Anomalous Weak Values Without Post-Selection

A weak measurement performed on a pre- and post-selected quantum system can result in an average value that lies outside of the observable's spectrum. This effect, usually referred to as an"anomalous weak value", is generally believed to be possible only when a non-trivial post-selection is performed, i.e., when only a particular subset of the data is considered. Here we show, however, that this is not the case in general: in scenarios in which several weak measurements are sequentially performed, an anomalous weak value can be obtained without post-selection, i.e., without discarding any data. We discuss several questions that this raises about the subtle relation between weak values and pointer positions for sequential weak measurements.Finally, we consider some implications of our results for the problem of distinguishing different causal structures.


INTRODUCTION
All quantum measurements are subjected to a fundamental trade-off between information gain and disturbance of the measured system. In particular, one can perform weak measurements that provide little information but only weakly perturb the system. A particularly interesting situation arises when weak measurements are combined with post-selection [1]. This can be conveniently described within the von Neumann model of quantum measurements, where the quantum system to be measured is coupled via a joint unitary operation to another quantum system, the pointer, which represents the measurement device. The measurement is then completed by performing a strong measurement of the pointer.
More formally, consider a system initially prepared (or pre-selected) in a pure state |ψ , and an observableÂ to be weakly measured on it. The system-pointer interaction is generated via a Hamiltonian of the form H = γÂ ⊗p, wherep denotes the momentum operator acting on the pointer. The latter is initially in a state |ϕ(0) , which we shall take here to be a Gaussian wave packet centred at a position x = 0 with spread σ. Assuming that we are in the weak measurement regime, with the coupling constant γ and interaction time ∆t such that g := γ∆t is small enough compared to the spread of the pointer, the global state after the coupling is given by e −iĤ∆t |ψ |ϕ(0) ≈ (1 1 − igÂp) |ψ |ϕ(0) (1) (where tensor products are implicit, and taking = 1).
For simplicity we will henceforth choose units so that g = 1; the strength of the measurement will then be controlled solely by the pointer spread σ, and the validity of the weak regime will depend only on this being sufficiently large (see Appendix for details). Next, the system is post-selected onto the state |φ (e.g. via a strong projective measurement). The final state of the pointer is then (up to normalisation) where is the so-called weak value of the observableÂ given the pre-selection in the state |ψ and post-selection in the state |φ [1]. The mean position of the pointer is thus displaced (via the displacement operator e −iA φ ψp ; see Appendix) to x ≈ ϕ(0)| e iA φ ψpx e −iA φ ψp |ϕ(0) = Re(A φ ψ ).
Note that the definition (3) of a weak value can be generalised to post-selections on a given result for any general quantum measurement [2, 3]. In particular, a trivial, deterministic measurement of the identity operator 1 1 amounts to performing no post-selection. This allows one to also consider a weak value with no post-selection, defined (see Appendix) as With this definition, Eq. (5) gives x = A 1 1 ψ = Re(A 1 1 ψ ): we recover the same relation as in Eq. (4), although now A 1 1 ψ is restricted to lie in [λ min (Â), λ max (Â)] since here it is simply equal to the expectation value ofÂ.
The phenomenon of a weak value outside the spectrum ofÂ is referred to as an "anomalous weak value" [1, 4, 5]. This has been observed in many experiments [6][7][8], and appears to be directly linked to various (a priori unrelated) areas such as tunnelling times [9] and fast light propagation [10,11]. In practice, anomalous weak values allow for the detection and precise estimation of very small physical effects [12][13][14][15], via a form of signal amplification. While astonishing at first sight, anomalous weak values can in fact be intuitively understood in terms of destructive interference of the pointer state, which occurs as a result of post-selection. With this in mind and given the rudimentary analysis above, it is rather natural to attribute the origin of anomalous weak values to the presence of post-selection; this opinion indeed seems to be widely shared in the community.
