A Scheme for Performing Strong and Weak Sequential Measurements of Non-commuting Observables

Quantum systems usually travel a multitude of different paths when evolving through time from an initial to a final state. In general, the possible paths will depend on the future and past boundary conditions, as well as the system's dynamics. We present a gedanken experiment where a single system apparently follows mutually exclusive paths simultaneously, each with probability one, depending on which measurement was performed. This experiment involves the measurement of observables that do not correspond to Hermitian operators. Our main result is a scheme for measuring these operators. The scheme is based on the erasure protocol [Phys. Rev. Lett. 116, 070404 (2016), arXiv:1409.1575] and allows a wide range of sequential measurements at both the weak and strong limits. At the weak limit the back action of the measurement cannot be used to account for the surprising behavior and the resulting weak values provide a consistent yet strange account of the system's past.


I. INTRODUCTION
The joint measurement of sequential observables in quantum mechanics is not canonically defined. A sequential measurement of two observables at different times can lead to incompatible results that depend on how the experiment is carried out [1]. Questions regarding sequential measurements are even more significant in systems with both past and future boundary conditions. The observable quantities that depend on past (pre-selected) and future (post-selected) boundary conditions are not limited to Hermitian operators (or POVM elements), but rather depend on the implementation of the measurement and how it affects the measured system (i.e. the observable transition amplitudes depend on Krauss operators). The Two-State Vector Formalism (TSVF) [2,3] provides a useful platform for asking questions about pre-and post-selected systems.
Aharonov, Bergman and Lebowitz (ABL) [2] derived a formula for calculating conditional probabilities for the outcomes of projective measurements performed on pre-and post-selected ensembles. Each set of probabilities predicted by the ABL formula is limited to a particular measurement strategy. This may lead to strange 'counterfactual' predictions regarding measurements which cannot be performed simultaneously [4]. Aharonov, Albert and Vaidman (AAV) introduced a new way to extract information from the system with negligible change of the measured state. The result of this weak measurement is a complex number called a weak value. Despite the 'weak' method used to obtain it, each 'weak value' is related to a particular 'strong' measurement strategy and can be used to understand some counterfactual probabilities obtained using the ABL formula [3]. Unlike standard observables, whose measurement result corresponds to a classical outcome, weak values can exceed the spectrum of the weakly measured operator and form an effective 'weak potential'. When we weakly couple a particle to an operator A describing a pre-and post-selected system, the weak value {A} w will enter the interaction Hamiltonian [5].
Weak values of sequential observables can shed light on, and test experimentally, fundamental questions such as Leggett-Garg inequalities [6,7] which are based on multiple-times observables. Mitchison et al. have previously used weak values of sequential observables to explain strange predictions regarding measurements in a double interferometer [8]. Here we extend these results and relate them to the original work on sequential observables [1]. Our main result is a paradox involving sequential measurements. Under the right pre-and postselection the system apparently traverses mutually exclusive paths, each with probability 1. Analysis in terms of weak values requires the introduction of a new measurement scheme, first described in [9]. Using the proposed scheme also helps in providing a detailed answer to the question 'where have the particles been?' [10][11][12]. It was suggested that in-between two strong measurements, the particles have been where they left a 'weak trace', i.e. a non-zero weak value. Our method let us track the particles in several places along their way and thus provides a richer notion of their past.
The outline of the paper is as follows. In section II we discuss measurements in the TSVF. In Sec. III we specialize to the case of sequential measurements, present the Deterministic-Path paradox and employ our previously developed method of erasure-based sequential weak measurements [9].

