"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.
