The minimal statistical interpretation is neither minimal nor statistical

[martinbn]

Well, you can make up your own mind whether the "ensemble" interpretation is powerful enough to change the mathematics of quantum theory.

IIRC, when @bhobba and I discussed, although he is a fan of Ballentine while I am not, bhobba's own Ensemble interpretation includes a postulate equivalent to state reduction, which is that one can at the end of decoherence in a measurement take the reduced density operator to be ignorance interpretable.

Also, the argument against state reduction is incomprehensible if one takes an operational view of quantum theory. In an operational view, who cares what absurdities happen to the state, since it is not necessarily real and just a tool to calculate the probabilities of measurement outcomes.

That @vanhees71 cares about "causality" (about which he is confused) shows that he thinks the quantum state is real, which would also make the objection to state reduction sensible. Very few people would consider such an interpretation to be minimal.

Not sure what this has to do with the discussion here! All I said is that I disagree with Demystifier, because the minimal statistical interpretation doesn't need the reduction postulate (and it doesn't need to be agnostic to be minimal). And I wrote how I think it should be phrased in the language of the interpretation. I don't claim that the interpretation is problem free (it seems that none is), nor that one should subscribe to it, nor that I do, just what it says and that it doesn't say that it has a reduction postulate. If you think I am wrong somewhere comment on that. But I don't see that point of vague references to different discussions or opinions.

 

[martinbn]

Also, the argument against state reduction is incomprehensible if one takes an operational view of quantum theory. In an operational view, who cares what absurdities happen to the state, since it is not necessarily real and just a tool to calculate the probabilities of measurement outcomes.

Of course, if you take a different point of view from the statistical, the statistical will seem incomprehensible. That applies to any two different points of view.

 

[bhobba]

which is that one can at the end of decoherence in a measurement take the reduced density operator to be ignorance interpretable.

Just to be clear what the ignorance in my interpretation is. At the end of decoherence you have a mixed state. If it is a proper one things are more sensible as far as everyday experience goes (ie it actually is in one of the possible pure states of the mixed state - but we do not know which one - only the probability) or an improper one where the act of measuring the mixed state itself, somehow gives the result. Trouble is at least so far nobody has figured how to tell the difference or even if it is worth worrying about.

Thanks
Bill

 

[Demystifier]

Are you asking me?! The point was that in the statistical interpretation you cannot say that.

That was certainly not my point.

 

[martinbn]

That was certainly not my point.

That was my point! I started by saying that I didn't agree with you. What is your point, besides the one in the first post? And what do you find not ok with my argument?

 

[atyy]

Just to be clear what the ignorance in my interpretation is. At the end of decoherence you have a mixed state. If it is a proper one things are more sensible as far as everyday experience goes (ie it actually is in one of the possible pure states of the mixed state - but we do not know which one - only the probability) or an improper one where the act of measuring the mixed state itself, somehow gives the result. Trouble is at least so far nobody has figured how to tell the difference or even if it is worth worrying about.

Wouldn't it make a difference for a selective measurement, where there are subensembles with conditional states labelled by measurement outcomes?

Whereas for a non-selective measurement (where the measurement result is discarded), certainly there is no difference between purely unitary decoherence and measurement with state reduction with respect to the observables that can be measured.

 

[Demystifier]

What is your point, besides the one in the first post?

None.

 

[vanhees71]

It is as it is. In case you leave the instrumentalist's point of view, you start to make "ontological statements". "The ensemble interpretation has the advantage that...": One can beat about the bush, but at the end these "ontological statements" must be proved by means of experiments.
Is there anything in the formalism of quantum theory which points into the direction of the "ensemble interpretation"? Nothing! To my mind, physicists should not insist on thinking about quantum phenomena with classical ideas. As V. A. Fock warns:

“The deeper reason for the circumstance that the wave function cannot correspond to any statistical collective [aka ‘ensemble’, LJ] lies in the fact that the concept of the wave function belongs to the potentially possible (to experiments not yet performed), while the concept of the statistical collective belongs to the accomplished (to the results of experiments already carried out).”

(V. A. Fock, “ON THE INTERPRETATION OF QUANTUM MECHANICS”, Czech J Phys (1957) 7: 643)

I don't care whether my statements or ontological or epistemic or whatever. These are just philosophical categories which don't mean much for the fact that we are able to observe things in Nature by our senses and using real-world apparati in our labs.