II. MEASUREMENTS IN THE TSVF
The TSVF is a time symmetric formulation of quantum mechanics, that can be used to calculate probabilities for measurements between a pre-selection |ψ and a post-selection φ|. It usually helps in acquiring better understanding of a system between two projective mea-surements.
The following setup is relevant for the rest of the paper: At time t 0 a system S is prepared in the state |ψ . Later, at t f the system undergoes a projective rank-1 measurement and is found to be in the state |φ . For clarity of notation we represent this post-selected state as φ|. Since quantum mechanics is time-symmetric it is possible to think of the state |ψ as evolving forward in time and on the state φ| as evolving backward in time. We assume the free evolution is the identity, so if there are no interactions at intermediate times, the state of the system is best described by the the two-state vector φ| |ψ . We want to predict the results of measurements made at the intermediate time interval t 1 to t 2 , where t 0 < t 1 < t 2 < t f . We consider two types of measurements, a strong (projective) measurement and a weak measurement. In this section we introduce the standard formalism for these two types of measurements. In the following sections we will extend them to the case of sequential measurements using two techniques: Modular measurements [1,13] and the erasure method [9].

A. Strong measurements
We use the term strong measurement to refer to a measurement that gives an unambiguous result. Let A be a measurement characterized by the projectors A k so A = k a k A k . Given the two-state vector φ| |ψ , the probability that an intermediate strong measurement of A yields a result corresponding to A k is given by the ABL formula [2,3]: We usually think of A as an Hermitian operator and A k are projectors onto the degenerate subspaces of this operator. However, as we will see below (see Sec. III D,) the formula can be extended to a more general case where A is a quantum channel and A k are the corresponding Krauss operators (see also [14]).

B. Weak measurements
For a system described by the two-state vector φ| |ψ , the weak measurement of an operator A produces an outcome {A} w = φ|A|ψ φ|ψ called the weak value of A [3,15]. The standard method for implementing this weak measurement is to couple the system to a meter with momentum P and position Q via the von Neumann interaction Hamiltonian H I = f (t)AP . This way, the shift in the pointer's position will be proportional to the weak value. The weakness (or strength) of the measurement is usually modified by varying the strength of the interaction g = f (t)dt and/or by changing the variance σ 2 of the initial state of the measuring device.
In general, the measurement process between t 1 and t 2 is an interaction between the system S and a meter M, such that at the end of the process the (change in the) meter's state corresponds to the measurement outcome. While the state of the system can be arbitrary, the state of the meter is specified according to the desired properties of the measurement. To avoid confusion, we use the subscript w for a meter in a weak measurement. The meter's initial state is denoted by |0 Mw . The weak measurement is a physical process parameterized by a weak interaction parameter g. In the standard weak measurement formalism g = τ 0 f (t)dt, where τ is the duration of the measurement. The weak measurement can be modeled as a map taking a pure product system-meter state ρ SMw 0 = |ψ ψ|⊗|0 0| to a joint state W g (ρ SMw 0 ), such that the following properties hold at g → 0 [9].
Property 1 -Non-disturbance -The probability of post-selecting a state φ|, given by P (2) Property 2 -Weak potential-After post-selection the meter state |0 Mw is shifted by a value proportional to the 'weak value' While most weak measurements have been realized via a coupling to continuous variables, there has been an increasing attention during the last few years to weak values appearing in couplings to qubit meters [13,[16][17][18].
In the current work we shall analyze the discret, qubit meter case. Properties 1,2 are valid also in the case of coupling to a discrete variable, as long as the coupling is weak enough [17]. Property 3 is easily achieved when the corresponding observables are dichotomic (as in the examples that follow), however one has to be careful in more general cases.
The case of dichotomic observables is of special interest and has an interesting property in the case of weak measurements [19].
This property follows simply from property 2 and the ABL rule, Eq. (1).

III. SEQUENTIAL MEASUREMENTS
As the first example of non-trivial sequential measurement Aharonov and Albert [1] showed an experiment where, by using a modular meter, it is possible to perform a correlation-type measurement of a two-time operator σ t2 x σ t1 z (that is, a measurement of σ z at time t 1 followed by a measurement of σ x at time t 2 ). Such a measurement can give qualitatively different results from correlations between a sequence of two measurements made on two devices, the first for σ z at time t 1 and the second for σ x at time t 2 . In a more recent approach [8], the path of a photon through a double interferometer was shown to exhibit strange behavior when questions about its location at a given point in time were asked. These were resolved using weak values of sequential observables. The weak values were, however, related to an indirect measurement procedure. If the observable at time t 1 is A 1 and the observable at time t 2 is A 2 then it is possible to extract the sequential weak value (A 2 , A 1 ) w by performing a weak measurements of A 1 using a meter q 1 and A 2 using q 2 . Using an equality initially derived by Resch and Steinberg [20] it can be shown that . So the weak value can be extracted although the procedure is not a weak measurement, i.e. it does not have properties 1-3 above.
Below we present a similar example of sequential measurements and show a scheme for performing these measurements directly. We will use this example to illustrate some of the features of weak and strong sequential measurements. To avoid confusion we use the notation P (A) to denote the probability of outcome corresponding to the operator A in a strong measurement and {A} w to denote the weak value of the operator A.

