Speakable and Unspeakable in Quantum Measurements

Quantum mechanics, in its orthodox version, imposes severe limits on what can be known, or even said, about the condition of a quantum system between two observations. A relatively new approach, based on so‐called “weak measurements”, suggests that such forbidden knowledge can be gained by studying the system's response to an inaccurate weakly perturbing measuring device. It goes further to propose revising the whole concept of physics variables, and offers various examples of counterintuitive quantum behavior. Both views go to the very heart of quantum theory, and yet are rarely compared directly. A new technique must either transcend the orthodox limits, or just prove that these limits are indeed necessary. Both possibilities are studied and orthodoxy is vindicated.


I. INTRODUCTION
In 1963 [1] ,1964 [2], and 1965 [3] Feynman and co-authors outlined what they considered the most general principles of the quantum method.At about the same time, the authors of [4] proposed what is known as the "Aharonov, Bergmann, and Lebowitz rule" for calculating the probabilities of a pre-and post-selected quantum system.Feynman's arguments are rarely revisited in a discussion of the quantum measurement theory.On the other hand, a recent development of the views expressed in [4] has given the reader the concept of "weak measurements", and a plethora of "quantum paradoxes" the concept helps to establish (see, for example, [6]- [14]).
The attitude to asking questions about the quantum method is often summed by Mermin's taunt "Shut up and calculate" [15].Calculations are, no doubt, useful, but understanding the method's limitations, could tell one something about the relation between a human observer and nature.
The purpose of this paper is to compare directly the orthodoxy of [1] (one can hardly be more orthodox than when appealing to a reputable undergraduate text) with recent developments in the quantum measurement theory [6]- [14], [16]- [21].Our aim is to see whether anything new has indeed been added to the views expressed in [1] - [3], or if at least some of these views have been disproved by the more recent research (a preliminary attempt of such a comparison was made in [22]).It is easy to see [23] that the original ABL rule is a particular example of the Feynman's rule for assigning probabilities [1].A further comparison is more involved.Feynman's reasoning is subtle, as one has to walk a "logical tightrope" in order to "interpret nature" [2].The centrepiece of the orthodox description is the Uncertainty Principle, best known in its Heisenberg form [1].According to the Principle, certain questions, easily answered in a classical context, should have no answer when dealing with a quantum system.In particular, very little may be said about the path taken by an unobserved particle in a double-slit experiment, [1], [2] and, by the same token, about the state of any system between two observations.The authors of [5] suggested that a meaningful description of a system between the measurements can be obtained by employing weakly perturbing detectors.(We refer the reader to Ref. [24] for a possible use of such detectors in order to surpass the Heisenberg limit in joint measurements of a particle's position and momentum.)Over the years, the approach proposed in [5] has lead to a number of spectacular claims [5]- [14], which we will submit to further scrutiny here.
The rest of the paper is organised as follows.In Sect.II we discuss consecutive quantum measurements, the need to produce records of the past outcomes, and a simplified description, possible if particular types of probes are used.Sect.III reviews the "orthodox" description of quantum behaviour, given in [1], [2], [3].In Sect.IV we use the description to analyse a quantum measurement analogy of the double-slit problem.In Sect.V we leave the narrative of [1], [2], [3], and discuss the "weak values", first introduced in [6], as well as their possible interpretations.In Sect.VI we introduce a three-slit ("three-box") system, and create our own "paradox", similar to that proposed in [9].In Sect.VII the "paradox" is dismissed for reasons explained in Sect.V.In Sect.VIII we revisit a four-slit (quantum Cheshire cat [12], [14]) problem.Sect.IX contains our conclusions.Appendices A and B contain the necessary technical details.To condense the presentation, we will bring in the proverbial Alice (an orthodox theorist), Bob (an experimenter), and Carol, a kind of "devil's advocate", who disagrees with Alice and asks her awkward questions.We apologise to the reader who may find these characters annoying.