Here we show, however, that this is not the case in general, and that anomalous weak values can in fact be obtained deterministically, without any post-selection at all. Specifically, we consider a situation in which two successive weak measurements are performed on a quantum system. The experiment thus involves two pointers, one associated to each weak measurement. Considering observables that are simply given by projectors, one expects to find the mean position of each pointer between 0 (the system's state being orthogonal to the projector) and 1 (the system's state aligning with the projector). Yet, we will see that the average of the product of the pointer positions can become negative. This may be understood in terms of the second measurement acting as an effective post-selection of the system, thus creating the desired interference. Importantly however, no data is discarded. Below, after discussing in detail a simple example of this effect, we provide more general insight and results on anomalous weak values and pointer positions obtained with no post-selection.

ILLUSTRATIVE EXAMPLE
To start with, let us consider a qubit system initially prepared in the state |0 , undergoing a sequence of two weak von Neumann measurements of the projection observables |ψ j ψ j | (j = 1, 2), where the states |ψ j and their orthogonal states |ψ ⊥ j are defined as To each measurement is associated a pointer in the state |ϕ j (x j ) , where x j is the mean position of the pointer wavefunction. The two pointers are initially independent, and both centred at x j = 0. The initial state of the system and pointers is therefore Following the von Neumann measurement procedure described earlier with interaction HamiltoniansĤ j = γ j |ψ j ψ j |p j , the average post-measurement position of the corresponding pointer is (with appropriate units so that γ j ∆t j = 1 as before) x j = 1 if the state of the system is |ψ j ; if the state is |ψ ⊥ j then the pointer does not move. The state of the system and pointers after the interaction with the first pointer is thus After interacting with the second pointer, it evolves to |ψ 2 |ϕ 1 (1) |ϕ 2 (1) + 3 4 |ψ 2 |ϕ 1 (0) |ϕ 2 (1) Tracing out the system, one finds that the joint pointer state is a mixture of the following two states, with probabilities p (1) = 5/8 and p (0) = 3/8 respectively. Finally the positions of the pointers are measured. The quantity of interest is the average of the product of the pointer positions, i.e., the expectation value x 1 ⊗x 2 . Since both states in Eq. (11) are product states of the two pointers it follows that, for each state in the mixture, x 1 ⊗x 2 = x 1 x 2 . As x 2 = 1 for |Φ (1) and x 2 = 0 for |Φ (0) , the expectation value of x 1 ⊗x 2 is thus simply p (1) Φ (1) |x 1 |Φ (1) .
Note that we have not yet specified the strength of either measurement. Considering Gaussian pointers with widths σ j for each measurement, we find (see Appendix) Notice that this quantity depends on σ 1 but not on σ 2 : the strength of the second measurement has no effect here.
Since both observables being measured are projectors with spectra {0, 1}, one would naturally expect an average value within the range [0, 1]. Independently of the strength of either measurement, each pointer, taken individually, indeed has an average position in [0, 1]: specifically, x 1 = 1/4 and x 2 = 5/8 (see Appendix). In the regime where the first measurement is strong (i.e. σ 1 → 0), Eq. (12) gives x 1 ⊗x 2 ≈ 1/16, which is consistent with the above argument. However, if the first measurement is sufficiently weak (i.e. σ 1 is large enough), the average value can become negative. In the limit σ 1 → ∞ we get This pointer reading is anomalous in that it gives an average value outside of the natural range. As we will discuss in more detail below, this result can be linked to an anomalous weak value without post-selection, (|ψ 2 ψ 2 | · |ψ 1 ψ 1 |) 1 1 0 := 0|ψ 2 ψ 2 |ψ 1 ψ 1 |0 (see Eq. (18) below); specifically, we have here We emphasise that this anomalous value is obtained despite the absence of post-selection. This effect can nevertheless be understood intuitively by considering that the second measurement acts as an effective post-selection on |ψ 2 , as the corresponding pointer moves only in this case. This becomes apparent upon rewriting the above weak value as which differs from the standard weak value (|ψ 1 ψ 1 |) ψ2 0 for a post-selection on |ψ 2 only by the factor | ψ 2 |0 | 2 , which is the probability that the projection of |0 onto |ψ 2 is successful. As it turns out, this factor ensures in particular that the anomalous weak value without postselection cannot be arbitrary large, a fact that we prove further below. For a sequence of two projection observ-ablesÂ andB (with eigenvalues 0 and 1), the above value of −1/8 for the real part is indeed the most anomalous value obtainable (see Appendix).

