Graduate Fundamental Theorem of Quantum Measurements

[Danny Boy]

The Fundamental Theorem of Quantum Measurements (see page 25 of these PDF notes) is given as follows:

Every set of operators $\{A_n \}_n$ where $n=1,...,N$ that satisfies $\sum_{n}A_{n}A^{\dagger}_{n} = I$, describes a possible measurement on a quantum system, where the measurement has $n$ possible outcomes labeled by $n$. If $\rho$ is the state of the system before the measurement, $\tilde{\rho}_n$ is the state of the system upon obtaining measurement result $n$, and $p_n$ is the probability of obtaining result $n$, then $$\tilde{\rho}_{n} = \frac{A_n \rho A_{n}^{\dagger}}{p_n}~~~~\text{where}~~~~p_n = \text{Tr}[A_{n}^{\dagger}A_n \rho].$$We refer to the operators $\{ A_n \}_n$ as the measurement operators for the measurement.

Query: Consider measurement operators $\{A_c \}_c$ where now the measurement outcomes $c$ is considered a continuous index rather than discrete as above. For example a Gaussian measurement of the form $$A_{c} = \int_{-\infty}^{\infty}\frac{e^{-(x'-c)^2}/(4V)}{(2 \pi V)^{1/4}}|x' \rangle \langle x' |dx'.$$
In this case of continuous measurement outcomes $c$, it would seem based on the theorem above that $p_c$ now gives the probability density rather than the probability. Hence am I correct in stating that the probability of getting an outcome from some subset $\mathcal{M}$ is given by $$Pr(c \in \mathcal{M}) = \int_{\mathcal{M}} p_c dc = \int_{\mathcal{M}} \text{Tr}[A_{c}^{\dagger}A_{c} \rho]dc$$ and that the final state should then be given by $$\tilde{\rho} = \int_{\mathcal{M}}p_c [A_c^{\dagger}\rho A_c]dc = \int_{\mathcal{M}}\text{Tr}[A_{c}^{\dagger}A_{c} \rho]A_c \rho A_c^{\dagger} dc $$

Please advise on this, thanks.

 

[Demystifier]

Your equations are not wrong, but you didn't explain their physical interpretation. Your equations correspond to a situation in which measurement of a continuous observable $c$ is sharp, but, for some reason, the observer does not know what that sharp value is. That's the meaning of the integration over $c$ in your last equation. By contrast, if the observer knows the value of $c$, then there is no integration over $c$ and the final state is given by the first equation with $n\rightarrow c$.

 

[Danny Boy]

@Demystifier Thanks for your response, this is what I was interested in discussing. My thinking was as follows: From the Fundamental Theorem of Quantum Measurement, for the discrete case, we obtain a probability mass function $Pr(n) = p_n = \text{Tr}[A_n A_n^{\dagger} \rho]$ with random variable $n$ (which is our measurement outcome). Whereas for the continuous case, where we have continuous outcomes $c$, the corresponding probability density function is $Pr(c) = p_c = \text{Tr}[A_c A_c^{\dagger} \rho]$, hence in the same way that we consider the probability in normal distribution by taking the integral over some subset (since the probability of a single value of normal distribution is zero, by definition of being a probability density function) we do the same here by taking the integral over some subset which I call $\mathcal{M}$. What do you think, does this make sense?

 

[Demystifier]

Yes, it makes perfect sense mathematically. But physically, I would say that possible measurement outcomes are never continuous in the first place.

 

[Danny Boy]

Are you saying that physically (or experimentally) we can consider the probability $p_c = \text{Tr}[A_c A_c^{\dagger} \rho]$ as the probability of obtaining measurement outcome $c$ regardless of whether $c$ is continuous or not?

 

[Demystifier]

Are you saying that physically (or experimentally) we can consider the probability $p_c = \text{Tr}[A_c A_c^{\dagger} \rho]$ as the probability of obtaining measurement outcome $c$ regardless of whether $c$ is continuous or not?

The equation $p_c = \text{Tr}[A_c A_c^{\dagger} \rho]$ is always correct, but the interpretation of $p_c$ depends on whether $c$ is discrete or continuous. In the former case $p_c$ is probability, while in the latter case $p_c$ is probability density.