If there is any interpretation consistent with experiments (there are no proofs in natural sciences only the agreement between a theory and all experiments done so far ;-)) it's the statistical interpretation: What QT claims to be "random" given a "preparation" has turned out to be "random" in experiment, and hitherto nobody has come up with a way to make what's "random" in QT "determined" in a new theory beyond QT. So we have to live with QT for the time being.

That Fock argues against any statistical interpretation is clear. The poor Russian physicists in soviet times had to make arguments against any implication of their theories being not in accordance with the (philosophical!) doctrine of dialectic materialism. It's my favorite example for the "value" of philosophy in the sciences. You can find an amusing (but at the time dead serious) philosophical discussion in Blokhintsev's book on QM.

 

[martinbn]

That was my point! I started by saying that I didn't agree with you. What is your point, besides the one in the first post? And what do you find not ok with my argument?

None.

So, you have not other point, and you find my argument ok. Did you change your mind? I guess not. Why not?

 

[Lord Jestocost]

Just to be clear what the ignorance in my interpretation is. At the end of decoherence you have a mixed state.

Decoherence cannot effect that a pure state evolves into a mixed state - the math is unambiguous. As Daniel Greenberger puts it - and that is - so to speak - "the state of the art":

"Decoherence merely scrambles up the phases so that it would be impossible, from a practical point of view, to unscramble them. But it does not solve the basic problem that a pure state does not evolve by the Schrödinger equation into a mixed one."

 

[Lord Jestocost]

Not sure i follow the argument...

One should not mix up statements about the post-measurement situation with statements about the pre-measurement situation. Maybe, the following might help. As Maximilian Schlosshauer puts it in “Decoherence, the measurement problem, and interpretations of quantum Mechanics”, Section B. 1. Superpositions and ensembles (https://arxiv.org/abs/quant-ph/0312059):

“Put differently, if an ensemble interpretation could be attached to a superposition, the latter would simply represent an ensemble of more fundamentally determined states, and based on the additional knowledge brought about by the results of measurements, we could simply choose a subensemble consisting of the definite pointer state obtained in the measurement. But then, since the time evolution has been strictly deterministic according to the Schrödinger equation, we could backtrack this subensemble in time and thus also specify the initial state more completely (“postselection”), and therefore this state necessarily could not be physically identical to the initially prepared state on the left-hand side of Eq. (2.1).“

 

[vanhees71]

Well, you can make up your own mind whether the "ensemble" interpretation is powerful enough to change the mathematics of quantum theory.

IIRC, when @bhobba and I discussed, although he is a fan of Ballentine while I am not, bhobba's own Ensemble interpretation includes a postulate equivalent to state reduction, which is that one can at the end of decoherence in a measurement take the reduced density operator to be ignorance interpretable.

Also, the argument against state reduction is incomprehensible if one takes an operational view of quantum theory. In an operational view, who cares what absurdities happen to the state, since it is not necessarily real and just a tool to calculate the probabilities of measurement outcomes.

That @vanhees71 cares about "causality" (about which he is confused) shows that he thinks the quantum state is real, which would also make the objection to state reduction sensible. Very few people would consider such an interpretation to be minimal.

The ensemble interpretation is just the math of quantum theory and no additional assumptions like collapse which is outside of the math of quantum theory, claiming that one needs another "mechanics" or "dynamics" for the interaction of a particle with a measurement device (or in some flavors any macroscopic object) than
what's described in QT. There's no hint that such a quantum-classical cut is needed though.

To take an argument FOR state reduction is incomprehensible if one takes and operational view of quantum theory. What quantum theory describes about the "operations" done in an experiment is precisely this: Some object is prepared in some specific way (represented by the statistical operator) and after some time a measurement of an observable of the system is done, giving some (usually random) value with a probability distribution predicted by the theory, and you can check this prediction on an ensemble of equally prepared system. There is no need to know, in which state the system after the measurement might be. Usually it's not even present anymore after the measurement in the previous form (e.g., a photon usually gets absorbed when detected).

I may be confused, but local relativistic QFTs are, by construction, causal. There cannot be a causal influence across space-like distances in Minkwoski space. That's a mathematical feature of the theory and directly contradicted by the state-reduction/collapse assumption.