ANALYSIS FOR ARBITRARY OBSERVABLES
In order to place the previous example in a more general framework, let us recall some facts about sequential weak measurements of noncommuting observables [16][17][18]. To this end, consider a system prepared in the pure state |ψ , which is subjected to a sequential weak measurement of the observablesÂ thenB, before being postselected onto the state |φ . The system-pointer interaction Hamiltonians areĤ 1 = γ 1Âp1 andĤ 2 = γ 2Bp2 . We will choose again, for simplicity, the coupling constants and interaction times such that γ j ∆t j = 1, and take Gaussian pointers initially in the states |ϕ 1 (0) and |ϕ 2 (0) with widths σ 1 and σ 2 , which dictate the measurement strengths.
However, while the notion of an anomalous weak value for single (non-sequential) weak measurements is intimately linked to the pointer displacement (and even justified) by the relation x = Re(A φ ψ ), the relationship between the mean pointer positions and (BA) φ ψ is more subtle for sequential weak measurements. In the presence of postselection, it has instead been shown [16,19] that within the weak regime (with large enough widths σ 1 and σ 2 ). This cautions that some care must be taken when linking (possibly anomalous) pointer positions to weak values.
Let us now generalise the sequential weak value of Eq. (6) to the case without post-selection, by defining, in a similar way to before, the sequential weak value with no post-selection as Connecting this to the pointer positions, we prove in the Appendix that, contrary to Eq. (17) (which was obtained with post-selection), we recover here the direct relation as anticipated already in Eq. (14), which holds as long as the first measurement is sufficiently weak. This justifies that our earlier illustrative example could indeed be interpreted as yielding an anomalous weak value without post-selection. Crucially, although for a single measurement without post-selection A 1 1 ψ is simply the expectation value ofÂ, no such interpretation can be given to (BA) 1 1 ψ sinceBÂ is only Hermitian -and thus defines an observable -ifÂ andB commute. In particular, this implies that (BA) 1 1 ψ need not be contained within the interval where Λ min(max) (Â,B) = min(max) k,ℓ λ k (Â)λ ℓ (B), as one one would naturally expect for the product of outcomes for a measurement ofÂ thenB.
Nevertheless, as we noted after Eq. (15), the value of (BA) 1 1 ψ cannot be amplified arbitrarily. It is possible to place a more quantitive bound on the values that it can in fact take. Using the Cauchy-Schwartz inequality, we indeed have (where · is the spectral norm). Thus, although one can obtain anomalous weak values without post-selection, their magnitude cannot be pushed outside what one can obtain using strong measurements. The bound above implies in particular that for observables with symmetric spectra, the real part of the weak value -and therefore the mean product of pointer positions, see Eq. (19) -cannot be anomalous; anomalous pointer positions are only obtained for observables with asymmetric spectra, such as projection observables. Nevertheless, one can also obtain complex weak values for observables with symmetric spectra, which can similarly be considered anomalous. Take, for example, a system initially prepared in the (+1)-eigenstate |0 of the Pauli matrixσ z , on which a sequential weak measurement of the Pauli observablesσ y andσ x is performed. One thus obtains (σ x σ y ) 1 1 0 = i. The imaginary part of the weak value here can be detected by measuring the pointer momenta [15,19] (see Appendix). We note again that such a complex anomalous weak value cannot be obtained without post-selection with only a single weak measurement.

