My next question is: what happens to the operator $A_{H}(t)$, after the measurement performed at time t? Does it change, and how?

To elaborate: The textbooks are silent about the description of measurement in the Heisenberg picture. I wonder if, upon measurement, the state vector does collapse in the Heisenberg picture, similarly to what happens in the Schrodinger picture; namely, if a system was prepared at time $t_0$ in a state $|\psi\rangle$, then at a later time $t > t_0$ the system is described by the same time-independent state vector $|\psi\rangle$, but if a measurement of an observable $A_{H}(t_1)$ is being performed on the system at a time $t_1 > t$, then, immediately after the measurement, the state vector of the system (in the Heisenberg picture) changes to $|a, t_1\rangle$, where $|a, t_1\rangle$ is the eigenvector of the operator $A_{H}(t_1)$ corresponding to an observed eigenvalue a (assumed non-degenerate), i.e., $$A_{H}(t_1) |a, t_1\rangle = a |a, t_1\rangle.$$

In the Heisenberg picture, this new (time-independent) state vector $|a, t_1\rangle$ continues to describe the state of the system at times $t > t_1$, until a new measurement is performed on the system.

The statement from the textbooks that the state vector does not change in time in the Heisenberg picture applies only to isolated systems, upon which no measurement is performed, but once the system is "measured", its state vector does change even in the Heisenberg picture.

I don't know if the measurement affects the time evolution of the operators representing the observables. Are they affected, and how?

My gut feeling is that the operators are not "abruptly" affected (i.e., "collapsed") by measurement, but continue to evolve continuously, according to the Heisenberg equation of motion. That is, for a time $t$, with $t_0 < t < t_1$, one has to solve the Heisenberg eq. $$\imath \hbar \frac{dA_{H}(t)}{dt} = \left[A_{H}(t), H\right]$$ with the initial condition $A_{H}(t_0)$ for $t = t_0$, and then, at precisely the measurement time $t_1$, the operator is $A_{H}(t_1)$, and finally, after the measurement, for $t > t_1$, one has to solve again the Heisenberg eq. $$\imath \hbar \frac{dA_{H}(t)}{dt} = \left[A_{H}(t), H\right]$$ with the initial condition $A_{H}(t_1)$ for $t = t_1$.

I would very much appreciate it if you could let me know as to whether my understanding of measurement in the Heisenberg picture, as sketched above, is the correct one, and if you could clarify as to what happens to the dynamical evolution of operators (representing observables) when measurement is involved.

This is clearly a philosophical question, so I will allow myself to express my personal opinion (and not just mine, to be honest).

Heisenberg picture is actually much more convenient for the purposes of describing measurements than Schrodinger picture. This is because it provides a nice separation between measurements and the unitary quantum evolution.

Consider, for example, a relativistic system (a field theory of some sort). How would you describe measurements in a Lorentz-invariant way? You would probably write down the Schrodinger equation, which will be (though not manifestly) Lorentz-invariant. But try describing the collapse in a Lorentz-invariant way and you will fail. The naive logic (aka measuring the position of the particle) is simply not compatible with Special Relativity.

The less naive point of view here is that we not only don't know what wavefunction collapse is and how it behaves, but we also are skeptical about it having any physical meaning. Measurements very well might be subjective (the so-called psi-epistemic point of view, e.g. Quantum Bayesianism). The question of which interprentation of wavefunction collapse is the correct one has a long and very sad history and is best left untouched since it is known to provoke long and meaningless discussions. We simply don't know how measurements are done, how they relate to space and time (do they happen in time or not?).

Heisenberg picture provides a great insight on how we could keep these strange and probably philosophical (rather than physical) questions separated from the actually important and falsifiable stuff like unitary evolution. Instead of wave functions, operators evolve in time. This is ingenious! Operators aren't affected by measurements/collapse, they are just there and their eigenvalues correspond to observable values of physical quantities.

States (or density matrices if you wish) on the other hand are given once and for all. They correspond (in QBism, for example) to the complete collection of information that we possess and therefore to our expectations of the world around us.

This allows measurements to be treated in any way (even to consider them to exist beyond space and time, meaning that I could easily talk about measurements in relativistic theories). I am no longer required to think of them happening in between of the stages of unitary evolution. They might not even "happen" since this word requires a background time axis to acquire its meaning. They are just there, that's it.

So in conclusion, my point is that Heisenberg picture allows a nice separation between unitary evolution and collapse to be made, which helps us a lot to distinguish between objective reality (quantum operators) and measurements (the nature and objectivity of which is a subject of a never-ending debate).