 

[Fra]

One should not mix up statements about the post-measurement situation with statements about the pre-measurement situation. Maybe, the following might help. As Maximilian Schlosshauer puts it in “Decoherence, the measurement problem, and interpretations of quantum Mechanics”, Section B. 1. Superpositions and ensembles (https://arxiv.org/abs/quant-ph/0312059):

“Put differently, if an ensemble interpretation could be attached to a superposition, the latter would simply represent an ensemble of more fundamentally determined states, and based on the additional knowledge brought about by the results of measurements, we could simply choose a subensemble consisting of the definite pointer state obtained in the measurement. But then, since the time evolution has been strictly deterministic according to the Schrödinger equation, we could backtrack this subensemble in time and thus also specify the initial state more completely (“postselection”), and therefore this state necessarily could not be physically identical to the initially prepared state on the left-hand side of Eq. (2.1).“

View attachment 276874

If this boils down to Ballentine's books (which i don't own), perhaps I miss some subtle point here.

Talking about mixing, my point was to not mix up how to infer a LAW vs how to apply it for expectations in a single situation. Fock seems to have an issues that the LAW that is inferred at statistical level, can't be applied for predicting the "potentially possible" in a single experiment. I still don't get the point.

Ensemble of "experiments already carried out" ~> statistical law that implies future distribution from past distribution (short time scales). This part of science creation is not about predictions, its about extracting laws from experimental data (timeless in this case). It's at this level that we can claim to have deterministic evolution of generalized probability distribution (represented via wavefunctions).

Next, applying this to NEW single experiments, similar to the situation in where the LAW was inferred, the task is not to extract new laws (we have it already), the task is just to predict the future, from the present, in an individual experiment; given the law. The rational expectation here of an individual expertiment, is just to used the probabilistic rules for the corresponding ensemble, right? This does not necessarily imply that the wavefunction has any ontological meaning for the system in a single individual experiment, one must not forget that the LAW is corroborated at the statistical level only. The "states" to which deterministic evolution applies, is a statistical state (represented mathematically by a vector). How can it not be a statistical state? what's the logic?

(The ponderings about subensembles and post-selection means we are considering a DIFFERENT system and different measurement setup and thus different observer in the abstract, that does not solve anything as i can see. From my perspective, it makes not sense to "transport" rules of reasoning between two different observers as if we are in a euclidian "reasoning space". This is done all the time, and is disturbing, and only makes the fog bigger. I don't think this has much to do with Focks objection?)

/Fredrik

 

[atyy]

There is no need to know, in which state the system after the measurement might be. Usually it's not even present anymore after the measurement in the previous form (e.g., a photon usually gets absorbed when detected).

Well, if there is no state, there is no unitary evolution, so this contradicts your claim that there is only unitary evolution of the quantim state.

 

[bhobba]

Decoherence cannot effect that a pure state evolves into a mixed state - the math is unambiguous.

Yes even the Microwave Background Radiation will do it. So valid point - if it was a proper mixed state it will soon be scrambled into a new mixed state - but is the new mixed state itself proper? That is how do you tell the difference between a proper and improper mixed state? I have no idea how to do that. If Atty can elaborate on his proposal it would help. Of course if it is a proper mixed state something else may also be going on we do not know at this epoch.

Thanks
Bill

 

[martinbn]

One should not mix up statements about the post-measurement situation with statements about the pre-measurement situation. Maybe, the following might help. As Maximilian Schlosshauer puts it in “Decoherence, the measurement problem, and interpretations of quantum Mechanics”, Section B. 1. Superpositions and ensembles (https://arxiv.org/abs/quant-ph/0312059):

“Put differently, if an ensemble interpretation could be attached to a superposition, the latter would simply represent an ensemble of more fundamentally determined states, and based on the additional knowledge brought about by the results of measurements, we could simply choose a subensemble consisting of the definite pointer state obtained in the measurement. But then, since the time evolution has been strictly deterministic according to the Schrödinger equation, we could backtrack this subensemble in time and thus also specify the initial state more completely (“postselection”), and therefore this state necessarily could not be physically identical to the initially prepared state on the left-hand side of Eq. (2.1).“

View attachment 276874

Just to stress that this is not about the ensemble interpretation of quantum mechanichs. It is about an ensemble interpretation of superposition.

 

[martinbn]

Well, if there is no state, there is no unitary evolution, so this contradicts your claim that there is only unitary evolution of the quantim state.

No, there is no state of the system! It is either absorbed, or it is no longer the same system. But the state of a given system evolves only unitarily.