MORE MEASUREMENTS
Eq. (20) might bound how anomalous a weak value can be without post-selection, but it is not generally tight. For two projection observablesÂ andB (with eigenvalues ±1), for example, it only implies a bound Re[(BA) 1 1 ψ ] ≥ −1; nevertheless, as we prove in the Appendix, the value of −1/8 obtained earlier for the real part of the weak value is the most negative value that one can obtain. Can one do better by considering longer sequences of successive weak measurements? Here we will see that this question has a subtle answer: the weak value itself can approach −1, but this will not mean the average product of the pointer positions does so as well.
For a sequence of n observablesÂ 1 , . . . ,Â n to be measured weakly on the state |ψ before a post-selection on |φ , the sequential weak value is defined (following, e.g., Ref. [16]) as When no post-selection is performed, this can be generalised to in analogy to the cases discussed earlier. As we show in the Appendix, a similar bound to Eq. (20) can be derived, namely For n projection observables, this implies the bound Re[(A n · · · A 1 ) 1 1 ψ ] ≥ −1. As it turns out, it is possible to obtain an anomalous sequential weak value without post-selection approaching −1 and thus saturating this bound in the limit n → ∞. To see this, take the initial state of the system to be |ψ = |0 and consider the sequence of n qubit projectorsÂ j = |a j a j | with |a j = cos( jπ n+1 ) |0 +sin( jπ n+1 ) |1 for j = 1, . . . , n. This sequence of weak measurements gives Note that for n = 2 this coincides precisely with the explicit two-measurement example we began with. As discussed above, for two sequential weak measurements in the absence of post-selection, the mean product of the pointer positions gives precisely the real part of the sequential weak value; see Eq. (19). However, for n > 2 measurements this direct relationship is broken and the mean product of the pointer positions corresponds instead to a mixture of sequential weak values for 2 n−2 different permutations of the observables (see the Appendix for an explicit expression). For example, for n = 3 we have, in the weak regime, The real part of (A 3 A 2 A 1 ) 1 1 ψ is thus not directly observed. However, as we show in the Appendix, its value (as well as the imaginary part) can nonetheless be deduced experimentally by measuring several different expectation values of the products of pointer positions and momenta [16].
Interestingly, by numerically minimising the mean product of the pointer positions for sequences of up to 5 projection observables, we were unable to obtain a value smaller than −1/8, and we conjecture that this is in fact the case for all n. Thus, although the weak value itself can be brought arbitrarily close to −1, it seems that additional sequential weak measurements may not lead to "more anomalous" pointer positions. This behaviour highlights oft-overlooked subtleties in the connection between anomalous weak values and pointer positions: for individual weak measurements, there is a direct correspondence between the pointer position and (the real part of) the weak value, and an anomalous weak value has an immediate physical relevance. For sequential weak measurements, a distinction must be made between anomalous weak values and anomalous pointer positions (with post-selection, this is already the case for two measurements; see Eq. (17) or Ref. [16]).
This divergence between weak values and pointer positions for sequential weak measurements means that, in general, it is more difficult to give a clear physical interpretation to sequential weak values, anomalous or not. Indeed, while some authors have argued that weak values for single weak measurements should be considered real properties of quantum states with direct physical meaning [20,21], it is unclear whether such arguments are justified for sequential weak values [16] given the lack of examples of physical scenarios where they play a direct, crucial role.

FURTHER DISCUSSION
Compare the situation of a sequential weak measurement of two observablesÂ andB with the alternative in which a bipartite system |ψ ab ∈ H a ⊗ H b is prepared andÂ andB are weakly measured on the two different substituent systems. One can view this either as a measurement of the joint observableÂ⊗B (with two different pointers, one coupled to each observable) or a sequential measurement of the commuting observablesÂ ⊗ 1 1 and 1 1 ⊗B. In the absence of any post-selection, one has (A ⊗ B) 1 1 ψ ab = ψ ab |Â ⊗B|ψ ab which, being just an expectation value, cannot lie outside the spectrum of the product observableÂ ⊗B. For tensor product measurements, an anomalous weak value is thus unobtainable without post-selection.
