Conservation of energy after measurement

[KleinMoretti]

TL;DR Summary

is energy conserved after a week measurement

so I thought that when a system was measured there could be an interaction between the measurement device or environment and the system but overall energy was conserved, but I came across these 2 articles which seem to imply this is not the case. https://link.springer.com/article/10.1140/epjs/s11734-021-00092-2.
https://journals.aps.org/prresearch/pdf/10.1103/PhysRevResearch.3.013247
I was hoping someone could tell me if my conclusion was right or wrong

 

[Demystifier]

Weak measurements can give a wide spectrum of weird counterintuitive results, possible violation of energy conservation is certainly not the weirdest.

 

[KleinMoretti]

but what does that mean for conservation laws, do they really stop holding in quantum physics?

 

[KleinMoretti]

"The conservation means that the measurable correlations (7) involving the conserved quantity q(t)corresponding to ˆQ(t)=ˆQ will not depend on t. It is true at the single average, where ⟨q(t)⟩=⟨ˆQ⟩. Interestingly, also for second-order correlations, the order of measurements has no influence on the result since ⟨q(t1)a(t2)⟩=⟨{Q,A(t2)}⟩/2 is independent of t1. However, the situation changes for three consecutive measurements (see Fig. 1), since in the last line of (7) the time order of operators matters, which has also been demonstrated experimentally [30]. Considering the difference of two measurement sequences Q→A→B and A→Q→B, we obtain the jump (which is absent in perfectly noninvasive classical measurements [26])


⟨{ˆQ,{ˆA(t2),ˆB(t3)}}−{ˆA(t2),{ˆQ,ˆB(t3)}}⟩=⟨[[ˆQ,ˆA(t2)],ˆB(t3)]⟩≡4⟨Δqa(t2)b(t3)⟩.

(9)​

This quantity will show up as jump Δq=q(t1)−q(t2) at t1=t2 when measuring ⟨q(t1)a(t2)b(t3)⟩. The jump will be nonzero for Q not commuting with A and B. Obviously, for superconserved quantities Q (commuting with every measurable observable), the jump is absent. The violation of the conservation principle is caused by the measurement of ˆA, not commuting with ˆQ, which allows transitions between spaces of different q with the jump size Δq not scaled by the measurement strength g (see Fig. 1). This difference is transferred to the detector, assuming that the total quantity (of the system and detector) is conserved regardless of the system-detector interaction. This observation can be compared to the WAY theorem, which applies to projective or general measurements. Here, we have shown that even taking the special limit of noninvasive measurement, the noncommuting quantity causes a jump in third-order (and higher) correlations. We can call it weak-WAY theorem, as both the input (the special construction of g-dependent measurements) and the output (correlations) are based on weak measurements. Note that imposing the condition that the jump (9) vanishes, equivalent conservation of ˆQ at the level of third-order correlation for an arbitrary state ˆρ (allowed by superselection rules, if any apply), namely,

[[ˆQ,ˆA],ˆB]=0,

(10)​

for all allowed observables ˆA and ˆB suffices to keep conservation also at all higher-order correlations. Then, ˆQ is not necessarily superconserved, it can commute with observables to identity, like momentum and position. This subtle difference between the weak-WAY and the traditional WAY theorem in sketched in Fig. 2.

FIG. 2.​

The difference between the WAY and the weak-WAY theorem. The former applies to general measurement and shows that the lack of coherence between eigenstates of ˆQ in the Kraus operators (1) leads to superconservation of the measured quantity. The latter applies to weak measurements (2), leading to a weaker condition for the observed conservation.
As an example, we can take the basic two-level system ( |±⟩ basis) with the Hamiltonian ˆH=ˆQ=ℏω|+⟩⟨+| and ˆA=ˆB=ˆX=|+⟩⟨−|+|−⟩⟨+|. Then, with ω>0 the ground state is |−⟩and the third-order correlation ⟨h(t1)x(t2)x(t3)⟩ for the ground state for t3>t1,2 reads ℏω(1−θ(t2−t1))cos(ω(t2−t3))/2. The jump is ⟨Δhx(0)x(τ)⟩=ℏωcos(ωτ)/2 for Δh=h(0−)−h(0+). The result can be generalized to a thermodynamical ensemble with a finite temperature T and reads (see Appendix A)

⟨Δhx(0)x(τ)⟩=ℏωcos(ωτ)tanh(ℏω/2kT)/2.

(11)​

For increasing temperature, the jump diminishes as illustrated in Fig. 3.