II. BOB WANTS TO MEASURE THREE QUANTITIES
Bob has a quantum system, and wants to measure, at times t 0 < t 1 < t 2 , the quantities A, B, and C, each taking certain number of discrete values M A,B,C ≤ N .He asks Alice the theorist to tell him how to do it, and what he is likely to see.
Alice tells him that at the end of the experiment, at t = t 2 , he would need three visible records of the measured values (the system itself cannot provide this information).He would then measure the frequency with which a particular series of records occurs after many trials.
Alice's mathematical task is to predict the said frequencies, and to do so she associates with the system an N -dimensional Hilbert space H S , and represents Bob's quantities by Hermitian operators, Â, B, and Ĉ, acting therein.(If a classical calculation were to be made, she would consider a phase space, and dynamical functions of positions and momenta).
There are three orthonormal eigenbases of Â, B, and Ĉ, {|a k ⟩}, {|b l ⟩}, and {|c n ⟩}.The first measured quantity, she tells Bob, must not have degenerate eigenvalues, so that if the first measurement yields A k , Alice knows that the system is "in a state |a k ⟩", and can proceed with her calculation.The second operator may have where ∆(X − Y ) = 1 if X = Y , and 0 otherwise.The same applies to the third operator, Ĉ, but for now we will assume that all C n are different.
Bob's records are to be produced and kept by three probes D A , D B , and D C , each with its own Hilbert space, H D A , H D B , and H D C , so the large Hilbert space which Alice will use for her calculations is a tensor product of the four, Alice also needs to construct a Hamiltonian ĤS+P (t), and a unitary evolution operator, Û S+P = exp[−i , also acting in H S+P .The first probe, D A , should briefly couple to the system at t = t 0 , in such a manner that any initial product state where |D A (0)⟩ is the probe's initial state, and its final condition |D A (A k )⟩ tells Bob that the outcome of the first measurement was A k .The action of the second probe at t = t 1 is defined similarly, whereas the third probe acts at t = t 2 according to (2).If the system has a non-zero time independent Hamiltonian, ĤS , between t 0 and t 1 , and t 1 and t 2 it undergoes its own unitary evolution with an operator Û S (t, t ′ ) = exp[−i ĤS (t − t ′ )] (see Fig. 1a).
It should be mentioned that at the beginning Bob has no prior knowledge of the system's state |ψ S ⟩ in Eq.( 2).So for him the experiment begins when the first probe reads A k .The necessary statistics can then be collected, e.g., by repeating the measurements on the same system, retaining only the cases where the first outcome is A k .For Alice the initial state of the composite is |D C (0)⟩|D B (0)⟩|D A (A k )⟩|a k ⟩, and the probability of having the entire sequence {C n ← B m ← A k } is given by (see Fig. 1a, the ⊗s henceforth omitted) The states |ϕ S j ⟩ form an arbitrary orthonormal basis, and the choice |ϕ S j ⟩ = |c j ⟩ simplifies the evaluation due to the appearance of a Kronecker δ jn (see Fig. 1b).
One can say that the operator Û S+P (t 2 , t 0 ) describes an unbroken unitary evolution of the composite {system + probes} from just after t 0 to just after t 2 .Or one can simply say that Alice needs to know matrix elements of a unitary operator Û S+P (t 2 , t 0 ) between particular states in the Hilbert space of the composite, and not mention the evolution at all [27].
Furthermore, the probabilities in Eq.( 4) can be obtained without mentioning the probes explicitly, since the couplings (2) and (3) are simple, and their actions are easily taken into account (details can be found, e.g., in [27]).In particular, one has for the probability of measuring the sequence (B m , C n ), provided the first detection gave A k , an equally valid form, FIG. 1: a) Unbroken unitary evolution of a composite {system + probes} from t 0 to t 2 [cf.Eq.( 4)]; b) a more detailed view of the evolution: the probes are coupled at t 0 , t 1 , and t 2 , and carry the records until the experiment is finished; c) broken unitary evolution of the system only [cf.Eq.( 6)].
if the measured eigenvalue of Ĉ is not degenerate.If it is, Ĉ = M C µ=1 C µ π(C µ ), an additional summation is needed, Thus, Alice has a choice.She may either consider an unbroken (except at t = t 2 ) unitary evolution of the composite {system + probes} in a larger Hilbert space H S+P [cf.Eq.( 4)], or she may deal with an evolution in a smaller space H S , interrupted at t = t 1 by insertion of the projector π(B m ) [cf.Eq.( 6)].This interruption is necessary to account for the action of Bob's probe D B at t = t 1 , and is by no means mysterious.This second option is clearly more practical.

III. A DIFFERENT VIEW OF THE SAME PROBLEM: THE ORTHODOXY
Alice can also reconsider her problem in the light of the general principles of the quantum theory as laid out in [1], [2].These principles, we recall, are: (a) The probability of an event is given by an absolute square of a complex valued probability amplitude.The amplitudes of two consecutive steps need to be multiplied.
(b) If the event can occur in different ways, and it is not possible to determine which of the alternatives is taken, the amplitudes for each way ought to be summed, and an interference pattern will emerge.
(c) If it is possible to determine, even in principle, whether one or the other alternative is taken, then the probabilities ought to be summed.
(d) The probabilities describe the state of the affairs when the experiment is "finished".
(e) One never sums the amplitudes for different final states [25].
(f) Uncertainty Principle: one cannot determine which of the alternatives has been taken without destroying the interference pattern.[26].
In Alice's case the event in question consists of preparing the system in a state |a k ⟩ at t = t 0 , and then finding it in a state |c n ⟩ at t = t 2 .At time t 1 , t 0 < t 1 < t 2 , the system can be deemed to be in one of the states |b l ⟩, so there are, in principle, N different ways (paths), {c n ← b l ← a k }, l = 1, ..., N , in which the event may occur.Notice that c n , b l and a k refer to states (lower case) and not to eigenvalues (upper case).By (a) the corresponding amplitudes are and if no attempt is made to determine the state at t = t 1 the amplitudes should be added Bob measures a B with degenerate eigenvalues, the amplitudes of the paths passing through the eigenstates of B corresponding to the same B m cannot be distinguished, and, by (b) and (c), Eq.( 5) follows.If some of the eigenvalues of Ĉ are also degenerate, the rule (e) prescribes adding the probabilities Eq.( 5) for all the eigenstates |c n ⟩ with the same eigenvalue C µ , which yields Eq.( 6).
According to [1], the essence (the "only mystery") of quantum mechanics is contained in the double-slit example.There a particle can reach a point on the screen via two holes, where the observed number of arrivals exhibits an interference pattern consisting of "bright" and "dark" fringes.Next we revisit the case in a slightly different context, and discuss it in some detail.