This observation raises some interesting implications. Consider for example a scenario in which two parties, Alice and Bob, each operate in a closed laboratory. Each receives a system, performs a weak measurement, and sends the resulting system out; they then come together to jointly measure their pointers. By repeating this many times (or on a large number of systems), they thus determine x 1 ⊗x 2 . If Alice and Bob have no knowledge of their causal relationship, they could unknowingly be weakly measuring the same system at different times (either Alice then Bob, or vice versa), or measuring different parts of a (potentially entangled) bipartite system. By observing an anomalous weak value they can differentiate between these two scenarios.
The problem of distinguishing these two causal structures for quantum systems -the former is known as a direct cause relationship, while the later a common cause relationship, since any correlations must be due to a (possibly quantum) common cause -has been the subject of recent interest; see, e.g., Refs. [22][23][24][25][26]. An anomalous weak value thus provides a novel way to witness a direct causal relationship and distinguish between these cases. We leave it as an open question whether such a witness can be found whenever Alice and Bob are capable of signalling to each other; i.e., if whenever they are connected by a quantum channel of nonzero capacity they can always find a pair of observables to measure that would generate an anomalous weak value without post-selection.
Recently, there has also been substantial interest in quantum processes that are not consistent with any definite (possibly stochastic) causal ordering [27], and practical approaches to witness such "indefinite" causal orders have been developed [28,29] and experimentally tested [30,31]. It would be interesting to see whether indefinite causal orders may also be witnessed by, for example, producing larger anomalous weak values than possible in a well-defined causal structure. [ The weak measurement regime can in general be defined for any type of von Neumann measurement interaction scheme, as introduced in the main text, by comparing the various parameters that describe it: the strength of the measurement interaction (γ), the time of the interaction (∆t), the width of the measurement pointer (σ), the eigenspectrum of the observableÂ being measured as well as the weak values to be considered. The relation between weak values and pointer positions can then be obtained by taking the appropriate limits.
As the point of our paper is to analyse specific cases of anomalous weak values and anomalous pointer positions, for simplicity we choose a specific form for the pointer states, namely, Gaussian states. By a Gaussian pointer, we mean a pointer whose state |ϕ(a) is described by a Gaussian wavefunction as follows: where {|x } x is a continuous eigenbasis of the pointer positionx. For a ∈ R, |ϕ(a) is properly normalised; for a complex value of a, its norm is e Im(a) 2 4σ 2 . The mean position of the pointer in the state |ϕ(a) (possibly after renormalisation) is x = Re(a) and its variance is x 2 − x 2 = σ 2 , while the mean value of the momentum operator Prior to the measurement, we always start with a = 0. Note that an operator of the form e −iαp (for any α ∈ C) acts as a displacement operator, such that e −iαp |ϕ(a) = |ϕ(a + α) .
Let us clarify here the conditions that define the standard weak measurement regime, under which the approximations of Eqs. (1)-(2) of the main text are valid. As everywhere in the paper, we choose units such that g = γ∆t = 1. By considering the spectral decompositionÂ = k a k |a k a k | of the observable under consideration and the completeness relation 1 1 = k |a k a k | (with {|a k } k an orthonormal basis of the system Hilbert space), one can write e −iÂ⊗p = k |a k a k | ⊗ e −ia kp and 1 1 − iÂ ⊗p = k |a k a k | ⊗ (1 1 − ia kp ), so that the difference between the left and right hand sides of Eq. (1) is The approximation of Eq. (1) is valid if for each k (for which | a k |ψ | is non-negligible), the norm of |δ k is small enough (compared e.g. to that of the lhs of Eq. (1), which is 1). Using Eq. (A1), one finds Similarly, the difference between the two lines of Eq.
(2) is (ignoring the common prefactor φ|ψ ) which is also small if Thus the weak regime is valid whenever the two conditions (A4) and (A6) are fulfilled.