• How will Heisenberg describe the state of his evolving operators immediately after an experimentalist performs a measurement on a quantum system? This was the main question by the OP, which I don't see it addressed in your answer. Commented Oct 3, 2015 at 8:00
• @Tarek I am not sure I follow. States don't evolve with time, operators do. But the same Born rule applies to measurements. Commented Oct 3, 2015 at 15:15
• I didn't mean the quantum state, I meant the state of the operators, or in other words, the operator matrices. Commented Oct 3, 2015 at 15:51
• @Tarek then it definitely was not what OP asked. The question was about measurements. Commented Oct 4, 2015 at 15:58
• Measurement is NOT Lorentz invariant because it drags the coordinates of the measurement device into the process. There is no need to try to make the theory Lorentz invariant because the physics of it isn't. Commented Apr 5, 2023 at 17:02

Good question and not merely philosophical or interpretational.

In short: the procedure in the OP gives the correct predictions (at least for unitary evolution) but it's not the only way to describe measurements in the Heisenberg picture. In a "fully Heisenberg" picture, states remain unchanged even after measurement, while observables get projected, taking care of composing projections and evolutions "backward in time".

Let's unpack this. First, let's recall that, besides any metaphysical consideration, the role of "collapse" is to provide correct predictions for sequences of measurements in time. Let us consider a scenario where we first evolve the state for some time, make a measurement, evolve again, and then make another measurement. It is more convenient to use density matrices, rather than vector states, and some notation can help. Let's define the (Schrödinger picture) evolution of a density matrix $$\rho$$ as $$\rho \mapsto \mathcal{E}(\rho) = U\rho U^{\dagger},$$ where $$U$$ is the unitary operator evolving the state for the desired amount of time. Let us then consider a projective measurement $$\{\Pi_a\}_a$$, where $$a$$ labels the result of the measurement (with $$\Pi_a = |\psi_a\rangle\langle \psi_a |$$ if the measurement is rank-1). This can be associated with an observable $$A=\sum_a \lambda_a \Pi_a$$ for some set of eigenvalues $$\{\lambda_a\}_a$$. Then we define the measurement map $$\rho\mapsto \mathcal{M}_a(\rho):= \Pi_a \rho \Pi_a.$$ Note that the state $$\mathcal{M}_a(\rho)$$ is not normalised. The "collapsed state" is given by $$\mathcal{M}_a(\rho)/P(a)$$, where $$P(a)= \mathrm{Tr} \mathcal{M}_a(\rho)$$ is the probability to get the projector $$\Pi_a$$ upon measuring the state $$\rho$$.

With all this, one can see that the joint probability to observe $$\Pi_a$$ at the first measrument and $$\Pi_b$$ at the second is \begin{align}P(a,b) &= \mathrm{Tr}\left(\Pi_b U_2 \Pi_a U_1 \rho U_1^{\dagger} \Pi_a U_2^{\dagger} \right)\\ &= \mathrm{Tr}\left[\Pi_b\circ\mathcal{E}_2\circ\mathcal{M}_a\circ\mathcal{E}_1 (\rho)\right], \end{align} where $$\mathcal{E}_{1,2}$$ denote evolution before and after the first measurement, respectively, and $$\circ$$ denotes function composition. The Heisenberg-picture version of this formula is simply obtained by cycling the operators inside the trace: \begin{align} P(a,b) &= \mathrm{Tr}\left(\rho U_1^{\dagger} \Pi_a U_2^{\dagger} \Pi_b U_2 \Pi_a U_1 \right)\\ &= \mathrm{Tr}\left[\rho\,\mathcal{E}^{\dagger}_1\circ\mathcal{M}^{\dagger}_a\circ\mathcal{E}^{\dagger}_2(\Pi_b)\right], \end{align} where the adjoints are defined as $$\mathcal{E}^{\dagger}(\rho) := U^{\dagger} \rho U$$ or, more generally, through the equation $$\mathrm{Tr}[\mathcal{E}^{\dagger}(A) B] = \mathrm{Tr}[A \mathcal{E}(B)]$$ for any pair of operators $$A$$, $$B$$.

As promised, the "fully Heisenberg" picture consists in taking the final measurement operator, $$\Pi_b$$, and evolving it "back in time", first with the second evolution map $$\mathcal{E}^{\dagger}_2$$, then with the measurement map $$\mathcal{M}^{\dagger}_a$$, and finally with the first evolution map $$\mathcal{E}^{\dagger}_1$$. The state $$\rho$$ remains untouched in this picture, even through the measurement.