SPEAKABLE
To create a kind of a double-slit case in the context of quantum measurements, one needs three successive observations, similar to those discussed in Sect.II.[Measuring just two operators, Â and B, won't do since, according to principle (e) in the previous section, there are no interesting interference effects.]Allowing B to have only two different values, B 1 and B 2 , and treating these alternatives as "slits", creates two virtual paths connecting the "source" |a k ⟩ with N "points on the screen", represented by |c n ⟩.The corresponding amplitudes are (m = 1, 2) The "intensities"

A. Closing one slit
In [1] Feynman pointed out the need for a "special language", required to describe the double-slit conundrum.Firstly, the slit chosen by the particle cannot be pre-determined by a "hidden" variable.Were it so, closing one of the slits could only decrease the number of particles arriving at a given point (see also [28]).This, however, would be a wrong prediction since the closure makes the particles arrive at a previously dark spot.Consequently, it is not of the "slits", and "points on the screen", respectively.b) a special choice eliminates two of the four virtual paths in a), and no interference is left in the system.c) similarly, ⟩ also reduces the number of paths to two.
possible to say that the particle goes through one slit, and not the other.The same happens in our simple example.Although we cannot literally block one of the slits, it is possible to redirect the system, passing via |b 1 ⟩ or |b 2 ⟩ to different destinations, e.g., by measuring B, and counting separately the cases where the result is B 1 or B 2 .By (c) of Sect.III, knowing the value of B at t 1 makes the system arrive at a previously forbidden state |c 2 ⟩, which would not be possible if the patch chosen by the system were pre-determined at t = t 0 .In the general case, in order to "walk the logical tightrope" [1], one cannot say that the system found in |c n ⟩ at t 2 was in one of the states |b m ⟩ at t B. Checking if the system passes through both "slits" at the same time Secondly, confirms [1], it is not possible to say the a particle divides and passes through both slits at the same time, since wherever one looks, one finds the entire particle, and not a fraction of it.To revisit the argument in the quantum measurements context, we assume, as before, that the initial and final states of our two-level system, |a 1 ⟩ and |c 2 ⟩, are accurately determined.For the probe D B we choose a von Neumann pointer [29] with position f .If the probe, prepared in a normalised state |G⟩, briefly couples to the system at t = t 1 via Ĥint = −i Bδ(t − t 1 ) ∂ f , the amplitude of finding a reading f just after t 2 is given by where G(f ) ≡ ⟨f |G⟩.Similarly, one can couple two probes to measure at t = t 1 the operators B = |b 1 ⟩⟨b 1 | and B′ = |b 2 ⟩⟨b 2 |.The amplitude of having readings f and f ′ becomes.
Apparently, there are no scenarios in which both pointers are displaced, or both retain their original position.If the displacement of the pointer is to indicate the presence of the system in its respective path, one can only conclude that the system cannot pass through both slits at the same time.An accurate measurement, where the width of G(f ) is much smaller than 1 will always register the presence of the entire system in one of the pathways, apparently confirming this conclusion.But if the accuracy is such to which both paths contribute simultaneously.One's inability to determine what happens to a quantum system in the presence of interference is reflected in the Uncertainty Principle (f) of Sect.III.

C. Fewer detections
In [1] Feynman gives the following illustration of the importance of the Uncertainty Principle [(f) of Sect.III].Suppose one determines the slit chosen in the double-slit experiment by sending a photon which is scattered only if the particle passes through a given slit.The destruction of the interference pattern on the screen is caused by the photon's collision with the particle.To minimise the disturbance, one may try to lower the intensity of light, but this will only reduce the number of photons, without diminishing the strength of each impact.As a result, all trials will fall into two categories: those in which the photon is scattered, and the particle's path is known, and those where one cannot even say that the particle went through one slit and not the other (cf.subsection A).The resulting pattern is a weighted sum of W (no interference) and W (interference).
This too has a simple analogy in our discussion.The pointer's initial state G(f ) in Eq.( 12) can be chosen to be a "rectangular window" (see Fig. 3a), The pointer becomes more accurate as ∆f → 0, and |G(f trial yields either f = B 1 or f = B 2 .For an inaccurate pointer, ∆f > |B 2 − B 1 |/2, there is a region where the two G's in (12) overlap (see Fig. 3b).If Bob finds a reading f inside that region, he can say nothing about the value of B (cf. subsection A).Otherwise he knows the path chosen by the system.As in the Feynman's example, the probability to find the system in a state |c n ⟩ at t = t 2 is a weighed sum where β = 2|B 2 − B 1 |/∆f , and W n (interference) and W n (no interference) are given by Eqs.( 9) and (10), respectively.Note that sending ∆f → ∞ is equivalent to introducing a new scaled variable f ′ = f /∆f , which changed the coupling Hamiltonian to Ĥint = −i Bδ(t − t 1 )/∆f ∂ f ′ .Broadening G(f ) is, therefore, equivalent to reducing the coupling between the system and the pointer.
It is instructive to see what happens in the case of a previously forbidden transition (Fig. 3c), where or [∆f /2, ∆f /2 + 1], and in each case Bob knows that B had either a value 0 (the system was in |b 1 ⟩), or 1 (the system was in |b 2 ⟩).The system reaches |c 2 ⟩ with a probability 2|A| 2 /∆f << 1, as the pointer has modified destructive interference between the paths, allowing it to reach, on a few occasions, the previously inaccessible final destination.For ∆f >> 1 , Bob needs to wait a long time for an arrival in |c 2 ⟩, but in each such arrival the pointer's displacement is very large, ∼ ±∆f /2.Note that this happens not because the pointer has experienced an unusually large shift, but because it could be found not far from there even before the measurement was made.