Relating weak values to pointer positions and momenta
The case of a single weak measurement It will be useful to relate here the formula for a weak value to the pointer position and momentum in a more general setting than that considered in the main text, where the initial state is not considered a priori to be pure, and where post-selection is conditioned on a given result of an arbitrary Positive-Operator Valued Measure (POVM) measurement on the system, rather than a projective measurement. In such a setting, Eq. (3) can be generalised to which now defines the weak value of the observableÂ, given the pre-selection in the state ρ and post-selection by the POVM element E. This definition was first proposed explicitly in Ref.
[2] (although earlier alluded to in Ref. [32]), and shown to be indeed the natural generalisation of the standard definition (3) of a weak value. Note already that this definition indeed reduces to Eq. (3) if the preparation is a pure state |ψ (i.e. for ρ = |ψ ψ|) and the post-selection is a projection onto another pure state |φ (for E = |φ φ|). It also allows one to define a weak value with no post-selection by taking a trivial POVM element E = 1 1, which indeed reduces to the definition of Eq. (6) in the case of a pure state ρ = |ψ ψ| -and which simply coincides here, in the case of a single observable, with the expectation value ofÂ; note in particular that, contrary to a general weak value, the weak value with no post-selection is linear in the pre-selected state.
In the von Neumann measurement scenario that we consider here, one thus prepares the density matrix ρ, weakly measuresÂ (= k a k |a k a k |) (with a pointer in a Gaussian state as described above), and finally post-selects an outcome corresponding to the POVM element E. The initial density matrix of the system and pointer is given by Under the interaction between the system and the pointer, the joint state evolves to Due to the post-selection upon E, the state of the pointer is then projected onto the (unnormalised) state Tr E |a k a k | ρ |a ℓ a ℓ | |ϕ(a k ) ϕ(a ℓ )| (A11) (where Tr S is the partial trace over the state of the system).
The expectation value of the position of the pointer, given that the post-selection was successful, is mn Tr E |a m a m | ρ |a n a n | ϕ(a n )|ϕ(a m ) .

(A12)
Evaluating the expressions in the fraction above for the Gaussian pointer of Eq. (A1) (with a k , a ℓ , a m , a n ∈ R), and taking the weak limit approximation in which σ ≫ |a k − a ℓ |, σ ≫ |a m − a n |, one finds, to the lowest order, so that we are left with x ≈ kℓ Tr E |a k a k | ρ |a ℓ a ℓ | (a k + a ℓ )/2 mn Tr E |a m a m | ρ |a n a n | = Tr(EÂρ) + Tr(EρÂ) /2 Tr(Eρ) = Re Tr(EÂρ) where we used the spectral decomposition ofÂ and the cyclic property of the trace, together with the fact that E,Â and ρ are Hermitian and thus Tr(EPÂ) = Tr((EPÂ) † ) * . Recalling the generalised definition (A7) of a weak value, we thus find that the mean position of the pointer (when the post-selection is successful) is, as in Eq. (4), One may also consider measuring the expectation value of the momentum of the pointer instead, conditioned again on a successful post-selection: mn Tr E |a m a m | ρ |a n a n | ϕ(a n )|ϕ(a m ) .

(A16)
The relevant quantity for the pointer states is, in the weak regime approximation, from which (together with Eq. (A13)) we find that the expectation of the pointer's momentum is mn Tr E |a m a m | ρ |a n a n | = 1 2σ 2 Tr(EÂρ) − Tr(EρÂ) /2i Tr(Eρ) = 1 2σ 2 Im that is, Note that, unlike the expression for x , this depends explicitly on the width of the pointer. The expectation value of the momentum is thus directly linked here to the imaginary part of the weak value. Using Eqs. (A15) and (A19), one may therefore recover both the real and imaginary parts of A E ρ from the expectation values of the pointer's position and momentum (in a regime where σ is large enough to ignore higher order terms, but not so small as to render the term that remains above, with the pre-factor 1 2σ 2 , unmeasurable).