Note that

• The same evolution applies to any final observable, not just a projector. In particular, since the whole expression is linear in $$\Pi_b$$, for $$A=\sum_b \lambda_b \Pi_b$$, we can define $$A_a(t) := \mathcal{E}^{\dagger}_1\circ\mathcal{M}^{\dagger}_a\circ\mathcal{E}^{\dagger}_2(A)$$. This can be interpreted as "the Heisenberg-picture observable $$A$$ at the final time $$t$$, when outcome $$a$$ is obtained in the intermediate measurement".
• The same expression works for an arbitrary evolution map $$\mathcal{E}$$ and measurement map $$\mathcal{M}$$, not just unitaries and projections.
• Even if this representation doesn't usually find its way in textbooks, it is certainly known in the literature. See for example this paper, where time evolution is further generalised to include an environment with memory.

Going back to the procedure described in the OP, it can be seen as a mixed Heisenberg-Schrödinger picture, where we treat unitary evolution in the Heisenberg picture but measurements in the Schrödinger picture. To see how this works (and check that it gives the right results) we can go back to the first expression for the joint probability and insert resolutions of the identity $$U_1U_1^{\dagger}=\mathbb{I}$$ after the first measurement: \begin{align} P(a,b) &= \mathrm{Tr}\left(\Pi_b U_2 U_1 U_1^{\dagger} \Pi_a U_1 \rho U_1^{\dagger} \Pi_a U_1 U_1^{\dagger} U_2^{\dagger} \right)\\ &= \mathrm{Tr}\left(\Pi_b U_2 U_1 \Pi_a(t_1) \rho \Pi^{\dagger}_a(t_1) U_1^{\dagger} U_2^{\dagger} \right), \end{align} where $$\Pi_a(t_1) := U_1^{\dagger} \Pi_a U_1$$ is the Heisenberg evolution of the projector $$\Pi_a$$. So, indeed, we can define the state collapsed by the Heisenberg-picture observable at time $$t_1$$, $$\rho_a(t_1) := \Pi_a(t_1) \rho \Pi^{\dagger}_a(t_1)/P(a).$$ By noting that the product $$U_2 U_1$$ is simply the total unitary time evolution, the conditional probability to find $$\Pi_b$$ at the second measurement, given that $$\Pi_a$$ was found at the first, can be written in the (partial) Heisenberg picture \begin{align} P(a|b) = P(a,b)/P(a) &= \mathrm{Tr}\left[\Pi_b(t) \rho_a(t_1) \right]. \end{align} As noted in the OP, the observable $$\Pi_b(t):=U_2 U_1 \Pi_bU_1^{\dagger} U_2^{\dagger}$$ is undisturbed by the measurement, as it evolves according to the unitary Heisenberg picture from the initial time (before the measurement) to the final time, in the same way as if no measurement was done. Note, however, that this description doesn't generalise well to arbitrary evolution maps $$\mathcal{E}$$, because in the derivation above reversibility played a crucial role when we plugged $$U_1U_1^{\dagger} = \mathbb{I}$$ into the expression.

We could also define other types of mixed Heisenberg-Schrödinger pictures, and also mix in other types of pictures, like the interaction picture. All that matters is that our expressions all agree in calculating probabilities and expectation values for sequences of measurements. The different pictures simply give different ways to represent the same physical statements, and can be more or less convenient depending on context and what one is trying to do.

• It doesn't matter how one does it in the theory, though, as these are all just sequences of linear operators. What is important is that the total energy, momentum and angular momentum in the system (and the ensemble) change every time we do a measurement. The problem with the way we teach quantum mechanics is that most students don't seem to be aware of that in either the Schroedinger or the Heisenberg picture. Commented Apr 5, 2023 at 17:10

In asking your question, you actually put your finger on the very incompleteness of Quantum Theory that so bothered Einstein. Analogous to a "code smell", you can think of it as a "theory smell", except that Einstein was never able to pin down the source of the odor. Your observation does: the Heisenberg Picture doesn't actually a measurement theory! You won't find anything in the literature. It's a bona fide gap.

Why has the gap persisted, and yet remained largely unnoticed? The answer boils down to the fact that we have a folklore result that states "the Heisenberg and Schroedinger Pictures are equivalent" - so that we don't really need to pay any attention to what's happening in the Heisenberg Picture, because "it's already accounted for in the Schroedinger Picture".

