# 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. – Tarek Oct 3 '15 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. – Prof. Legolasov Oct 3 '15 at 15:15
• I didn't mean the quantum state, I meant the state of the operators, or in other words, the operator matrices. – Tarek Oct 3 '15 at 15:51
• @Tarek then it definitely was not what OP asked. The question was about measurements. – Prof. Legolasov Oct 4 '15 at 15:58