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.
 A: 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).
A: 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 fratricidal 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.
