Sequential weak measurement
The notion of weak measurement provides a formalism for extracting information from a quantum system in the limit of vanishing disturbance to its state. Here we extend this formalism to the measurement of sequences of observables. When these observables do not commute, we may obtain information about joint properties of a quantum system that would be forbidden in the usual strong measurement scenario. As an application, we provide a physically compelling characterisation of the notion of counterfactual quantum computation.
Quantum mechanics is still capable of giving us surprises. A good example is the concept of weak measurement discovered by Aharonov and his group Aharonov et al. (1988a); Aharonov et al. (1991), which challenges one of the canonical dicta of quantum mechanics: that non-commuting observables cannot be simultaneously measured.
Standard measurements yield the eigenvalues of the measured observables, but at the same time they significantly disturb the measured system. In an ideal von Neumann measurement the state of the system after the measurement becomes an eigenstate of the measured observable, no matter what the original state of the system was. On the other hand, by coupling a measuring device to a system weakly it is possible to read out certain information while limiting the disturbance to the system. The situation becomes particularly interesting when one post-selects on a particular outcome of the experiment. In this case the eigenvalues of the measured observable are no longer the relevant quantities; rather the measuring device consistently indicates the weak value given by the AAV formula Aharonov et al. (1988a); Aharonov and Rohrlich (2005):
where is the operator whose value is being ascertained, is the initial state of the system, and is the state that is post-selected (e.g. by performing a measurement). The significance of this formula is that, if we couple a measuring device whose pointer has position coordinate to the system , and subsequently measure , then the mean value of the pointer position is given by
where denotes the real part. This formula requires the initial pointer wavefunction to be real and of zero mean, but these assumptions will be relaxed later. The coupling interaction is also taken to be the standard von Neumann measurement interaction . The coupling constant is assumed to be small, but we can determine to any desired accuracy if enough repeats of the experiment are carried out.
The formula (1) implies that, if the initial state is an eigenstate of a measurement operator , then the weak value post-conditioned on that eigenstate is the same as the classical (strong) measurement result. When there is a definite outcome, therefore, strong and weak measurements agree. However, weak measurement can yield values outside the normal range of measurement results, eg spins of 100 Aharonov et al. (1988b). It can also give complex values, whose imaginary part correspond to the pointer momentum. In fact, the mean of the pointer momentum is given by
where denotes the imaginary part and is the variance in the initial pointer momentum.
The fact that one hardly disturbs the system in making weak measurements means that one can in principle measure different variables in succession. We follow this idea up in this paper.
Ii A new paradox
Weak measurement has proved to be a valuable tool in analysing paradoxical quantum situations, such as Hardy’s paradox Hardy (1992); Aharonov et al. (1991). To illustrate the idea of sequential weak measurement and its potential applications we first construct a new quantum paradox. Consider the double interferometer, the optical circuit shown in Figure 1, where a photon passes through two successive interferometers. This configuration has been considered previously by Bläsi and Hardy Bläsi and Hardy (1995) in another context. Using the labels of the paths shown in the figure, and denoting the action of the -th beam-splitter by , the system evolves as follows:
(The signs here are determined by the fact that reflection on the silvered outer surface of a beam-splitter gives a phase of whereas transmission or reflection by the inner surface gives zero phase.)
Suppose now that we select a large number of successful runs of our experiment, i.e. those runs where the photon is detected by the detector .
We can now make the following statements about this situation:
(1) All photons go through path .
Indeed, equations (4) and (5) tell us that if a photon is injected along path A, it must exit the first interferometer along path . Consequently, if we measure the observable , the projector for path , we find the total number of photons detected is with certainty.
(2) All photons go through path .
Indeed, the second interferometer is arranged in such a way that any photon entering along path will end up at . Hence, a very simple calculation shows that if, instead of measuring , we measure , the number of photons going along path in all runs of the experiment, we will obtain with certainty .
(3) When photons go through path , a subsequent measurement reveals that half of them must go through path and half through path .
Indeed, if we measure the position of the photons in the first interferometer and find that all go via , then a subsequent measurement of and must yield in each case, up to statistical fluctuations. (In fact this is true regardless of whether or not all photons end up eventually at ).
(4) When photons go through path , a subsequent measurement reveals that half of them must have come via path and half via path .
This last statement is similar to point (3) above.