Two sequential weak measurements
Let us now turn to the sequential measurement of two observablesÂ andB. The sequential weak value (BA) φ ψ for a system prepared in the pure state |ψ and post-selected in |φ , defined in Eq. (16), was introduced in Ref. [16]. As in the previous section, one may consider a more general setting where the initial state ρ is not necessarily assumed to be pure, and the post-selection is conditioned on a general POVM element E. In such a case, the definition of Eq. (16) can, similarly to Eq. (A7), naturally be generalised to (BA) E ρ := Tr(EBÂρ) Tr(Eρ) . (A20) One can indeed verify that one recovers Eq. (16) for ρ = |ψ ψ| and E = |φ φ|. As before, this definition also allows one to define a sequential weak value with no post-selection by taking the trivial POVM element E = 1 1, as in Eq. (18) for the case of a pure state ρ = |ψ ψ|. Again, and contrary to a general sequential weak value, the sequential weak value with no post-selection is linear in the pre-selected state; note, however, that it no longer coincides with an expectation value, as in general the productBÂ is not Hermitian, and thus does not define a valid observable.
We consider here a sequential von Neumann measurement scenario where two separate Gaussian pointers (labelled by the subscripts j = 1, 2) are used to measureÂ (= k a k |a k a k |) andB (= m b m |b m b m |) on a system prepared in the state ρ and post-selected on a POVM element E. Similarly to the analysis in the previous section, the final (unnormalised) state of the two pointers after the post-selection is given by (with implicit identity operators) Using the weak regime approximations of Eq. (A13), for both the weak measurement ofÂ and ofB, we find that the expectation value of the product of the pointer positions, given that the post-selection was successful, is The real and imaginary parts of the weak value (BA) E ρ = Tr(EBÂρ) Tr(Eρ) are thus not directly given by the mean values of the pointer positions and momenta (conditioned on a successful post-selection), as observed previously in Ref. [16,19], but can still easily be recovered by combining the mean values above as follows: Nevertheless, if no post-selection is made (E = 1 1), then the two summands in Eq. (A22) and in Eq. (A23) are equal, and one again directly obtains (as in Eq. (19) for x 1 ⊗x 2 ). Moreover, these expressions hold as long as the weak regime is applicable for the first measurement, irrespective of the strength of the second. To see this, note that, in the absence of post-selection, Eq. (A21) reduces to Eq. (A13) thus holds with exact equalities for the terms in Tr(x 1 ⊗x 2 η) and Tr(p 1 ⊗x 2 η) corresponding to the second pointer, and the weak regime approximation is thus only required for the first measurement. This also justifies further the claim that the second measurement can be seen as performing an effective post-selection, since it can be taken to be arbitrarily strong.
The example of an imaginary anomalous weak value without post-selection using Pauli observables described in the main text, for which (σ xσy ) 1 1 0 = i, can, from Eq. (A27), thus be observed by measuring p 1 ⊗x 2 .
Generalisation to n sequential weak measurements Considering now n sequential weak measurements, the previous definition of a weak value, for a system prepared in the (possibly mixed) state ρ and post-selected on a POVM element E can be generalised to (A n · · · A 1 ) E ρ := Tr(EÂ n · · ·Â 1 ρ) Tr(Eρ) .
which concludes the proof.
Note, furthermore, that by the linearity of the weak value with no post-selection with respect to the pre-selected state, (BA) 1 1 ρ = i q i (BA) 1 1 ψi for a mixed state ρ = i q i |ψ i ψ i |, which implies that the bound above also holds for (BA) 1 1 ρ ; that is, Re[(BA) 1 1 ρ ] ≥ −1/8 for any two projection observablesÂ andB and any mixed state ρ.
We further note that by using the linearity of the weak value with no post-selection and the triangle inequality, it is easy to see that Eq. (23) also holds for a preparation in any mixed state ρ = i q i |ψ i ψ i |: