TIME (A)SYMMETRY

Before publishing in New Journal of Physics, our result underwent long peer-review, with several rejections, including New journal of Physics itself (!). Here I present excerpts from the reports, with comments. Thanks to all referees our main conclusion remained unchanged but is now much better supported. If you are interested in full correspondence, it is available on request by email: abednorz[at]fuw.edu.pl.

1. Nature and Nature Physics(rejected editorially)

Comment:
The submitted version was close to the first arXiv version. The editors send only a small fraction of submissions to referees. Ours was obviously to reject.

2. Physical Review Letters (rejected after 4 rounds and appeal)

Referee A:
"The weak measurement in the double-well system considered in the manuscript was analyzed in detail over a decade ago. And I do not see anything really interesting, which would go beyond that analysis."

Comment: (from our reply)
So we should guess the reference. See also below.

Referee B (positive!):
"this paper is another manifestation of the disturbing nature of weak measurements in the weak limit, bringing out another aspect of this phenomenon, applied to the novel question of time-reversal symmetry of series of weak measurement correlation functions. The phenomenon is sufficiently interesting."

Referee C:
"Consider a system in pure state |ψ>. Let us perform a weak measurement tracing out the pointer will result in a mixed state for the system. Hence the state has been perturbed in an irreversible manner, resulting in an asymmetry under time-reversal. It is not clear to me whether the effect described by the authors is simply the cumulative effect of the three successive weak measurements, or if there is a more subtle process involved here"

Comment:(from our reply)
"Then the asymmetry should vanish asymptotically, which does not happen. Both pointer correlations and disturbance vanish but the disturbance effect vanishes faster. Therefore the asymmetry is not a cumulative effect of residual disturbance. There is no room for a more subtle process within this framework."
See also below.

Referee D:
"One can use a different, mathematically simpler method to analyze the correlations of three weak measurements, and I do this below. A single weak measurement of an observable A on a system with initial density matrix ρ is given by a single time-step of a stochastic Schrödinger equation. The measurement result is

Here dW is a Gaussian random number with mean zero and standard deviation equal to √dt. The time-step is dt, and it is this that scales the strength of the measurement. As dt → 0 the measurement becomes infinitely weak. The constant k can be regarded merely as converting units where necessary. Note that the noise term dW/dt has variance 1/dt, and thus goes to infinity as t → 0: the noise swamps the mean value as the measurement gets very weak. From the above equation, we see that a weak measurement makes only an infinitesimal change to the quantum state. The result of a subsequent measurement only knows about the result of the first measurement by the change it has made to the state, and thus any correlation between the two will necessarily be infinitesimal. By using the above equations to describe three measurements on a system, and by keeping only terms that cause correlations to leading order in dt, we obtain the following expression for the product of the three measurement results:

The only term that can generate 3-way correlations is the once with all three operators A;B; and C. The authors set Tr[Cρ] = Tr[Bρ] = Tr[Aρ] = 0. But I think it is educational to leave them nonzero. Now multiplying out and taking the expectation value we have

Now we see how the correlations are generated. The change in the state caused by each measurement is proportional to dW, and thus tends to zero in the weak limit. However, the correlation that is generated between two weak measurements is be- tween the infinitesimal change picked up by the mean of the second measurement (caused by the first), and the large noise of the first measure- ment, the latter going to infinity in the weak limit. The infinitesimal and infinite cancel, and we obtain a finite correlation. The above result is surprising, because no matter how weak the measurements, the correlations between them remain finite. Since they remain finite, they can then be time-asymmetric because of the non-commutivity of quantum observables. Once one sees why the correlations remain non-zero, it not surprising that they are time asymmetric. I don't think it is sufficiently fundamental, or of sufficiently broad interest"

Divisional Associate Editor:
"Two referees (A,C) have difficulties fully appreciating the results of the work, while one referee (B), who is clearly knowledgeable on the issue of weak value measurements recommends publication. A fourth referee D completely clarifies the issues and resolves the paradoxical or disturbing features. The result is of a less fundamental nature"

Comment:
The referee D is so clear that should have written the PRL instead of us. All PRLs are more fundamental than ours, but fortunately not all NJPs.