The above four statements seem to imply a paradoxical situation. On the one hand, statement (2) tells us, when we pool all the results, that all photons go via path ; together with statement (3) this implies that the number of photons that go along path must be . On the other hand, statement (1) tells us that all photons actually go along path ! A similar contradiction arises in connection with the number of photons going along path . On the one hand, statement (1) tells us that all photons go via ; together with statement (4) this implies that the number of photons that go along path must be only . On the other hand, statement (2) tells us that all photons actually go along path !
The usual way of resolving this paradox is to say that the above statements refer to measurements that cannot all be made simultaneously. Indeed, it is true that if we measure we find it is 1 with certainty, but only if we do not also measure . If we also measure in the same experiment, then it is no longer the case that . Similarly, it is true that with certainty, but only if we do not also measure . If we also measure in the same experiment, then it is no longer the case that . So, we are told, the statements (1)-(4) above have no simultaneous meaning, for they do not refer to the same experiment. Hence there is no paradox: In formulating the paradox presented above we made use of facts that are not all simultaneously true.
On the other hand, as is emphasised in Aharonov and Rohrlich (2005), one should not dismiss such paradoxes too lightly. Indeed it is possible to make a trade-off: By accepting some imprecision in measuring , , etc., we can limit the disturbance these measurements produce. The way to do this is to weaken the coupling of the measuring devices to the photons.
Since the disturbance is now small, we can make all the measurements in the same experiment, and we expect all the statements (1)-(4) to be true. Hence we expect , and obviously and . On the other hand, we also expect that , and , the total numbers of photons that went along and subsequently along or , respectively, should both be equal to ; this is because all the photons go via and half of them should continue along and half along . Also we expect , the number of photons that went along and subsequently along , to be . Similarly we expect that and should both be , since all photons go along and half of them must come via and half via .
While all the above predictions seem reasonable, here is the surprise: Overall we have only photons. They could have moved along four possible trajectories: , , or . Since and since it must be the case that ! Furthermore, our prediction has a remarkable internal consistency. We know that the total number of photons that go along must be zero. They can arrive at in two ways, either by or . Thus . As noted above, , but no photons are supposed to go through . This is due to the fact that is negative, i.e. .
The above predictions seem totally puzzling, no less puzzling than the original paradox. However, what we have now is not a mere interpretation that can simply be dismissed. These are now predictions about the results of real measurements - in particular the weak measurement of the number of photons that passes along path B and then along path F. This is a two-time measurement.
In general, by ensuring that the measurement interaction is weak, we can consider sequences of measurements. Describing such measurements is the main subject of our paper. In the process, we will formally derive the strange predictions made above for the double interferometer, and will discuss the interpretation of weak measurements. Finally, we apply these ideas to counterfactual computation, which is a catch-all for numerous counterfactual phenomena including, for example, interaction-free measurement Elitzur and Vaidman (1993).
Iii Sequential weak measurements
The situation we shall consider is where a system evolves unitarily from an initial state to a final post-selected measurement outcome . At various points, observables may be measured weakly. Here we consider the scenario where there is a single copy of the system, with the measuring device weakly coupled to it. Generally, reliable information will only be obtained after many repeats of the given experiment.
In the simplest case where there is just one observable, say, we assume the evolution from to the point where is measured is given by , and from this point to the post-selection the evolution is given by . Then we can rewrite (1) as:
and the mean of the pointer is given by (2) as before.
Consider next the case of two observables, and , measured at different times on a system . We assume the system evolves under from to the point where is measured, then under to the point where is measured, and finally under to . Our strategy is to use two measuring devices for measuring and . Let the positions of their pointers be denoted by nd , respectively. We couple them to the system at successive times, measure and , and then take the product .
We begin, therefore, with the weak coupling of system and pointers, with the usual von Neumann-type Hamiltonians for measuring and . The state of system and pointers after this coupling is:
where and are the two pointer momenta (the label refers to the system and , to the pointers). Here is the initial pointer distribution, and we have assumed, for simplicity, that the two pointers have identical initial distributions and equal coupling constants . Post-selecting on gives the state of the pointers as
As is small, we can approximate the state as:
Putting , we get
where , , , , and is defined by
Following measurement of and , the expected value of their product is given by
For simplicity, let us make the following assumption (we will discuss
the general case later):
Assumption A: The initial pointer distribution is real-valued, and its mean is zero, i.e. .