 

[bhobba]

What is the operational interpretation of probability in QM? Choose whatever interpretation you favor.

Formally if you read a math book on QM it is the Kolmogorov axioms eg:
https://www.amazon.com/dp/0387493859/?tag=pfamazon01-20

As far as the Ensemble interpretation goes operationally it is frequentist.

For Copenhagen it doesn't really specify, but it fits naturally with the Bayesian interpretation. And there is an interpretation called Bayesian which obviously uses the Bayesian view:
https://www.quantamagazine.org/quantum-bayesianism-explained-by-its-founder-20150604/

Thanks
Bill

 

[bhobba]

Well, if there is no state, there is no unitary evolution, so this contradicts your claim that there is only unitary evolution of the quantim state.

Well if you take Gleason literally the state is just an aid to calculation. It is taken as unitarty because then it is normalised when you calculate probabilities making things easier. An actual pure state is really an operator |u><u|. It is just usually mapped to |u> again for convenience - but the 'phase' is irrelevant.

Thanks
Bill

 

[Demystifier]

So, you have not other point, and you find my argument ok. Did you change your mind? I guess not. Why not?

If you say that I find your argument OK, who am I to question it?

 

[martinbn]

If you say that I find your argument OK, who am I to question it?

If you don't find it ok, can you please tell me what you think is not ok.

ps: This is a discussion thread started by you. I would have thought that you'd be interested in the topic. So why do I have to pull one word answers from you. You are not a spy and this is not an interogation. If you are not interested fine. But it is hard for me to take siriously short "witty" responses with lots of emojies.

 

[vanhees71]

Well, if there is no state, there is no unitary evolution, so this contradicts your claim that there is only unitary evolution of the quantim state.

We argue in circles again. Unitary time evolution applies to the dynamics of a CLOSED system. For open systems you get an effective description, which is often even a classical theory (Kadanoff-Baym equations, (quantum-)transport or Langevin equations for particles and/or quasiparticles, hydrodynamics, point-particle mechanics...).

In classical dynamics (point particles and/or fields) for open system also the fundamental dynamical laws, i.e., the Euler-Lagrange equations of the corresponding action principle, are substituted by corresponding effective descriptions. So this is nothing specific to quantum theory.

 

[atyy]

We argue in circles again. Unitary time evolution applies to the dynamics of a CLOSED system. For open systems you get an effective description, which is often even a classical theory (Kadanoff-Baym equations, (quantum-)transport or Langevin equations for particles and/or quasiparticles, hydrodynamics, point-particle mechanics...).

In classical dynamics (point particles and/or fields) for open system also the fundamental dynamical laws, i.e., the Euler-Lagrange equations of the corresponding action principle, are substituted by corresponding effective descriptions. So this is nothing specific to quantum theory.

But wouldn't you say that if you enlarge the system you would still get a closed system, and by partial tracing you can get the quantum state of the subsystem?

 

[Fra]

Is this what Demystifier means...?

"because the probability (conditional probability, to be more precise) suddenly changes when new information (about the measurement outcome) arrives."

But this conditional probability (the one that collapses), belongs not to the whole ensemble, but to the agent that is part in the interaction in one specific experiment. Is this what is corroborated? I woud say no.

"Therefore the state, which is nothing but the probability amplitude, suddenly changes as well, and this sudden change is called collapse (or state reduction)."

I this is your point, then i say the whole ensemble does not "collapse" just because of ONE experiment. To ask what happens after the measurement, means we have a new system. QM only describes what happens "in between" measurement, which is also when its "closed. We "open" the system for masurement, then close it, to open it later.

An ensemble is robust by construction, but an agent doing a randow walk will face "brownian like" collapses all the time, its part of the game. The only problem is that this interpretation where the probabiity is attached to one individual agent participating and updating his information, is sensible and plausible and IMO what a measurement theory SHOULD describe, but its when you think deeper about this I've realized that the mathematics of QM does not describe this. At best it describes a crippled picture with dominat agents sitting at infinit observing but not needing to face backreactions.

There are several ways to suggest how this situation can be improved but that involves radical speculation and explicit modification (or reconstruction) of QM. But until the, I think the statistical interpretation is the most accurate one. The problem for me is not the interpretation, but the theory itself when it comes to unification. Once the theory is reconstruted, obviously we need a "new interpretation" as well, and in my fantasies then the preferred one is the "inside agent" interpretation. But that interpretation fails for current formalism.