You can see the circular reasoning, there, being committed by that tacit line of argumentation, that leads to the gap.

Let's take a look, more closely, at the equivalence result - with a particular mind to answering the question of what's actually included in the equivalence, and what's not.

Von Neumann laid out a formulation for Quantum Theory that posed two axioms: the Evolution Postulate and the Projection Postulate. The former is where you find the dynamics of quantum theory, while the latter encapsulates the measurement theory. Invariably, this axiomatic formulation is framed in the Schroedinger Picture, given the above-mentioned bias. In it, a quantum system is described by a state that evolves in an "historical time", with this evolution given by the Schroedinger Equation.

The various quantities that describe a physical system are rendered, in this framework, as operators that act on the state, and are framed in timeless form. The time-dependence of the quantities represented by those operators is inherited from the time-dependence of the states they act on.

This way of laying out the foundation sets up the Projection Postulate, which states that the measurements, of a quantity, which are done on a system are represented by the action of the operator on the system, in such a way that the outcome yields an eigenstate of the system and cranks out the corresponding eigenvalue. That is, each measurement is associated with a "projection".

A standard representation for this projection is the Born Rule.

How and why this comes about (and the status of the Born Rule, itself) is the central topic of measurement theory, and we won't concern ourselves with the details here, because it is tangential to the fact that there has to be something there, and that this something is not (and can never be) fully accounted for by the Evolution Postulate alone.

One of the replies you already received "who didn't answer the question" correctly noted how clear this discrepancy becomes when you render it in the Heisenberg Picture. It's actually more clear than they're leading on to: namely that there's an outright gap in the Heisenberg Picture, that shows that we're missing something important and that quantum theory is incomplete.

Much is made of the issue that the "historical time" of quantum theory is totally at odds with the view of time as a "block universe" or "all-there" time that relativity seems to advance. But this view is actually a red herring and the schism has been both misplaced and misidentified.

The "historical time" is not a feature of quantum theory, itself, but of the Schrodinger Picture; and the schism entailed by the discrepancy in how time is to be regarded is not a schism between quantum theory and relativity at all, but is actually a schism internal to quantum theory, itself - between the Heisenberg and Schroedinger pictures! For, the Heisenberg Picture actually treats time as "block time", and agrees with relativity on that account.

In the Heisenberg Picture, states are timeless. A state represents the entire history of a system. Instead, it's the operators that represent physical quantities that contain this time-dependence - but with a notable difference: the way in which they contain that dependence is sufficiently well in accord with how they also contain spatial dependence that all the coordinates can be treated on an equal footing. It's an "all there" time, like relativity.

The equivalence between the Schroedinger and Heisenberg Picture pertains to the Evolution Postulate only. The dynamics of a quantum system are represented, in the Heisenberg Picture, by the Heisenberg Equation, and the equivalence is between it and the Schroedinger Equation. However, in the Heisenberg Picture, the system does not "evolve" in time. Instead, the Heisenberg Equation is more properly thought of as describing the unfolding of the system in space-time, since all the coordinates are on equal footing.

The reason this folklore "equivalence" result is misleading and misapplied is that there is no equivalence with respect to the Projection Postulate, as there is no Born Rule, nor any measurement theory, in the Heisenberg Picture. The two pictures are not equivalent, because one of them is incomplete - the Heisenberg Picture.

It needs to be lifted to a higher form with additional infrastructure. Moreover, when that infrastructure is added, thereby raising the Heisenberg Picture to an expanded version sufficiently endowed to house the Born Rule, the resulting addition goes beyond what's present in the Schroedinger Picture, thereby showing the gap that it, too, has.

So, what more is that that we need? To answer this, let's take a closer look at the Projection Postulate.

The Projection Postulate is not just there to provide a connection to and grounding in the world, to allow empirical statements to be extracted. The real point of the postulate is to also enforce a dependency between projections. If projection $$a$$ comes after projection $$b$$ - particularly if the measurements associated with them are mutually non-commuting - then we need the outcome of $$b$$ first before we apply $$a$$. Projections feed into other projections. When such a dependency relation occurs, we will designate it as $$b → a$$.

If you try to formulate this in a timeless fashion, the closest equivalent that you'll get to what we want is the mathematics used in the Consistent Histories formulation. The mathematics used for its version of the Born Rule is essentially the same as what we want here, but the formulation itself takes a different route from what we want and need.