We also assume, without loss of generality, that is normalised so that . With these assumptions, all the terms in (13) of order 0 and 1 in vanish, and we are left with
where bars denote complex conjugates. Integration by parts implies , so we get the final result
Here is the sequential weak value given by (12); note the reverse order of operators, to fit with the convention of operating on the left.
Iv The sequential weak value
In the section above we considered two measurements – a measurement of at time and of at – and we looked at the product of the outcomes in the limit when the coupling of the measuring devices with the measured system was weak. This procedure was motivated by our example of the double interferometer: we wanted to check whether the photon followed a given path, say the path that goes along in the first interferometer and then along in the second interferometer. In that case the variables of interest are , the projector on path C and , the projector on path . When the photon follows this path, the value of the product of these projectors is while in all other situations the product is . We wanted to see what the behavior of the photon was when the measurements did not disturb it significantly.
Since measures and measures , it seems obvious that the quantity that represents the product of the two observables is given in (14) above. However, the situation is more subtle, as we show below.
Consider the simpler case of two commuting operators and , and suppose we are interested in the value of the product at some time . (Note that we are now talking about operators at one given time, not at two different times.) We can measure this product in two different ways. First, we can measure the product directly, by coupling a measuring device directly to the product via the interaction Hamiltonian . When we make the coupling weaker, we find that the pointer indicates the value
This is straightforward: it is simply the weak value of the operator . On the other hand, we could attempt to measure the product in the same way that we measured the sequential product. That is, we can use two measuring devices with pointer position variables and , couple the first measuring device to and the second to , and then look at the product . The latter method was proposed by Resch and Steinberg Resch and Steinberg (2004) for the simultaneous measurement of two operators. They showed that in this case
We see that the value indicated by is not equal to the weak value of the product, but contains a supplementary term, . In other words, although we expected the two methods to be equivalent, it is not the case. To obtain the true weak value of the product we must subtract this second term. This second term is an artifact of the method of using two separate measuring devices rather than coupling one measuring device directly to the product operator.
In the case of sequential measurement there is no product operator to start with, for we are interested in the product of the values of operators at two different times. Hence the first method, of coupling directly to the product operator, makes no sense, and we must use two independent couplings. In order to obtain the quantity of interest, i.e. the quantity that is relevant to situations such as the double interferometer of Section II, we must subtract the term from (15). We thus conclude that the quantity of interest is the sequential weak value given in (12).
V General sequential weak measurement
Sequential weak measurement can be easily extended to measurements of Hermitian operators with intervening unitary evolution steps . The weak values are given by
and the expected values can be expressed in terms of these weak values. For example, with Assumption A
and the case of general is given in the Appendix. Similarly, we can express expected values for products of momenta in terms of the weak values (see Appendix). For instance
Mixed products of positions and momenta give similar formulae. For instance
The foregoing examples illustrate a general pattern, which is that expectations of products of ’s and ’s depend on the real part of sequential weak values if there is an even number of ’s in the product and on the imaginary part if there is an odd number of ’s.
The sequential weak values satisfy the following rules:
1) Linearity in each variable separately:
for any .
2) Agreement with strong measurement:
Suppose that, with preselection by and post-selection by , strong measurements of , , , always give the same outcomes ; then .
3) Marginals: If is the identity operator at location :
We can illustrate some of these rules with the double interferometer experiment (figure 1). The measurements we consider are projectors that detect the presence of a photon on various edges; for instance, the projector indicates whether a photon is present on the edge . For simplicity we write for the weak value , etc., and we use the same convention for sequential weak values. Then using (7) we find , , and . Using (12) we find , , and . Since , rule 1) implies , and then rule 3) implies . Thus we expect , which holds if we substitute the values above. Similarly , and so on. As for rule 2), we have seen (Section II) that strong measurement of and yields 1, so we expect the weak values to be the same, as is the case.
There is a further rule that applies when one of the operators being measured is a projector. We illustrate it with the double interferometer. We can write
Here and in the final ratio are calculated assuming that , in other words, as though we were calculating weak values for the second interferometer treated separately from the rest of the system, with initial state and post-selection by (Figure 3). If we only knew the single-measurement weak values , and , we could calculate and using this rule and the relationship derived above.