FIG. 3.​

The nonconserving jump for τ=0+ (thick lines) compared to the average energy (thin lines) for the two-level system with level spacing ℏω (red) and the harmonic oscillator with eigenfrequency ω (blue). At high temperatures, the jump becomes unobservable and the classical conservation is restored. All quantities are normalized to ℏω.
Another basic example is the harmonic oscillator with ˆH=ˆQ=ℏωˆa†ˆa with [ˆa,ˆa†]=1. Taking the dimensionless position √2ˆX=ˆa†+ˆa=ˆA=ˆB, we find for the jump ⟨Δhx(0)x(t)⟩=−ℏωcos(ωt)/4, independent of the state of the system (see Appendix A). As illustrated in Fig. 3, the jump becomes unobservable at high temperatures since the average energy ⟨h⟩=ℏω/[exp(ℏω/kT)−1] increases with temperature.
The previously discussed very simple examples illustrate the fundamental finding of our paper. If one tries to verify the conservation of energy while measuring another observable that is not commuting with the Hamiltonian, it is possible to find a violation of the energy conservation. It constitutes a pure quantum effect since it vanishes at high temperature, where the classically expected conservation holds. One could object that performing a series of measurements already breaks time-translational symmetry and, therefore, the total energy is not conserved. However, one can keep the time symmetry by replacing the detector-system interaction by a clock-based detection scheme [9] (see Fig. 4). The total Hamiltonian reads

ˆH+ˆHx+ˆHz+ˆHI,

(12)​

where ˆH is the system's part, ˆHx is the detector's part, ˆHz is the clock's part, and ˆHI is the interaction between the clock, the system, and detector. Each part is time-independent so the time-translation symmetry is preserved. Both the detector and the clock can be represented by single real variables, x and z. Now, to measure the system's ˆA at time t1, we set ˆHx=0 and

ˆHz=vˆpz,ˆHI=gˆAδ(ˆz)ˆpx,

(13)​

where ˆpx,z are conjugate (momenta), i.e., ˆpx=−iℏ∂/∂x and g→0 is a weak coupling constant. The initial state (at t=0) reads ˆρˆρxˆρz, where both ˆρx,z=|ψx,z⟩⟨ψx,z∣∣ are taken as Gaussian states

ψz(z)=(2πσ)−1/4exp(−(z+vt1)2/4σ),ψx(x)=(π/2)−1/4exp(−x2),

(14)​

respectively. For small g and σ, the interaction effectively occurs at time t=t1 and, in the end (after the clock decouples the system and the detector again) to lowest order we find (see details in Appendix B)

⟨x⟩≃g⟨ˆA⟩=g⟨a(t1)⟩.

(15)​

For sequential measurements, one simply adds more independent detectors and clocks, obtaining in the lowest order of g

⟨xAxB⟩≃g2⟨a(t1)b(t2)⟩,⟨xAxBxC⟩≃g3⟨a(t1)b(t2)c(t3)⟩,⟨xAxBxCxD⟩≃g4⟨a(t1)b(t2)c(t3)d(t4)⟩,

(16)​

with the right-hand sides are given by the quantum expressions (7).

FIG. 4.​

Detection model based on a clock. The clock is a localized particle traveling with a constant speed v. The interaction between the detector and system takes place only when the clock is passing the interaction point.
Although the above detection model is based on time-invariant dynamics, the initial state of the clock spoils the symmetry. The time-invariant state would require a constant flow of particles or field at a constant velocity, so that the position on the tape imprints time of measurement (see [31] for detailed construction). However, such a constant interaction between the detector (clock) and the system leads to a backaction and makes the measurement invasiveness growing with time, which needs to be reduced by additional resources, e.g., additional coupling to a heat bath.
In order to show that the nonconservation can also occur independently from the time-translation asymmetry present either intrinsically or induced by a quantum clock, one can look at other quantities that are conserved, e.g., due to spatial symmetries. As an example, we will use one component of the angular momentum in a rotationally invariant system in the following."


if understood this correctly they say that while there may be an apparent violation of conservation of energy after 3 consecutive measurements the total energy of the system+detector is conserved, but then they conclude the paper with this.

"We have shown that conservation laws in quantum mechanics need to be considered with care since their experimental verification might depend on the measurement context even in the limit of weak measurements. The conservation is violated if extracting objective reality from the weak measurements. It means that either (i) weak measurements cannot be considered noninvasive, or (ii) the conservation laws do not hold in quantum objective realism."

so now I don't if I understood correctly or not.

 

[PeterDonis]

what does that mean for conservation laws, do they really stop holding in quantum physics?

I would say it means that quantum experiments do not always give a way of even checking to see if conservation laws hold, since the measurements involved might be of observables that do not commute with the observables corresponding to the conserved quantities (for example, the Hamiltonian for energy).

