On the Heisenberg uncertainty relation

[A. Neumaier]

Are there fundamental limits on the accuracy for measuring both position $q$ at time $t$ and momentum $p$ at time $t+\Delta t$, with tiny $\Delta t$?

If yes, why?

If no, why can't one then measure (in principle) both $q$ and $p$ arbitrarily well at the same time $p$ (which is not allowed by Heisenberg's uncertainty relation), by taking $\Delta t$ sufficiently small and noting that any measurement takes time?

 

[Mordred]

Inherent fuzziness as the state of a particle isn't distinct as defined by the no go theorem.

It is impossible to prepare states in which position and momentum are simultaneously arbitrarily well localized. (B) It is impossible to measure simultaneously position and momentum. (C) It is impossible to measure position without disturbing momentum, and vice versa.

Quote from the following arxiv.

http://www.google.ca/url?sa=t&source=web&cd=&ved=0ahUKEwiCxae-wdbRAhVBMGMKHSkeBNMQFggfMAE&url=https://arxiv.org/pdf/quant-ph/0609185&usg=AFQjCNG28OmlNvKNJNh9wvuHpl1xACAFdQ&sig2=f_lOW6UGLyNNpNTpij_RHA

Its one of the more rigorous treatments but I'll let you judge it on its own merits. Particularly since your far more skilled at QFT than I am. Though I'm also not sure if you've blocked me or not.

Another good article I enjoyed is

Violation of Heisenberg’s Measurement-Disturbance Relationship by Weak
Measurements.

http://www.google.ca/url?sa=t&source=web&cd=1&ved=0ahUKEwjvotv309bRAhUmqVQKHZQaA00QFggaMAA&url=https://arxiv.org/pdf/1208.0034&usg=AFQjCNFWXC9CpbV3u5awr5yxMyWsgBS5IQ&sig2=vq1dBFKge9UUa91xtDArfQ

I haven't heard of any further means to reduce the HUP beyond the last link. I do recall the last paper was discussed previously on this forum back when I was active a few years back

 

[Strilanc]

The qubit analogy of what you're asking is "What if I prepare a qubit to point upward, then very quickly turn it to point rightward? Does that give the qubit a well-defined X and Z axis values at the same time, even though those two measurements don't commute and there's no unit vector with $x=z=1$?".

The answer is no. Even if you could do the rotation literally infinitely fast after $t=0$, the qubit would just be pointing upward for times satisfying $t \leq 0$ and pointing rightward for times satisfying $t > 0$. You moved the uncertainty around really quickly, but it's still there. There is no time $t$ where you know with certainty what both the X and Z axis measurements would be.

Analogously, even if you could transform the position state $q=0$ into the momentum state $p=0$ instantaneously after $t=0$, the uncertainty relationship is still satisfied at every time $t$.

 

[Mentz114]

Are there fundamental limits on the accuracy for measuring both position $q$ at time $t$ and momentum $p$ at time $t+\Delta t$, with tiny $\Delta t$?

If yes, why?

If no, why can't one then measure (in principle) both $q$ and $p$ arbitrarily well at the same time $p$ (which is not allowed by Heisenberg's uncertainty relation), by taking $\Delta t$ sufficiently small and noting that any measurement takes time?

I think perhaps you are having a little joke, because you refer to the publication

P. Busch, P.J. Lahti and P. Mittelstaedt, The quantum theory of measurement,
2nd. ed., Springer, Berlin 1996.

in your 2003 paper with the axiomatic construction ( which I am still sudying and enjoying). I imagine some of the conclusions in the paper cited above by @Mordred will be in this book?

The 2007 measurement paper confirms my prejudices, so naturally, I think it's great.

 

[rubi]

One way to discuss this is by looking at Heisenberg observables $\hat O(t)=e^{i t \hat H} \hat O e^{-i t \hat H}$. Then ($\hbar = 1$, $\hat H = \frac {p^2} {2m} + V$, sloppy application of Hadamard rule):
$$\left[\hat x(t),\hat p(t+\Delta t)\right] = \left[\hat x(t), \hat p(t) + \left[i\Delta t \hat H, \hat p(t)\right]\right] + O(\Delta t^2) = i + O(\Delta t^2)$$
So for small enough $\Delta t$, the uncertainty bounds don't change much. Corrections first appear in quadratic order. But even if you include many orders, the dynamics will usually not evolve an $\hat x$ eigenstate into a $\hat p$ eigenstate, not even after a long time.

 

[A. Neumaier]

simultaneously arbitrarily well localized.

I explicitly asked about the non-simultaneous version, where there is a slight time difference.

 

[A. Neumaier]

The qubit analogy of what you're asking is "What if I prepare a qubit to point upward, then very quickly turn it to point rightward? Does that give the qubit a well-defined X and Z axis values at the same time, even though those two measurements don't commute and there's no unit vector with $x=z=1$?".
The answer is no.

You are answering an unrelated question.

The correct qubit analogue of my question is: I have a qubit in an arbitrary state. I measure its up-ness, and after a very short time, I measure its right-ness. In between there is presumably a collapse of the state (or whatever you wish to assume in your measurement model).

According to the eigenvalue-measurement link, I should in both measurements obtain a 100% exact answer (up or down in the first, right or left in the second). Thus shouldn't the answer be yes?

Fundamental limitations can only come from an argument showing that it is impossible to make the two measurements.

 

[A. Neumaier]

One way to discuss this is by looking at Heisenberg observables $\hat O(t)=e^{i t \hat H} \hat O e^{-i t \hat H}$. Then ($\hbar = 1$, $\hat H = \frac {p^2} {2m} + V$, sloppy application of Hadamard rule):
$$\left[\hat x(t),\hat p(t+\Delta t)\right] = \left[\hat x(t), \hat p(t) + \left[i\Delta t \hat H, \hat p(t)\right]\right] + O(\Delta t^2) = i + O(\Delta t^2)$$
So for small enough $\Delta t$, the uncertainty bounds don't change much. Corrections first appear in quadratic order.

But the Heisenberg equation is not valid at all times as the measurement induces an uncontrolled perturbation of the system at the moment of measurement. Or do you argue in a framework where there is no collapse?

 

[A. Neumaier]

the dynamics will usually not evolve an $\hat x$ eigenstate into a $\hat p$ eigenstate, not even after a long time.

This is not needed as a system is never in a position or momentum eigenstate (these are not normalizable, hence invalid states). Position and momentum measurements can nevertheless be made.

 

[dextercioby]

The HUP is nothing but a mathematical statement (whose rigorous deduction within the framework of functional analysis requires care) about the statistical spread of expectation values around the mean. I like an interpretation of QM in terms of virtual statistical ensembles of identically prepared systems. There one can easily relate expectation values to experimentally accurate values of measured observables.

 

[A. Neumaier]

I like an interpretation of QM in terms of virtual statistical ensembles of identically prepared systems. There one can easily relate expectation values to experimentally accurate values of measured observables.

What do you mean by ''virtual statistical ensembles''?

 

[dextercioby]

The comment was off the top of my head, as I'm not home. I have to review the notes from college from 13 years ago. It's related to https://en.wikipedia.org/wiki/Statistical_ensemble_(mathematical_physics)

 

[A. Neumaier]

The comment was off the top of my had, as I'm not home. I have to review the notes from college from 13 years ago. It's related to https://en.wikipedia.org/wiki/Statistical_ensemble_(mathematical_physics)

That applies as stated only to the statistical mechanics of macroscopic systems. Generalizing it to arbitrary quantum systems leads to my thermal interpretation of quantum mechanics.

But how is your comment related to my question in post #1?

 

[rubi]

But the Heisenberg equation is not valid at all times as the measurement induces an uncontrolled perturbation of the system at the moment of measurement. Or do you argue in a framework where there is no collapse?

After a measurement of $\hat x(t)$, the system will be in a state $P_{\hat x(t)}(B)\Psi$, where $P_{\hat x(t)}$ is a projector of $\hat x(t)$. If you perform a measurement on this state at time $t'$, the question is whether the projectors $P_{\hat p(t')}(O')$ of $\hat p(t')$ commute with $P_{\hat x(t)}(B)$, because otherwise, the state after the second measurement will not lie in the subspace onto which $P_{\hat p(t)}(B)$ projects. So the question is whether the projectors of $\hat x(t)$ and $\hat p(t')$ commute and (up to technicalities about unbounded operators) this is equivalent to asking whether $\hat x(t)$ and $\hat p(t')$ commute.

(This is already without loss of generality, since a POVM measurement can be realized (in a bigger Hilbert space) as a PVM followed by unitary evolution and we can assume this extra unitary evolution to be included in $e^{-i\Delta t \hat H}$.)

This is not needed as a system is never in a position or momentum eigenstate (these are not normalizable, hence invalid states). Position and momentum measurements can nevertheless be made.

Well, I ignored these technicalities, because it doesn't really change the conceptual problem. Let's make a concrete example. Let's say, we have a Hamiltonian $H=\frac{p^2}{2m}$ and we have just measured the position. The quantum state will be sharply peaked on some definite position (let's say $0$). We know that sharply peaked functions correspond to broadly spread functions in momentum space. If the wave function is a Gaussian $\Psi(x,t=0)=e^{-\frac{x^2}{4\sigma_x}}$, then the momentum wave function will be a Gaussian too with $\sigma_x \sigma_p \approx 1$. Now, our concrete Hamiltonian will just evolve $\Psi$ into a Gaussian with $\sigma_x(\Delta t)=\sigma_x + \Delta t \xi$, hence after a short time $\Delta t$, the Gaussian will still be sharply peaked. But this means that in momentum space, it will still be broadly spread: $\sigma_p(\Delta t)\approx\frac{1}{\sigma_x+\Delta t\xi}\approx\frac 1 {\sigma_x}-\frac{\Delta t\xi}{\sigma_x^2}$. So after a short time, the state won't have evolved into a state of definite momentum.

 

[A. Neumaier]

because otherwise, the state after the second measurement will not lie in the subspace onto which [the projector] projects.

But why should this be needed? After the second measurement we simply have the state $P'P\psi$, which is as good as any other.

Thus your argument tells nothing about how accurate $q$ and $p$ are measured - which are properties encoded into $P$ and $P'$.

From this perspective, there seems to be no restriction on the precision with which $q$ at time $t$ and $p$ at a slightly later time $t'$ can be measured.

 

[A. Neumaier]

So after a short time, the state won't have evolved into a state of definite momentum.

Even after a long time it won't have evolved into one, assuming the free motion. But surely we can make a macroscopic time later a momentum measurement and will get a definite measurement result approximating the momentum.

 

[rubi]

What I'm trying to explain is the following: If you have measured position with an accuracy $\sigma_x(0)$, then the state of the system is $P\Psi$. Now the question is the following: What is the variance $\sigma_p^2(\Delta t)$ of momentum in the state $e^{-i\Delta t H}P\Psi$? In order for the variance to be small, the state must be close to being a momentum eigenstate. But if $\sigma_x(0)$ is small, then $\sigma_p(\Delta t)$ will not be small in general. Hence, we get a trade-off between position uncertainty $\sigma_x(0)$ and momentum uncertainty $\sigma_p(\Delta t)$. There is $c$ with $\sigma_x(0)\sigma_p(\Delta t)\geq c$.

 

[Strilanc]

You are answering an unrelated question.

The correct qubit analogue of my question is: I have a qubit in an arbitrary state. I measure its up-ness, and after a very short time, I measure its right-ness. In between there is presumably a collapse of the state (or whatever you wish to assume in your measurement model).

According to the eigenvalue-measurement link, I should in both measurements obtain a 100% exact answer (up or down in the first, right or left in the second). Thus shouldn't the answer be yes?

Fundamental limitations can only come from an argument showing that it is impossible to make the two measurements.

If you're picturing a situation where the measurements take time, then once $\Delta t$ is lower than that time your results will become junk. The specific way in which they become junk depends on how you're doing the measurement. There are many ways to model this, creating many varieties of junk.

For example, suppose we're on a quantum computer that performs up/down measurement by performing a continuous controlled-NOT of the target qubit onto a fresh spare qubit over $1us$ (then it properly measures the spare qubit using some slower process). Furthermore, the computer performs left/right measurement by applying a Hadamard operation arbitrarily fast (rotating the qubit so its old left/right axis is its new up/down axis), then performing an up/down measurement, then rotating the qubit back with another Hadamard.

If we start the up/down measurement at $t_0$ and the left/right measurement at $t_0 + \text{us}/2$ then we are effectively doing half of a CNOT onto an ancilla, Hadamarding, then the other half of the CNOT as well as a full CNOT onto a second ancilla, then Hadamarding, and taking our time to measure the ancilla properly. Like this circuit:

If you pass an upward or downward qubit into this circuit, the qubit ends up leftward or rightward and you don't really learn that the qubit was upward or downward. The circuit does something, but that something certainly isn't violating the HUP. I won't go into the various other behaviors of this particular circuit (I will mention that what's happening the the entangled partner is kinda neat). The main points I wanted to make with it are:

1. You can make explicit models of the "measurements happening so close that they overlap" situation.
2. Those models fail to distinguish between upward and rightward with certainty.

If your measurements are arbitrarily fast, then having them happen near each other isn't different from doing one then the other. If your measurements aren't arbitrarily fast, then they will overlap and generally break each other's effects. The way they break depends on how the measurements are implemented.

 

[Mordred]

Doesn't lemma c I posted state that you disturb the system when you make the first measurement? So when you make the make the second measurement the system is not the same?

I don't know about anyone else but I would call that a fundamental limit.

(C) It is impossible to measure position without disturbing momentum, and vice versa.

So now you have induced additional inaccuracy before you take the second measurement

 

[A. Neumaier]

What I'm trying to explain is the following: If you have measured position with an accuracy $\sigma_x(0)$, then the state of the system is $P\Psi$. Now the question is the following: What is the variance $\sigma_p^2(\Delta t)$ of momentum in the state $e^{-i\Delta t H}P\Psi$? In order for the variance to be small, the state must be close to being a momentum eigenstate. But if $\sigma_x(0)$ is small, then $\sigma_p(\Delta t)$ will not be small in general. Hence, we get a trade-off between position uncertainty $\sigma_x(0)$ and momentum uncertainty $\sigma_p(\Delta t)$. There is $c$ with $\sigma_x(0)\sigma_p(\Delta t)\geq c$.

But this is applying different standards to the two measurements. You consider the accuracy of the measured position but the accuracy of the predicted momentum. Whereas I was asking about limitations in the accuracy of the measured position and the measured momentum.

The usual HUP is conventionally derived as (i) a constraint on the predicted uncertainty of both position and momentum at the same time, but then interpreted (with somewhat vague arguments) as (ii) a constraint on the measured uncertainty of both position and momentum in a simultaneous measurement. I am interested in the (ii)-part - except with the time delay.

 

[A. Neumaier]

you disturb the system when you make the first measurement? So when you make the make the second measurement the system is not the same?

The system is the same, only the state changes, being multiplied by a subunitary matrix, often (for simplicity) taken to be a projector.

 

[A. Neumaier]

http://www.google.ca/

You should give the arxiv links and not the google links!

 

[Mordred]

Using a phone I would prefer that but I tend to get auto downloads when I try. (sorry for the inconvenience) My laptop crashed on me. On the previous comment Doesn't lemma c specify your influencing momentum and position when you take a measurement

 

[A. Neumaier]

your influencing momentum and position when you take a measurement

Influencing, yes. The question is whether this implies limitations of the quality of the measurement results.

 

[Mordred]

ah kk gotcha

 

[PeterDonis]

I was asking about limitations in the accuracy of the measured position and the measured momentum.

How are you defining "accuracy" of the measurements?

If you prepare a large ensemble of particles in the same state, and then subject them, one by one, to a position measurement followed very rapidly by a momentum measurement, then if you take a subset with a narrow spread in the position results, that subset will have a very wide spread in the momentum results. (At least, that is what I take to be the standard QM prediction.) Does that mean the momentum measurements were "inaccurate"? Or just that the spread in the results is very large? If the latter, how would you define the "accuracy" of the momentum measurements in this scenario?

 

[Mordred]

Is there a way to interfere without being either constructive or destructive?
I'm not aware of any

 

[A. Neumaier]

How are you defining "accuracy" of the measurements?

I asked the question because I was fed up with the many discussions on the foundations that completely ignore the issues that I see make up the measurement problem. (See here for some of these problems.) I am not even sure what it means to have measured position, let alone how to specify the accuracy of the measurement. If someone knows (with authoritative references) I'd like to be enlightened. My question is kind of fishing in the dark, hoping to discover during the discussion what really matters and what the words so often used in foundational discussions actually mean.

If you prepare a large ensemble of particles in the same state, and then subject them, one by one, to a position measurement followed very rapidly by a momentum measurement, then if you take a subset with a narrow spread in the position results, that subset will have a very wide spread in the momentum results. (At least, that is what I take to be the standard QM prediction.)

Yes, I agree, if the position measurements project to a state with a narrow spread in position.

Does that mean the momentum measurements were "inaccurate"? Or just that the spread in the results is very large?

On the surface, it means only the second. Because assigning a truth value to the first statement makes sense only if we know what it means to have made a momentum measurement of a certain accuracy.

If the latter, how would you define the "accuracy" of the momentum measurements in this scenario?

Taking rubi's explanation

we have just measured the position. The quantum state will be sharply peaked on some definite position (let's say 000).

for what it implies to have done an accurate position measurement as a temporary definition, and demand the same for momentum, an accurate momentum measurement should imply that the state is projected to a quantum state sharply peaked around the measured momentum.
The measurement is presumably the more accurate the more narrow the peak. With this definition, your second possibility offered definitely does not imply the first.

 

[dextercioby]

Hi Arnold, I found the section about the virtual statistical ensembles from my QM course I took part in Nov-Dec. 2003. It is all written in Romanian, so I need to translate it to German or English. Leave me some time, please. :) [to be continued in the dedicated thread].

 

[Jilang]

I think the issue is in the timing. Isn't there an inherent uncertainty in that?

 

[bhobba]

How are you defining "accuracy" of the measurements?

That would be my question as well.

The HUP applies to idealized measurements of exact accuracy and is a statement about the statistical spread of such.

We all know such idealized measurements do not exist. That's why a state of exact position is modeled by a Dirac Delta function which is a mathematical abstraction - nothing can have zero width and an infinite height. It took a long time for mathematicians to develop a rigorous theory of such involving such luminaries as the great Grothendieck. This I suspect was at least in part from scathing criticisms Von-Neumann made in his classic - Mathematical Foundations of QM. I was first told about it when it was introduced in my class on applied differential equations - I read it (its in the first few pages) and thought - now that's a criticism. My teacher never did tell me the solution - I asked - he just smiled and said - Bill you can look that one up yourself.

Thanks
Bill

 

[bhobba]

Hi Arnold, I found the section about the virtual statistical ensembles from my QM course I took part in Nov-Dec. 2003. It is all written in Romanian, so I need to translate it to German or English. Leave me some time, please. :) [to be continued in the dedicated thread].

Its also in Ballentine.

Thanks
Bill

 

[Mordred]

Wouldn't most of this be under the Von-Neumann measurement problem?

this article has a couple of good sections ie section 2.1
"The probability of detecting a probe position z as a function of time"

http://www.johnboccio.com/research/quantum/notes/13117649.pdf

looks like it has decent details on the Stern_Gerlach experiment and the width aspects under section 3.1
for the probability density of Q after interaction.

Not sure if this is what your after but if it is I saw a decent chapter on this in one of the QFT manuals I've been reading the past week

 

[rubi]

Let's approach it in a deductive way. We define the uncertainty of an observable $A$ in a state $\Psi$ as $\sigma_A^\Psi =\sqrt{\left<\Psi,A^2\Psi\right>-\left<\Psi,A\Psi\right>^2}$. Without loss of generality, let $\left<\Psi,A\Psi\right>=0$ (otherwise, use $A'=A-\left<\Psi,A\Psi\right>$). We are interested in bounds on the product $\sigma_A^\Psi \sigma_B^{U(t)\Psi}$, where $U(t)$ is the time evolution operator. We find:
$$\sigma_B^{U(t)\Psi}=\sqrt{\left<U(t)\Psi,B^2 U(t)\Psi\right>} = \sqrt{\left<\Psi,U^\dagger(t) B U(t) U^\dagger(t) B U(t) \Psi\right>} = \sqrt{\left<\Psi,(U^\dagger(t)BU(t))^2\Psi\right>}=\sigma_{U^\dagger(t) B U(t)}^\Psi=\sigma_{B(t)}^\Psi$$
We have defined the Heisenberg observable $B(t)=U^\dagger(t)B U(t)$. Now, by the usual uncertainty relation, we find
$$\sigma_A^\Psi \sigma_B^{U(t)\Psi}=\sigma_A^\Psi \sigma_{B(t)}^\Psi \geq \frac 1 2 \left|\left<\Psi,\left[A,B(t)\right]\Psi\right>\right|$$

Right now, this is just a mathematical bound on the product of uncertainties $\sigma_A^\Psi \sigma_B^{U(t)\Psi}$, but how does it relate to measurement?
If we decide to measure the operator $A$ at time $t=0$ and we have designed the measurement apparatus to measure at a certain accuracy $\epsilon_A$, which means that the statistical uncertainty in the measurement results will be $\epsilon_A$, then after a measurement, the state $\Psi$ will have collapsed onto a state $\frac{P_A(S)\Psi}{\lVert P_A(S)\Psi\rVert}$ with uncertainty $\sigma_A^{\frac{P_A(S)\Psi}{\lVert P_A(S)\Psi\rVert}} \approx \epsilon_A$. Thus, by our uncertainty relation at different times, we find
$$\sigma_A^{\frac{P_A(S)\Psi}{\lVert P_A(S)\Psi\rVert}}\sigma_B^{U(\Delta t)\frac{P_A(S)\Psi}{\lVert P_A(S)\Psi\rVert}}\geq \frac 1 2 \left|\left<\frac{P_A(S)\Psi}{\lVert P_A(S)\Psi\rVert},\left[A,B(\Delta t)\right]\frac{P_A(S)\Psi}{\lVert P_A(S)\Psi\rVert}\right>\right|=:c_\Psi^{\epsilon_A}$$
Thus, quantum mechanics predicts that if you have measured $A$ with accuracy $\epsilon_A$ at $t=0$, then you can measure $B$ with accuracy at most $\frac{c_\Psi^{\epsilon_A}}{\epsilon_A}$ at time $t=\Delta t$. If you perform the experiment a large number of times, then no matter how many digits your measurement apparatus can display, if you calculate the uncertainty $\sigma_B$ afterwards, you will get a number that exceeds $\frac{c_\Psi^{\epsilon_A}}{\epsilon_A}$. You can't build a device that can produce a smaller uncertainty. This is not because you are a bad engineer, but because quantum mechanics limits your ability to prepare a state in such a way that it produces a statistically robust definite outcome for $B$ at time $t=\Delta t$ (assuming you first measured $A$ with accuracy $\epsilon_A$). (See post #43 for more.)

 

[ShayanJ]

Thanks @rubi, incredible post!
I just have one question. It seems to me that there is no difference between $\epsilon$ and $\sigma$. Why do you use different names for them?
The reason I say this, is that we don't assume the quantum mechanical observables have any "real" value before measurement so it doesn't make sense to talk about the difference between the measured value and the real value. So there is nothing called the accuracy of a single measurement. All we can talk about is the uncertainty $\sigma$. And it seems to me that you're using them as the same quantity too. Because from $\sigma_A \sigma_B \geq c$, you get $\epsilon_B \geq \frac{c}{\epsilon_A}$.

EDIT: Unless $\epsilon$ is interpreted as some kind of a resolution, i.e. the ability of the measuring device to distinguish between two values of the observable. Then it makes sense but again, how do you get $\epsilon_B \geq \frac{c}{\epsilon_A}$ from $\sigma_A \sigma_B \geq c$? The resolution should be a property of our measurement device. How can it be related to any other measurement or whether we have a lot of systems or not?