/Fredrik

 

[Morbert]

As an aside I think the minimal ensemble interpretation can be readily extended to successive measurements without having to worry about conceptualising new ensembles throughout. E.g. If we have a preparation $\rho$ and carry out a measurement of observables $\hat{A}=\sum a_i\Pi_{a_i}$ and $\hat{B}=\sum b_i\Pi_{b_i}$ at times $t_1$ and $t_2$ respectively, we can construct a compound observable $$\hat{C} = \sum_{i,j}a_ib_j\Pi_{a_i}(t_1)\Pi_{b_j}(t_2)$$The user can then compute probabilities and conditional probabilities like
\begin{eqnarray*}
p(a_i\land b_j) &=& \mathrm{tr}\{ \rho \Pi_{a_i}(t_1)\Pi_{b_j}(t_2) \}\\
p(b_j | a_i) &=& \frac{\mathrm{tr}\{ \rho \Pi_{a_i}(t_1)\Pi_{b_j}(t_2) \}}{ \mathrm{tr}\{ \rho \Pi_{a_i}(t_1) \} }\\
\end{eqnarray*}Without having to conceptualise any intermediate ensemble $\rho(t)$.

 

[martinbn]

As an aside I think the minimal ensemble interpretation can be readily extended to successive measurements without having to worry about conceptualising new ensembles throughout. E.g. If we have a preparation $\rho$ and carry out a measurement of observables $\hat{A}=\sum a_i\Pi_{a_i}$ and $\hat{B}=\sum b_i\Pi_{b_i}$ at times $t_1$ and $t_2$ respectively, we can construct a compound observable $$\hat{C} = \sum_{i,j}a_ib_j\Pi_{a_i}(t_1)\Pi_{b_j}(t_2)$$The user can then compute probabilities and conditional probabilities like
\begin{eqnarray*}
p(a_i\land b_j) &=& \mathrm{tr}\{ \rho \Pi_{a_i}(t_1)\Pi_{b_j}(t_2) \}\\
p(b_j | a_i) &=& \frac{\mathrm{tr}\{ \rho \Pi_{a_i}(t_1)\Pi_{b_j}(t_2) \}}{ \mathrm{tr}\{ \rho \Pi_{a_i}(t_1) \} }\\
\end{eqnarray*}Without having to conceptualise any intermediate ensemble $\rho(t)$.

What about the evolution? Say in the Schrodinger picture, how does $\rho(t)$ evolve over time?

 

[atyy]

As an aside I think the minimal ensemble interpretation can be readily extended to successive measurements without having to worry about conceptualising new ensembles throughout. E.g. If we have a preparation $\rho$ and carry out a measurement of observables $\hat{A}=\sum a_i\Pi_{a_i}$ and $\hat{B}=\sum b_i\Pi_{b_i}$ at times $t_1$ and $t_2$ respectively, we can construct a compound observable $$\hat{C} = \sum_{i,j}a_ib_j\Pi_{a_i}(t_1)\Pi_{b_j}(t_2)$$The user can then compute probabilities and conditional probabilities like
\begin{eqnarray*}
p(a_i\land b_j) &=& \mathrm{tr}\{ \rho \Pi_{a_i}(t_1)\Pi_{b_j}(t_2) \}\\
p(b_j | a_i) &=& \frac{\mathrm{tr}\{ \rho \Pi_{a_i}(t_1)\Pi_{b_j}(t_2) \}}{ \mathrm{tr}\{ \rho \Pi_{a_i}(t_1) \} }\\
\end{eqnarray*}Without having to conceptualise any intermediate ensemble $\rho(t)$.

"... can be seen as a consequence of the wave packet reduction postulate of quantum mechanics, since we obtained it in this way. But it isalso possible to take it as a starting point, as a postulate initself: it then provides the probability of any sequence of measurements, in a perfectly unambiguous way, without resorting, either to the wave packet reduction, or even to the Schroedinger equation itself. The latter is actually contained inthe Heisenberg evolution of projection operators, but it remains true that a direct calculation of the evolution of|Ψ> is not really necessary. As for the wave packet reduction, it is also contained in a way in the trace operation of (37), but even less explicitly."
https://arxiv.org/abs/quant-ph/0209123

 

[Morbert]

What about the evolution? Say in the Schrodinger picture, how does $\rho(t)$ evolve over time?