 

[A. Neumaier]

Yes, it makes perfect sense mathematically. But physically, I would say that possible measurement outcomes are never continuous in the first place.

So you doubt the validity of Born's rule for continuous observables?

 

[Danny Boy]

@Demystifier Yes agreed (this is what I stated in my original post). What I am asking is what is the most sensible way of representing the final state $\tilde{\rho}$. I have already suggested weighting by the probabilities $p_c$:
$$\tilde{\rho} := \int_{\mathcal{M}} p_{c}[A_{c}^{\dagger}\rho A_c]dc.$$ Another idea is $$\tilde{\rho} := \frac{\int_{\mathcal{M}}A_{c}^{\dagger}\rho A_c}{pr(c \in \mathcal{M})} = \frac{\int_{\mathcal{M}}A_{c}^{\dagger} \rho A_c}{\int_{\mathcal{M}}\rho_cdc}.$$

 

[Demystifier]

So you doubt the validity of Born's rule for continuous observables?

No, I doubt the physical existence of continuous observables.

 

[Demystifier]

@Demystifier Yes agreed (this is what I stated in my original post). What I am asking is what is the most sensible way of representing the final state $\tilde{\rho}$. I have already suggested weighting by the probabilities $p_c$:
$$\tilde{\rho} := \int_{\mathcal{M}} p_{c}[A_{c}^{\dagger}\rho A_c]dc.$$ Another idea is $$\tilde{\rho} := \frac{\int_{\mathcal{M}}A_{c}^{\dagger}\rho A_c}{pr(c \in \mathcal{M})} = \frac{\int_{\mathcal{M}}A_{c}^{\dagger} \rho A_c}{\int_{\mathcal{M}}\rho_cdc}.$$

The first one is the correct one. But for practical purposes the second one can also be a good approximation, assuming that $\mathcal{M}$ is sufficiently small so that $p_c$ does not vary much within $\mathcal{M}$.

 

[A. Neumaier]

No, I doubt the physical existence of continuous observables.

So position does not exist as an observable?

 

[bhobba]

So position does not exist as an observable?

Of course it does - but what does not exist is an eigenstate of position - its not even square integrable. That's where Rigged Hilbert Spaces are required.

I have a my own view on this I know you do not agree with - so will leave it at the above comment. These thing's are just mathematical fictions introduced for mathematical convenience.

Thanks
Bill

 

[vanhees71]

I've no clue what you are debating about ;-)). It's my understanding that the issue of continuous spectra of essentially self-adjoint operators on the separable Hilbert space is well formulated both in the traditional way a la von Neumann as well as in the modern formulation of rigged Hilbert space, which is more to the taste of physicists' "robust approach" to this issue. It's clear that the position of a particle can never be sharply determined since there is no position eigenvector. In other words (within non-relativistic physics!) you can localize a particle only within a finite volume, and that volume can be made arbitrarily small. Further, relativistic QT even weakens the localizability further, because to confine a particle in an ever smaller volume you need to apply ever stronger fields, leading to creation-annihilation processes of more quanta. A very readable and physical discussion of this can be found in vol. IV of the textbook series by Landau and Lifshitz.

 

[Demystifier]

So position does not exist as an observable?

Nobody ever measured position with an infinite precision, ergo physical position observable is not continuous.

 

[A. Neumaier]

Nobody ever measured position with an infinite precision, ergo physical position observable is not continuous.

But it is modeled as such in classical mechanics, in Bohmian mechanics, and (according to Born's original probability interpretation of the wave function) also in orthodox quantum mechanics.

 

[Demystifier]

But it is modeled as such in classical mechanics, in Bohmian mechanics, and (according to Born's original probability interpretation of the wave function) also in orthodox quantum mechanics.

So? Theoretical physics (or any theoretical science) deals with idealizations that only approximately correspond to actual observations. Theoretical physics is ... see my signature.

 

[A. Neumaier]

Theoretical physics is ... see my signature.

But so is all language, including that used in our discussion. We can only think in idealized terms. All notions of experimental physics, including the notion of an ''actual observation'' are idealizations, too, that only approximately corresponding to reality. Thus your comments empty everything from having logical force.