So, the Heisenberg Picture needs lifted to a higher form with more infrastructure that includes - at a bare minimum - the assumption that there exists a set of "projections". A quantum system is described not just by the variables making it up (which are represented by operators) and the dynamics describing its unfolding, but also a delineation of all the projections that the system is subject to.

Call this set $$C$$. In order to have something coherent, we may require that the projections in $$C$$ form a partial order under the relation $$b → a$$ - no dependency loops. No time travel for projections.

Note, however, this need not preclude the existence of time travel for the coordinate "block time" of the Heisenberg Picture!

Since $$C$$ is partially ordered, then there exists a large number of ways to partition it into two subsets $$A, B ⊆ C$$ (the "after" and "before" subsets), such that (1) $$C = A∪B$$, (2) $$A∩B = ∅$$, (3) for no $$a∈A, b∈B$$ is it the case that $$a→b$$.

Denote the set of all such partitions of $$C$$ as $$P(C)$$.

Each such partition $$(A,B) ∈ P(C)$$ divides projections of $$C$$ into those that "already happened" (the subset $$B$$) and those "which haven't yet happened" (the subset $$A$$) - and as such, encodes a concept we are already quite familiar with - a concept of Now or of The Present. Each partition $$\left(A,B\right)$$ is a Now. A semblance of "historical time" thus emerges, with the extra infrastructure in place.

However, there is a notable difference: these Now's do not form a linear progression (except in non-relativistic quantum theory). Instead, they are themselves partially ordered $$\left(A_0,B_0\right) → \left(A_1,B_1\right)$$ if and only if $$A_0 ⊇ A_1$$ and $$B_0 ⊆ B_1$$. The first Now contains more After's and fewer Before's than the second one. So, the $$→$$ relation on $$C$$ generates a similar relation $$→$$ over $$P(C)$$, itself.

Of particular importance is the relation of immediate succession, which we'll denote $$⇒$$. This occurs where only one projection, $$c∈C$$, lapses from Before to After, and we write $$\left(A_0,B_0\right) ⇒ \left(A_1,B_1\right)$$ if and only if $$A_0 = A_1∪\{c\}$$ and $$B_1 = B_0∪\{c\}$$.

With this additional infrastructure, we're also able to encode the concept of an Observer. Though the family $$P(C)$$ may be partially ordered, it also contains a large number of maximal linear suborders, each of which corresponds to a linear progression of Now's, each one being an immediate successor of the previous one. As this corresponds to how observers perceive the world, then we may think of each physical observer as not merely residing on a maximal linear chain, but to actually be part the chain that they reside on.

This goes beyond the usual notion of observer, which usually assumes that it has a start and finish time. An observer of finite duration can be identified with the entire set of maximal linear chains that contain the observer's Now's. So, we draw a distinction between eternal observers and finite observers, the latter being treated a bundle of the former.

So, now with this extra structure, we have enough to state a Heisenberg Picture version of the Born Rule. First, we have to expand the notion of a state. As you already noted, a measurement ought to "change" a Heisenberg state. We'll directly encode this by now treating a state in the expanded Heisenberg Picture as a map $$Ψ$$ from $$P(C)$$ to Heisenberg Picture states. Not all maps are admissible. A central requirement is that $$\left(A_0,B_0\right) ⇒ \left(A_1,B_1\right)$$ with the projection $$c$$ being the one that lapsed from Before to After, then $$Ψ_0 = Ψ(A_0,B_0)$$ should be connected to $$Ψ_1 = Ψ(A_1,B_1)$$ by an application of the Born Rule. The Born Rule may be applied by transforming the states $$Ψ_0$$ and $$Ψ_1$$ to the Schroedinger Picture for that one projection $$c$$ alone and applying the Schroedinger Picture version of the Born Rule to it. In accordance with the Born Rule, this associates a probability with the transition from $$Ψ_0$$ to $$Ψ_1$$, with $$Ψ_1$$ being associated with an eigenstate of the measurement identified by the projection $$c$$. A map $$Ψ$$ on $$P(C)$$ is admissible if every two states associated with two Now's in immediate succession satisfy a condition of this form.

So, now we've lifted the Heisenberg Picture to a form that allows a version of the Born Rule to be stated; and almost by magic, we obtain the encoding of a several other concepts that heretofore have been lurking behind the scenes as nebulous ideas, defying precise definition.