D. Fuzzy detections
Thus, continues Feynman [1], the way to observe the particles without disturbing them considerably may be to increase the photon's wavelength, rather than decrease their number.
A red coloured photon carries a smaller momentum than a blue one, so the jolt produced on the particle should also be smaller.However, if the wavelength λ were to exceed the distance between the slits, d, one would only see a "fuzzy" flash, unable to decide which of the two slits it came from.This is another illustration of the Uncertainty Principle (f): "an apparatus, capable of determining which way a particle goes, cannot be so delicate so as not to disturb the interference pattern in an essential way [1]." Such a fuzzy detection can be mimicked by using a pointer prepared in a Gaussian (rather that in a rectangular) state, where Even though there are two scenarios the system may follow, the last interference term clearly prevents one from knowing which of them has actually been realised if the measurement is fuzzy.The term disappears if the measurement is sharp ∆f << |B 2 − B 1 |, and the system is always seen to choose one of the two paths.
One might object by pointing out that we kept changing the subject.By sending ∆f → 0 Bob perturbs the system considerably, while one only wished to know what happened to the unperturbed system.There is an orthodox answer to that too.The wish cannot be granted, or the Uncertainty Principle (f), necessary for the very existence of the quantum theory [1], would be violated.It appears that the "which way?" question simply cannot be answered in the presence of interference.
We could finish the discussion with the adaptation of the general rules of [1] to the needs of the measurement theory, were it not for one additional circumstance.

V. THE "WEAK MEASUREMENTS"
In an attempt to study the properties of an (almost) unperturbed system, Alice may send ∆f in Eq.( 15) to infinity.The first thing she would notice is that if the final position of the pointer is representative of the value of the measured B, this value is, indeed, indeterminate, since f can lie almost anywhere.The answer to a question which should have no answer can be "anything at all".She can also expect that no other useful information can be extracted from the very broad distribution of the pointer's reading.It is only reasonably to expect that the final mean reading of an infinitely inaccurate pointer would also be zero, just as was the initial one.This is, however, not the case.In [6] the authors have shown (using a different language and notations) that the average pointer's reading in the limit ∆f → ∞ is given by where is the path amplitude, renormalised to a unit sum over the paths leading to the same destination |c 2 ⟩, l α l = 1.The complex valued quantity ⟨ B⟩ W was called "the weak value (WV) of B".Its imaginary part, Im[⟨ B⟩ W ], can be obtained if the pointer's mean final momentum is measured instead [6], (see also [30]).In practice this means that after repeating his measurements K >> 1 times, Bob will be able to deduce the value in the r.h.s. of Eq.( 17) sufficiently accurately, even if the uncertainty of each trial, ∆f , is as large as he wants, and the interference between the system's paths is affected as little as he wants.
[Bob's task, although possible in principle, will rapidly become impractical.Indeed, by the Central Limit Theorem [31], the number of trials (copies of the system) will have to grow This result should be of interest to anyone wishing to challenge the conclusions of Refs.[1], [2], and [3].Contrary to one's expectations, a concrete information clearly related to the intermediate condition of an (almost) unperturbed system can be obtained experimentally.Does measuring ⟨f ⟩ in this manner allow Bob to speak about the things Feynman thought unspeakable?

A. "Weak values" and the Uncertainty Principle
Apparently not.The Uncertainty Principle of [1] does not forbid knowing the values of the probability amplitudes.It does, however, forbid to make assumptions about the system's past where the scenarios one wishes to distinguish continue to interfere.Nothing in the Principle stops Bob from learning the values of either the amplitudes (18), or of their various combinations (17).The question is, what conclusions is he allowed to draw from the "weak values" once they have been measured?
Before addressing it, it may be useful to demonstrate that there always exists a transition in which the weak value of any chosen operator B equals any desired complex number Z.
Indeed, there are infinitely many choices of α l = A l / l ′ A l ′ , such that ⟨ B⟩ W = ℓ B l α l equals Z.If a particular set α i (Z) has been chosen, the path amplitudes A l must satisfy Since N l=1 α l = 1, the matrix in the square brackets is degenerate, and the system (19) has a particular solution A l = α l /α 1 .
If only three quantities are measured, the amplitudes are given by and for any |a k ⟩ one finds a (yet unnormalised) state |c n (t 1 )⟩, ⟨b l |c n (t 1 )⟩ = α * l /[α 1 ⟨b l |a k (t 1 )⟩] * .Normalising |c n ⟩ to unity yields a final state |c n ⟩ such that, for a given initial state |a k ⟩, ⟨ B⟩ W equals the desired value Z, The indeterminacy implied by the Uncertainty Principle is restored, albeit at a higher level, where all possible transitions are considered.The weak value of a z-component of a spin-1/2 can be found to have a value ⟨σ z ⟩ W = 1/2, as well as ⟨σ z ⟩ W = 1.2345 * 10 19 − 6.789 * 10 28 i.
Both results are equally meaningful or, if one prefers, equally meaningless, as we discuss next.