3. New Journal of Physics (first submission, rejected after single round and appeal).

First Referee: (positive!)
"the paper presents interesting fundamental results"

Second Referee:
"The sequence of weak measurements of an observable A undertaken at successive separated times is described by breaking the unitary evolution operator U(t₀, t_f) = exp{-iH(t_f − t₀)} acting on the initial density matrix ρ of the system into factors corresponding to each of the intervals, interspersed by the interaction operators exp{-i∫ gε(t) p_j A dt}. Here ε(t) is a compact function of time, peaked around t_j, and vanishing outside a narrow range much smaller than |t_j-t_j-1| and |t_j+1-t_j|. Also, p_j is the momentum operator acting on the j-th ancilla, and g is the corresponding interaction strength, the interaction to be eventually considered in the g → 0 limit. It is convenient to consider each of the evolution operators in the eigenbasis of A. When projective measurements are performed on each of the n ancillae, the density matrix acquires the following factor for the j-th ancilla (keeping as much as possible close to the authors' notation):

where a_j is the value of A read off after projective measurement of the j-th ancilla. The measurement could be viewed as non-invasive if the dependence of the ancilla wave functions φ on the intermediate states a_j' and a_j'' does not significantly affect the corresponding integrals so that the "fragments" of U get reassembled into the complete evolution operator, thus corresponding to unperturbed time evolution of the density matrix of the system. The authors substitute a weaker definition of non-invasiveness, effectively requiring this "re-assembling" to happen not for arbitrary a_j, but only for the evolved density matrix integrated over a_j. The difficulty with this definition is that, in the absence of any decoherence mechanism, the ancillae stay entangled with the measured system, and hence the nature of the projective measurement of the ancillae affects the density matrix. In a sense, integrating over all a_j is the least "invasive" act, as it effectively erases all the information gained from the measurement. The "residual factor" left behind after this integration is indeed weakly dependent on a_j' and a_j'' in the g → 0 limit (as exp{-g²(a_j'-a_j'')²} for the Gaussian wave functions used by the authors), allowing to claim non-invasiveness (...) as g → 0. However, if one instead measures the first moment of a₁, the integration leaves behind a factor of a_j'+a_j'' which stays finite in the g → 0 limit. Consequently, the effective evolution operator acting on the density matrix of the system is no longer U(t₀, t_f), and non-invasiveness gets violated. More generally, if one "reads off" from the projective measurement of the ancillae the generating function exp{i k a_j, the result (again for Gaussian φ) contains a factor of

which cannot be neglected for any g if k is finite, hence materially affecting the time evolution of the underlying system. In summary, there is no contradiction between non-invasiveness of the "null measurement" implied by integrating over a_j (...), and perturbed evolution of the density matrix produced by measurements of the moments of a_j, and hence no paradox in the lack of invariance under time reversal. "

Comment: (from our appeal)
"The initial random probability distribution is changed by a measurement because once we read off the value of the initially random variable it will collapse to the Dirac delta at the measured value. Does it prove invasiveness? No! This is not a physical collapse - like in quantum projection - but only informational: we learn the actual value but do not disturb anything. Only if the integration over the read-off results gives a different probability distribution, one can claim invasiveness. By the referee's reasoning one would prove that all classical measurements are invasive, too, also in the limit g → 0, while the time reversal symmetry is maintained if considering noninvasiveness in our sense. The referee seems to have a wrong notion of invasiveness. His/her requirements for noninvasiveness could not be met by any measurement scheme, classical or quantum."

Adjudicator:
"the measurement processes are responsible for the violation of time reversal symmetry, just as they are in classical statistics. The amount of entanglement required to achieve a useful measurement signal seems to undermine the notion of a non-invasive measurement. In a series of N measurements, the correlation has a signal of g^N, but the disturbance caused by the measurements has a magnitude of Ng². This means that the disturbance can only be neglected for N ≠ 2."

Comment:(from our appeal)
"The reversal of the classical dynamics makes the correlations of noninvasive measurements time-symmetric. The amount of entanglement, and so the disturbance of the density matrix, is negligible. The correction to the signal due to disturbance is of the order Ng^N+2, negligible in the limit of g → 0."

Second Adjudicator (on appeal):
"for a Gaussian measurement of width σ and state-dependent splitting g, the density matrix element ρ₊₊ will be enhanced relative to its previous value by a factor exp(zg/σ²), while the ρ₋₋ element will be suppressed by the inverse factor. Notice this does not necessarily go away as g/σ → 0, since for very large values of z, there can still be a sizable disturbance. I am not against the publication."

Comment:
So all classical measurements are invasive, too, see also our published paper, end of section 4 (but before 4.1).
The Second Adjudicator, ignoring Second Referee and First Adjudicator, encouraged us to resubmit the paper once again to NJP, including rebuttal of the criticism of Second Referee and both Adjudicators

4. New Journal of Physics (second submission, accepted after two rounds, for better presentation).

First Referee:
"This paper concerns the topics of weak quantum measurement and presents a very interesting conclusion: the non-invasive classical and quantum measurement differ in property the authors call time-symmetry, this property can be quantified by accessing the statistics of measuring results."

Second Referee:
"The manuscript reports a study of weak measurements and comes to the conclusion that a non-invasive quantum-mechanical measurement is generically not time-symmetric. The authors proceed by suggesting an experiment in which this symmetry violation can be measured. This is an important conclusion"

Editorial Board Member:
"The authors are making quite a subtle and fundamental point about quantum mechanics. The paper has an important message."

Comment: Thanks a lot!!!