The most significant features of the addition are that (1) a distinction is drawn between coordinate time and historical time, in a sense it is not the three dimensions of space that "evolve" in time, but the entire spacetime continuum that does(!); (2) historical time is a partial order that contains many different linear progressions; (3) a concept of Now emerges, as does a concept of Observer.

All of this is necessary to merely be able to write down Born in Heisenberg, but is absent from the Schroedinger Picture - thereby showing that both it, and quantum theory, are incomplete, just as Einstein said. That incompleteness centers on the very question you asked.

I think this helps pin down what the incompleteness is, and what needs to be added.

• You are overthinking, like most theoretical physicists who have thought about the problem. "measurement" requires an irreversible energy transfer process, which breaks Lorentz invariance, energy, momentum and angular momentum conservation with regards to the free system. There is no "incompleteness" in the standard Copenhagen interpretation. Copenhagen spells it out very explicitly for us. Many theorists simply don't take what it does seriously enough. That the Born rule is so "disruptive" is a direct consequence of a physical measurement being disruptive. Commented Apr 5, 2023 at 17:05
• "there is no 'incompleteness'" (said in direct reply to what that incompleteness is). It is illogical to state a response in direct reply to its own refutation. "like most theoretical physicists who have thought out the problem" [of measurement theory in the Heisenberg Picture]. Almost none have. Those who have, have noted the gap and the need to resolve it; e.g. "Everett formulated his construction in the Schrödinger picture, [...] But the construction has never been satisfactorily expressed in the Heisenberg picture. This is potentially problematic for Everettian quantum theory." - Deutsch Commented Apr 9, 2023 at 19:10
• Kuypers S. and Deutsch D., 2021, ‘Everettian Relative States in the Heisenberg Picture’, Proceedings of the Royal Society A, 477(2246): 20200783. and preprint "Everettian relative states in the Heisenberg picture" arxiv.org/pdf/2008.02328.pdf Commented Apr 9, 2023 at 19:10
• You are welcome to read Everett's thesis. It's already wrong in the second sentence by misidentifying the state of the ensemble for the state of the individual system. I don't know if he is directly quoting von Neumann's book there or not, but if he is, then he simply took a physically false statement by a mathematician and ran with it without questioning it at all. Everettianism is exactly what I said: it's overthinking without thinking, at all, about what the Copenhagen interpretation tells us about physics. Commented Apr 9, 2023 at 22:21

In what follows I will assume that quantum theory is an accurate description of how the world works. The standard Copenhagen interpretation of quantum theory claims those equations are false but doesn't give a clear description of what theory should be used instead.

A measurement is an interaction that produces a record that can be copied of some physical quantity. Performing a perfect measurement of an observable $$\hat{A}$$ of a system $$S$$ with eigenstates $$|a\rangle_S$$ in the Schrodinger picture is represented by the evolution: $$|a\rangle_S|0\rangle_M\to|a\rangle_S|a\rangle_M$$ which is described by a unitary operator $$U=\sum_a|a\rangle_S|a\oplus b\rangle_{M}\langle a|_S\langle b|_M$$ where $$\oplus$$ is addition modulo the number of eigenvalues of $$\hat{A}$$.

If you measure a state in which the observable is unsharp you get: $$\sum_a\alpha_a|a\rangle_S|0\rangle_M\to\sum_a\alpha_a|a\rangle_S|a\rangle_M$$ and you don't see records of multiple outcomes so there must be something that prevents you from seeing more than one outcome. The process of copying information out of a system suppresses interference: a process called decoherence. As a result of decoherence the different possible measurement results can't interact so you can only see one of them:

https://arxiv.org/abs/1111.2189

https://arxiv.org/abs/0707.2832

So as a result of the measurement you see some particular measurement outcome and you change your relative state to reflect that outcome using the standard update rule.

Most measurements aren't perfect and can be described by somewhat messier models of decoherence but the basic picture doesn't change much:

https://arxiv.org/abs/0712.0149

Now, the evolution operator of the Schrodinger picture can be written down as a function of the static Schrodinger picture observables. The corresponding Heisenberg picture evolution can be represented by a unitary operator that is of the same form with the Schrodinger picture observables swapped with the corresponding Heisenberg picture observables, as described in this paper:

https://arxiv.org/abs/quant-ph/9906007

To describe the measurement you can then evolve the Heisenberg picture observables using these operators. Decoherence can be described in terms of the Heisenberg picture as in this paper:

https://arxiv.org/abs/0903.1802

There is also a treatment of how to update the relative state in the Heisenberg picture:

https://arxiv.org/abs/2008.02328