B. Interpretation of the "weak values"
This is the most contentious part of our discussion.Enter Carol who, unlike Alice, suspects that Feynman's narrative [1] it too restrictive, and there is more to be learnt through the "weak measurements" described above.
C: You forbid saying anything about the value of B half way through the transition.I put it to you that this weak (WV) value is given by Eq.( 17).The authors of [6] have shown that Bob can measure a component of a spin-1/2 and give you a number 100.Here is the answer to a question which you thought could have no answer.
A: By the same logic, the WV of a projector B = |b 1 ⟩⟨b 1 | should be the probability of the system taking that path.I can always find a transition [cf.Eq.( 21)] where ⟨ B⟩ W = −5i.This is not a probability.A probability is related to the number of occurrences, and always has value between 0 and 1.I do, however, have a different name for this result: it is a probability amplitude (18).C: All right, one cannot substitute the relative amplitudes for probabilities, but maybe they be interpreted as "occupation numbers", as was suggested in [8].Let us say that the "weak value" of a projector B = |b l ⟩⟨b l |, ⟨ B⟩ W = α l , represents, in some sense, the "number of particles" in the l-th path (the l-th arm of an interferometer, if an optical realisation of the experiment is considered).These weak values can be measured simultaneously, obey a "self-consistent logic", and should lead to a "deeper understanding of the nature of quantum mechanics" according to [8].
A: I am afraid, you would have the same problem: −5i is by no stretch of imagination a valid number of particles (or particle pairs as proposed in [8]).It would be even worse for a "dark" final state, where l A(c 2 ← b l ← a 1 ) = 0, unless you want to tell Bob that infinitely many particles in each pathway conspire so that nothing arrives at the final destination |c 2 ⟩.Besides, a WV of a projector is not a good local indicator, and should not be used to quantify what happens in a particular arm of the interferometer (see Appendix A).The WV of B = |b l ⟩⟨b l | depends on the amplitudes of all pathways leading to the same final state [cf.Eq.( 49)], in a way its accurate "strong" value does not [cf.Eq.( 47)].And, by the way, the "self-consistent logic" [8] enjoyed by the weak values is but the property that the relative amplitudes (18) always add up to unity.
C: Let us forget about the numerical values of the WVs, be they real or imaginary, and simply use them as indicators of the particle's presence in particular part of the system, as was suggested in [11].One couples a weak pointer to a particular system's state (or a part of the interferometer).Surely, if the pointer does not move (the weak value of the corresponding projector is 0), the system never visits that part of the Hilbert space?If, on the other hand, it did move, the system must have been there.What can possibly go wrong with relying on such a "weak trace"?A: This is the same as saying that the system does not use a pathway whose amplitude is zero.But the pathways can be combined, and you will be in danger of finding the particle in parts while no finding it in the whole.This is not too different from Feynman's dilemma of Sect.3A, whereby one finds more particles arriving at the screen if one slit is closed.We will discuss this case next.

VI. THE THREE-SLIT CASE.
With three rather than just two dimensions it comes a possibility of measuring operators with degenerate eigenvalues, which we will explore next.There are now three paths, connecting |a 1 ⟩ with |c 1 ⟩, which, to shorten the notations, we will denote {l} ≡ {c 1 ← b l ← a 1 }, and three amplitudes, A l ≡ A(c 1 ← b l ← a 1 ), l = 1, 2, 3 (see Fig. 4a).We will consider two commuting projectors, designed to question the presence of the system, at t = t 1 , in the state |b 1 ⟩, and in the sub-space spanned by |b 1 ⟩ and |b 2 ⟩, respectively.Following [9], we tune the system to ensure If no measurements are made, the probability of the system arriving at |c 1 ⟩ at t = t 2 is

A. No paradox so far
An accurate measurement of B always finds the system in the state |b 1 ⟩ (we omit the dependence on |a 1 ⟩ and |c 1 ⟩) However, an accurate measurement of B′ never finds it in the sub-space containing |b 1 ⟩, In both cases the probability of a successful post-selection in |c 1 ⟩ is, as before, Although the authors of [9] found this "paradoxical", there is nothing particularly surprising.
In Sect.II we went to some length stressing the need for employing a probe which would carry the record of the outcome obtained at t = t 1 until the experiment is finished just after t 2 .Measuring B and B′ requires different probes (−i∂ f B ̸ = −i∂ f B′ ) which affect the system, expected later to reach the same final state, in different ways.The resulting statistical ensembles are, therefore, different, and there is no conflict between Eqs.( 24) and ( 25) [32].
Measuring B and B′ one after another also shows nothing unusual.With two intermediate measurements there are 9 paths, of which only three have non-zero amplitude (we put ĤS = 0) There are also four possible outcomes, B = 0, 1, B ′ = 0, 1.The outcome (B = 1, B ′ = 0) would indeed be problematic, indicating the presence of the system in the path 1 but not in the union of the paths 1 and 2. However, this scenario is never realised, as the corresponding probability vanishes, P (B = 1, B ′ = 0) = 0 (see Fig. 4b).Each of the three remaining outcomes allow to identify the paths taken by the system, and by (c) of Sect.III, one has We note also that, just like in Feynman's two-slit example of Sect.4A, determination of the path taken by the system increases the odds on its arriving in the state |c 1 ⟩, One cannot, therefore, say that an unobserved system is in one of the states |b l ⟩, and not in the other two.By the same token, whenever some of the paths are not resolved by a measurement (the paths {1} and {2} in our example), it should be impossible to say that the system chooses one of them and not the other [cf.Eqs.( 24) and ( 25)].Next, as promised, we will bring the "weak values" into the discussion.