I don't think a minimalist would be concerned with a question like that. The only thing they are interested in is starting with a preparation, dynamics, and possible outcomes $\mathcal{T} = (\rho, U, \{o_i\})$ and computing the probabilities $\mathcal{P} = \{p(o_i)\}$. Any computational technique used to get from $\mathcal{T}$ to $\mathcal{P}$ is just that: a computational technique.

 

[atyy]

I don't think a minimalist would be concerned with a question like that. The only thing they are interested in is starting with a preparation, dynamics, and possible outcomes $\mathcal{T} = (\rho, U, \{o_i\})$ and computing the probabilities $\mathcal{P} = \{p(o_i)\}$. Any computational technique used to get from $\mathcal{T}$ to $\mathcal{P}$ is just that: a computational technique.

Which is why I find @vanhees71's objection incomprehensible from the minimalist point of view. Why should one be bothered by state reduction? It's just a computational technique. That he is bothered by it suggests to me he is not a minimalist.

 

[martinbn]

I don't think a minimalist would be concerned with a question like that. The only thing they are interested in is starting with a preparation, dynamics, and possible outcomes $\mathcal{T} = (\rho, U, \{o_i\})$ and computing the probabilities $\mathcal{P} = \{p(o_i)\}$. Any computational technique used to get from $\mathcal{T}$ to $\mathcal{P}$ is just that: a computational technique.

But what do I put in the formulas for the probabilities? Either the state or the observables are subject to differential equations, the initial conditions are not enough.

 

[Lord Jestocost]

Just to stress that this is not about the ensemble interpretation of quantum mechanichs.

With all due respect, there is no need for beating about the bush. As D. Home and M.A.B. Whitaker put it in “Ensemble interpretations of quantum mechanics. A modern perspective” (Physics Reports, Volume 210, Issue 4, January 1992, Pages 223-317):

“Ensemble interpretations of quantum theory contend that the wave function describes an ensemble of identically prepared systems. They are thus in contrast to ‘orthodox’ or ‘Copenhagen’ interpretations, in which the wave function provides as complete a description as is possible of an individual system.”
[Italics in original, LJ]

 

[martinbn]

With all due respect, there is no need for beating about the bush. As D. Home and M.A.B. Whitaker put it in “Ensemble interpretations of quantum mechanics. A modern perspective” (Physics Reports, Volume 210, Issue 4, January 1992, Pages 223-317):

“Ensemble interpretations of quantum theory contend that the wave function describes an ensemble of identically prepared systems. They are thus in contrast to ‘orthodox’ or ‘Copenhagen’ interpretations, in which the wave function provides as complete a description as is possible of an individual system.”
[Italics in original, LJ]

Yes, but your other quote was about something different. It was about whether one can interpret a superposition as lack of information just like one does in classical mechanics. That is incomaptible with QM and is not what the statistical interpretation says.

 

[AlexCaledin]

- but that "individual system" (unless it's the whole universe) might be merely a thinking tool! So that quote seems presenting a complementarity rather than contradiction...

 

[Demystifier]

If you don't find it ok, can you please tell me what you think is not ok.

ps: This is a discussion thread started by you. I would have thought that you'd be interested in the topic. So why do I have to pull one word answers from you. You are not a spy and this is not an interogation. If you are not interested fine. But it is hard for me to take siriously short "witty" responses with lots of emojies.

Sorry, but you are never satisfied with my answers, so it's hard to find a motivation for a serious answer.

 

[Morbert]

Yes, but your other quote was about something different. It was about whether one can interpret a superposition as lack of information just like one does in classical mechanics. That is incomaptible with QM and is not what the statistical interpretation says.

So e.g. say we want to carry out a measurement of $\hat{A}$ and $\hat{B}$ at times $t_1$ and $t_2$ respectively. This gives us $(\rho,U,\{\sigma_{ij}\})$ where $\rho$ is the preparation, $U$ is the dynamics, and $\{\sigma_{ij}\}$ are possible outcomes where $\sigma_{ij} = \Pi_{a_i}(t_1) \Pi_{b_j}(t_2) = U^\dagger(t_1)\Pi_{a_i}U(t_1)U^\dagger(t_2)\Pi_{b_j}U(t_2)$, then the probability $p(\sigma_{ij}) = \mathrm{tr}\{\rho\sigma_{ij}\}$