 

[Danny Boy]

@Demystifier Given that I consider the idea above: $$\rho \mapsto \tilde{\rho} := \frac{\int_{\mathcal{M}} A_{c}^{\dagger} \rho A_c}{Pr(c \in \mathcal{M})} = \frac{\int_{\mathcal{M}} A_{c}^{\dagger} \rho A_c}{\int_{\mathcal{M}}Tr[A_c^{\dagger}A_c \rho]}$$ What do you think I should consider $\mathcal{M}$ to be given some choice of $c$ (hence $\mathcal{M}$ would serve as some neighborhood of $c$ which is an open interval containing $c$)? I assume that this would be apparatus dependent? But maybe there is a general idea that seems reasonable, what do you think?

 

[Demystifier]

@Demystifier Given that I consider the idea above: $$\rho \mapsto \tilde{\rho} := \frac{\int_{\mathcal{M}} A_{c}^{\dagger} \rho A_c}{Pr(c \in \mathcal{M})} = \frac{\int_{\mathcal{M}} A_{c}^{\dagger} \rho A_c}{\int_{\mathcal{M}}Tr[A_c^{\dagger}A_c \rho]}$$ What do you think I should consider $\mathcal{M}$ to be given some choice of $c$ (hence $\mathcal{M}$ would serve as some neighborhood of $c$ which is an open interval containing $c$)? I assume that this would be apparatus dependent? But maybe there is a general idea that seems reasonable, what do you think?

Yes, that's apparatus dependent.

 

[Demystifier]

But so is all language, including that used in our discussion. We can only think in idealized terms. All notions of experimental physics, including the notion of an ''actual observation'' are idealizations, too, that only approximately corresponding to reality.

I agree.

Thus your comments empty everything from having logical force.

I don't even understand what is that supposed to mean.

 

[vanhees71]

Nobody ever measured position with an infinite precision, ergo physical position observable is not continuous.

This argument doesn't make sense. The possible outcomes of measurements of position are continuous since the spectrum of the position operator is continuous. Of course, there's no state where any observable is precisely determined to have a value in the continuous part of the spectrum of the corresponding representing operator, but that doesn't mean that the observable is not continuous.

 

[Demystifier]

This argument doesn't make sense. The possible outcomes of measurements of position are continuous since the spectrum of the position operator is continuous. Of course, there's no state where any observable is precisely determined to have a value in the continuous part of the spectrum of the corresponding representing operator, but that doesn't mean that the observable is not continuous.

By "physical observable", I didn't mean a mathematical object of theoretical physics. I meant an object that has a physical existence in the real laboratory. In that sense, even though a continuous observable exists as a mathematical object, it does not exist in a real laboratory.

But I am a theoretical physicist, so why do I take such an experimental perspective in this context? Because I think that quantum-mechanical observable is nothing more but a tool in instrumental interpretation of QM, which does not make any sense outside of the laboratory. This should be contrasted with the concept of beable, which is an ontic object supposed to be there even if we don't measure it.

 

[vanhees71]

Well, then it's even easier to argue: There's no empirical hint at any discreteness of position, at least not down to subatomic scales covered by the Standard Model. At least the argument that only because you cannot determine accurately the position position is not a continuous observable, is in no way convincing.

 

[Demystifier]

Well, then it's even easier to argue: There's no empirical hint at any discreteness of position, at least not down to subatomic scales covered by the Standard Model. At least the argument that only because you cannot determine accurately the position position is not a continuous observable, is in no way convincing.

I think you haven't understood what do I claim. You make valid arguments against a non-convincing thesis, but this non-convincing thesis is not a thesis that I defend. I defend a different thesis, with which you would agree if you only understood what this thesis claims.

My thesis is very trivial. Experiments cannot determine position (or any other observable) with perfect precision, therefore there is some finite resolution $\Delta x$ in the experiments, therefore the measured position is of the form $x=n\Delta x$ with integer $n$. I don't see how could anybody find this argument controversial.

 

[vanhees71]

No, as I said, this argument doesn't make any sense, and I understood your argument indeed as you've just detailed. The measured position is of the form $x = x_0 \pm \Delta x$ (with $\Delta x$ defined as the standard deviation with some given confidence leven and $x_0$ the average value). There's no discreteness for $x_0$, at least not according to standard QT. There's also not absolute limit for $\Delta x$.

Of course, there are some theoretical ideas on "quantized spacetime", but that's not very convincing to me yet since there's no empirical evidence for it nor a convincing solution of the notorious problem of quantizing gravity using such an approach either.

 

[Demystifier]

No, as I said, this argument doesn't make any sense, and I understood your argument indeed as you've just detailed. The measured position is of the form $x = x_0 \pm \Delta x$ (with $\Delta x$ defined as the standard deviation with some given confidence leven and $x_0$ the average value). There's no discreteness for $x_0$, at least not according to standard QT. There's also not absolute limit for $\Delta x$.

Of course, there are some theoretical ideas on "quantized spacetime", but that's not very convincing to me yet since there's no empirical evidence for it nor a convincing solution of the notorious problem of quantizing gravity using such an approach either.

You still don't understand what I am talking about. You define $\Delta x$ as standard deviation, but I define $\Delta x$ as a resolution of the measuring apparatus. That's not the same. Standard deviation is obtained from a statistics of many measurements. Resolution is a property of a single measurement. For instance, if one performs a series of measurements and obtains the values: 3.01, 3.12, 2.88, 3.22, 2.91, ... then it is clear that the resolution is 0.01, even though the standard deviation is significantly larger. With resolution 0.01, the possible measurement outcome of a single measurement is $n\cdot 0.01$ with integer $n$.

 

[Demystifier]

Related to this, I have an intriguing question for experimentalists. Suppose that a measuring apparatus has the resolution 0.01. And suppose that one performs a series of measurements with the results: 3.17, 3.17, 3.17, 3.17, 3.17, 3.17, 3.17, 3.17. What is the final result, is it
$$3.17\pm 0.00$$
or is it something like
$$3.17\pm 0.005$$
?

 

[vanhees71]

You still don't understand what I am talking about. You define $\Delta x$ as standard deviation, but I define $\Delta x$ as a resolution of the measuring apparatus. That's not the same. Standard deviation is obtained from a statistics of many measurements. Resolution is a property of a single measurement. For instance, if one performs a series of measurements and obtains the values: 3.01, 3.12, 2.88, 3.22, 2.91, ... then it is clear that the resolution is 0.01, even though the standard deviation is significantly larger. With resolution 0.01, the possible measurement outcome of a single measurement is $n\cdot 0.01$ with integer $n$.

Even in this sense position is a continuous quantity. What you are talking about is the binning in the sense of the resolution, but that doesn't make the measured quantity continuous or discontinuous.

Also you don't make a quantitiy continuous only when binning it when measuring it: If you measure the angular momentum of a quantum system you can also bin it within wide bins. Angular momentum is discrete according to QT, and the binning doesn't make it continuous.

 

[stevendaryl]

Related to this, I have an intriguing question for experimentalists. Suppose that a measuring apparatus has the resolution 0.01. And suppose that one performs a series of measurements with the results: 3.17, 3.17, 3.17, 3.17, 3.17, 3.17, 3.17, 3.17. What is the final result, is it
$$3.17\pm 0.00$$
or is it something like
$$3.17\pm 0.005$$
?

It seems to me that the notation x \pm \Delta x is used both for standard deviation and for resolution limitations. I guess you have to read the report to know which is meant. According to this paper: http://www2.ece.rochester.edu/courses/ECE111/error_uncertainty.pdf, if you are measuring positions with a resolution of 0.01 meters, then you should report it as 3.17 \pm 0.001. The error can never be less than 1/10 the resolution.

The use of x \pm \Delta x when \Delta x is a standard deviation is kind of misleading, because a measured value can be off from the true value by more than one standard deviation.

 

[vanhees71]

Related to this, I have an intriguing question for experimentalists. Suppose that a measuring apparatus has the resolution 0.01. And suppose that one performs a series of measurements with the results: 3.17, 3.17, 3.17, 3.17, 3.17, 3.17, 3.17, 3.17. What is the final result, is it
$$3.17\pm 0.00$$
or is it something like
$$3.17\pm 0.005$$
?

Of course neither the former nor the latter. You cannot use the rules of statistical-error analysis to deal with sytematical errors. If the resolution of an apparatus (systematic uncertainty!) is 0.01, the systematic error is 0.01, not 0.000 or some smaller value due to averaging as for independent measurements with purely statistical errors.