Vi The meaning of weak values
Consider some experiment in which we inject some kind of particle and weakly measure the projector onto some location . Suppose we collect some large number of runs of the experiment that satisfy the post-selection criterion. We interpret the fact that the projector at has weak value to mean that, for any appropriate physical property we test, due for instance to the charge, gravitational field, etc. of the particle, it is as though particles (up to a binomial distribution error) passed along . Thus in the double interferometer experiment we expect all physical tests to give outcomes appropriate to there being, in all runs of the experiment, a total of photons passing along , photons passing along then , and so on.
Can we justify the foregoing interpretation of weak values? For weak measurements of a single operator, there is a body of work showing that weak values, even when they lie in an unexpected range, can be treated as though they were the actual values in the underlying physical theory and will then yield correct predictions. Examples of this include weakly measured negative kinetic energies when a particle is in a classically forbidden region Rohrlich et al. (1995), and weakly measured faster-than-light velocities that are associated with Cerenkov radiation Rohrlich and Aharonov (2002). If a measure is entirely consistent with physics in this fashion, then we are entitled to say that it is telling us a true physical fact. For sequential weak values, we can make a similar argument. The physical meaning of sequential weak values needs to be explored in many physical situations to give the kind of justification that single weak values enjoy. However, the internal consistency is already clear from the double interferometer example, and, more generally, from the rules in Section V.
Vii Broadening the concept: weak interactions
So far, we have considered ideal weak measurements, in which the pointer distribution is real and has zero mean (Assumption A). If we drop these assumptions, we find in place of (2) that
where , with , .
The expectation for a general initial pointer distribution, where each is either or , is a very complicated expression, but, so far as the system goes, depends only on the real and complex parts of sequential weak values up to . Thus we can write
for some polynomial function . The coefficients in are themselves polynomials in expectations for polynomials , as we see in the case of equation (23), where has this form.
In the next section, we shall want to consider the most general possible type of weak interaction which allows any sort of (suitably weak) coupling between the system and an ancilla followed by any further evolution or measurement of the ancilla alone (the pointer in our previous discussion and its von Neumann measurement interaction will be a special case of such an ancilla and weak interaction). Our notion of general weak interaction is the following: Consider the system and ancilla initially in product state . Let be any Hamiltonian of the joint system, and a coupling constant. For a single interaction event, and to first order in , the state becomes
Any joint Hamiltonian may be expressed as a sum of products of individual Hamiltonians
Post-selecting the system state in equation (25) with gives
So the system Hamiltonians have been effectively replaced by their weak values . The important point here is that all subsequent manipulations of the ancilla will depend on the pre- and post-selected system only through weak values of suitably chosen observables. A similar result clearly holds for any sequential weak interactions and suitably associated sequential weak values, and also for terms of any higher order in .
As a simple illustrative example, suppose that the ancilla is the pointer system of a von Neumann measurement interaction with Assumption A in force, and that this same pointer is weakly coupled twice for the sequential measurement of both and . If this pointer has position and momentum , the pointer state after post-selection is
The effect in this instance is therefore the same as adding the individual post-measurement results, and it depends on the system only through associated weak values.
Viii Counterfactuality and weak measurement
Counterfactual computation Jozsa (1999); Mitchison and Jozsa (2001) provides a general framework for looking at counterfactual phenomena, including interaction-free measurement as a special case. We consider arbitrary protocols, at various points of which a quantum computer can be inserted. The computer has a switch qubit (with =off and =on) and an output qubit. A special case of this formalism is where the protocol is represented by an optical circuit, and a computer insertion means that the computer (or a copy of it) is placed in some path of the circuit and is switched on by a photon passing along that path.
We assume that the computer is programmed ready to perform a computational task with answer or which will be written into the output qubit if the switch is turned on. In addition to the switch and output qubits, the protocol will in general have additional qubits, and will involve some measurements. We say that an outcome of these measurements determines the computer output if that outcome only occurs when the computer output has a specific value, or . Such an outcome is said to be counterfactual if its occurrence also implies that the computer was never switched on, i.e. its switch was never set to , during the protocol.
To make this precise, note first that one can always produce an equivalent protocol in which the state is entangled with extra qubits and the measurement deferred to the end of the protocol. Thus the protocol can be assumed to consist of a period of unitary evolution followed by a measurement, which can be assumed (again by adding extra qubits) to be a projective measurement. Let be the initial state of the protocol, and let be a measurement outcome that determines some specific computer output, in the sense defined above. Suppose the computer is inserted times. Let (for “oFf”) denote the projection onto the off value of the computer switch and (for “oN”) denote the complementary projector , and let be one of the possible strings of ’s or ’s of length ; we call this a history. Let denote the unitary evolution in the protocol between the th and th insertions of the computer.
Definition VIII.1 (Counterfactuality by histories Mitchison and Jozsa (2001)).
The measurement outcome is a counterfactual outcome if
1) determines the computer output.
2) The amplitude of any history containing an vanishes. In other words, for all histories other than the all- history, .
One may question whether this is the “correct” definition of a notion of counterfactual computation or whether alternative definitions might be convincingly plausible. Condition 1) is uncontroversial but condition 2) might seem less immediately compelling. It is evidently equivalent to obtaining a null result if we carry out a strong non-demolition measurement of N at each computer insertion. However the disturbance that such a measurement causes might lead one to question the suitability of this condition. Indeed recently Hosten et al. Hosten et al. (2006) proposed an alternative definition of counterfactual computation that violates condition 2) of definition VIII.1 and sparked a controversy Mitchison and Jozsa (2006) over the relative merits and validity of the two notions. We will now develop some alternative characterisations of our definition VIII.1 in terms of weak measurements, thereby addressing the disturbance issue. We will argue that these new characterisations considerably strengthen the credibility of the original definition as the “correct” one.
Let us therefore consider carrying out a weak measurement of at each insertion. A non-zero weak value implies that there is a detectable physical effect that can only occur if the computer is switched on. Vaidman’s treatment of the three-box paradox Vaidman (2006) gives a good example of this reasoning.
Our two-interferometer example shows that it does not suffice to consider the individual weak values at each insertion. For suppose the computer is inserted in paths and , as shown in Figure 4. Then we have seen that the weak values and are zero, yet the sequential weak value is non-zero. The non-vanishing of the sequential weak value implies that a photon passes along both path and , since there is a physical effect that causes correlated deflections of pointers at both sites.
There is a subtlety here, because it could be argued that, because sequential pairwise weak measurements give second-order effects in (see (15)), we might detect a departure from zero in the weak measurements for each operator individually, i.e. in the deflections of the pointers at and , if we looked at second or higher order terms in . However, if is any projector and , then the von Neumann interaction reduces to , which is the identity to all orders in in the weak measurement calculation. Thus we truly need to carry out the sequential weak measurement here to identify the physical effect due to the photon.
In general, we need to consider all possible sequential weak measurements to obtain an adequate test of counterfactuality. This is why we must use weak rather than strong measurements. As we have seen in Section IV, there is no strong measurement corresponding to sequential weak measurements.
We therefore propose the following:
Definition VIII.2 (Counterfactuality by weak values).
The measurement outcome is a counterfactual outcome if
1) determines the computer output.
2) , for any , where is the number of insertions of the computer.
By (18), conditions 2) for VIII.1 and VIII.2 are equivalent, using the fact that together with the linearity and marginal rules. For instance, with two insertions of the computer, condition 2) of Definition VIII.1 amounts to , and , and these imply , and , which constitute condition 2) for Definition VIII.2.
We can try to strengthen the requirements for counterfactuality by demanding that a zero response is obtained for any conceivable weak interaction, in the sense of the preceding section. In our present application we must further restrict the weak interaction to take place only if the switch has the property of being ”on”, i.e. the interaction Hamiltonian must have the form . We say that such an interaction is a weak interaction involving the projector . Since is a one-dimensional projector, this implies that the interaction Hamiltonian has the form . In a more general scenario the projector for counterfactuality (analogous to the switch being ”on”) may have rank larger than 1 and then the interaction Hamiltonian may have the more general form for any Hermitian . For example, the switch may be a photon with both path and polarisation properties. Then a weak interaction restricted to its presence on a path would correspond to a two-dimensional projector on its polarisation state-space associated to that path.
Definition VIII.3 (Counterfactuality by general weak interactions).
The measurement outcome is a counterfactual outcome if
1) determines the computer output.
2) Any possible weak interaction involving the projections yields a null result.
By a null result, we mean the same result that would be obtained for . It is not difficult to show that this apparently much broader concept is in fact equivalent to Definition VIII.2. In one direction, we know from the last section that any expectation depends only on the sequential weak values, involving the projectors , so when these weak values vanish we obtain a null result. In the other direction, we have only to show that we can choose particular weak interactions whose null results will imply the vanishing of all sequential weak values. However, if we first obtain a null value of and for the standard von Neumann measurement weak interaction for every , then we know by (2) and (3) that both real and imaginary parts of all the weak values are zero. Then by obtaining null values of and for all , we infer from (15) and (21) that the real and imaginary parts of all are zero. We continue this way, using the fact that expectations of products of ’s and ’s with an even number of ’s depend on the real part of sequential weak values, whereas those with an odd number of ’s depend on their imaginary parts (see Appendix).
We have therefore proved:
Sequential weak values are a natural generalisation of the weak value of a single measurement operator Aharonov et al. (1988a). Resch and Steinberg’s simultaneous measurement of two operators Resch and Steinberg (2004) gives the same result in the special case where these operators commute, but it does not address the case where we have a succession of measurements with unitary evolution between them.
One can argue that both single and sequential weak measurements tell us what the physical situation is. In the double interferometer, for instance, really means that all the photons go via , and really means that approximately half the photons go via followed by . This is of course a matter of interpretation, and may be disputed; but at least it seems to be true that weak values can be fitted into the framework of physics without contradiction, and give illuminating explanations of many phenomena.
Our application of weak measurement to counterfactuals does not depend on the foregoing interpretation. The most straightforward part of our claim is that, if a weakly coupled measuring device indicates a displacement of pointers in some region of an apparatus, then one cannot claim that the state of the system was unaltered in that region; for example, in the case of an optical device, such a shift would indicate that a photon was present. The importance of sequential weak measurements in this context is illustrated by the double interferometer (Figure 1). If two pointers are coupled to the paths and in this apparatus, each pointer individually will show no displacement on average after many runs of the experiment. However, the product of the positions of the pointers will show a shift. Thus the photon reveals its presence only when information from both pointers is suitably combined.
The other part of our claim about counterfactuals can be summed up by what we might call the principle of weak detectability:
An event that cannot be detected by any possible weak interaction does not take place.
This means that we learn a fact about an event counterfactually from a certain experiment if (1) the outcome of the experiment implies , and (2) no possible weak interaction can detect the occurrence of this event during the experiment. It seems as though part (2) might be hard to confirm, because there is a great variety of possible weak interactions. However, this condition proves to be equivalent to the vanishing of all sequential weak values associated to the event in question, and this will often be much easier to check.
Finally, we mention the striking fact that sequential weak values are formally closely related to amplitudes. Consider the case where we measure projectors that define a path between the initial and post-selected states and , respectively. We can write
where runs over all paths between and . Nonetheless, weak values are like measurement results rather than amplitudes! This way of looking at sequential weak values suggests a close connection with path integrals that remains to be explored.
We thank L. Vaidman for helpful comments on an earlier version. GM acknowledges support from the project PROSECCO (IST-2001-39227) of the IST-FET programme of the EC. RJ and SP are supported by the EPSRC QIPIRC and EC networks QAP and QICS, and SP also acknowledges support from the EPSRC grant GR/527405/01.
Appendix A Calculation of general correlations
With Assumption A, we show here that the general version of (15) is
where the weak values in this formula are given by 18. In (30) the sum is over all ordered indices with for , and ordered indices that make up the complement of in the set of integers from to , i.e. that satisfy and . We include the empty set as a possible set of indices. In order not to count indices twice, we require , and when we require .
For instance, with , the possible indices are , ; , , which yields
Assumption A implies that only the terms in in need to be taken into account in calculating
and this leads to (30).
We can also calculate , the product of the momenta of the pointers. To do this, it is convenient to move to the momentum basis, replacing by its Fourier transform and carrying out an expansion in the :
Assumption A implies that only the terms in in need be considered in calculating
It is simplest to treat the cases of even and odd separately. For the even case we have
and for the odd case:
The case of mixed products of positions and momenta are treated similarly, and they depend only on the real or imaginary parts of the sequential weak values given by (18). For example, to calculate we express the first variable in the position basis and the second in the momentum basis:
which yields (21). For these mixed products, since there is a factor of for each in the product, we take the imaginary part of weak values when there is an odd number of ’s present and the real part otherwise.
Thus all possible expectations of products of position or momentum can be obtained from the sequential weak values.
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