 

[KleinMoretti]

I would say it means that quantum experiments do not always give a way of even checking to see if conservation laws hold, since the measurements involved might be of observables that do not commute with the observables corresponding to the conserved quantities (for example, the Hamiltonian for energy).

what do you mean, Im even more confused, also did you see my other answer, did I understand what they are saying incorrectly

 

[PeterDonis]

what do you mean

In order to check whether energy (for example) is conserved, you have to measure energy. If you don't measure the energy, you can't check whether it's conserved. So, for example, if you want to check to see whether the total energy of system + detector is conserved, you would have to measure the energy of both the system and the detector, before and after whatever else you are going to do with them, and see if the total is the same.

And if you measure some other observable that doesn't commute with the energy observable (the Hamiltonian), then you can't say anything meaningful about the energy after that measurement because the system won't be in an eigenstate of energy, so its energy isn't even well defined. So even if you do measure the energy of system + detector before and after your experiment, if you measure some non-commuting observable in between, you can't say the energy was conserved during the experiment, even if you measure it before and after and get the same result, because during the experiment the system wasn't in an eigenstate of energy.

All of this goes for any type of measurement; weak measurements add even more complexities.

 

[KleinMoretti]

In order to check whether energy (for example) is conserved, you have to measure energy. If you don't measure the energy, you can't check whether it's conserved. So, for example, if you want to check to see whether the total energy of system + detector is conserved, you would have to measure the energy of both the system and the detector, before and after whatever else you are going to do with them, and see if the total is the same.

And if you measure some other observable that doesn't commute with the energy observable (the Hamiltonian), then you can't say anything meaningful about the energy after that measurement because the system won't be in an eigenstate of energy, so its energy isn't even well defined. So even if you do measure the energy of system + detector before and after your experiment, if you measure some non-commuting observable in between, you can't say the energy was conserved during the experiment, even if you measure it before and after and get the same result, because during the experiment the system wasn't in an eigenstate of energy.

All of this goes for any type of measurement; weak measurements add even more complexities.

Ok i think i understand what you are saying, however in the case considered in the paper i provided they are saying the total energy is conserved no?
"This quantity will show up as jump Δq=q(t1)−q(t2) at t1=t2 when measuring ⟨q(t1)a(t2)b(t3)⟩. The jump will be nonzero for Q not commuting with A and B. Obviously, for superconserved quantities Q (commuting with every measurable observable), the jump is absent. The violation of the conservation principle is caused by the measurement of ˆA, not commuting with ˆQ, which allows transitions between spaces of different q with the jump size Δq not scaled by the measurement strength g (see Fig. 1). This difference is transferred to the detector, assuming that the total quantity (of the system and detector) is conserved regardless of the system-detector interaction."

 

[PeterDonis]

in the case considered in the paper i provided they are saying the total energy is conserved no?

They are assuming that it is. Note that that word occurs explicitly in the last sentence of what you quoted. They are not demonstrating that it is, or providing experimental evidence that it is. They are just assuming it.

 

[KleinMoretti]

They are assuming that it is. Note that that word occurs explicitly in the last sentence of what you quoted. They are not demonstrating that it is, or providing experimental evidence that it is. They are just assuming it.

that is where my confusion is, if i understand what they are saying correctly, they are showing that a violation of conservation of energy appear in third order time correlations but that doesnt affect the assumption that the total energy is conserved.

 

[PeterDonis]

that is where my confusion is, if i understand what they are saying correctly, they are showing that a violation of conservation of energy appear in third order time correlations but that doesnt affect the assumption that the total energy is conserved.

That's because they are assuming that whatever differences in energy show up in their numbers obtained from measurements on the quantum system are compensated for by changes in the energy of the detector--which they don't measure.

 

[KleinMoretti]

That's because they are assuming that whatever differences in energy show up in their numbers obtained from measurements on the quantum system are compensated for by changes in the energy of the detector--which they don't measure.

my question was whether or not they are claiming that a violation of conservation in third order time correlations implies a violation of the conservation of the total energy which if i understand what they are saying correctly, the assumption is still that the total energy is conserved meaning that they are not implying a violation in third order time correlations means total energy isn't conserved.

 

[KleinMoretti]

That's because they are assuming that whatever differences in energy show up in their numbers obtained from measurements on the quantum system are compensated for by changes in the energy of the detector--which they don't measure.

now that i read this ( https://link.springer.com/article/10.1140/epjs/s11734-021-00092-2.) again, they have a specific section "total conservation" where i think they say that the total energy is conserved. again i may be misunderstanding what they are saying.