B. An unlikely "cat"
To add panache to the story, we will refer to the system as a "cat", promote the paths {1} and {2} to a "one-bedroom flat", and call the remaining pathway {3} the "yard".Our aim, we recall, is to divine the behaviour of an unobserved and, therefore, unperturbed cat.To do so we will relay on the existence or absence of the "weak traces", non-zero "weak values" b) of the corresponding projectors.Using Eq.( 23) we find that the weak pointer looking for the cat in the bedroom {1} has indeed "moved", The weak pointer looking for it in the entire flat, however, did not (see Fig. 4c) and we have a "paradox": A quantum cat can be in a part of the flat, and yet not in the flat as a whole.
the readings consists of two disjoint parts, ρ(f, f where The first part corresponds to the cat not found in the flat, and shows no trace of the cat in the bedroom, ⟨f ⟩ 1 = f ρ 1 (f, f ′ )df df ′ = 0.The second part finds the cat in the flat, and there is also a trace of it in the bedroom (the meters are Gaussian), ⟨f 1/2.However, the probability of the latter outcome, which occurs only because the second Bob's pointer slightly upsets otherwise perfect cancellation between the paths {1} and {2}, vanishes as ∆f , and Bob only gets the outcomes where neither of the two pointers "has moved".

Alice and Carol again:
C: A quantum system can be absent from a subspace of its Hilbert space, yet simultaneously present in a smaller part of that subspace.In an optical experiment a photon can be present in one arm of the interferometer, but absent from a larger part of the setup, containing the arm in question.
A: And how do you know that?
C: A weak pointer moves (albeit on average) if I look in a smaller subspace, but not if I look in the larger one.
A: Your "paradoxical" conclusion is predicated on the link between the movement of the pointer and the actual presence of the system at the specified location.But you can also look at your result differently.You are simply saying that the system is simultaneously present in all the pathways were the corresponding amplitudes, A l , does not vanish.If so, in a double-slit experiment a particle passes via both slits at the same time.In the orthodox view [1], [2], [3] you can always say that an A l ̸ = 0.You cannot, however, conclude from this that the system, or a part of it, is there.
C: And why is that?
A: Because a more accurate measurement may find the entire system elsewhere, or destroy the "paradox" as was discussed in above example.Your prediction will be proven wrong.
C: But you said yourself that an accurate measurement will disturb the system, and I am taking about the experimental evidence of an unperturbed system's behaviour.Perhaps the weak measurements, unknown to the authors of [1], [2], [3] at the time, conveniently filled the gap left in their reasoning?
A: The authors of [1], [2], [3] have an answer to this.The fact that one can deduce the values of the amplitudes experimentally is of no fundamental importance.The amplitudes are always known to a theorist, and the orthodox contention is that they cannot be used beyond what is covered by the rules of Sect.III.In [2], Feynman admits that one can say something about what happens to the system in the presence of interference, but needs "to stop thinking immediately and make no deductions from it".You can say that there are particles passing through each slit yet never arriving at a dark fringe on the screen.You can say that photons have discontinuous trajectories [11], or that a cat both is and is not inside the flat, but that is all."Physicists prefer not to say it , rather than stop thinking at the moment."[2].

VIII. THE FOUR-SLIT CASE. "QUANTUM CHESHIRE CATS"
Adding the fourth dimension allows one to consider a pair of two-level systems.Following [12] call the first one a "Cheshire cat" (sic!), and the second one the cat's "grin".The cat can be either "on the right", or "on the left", i.e., in states |L⟩ or |R⟩, respectively.The grin can be "up" or "down".The four possible states of the joint system, therefore, are There are also two operators, diagonal in the basis (32), B = |L⟩⟨L|, with the eigenvalues (1, 1, 0, 0), and B′ = |R⟩⟨R| ⊗ (|up⟩⟨up| − |down⟩⟨down|), with the eigenvalues (0, 0, 1, −1), where the operator |up⟩⟨up| − |down⟩⟨down|, akin to the z-component of a spin-1/2, σz , represents the "cat's" "grin".One can always choose (see Sect.VA) the four amplitudes We leave it to the reader to check what happens if the above operators are measured accurately, together or separately, and note only that the "weak values" of both operators vanish, To follow the story one has (a) to agree that a null "weak value" of B means that the cat is not on the left , and (b) to accept that a null "weak value" of B′ means that the cat's grin is not on the right.There is also a tacit assumption that if something is not in one place, it must be in the other, since neither the cat nor the grin are expected to disappear.Thus, one may conclude that the cat (on the right) has been "separated" from its grin (on the left), and consider other possible "separations" mentioned in the second of our epigraphs.Further discussion of this "paradox" can be found in [12], [14].
Our purpose is to question the principle.In Eq.( 34) ⟨ B⟩ W vanishes because B 1 = B 2 and The story "told by the simultaneous weak values" is now as follows: (i) there is no cat on the left hand side of the setup, (ii) yet there is a cat there, provided its grin is "up", (iii) there is no grin on the right hand side, (iv) yet there is a grin, provided it is "up".

Alice and Carol again.
A: You told Bob that there is no grin on the right hand side, but he finds a part of it there.

Another wrong prediction?
C: Au contraire, you just discovered another quantum mechanical paradox: Bob can have the cat in one place, the cat's grin elsewhere, and still a bit of the grin where the grin is not.Have you two thought of a publication?
A: All we know is that Bob's "weak measurements" have confirmed the relations (33) between the four amplitudes in question.But there is no surprise, Bob has set up his experi-ment ensuring that α 1 = −α 2 = α 3 = α 4 .Your "paradox" disappears the moment you stop interpreting Eqs.(33) the way you do.

C: And how should I interpret them?
A: You should not interpret them at all, lest you create spurious "paradoxes".

C: What if I like paradoxes?
The next Section contains our conclusions.

IX. CONCLUSIONS AND DISCUSSION
In summary, our aim was compare two apparently contradictory approaches to the description of quantum behaviour.The first one, which we called the orthodox, can be applied to quantum measurements as follows.An experimentalist wants to measure three (or more) quantities one after another.He/she needs probes (pointers) to read off the past measured values at the end of experiment.A theorist describes the system in terms of virtual paths connecting the states in the system's Hilbert space H S , and the corresponding probability amplitudes A l .If the equipment allows to determine the path taken by the system in each trial, absolute squares of the amplitudes, |A l | 2 , predict the relative frequency (probability) with which a particular path will occur after many trials.A quantity B is represented by an operator B acting in H S .Its mean value, ⟨B⟩, is then obtained by multiplying each path probability by the corresponding eigenvalue of the operator B, adding up the results, and normalising the sum as appropriate.
If the paths taken cannot be told apart, the amplitudes must be added, and one can neither say that a particular scenario was realised, nor that all the scenarios were realised at the same time.According to [1], [2], [3], either assumption about the system's past would lead to wrong predictions, or contradictions.In particular, not even the mean value of B can be determined at a time between two observations.Moreover, there is no deeper explanation of the law of adding amplitudes, which must be accepted with all restrictions it imposes upon what can be known.These restrictions amounts to the Uncertainty Principle, a centrepiece of quantum description, without which "quantum mechanics would collapse" [1].
The Principle appears to leave an unpleasant void at the heart of the theory, and a desire to fill it can only be natural.The task is, however, difficult and, if attempted, must end either in an important new development, or in a misunderstanding of similar proportions.
Feynman, for his part, was in no doubt that it would end in a "blind alley, from nobody has yet escaped" [2].
A different approach aims at describing a quantum system between measurements [5] and, can, therefore, be seen as an attempt to fill the gap left by the Uncertainty Principle.For the purpose of our discussion it suffices to note that the method relies on experimentally accessible "weak value" (WV), ⟨B⟩ W , a sum of the probability amplitudes, weighted by the eigenvalues of the measured operator B, (more technical details are given in Appendix A).The amplitudes α l are renormalised to a unit sum, and there always exists a transition, in which ⟨B⟩ W takes any desired complex value (cf.Sect.V).As yet, there is no contradiction with the Uncertainty Principle which allows knowing probability amplitudes regardless of whether or not the system is being observed.It remains, however, to clarify the meaning of the result (35).Different propositions have been put forward by the authors of [6]- [14].
Could ⟨B⟩ W yield the elusive value of B in the presence of interference?If so, one can have a component of a spin-1/2 equal to 100 [6], a negative kinetic energy, a system "disembodied from its properties" [12], a flux of "disembodied conserved properties" [14], or a particle's velocity exceeding the speed of light [10].The question is whether these results provide a further insight into the "machinery behind the law" [1].As always, the double slit case provides for a simple test.Let B measure the number of the slit chosen by the system, B 1 = 1 and B 2 = 2. Finding ⟨B⟩ W = −100, one must conclude that, with only two slits available, the system passes on average, through a slit number −100.
Could a WV be the "occupation number", counting the number of particles, or particleantiparticle pairs, as was suggested in [8]?Again, in a double-slit experiment where two paths lead to a dark fringe, A 1 = −A 2 , the "occupation numbers" given by WV's of the projectors B = |b 1 ⟩⟨b 1 | and B = |b 2 ⟩⟨b 2 | are ∞ and −∞, respectively.The fact that the system never arrives in its final state appears to rely on the presence (in some strange sense) of infinitely many copies of the system in each of the paths leading to it.
Finally, and at first glance most reasonably, is simply to say that if the WV of a projector is 0 (the weak pointer has not moved), the system has definitely not taken the chosen path [11], [17].Conversely, if the pointer has moved, the system must have been present in the not to say it at all.The barely speakable is, in practice, rather unspeakable.
Finally, it is remarkable the "weak measurements" approach, which purports "to revisit the whole quantum mechanical notion that physical variables are described by Hermitian operators" [34] has not, until now, been measured against the textbook orthodoxy of [1], [2], [3].Above we have tried to do just that, and even if the reader remains unconvinced by our arguments, he/she may be interested in revisiting the subject and drawing his/her own conclusions.
X. APPENDIX A. MEASUREMENTS: "STRONG" AND "WEAK", LOCAL AND NON-LOCAL.
Suppose Bob makes three measurements of Â, B and Ĉ. Bob will use three von Neumann pointers [29] with positions f A , f B , and f C , prepared in the initials states The functions G(f I , ∆f I ), whether complex or real valued, have a property [δ(f ) is the Dirac delta] and become very broad and slow-varying as ∆f I → ∞ .The pointers, which have no own dynamics, can be briefly coupled to the system by means of Ĥint I = −i∂ f A Îδ(t − t I ), t I = t 0 , t 1 , t 2 , so that each interaction entangles a pointer with the system.For example, we The probability of having at t = t 1 a reading f B , while measuring A k and C n at t = t 0 and t = t 2 , respectively, is, therefore, given by

C. Accurate "strong" measurements and locality
If the second measurement is also accurate, ∆f B → 0, there are only two possible readings, f B = 1 and f B = 0, Notably the likelihood of the system being seen to take a path {c n ← b 2 ← a k } depends only on the corresponding amplitude, A S (c n ← b 2 ← a k ).This can be seen as a "local property".
In an optical realisation of the experiment, the probability of arriving at a detector via one arm of the interferometer would not depend on what happens elsewhere in the setup.This dependence does, however, appear if one calculates the relative probabilities of arriving at the same detector via different arms.Indeed, given that the system arrives in |c n ⟩, for the frequency with which a path {c n ← b 2 ← a k } is chosen one has This is, however, still a "local" result since two probabilities in the denominator are evaluated separately for each of the routes shown in Fig. 5.The same can be said about the mean pointer reading, also given by D. Inaccurate "weak" measurements and non-locality In the opposite limit, where the measurement of B is highly inaccurate ("weak"), ∆f B → ∞, the interference between the paths in Fig. 5  Evaluating the integrals we obtain and

FIG. 2
FIG. 2: a) The simplest double-slit problem.Intermediate, of |b 1,2 ⟩, and final, |c 1,2 ⟩, play the role 1 .[The only exceptions are the cases where the choice of |b m ⟩ leaves only one path connecting the initial and final states since the other amplitude vanishes, as shown in Figs.2b and 2c.] FIG. 3: a) The initial pointer's state G(f ) in Eq.(13); b) Probability distribution of the pointer's readings, W (f ) = |Ψ(f )| 2 , for a system found in |c 2 ⟩ at t = t 2 (schematic, arbitrary units).If a reading f lies in regions I or II, Bob knows that B's value was B 1 or B 2 , respectively.Nothing can be said about the value of B if the reading lies in the region II.c) A projector B = |b 2 ⟩⟨b 2 | with the eigenvalues 0 and 1 is measured by a weakly coupled pointer, ∆f >> 1, in the case that the transition is forbidden, A(c 2 ← b 1 ← a 1 ) = −A(c 2 ← b 2 ← a 1 ).The readings are unusually large, |f | >> 1, and Alice always knows the value of B at t = t 2 .
∆f exceeds the difference of the measured values, and the condition λ > d is replaced by ∆f > |B 2 − B 1 |.Consider again the situation where one measures an operators B = |b 1 ⟩⟨b 1 |.The probability of finding a reading f , conditional on post-selection in |c 2 ⟩, is given by

FIG. 4 :
FIG. 4: a) Three virtual pathways with the amplitudes A 1 = −A 2 = A 3 .At t = t 1 Bob measures the operator B in Eq.(22), and at t = t ′ 1 , immediately after, the operator B′ .What happens depends on the accuracy of Bob's pointers.b) The distribution of the pointer readings, ρ(f, f ′ ) when both pointers are accurate (strong), ∆f = ∆f ′ = 0.1.c) The same distribution when both pointers are inaccurate (weak), ∆f = ∆f ′ = 10.d).The first pointer is accurate, ∆f = 0.1, and the second one is weak, ∆f ′ = 10.
This situation was considered in some detail in the previous two Sections.The case of ⟨ B′ ⟩ W = 0 is slightly different.Now the path amplitudes are the same, A 3 = A 2 , but the two eigenvalues have opposite signs, B ′ 3 = −B ′ 4 .To clarify the situation, Bob (aware of Carol's objections to using strong measurements) may decide to add yet another "weak" pointer, measuring B′′ = |L⟩⟨L| ⊗ |up⟩⟨up|.Whoever agrees with (a) and (b) must also accept that ⟨ B′′ ⟩ W indicates the presence of an "upwards grinning" cat at the left location.Having been told that there is no cat on the left, Bob expects to obtain ⟨ B′′ ⟩ W = 0, but finds instead ⟨ B′ ⟩ W = 1/2.Similarly, weakly measuring B′′′ = |R⟩⟨R| ⊗ |up⟩⟨up|, he finds the "up" component of the grin on the right, since ⟨ B′′′ ⟩ W is also 1/2.

44 )(
In the case of a degenerate eigenvalue C m one would need to sum Eq.(44) over all |c n ⟩ corresponding to the same C m .)Asan example, consider measuring at t = t 1 a projector B = |b 2 ⟩⟨b 2 | , with a non-degenerate eigenvalue 1, and (N − 1)-fold degenerate eigenvalue 0. This is a double-slit problems with N final states, sketched in Fig.5.

FIG. 5 :
FIG. 5: A "double-slit problem" arising from measuring a projector B = |b 2 ⟩⟨b 2 | in N = 4 dimensions.There are two virtual routes to a final state |c 1 ⟩, one passing through |b 2 ⟩ at t = t 1 , and the other not.The corresponding amplitudes are A 2 (1) and l̸ =2 A l (1), respectively.[A l (n) ≡ ⟨c n | Û S (t 2 , t 1 )|b l ⟩⟨b l | Û S (t 1 , t 0 )|a 1 ⟩.]By measuring both B and Ĉ accurately, Bob can determine the route chosen by the system.
corresponding to being in state |a k ⟩, measuring the value B m , and finally being in state |c n ⟩, may or may not show an "interference 1 ⟩ and |c 2 ⟩ (⟨c 2 |c 1 ⟩ = 0) [see Fig 2a].For an initial state |a 1 ⟩, special choice |c 1 ⟩ = Û S (t 2 , t 0 )|a 1 ⟩, makes |c 1 ⟩ and |c 2 ⟩, the bright and dark fringes, respectively, since