A. Setup
Our example is based on the following scenario (see Fig. 1.a): A system S is prepared in the state |ψ = cos θ |0 + sin θ |1 and post-selected in the state where we use the standard notation σ z |0 = |0 σ z |1 = − |1 and σ x |± = ± |± .
These correspond to the following four operators: that add up to the identity A + B + C + D = I. We also note that The four transition amplitudes are

B. A sequence of two strong measurements
If we use a sequence of two projective measurements (i.e. we measure Z, read out the result and then measure X) we get the probabilities  This type of measurement represents the question 'which path did the system follow?'. We now turn our attention to two other ways to measure the observables A, B, C, D . The first of these (figure 2.a) is a measurement that can only distinguish between pairs of the results in Eq. (8), by coupling to a single meter. In the final setup ( figure 2.b), each measurement distinguishes one result from the other three, marking a distinct path from |ψ to |φ . Our choice of pre-and post-selection is inspired by the three box paradox [19] and produces a situation where we can deterministically observe the system going through two mutually exclusive paths.

C. Sequential measurement using modular values
In the spirit of [1] we will now study a measurement of a sequential observable σ ZX = σ t2 x σ t1 z . A result of +1 corresponds to both (0, +) and (1, −) in the Z, X measurements, while a −1 result corresponds to (0, −) and (1, +). This is the same as measuring the operators A + D and B + C respectively (see Fig. 2.a).

The probabilities for a strong measurement
The probabilities for a strong measurement are The difference between this measurement and a scheme using two separate measurements is apparent when we  19) we need to set θ = 0, π/2 and φ = 0, π/2. In contrast, for the modular measurement, we need to set either θ = −φ to obtain b + c = 0 or θ = π/2 − φ to obtain a + d = 0.

The corresponding weak values
To calculate the weak values we can use the fact that weak values are additive, hence We can now see the difference between the four outcome measurement of Eq. (19)  To experimentally measure the weak and strong values for this setup it is possible to use the modular values method of Kedem and Vaidman [13], or the erasure method [9] (see Sec. III D 3 below).

D. The Deterministic Path paradox
In the second example the measurement device is set in such a way (see III D 3 below) that it clicks only for one of the four possible outcomes, say C (see Fig. 2.b). In this case we have four different measurement settings A, B, C, D. Each operator is dichotomic, a result of 1 (or 'click') corresponds to the systems going through the path and a result 0 (or 'no click') corresponds to the system not going through the path. Using the ABL rule we arrive at the following probabilities for a 'click' in each experiment.
These probabilities are very different from the sequence of 4 measurements described in Eq. (19).

Derivation of the 'paradox'
We want to find θ and φ such that B and C click with certainty, if the paths are projectively measured. Using Eq. (24) the conditions for P (B) = P (C) = 1 are Subtracting these we get c = b, which translates to cot(θ) = cot(φ). Using this, and dividing the first equation by sin θ sin φ = 0, we get and the solution is The apparent paradox can be described as: 1. The paths B and C are mutually exclusive. 2. The system traveled through path B with certainty. 3. The system traveled through path C with certainty. However, statements 2 and 3 are counterfactual, we cannot ascertain them simultaneously. Or can we?

The weak value of a path
As in the three box paradox [19], weak measurements allow us to make sense of counterfactual statements. While the measurements of B and C cannot be performed simultaneously, their weak counterparts can. The results are consistent with the apparent 'paradox'. For general θ and φ the weak values are: For the deterministic case, using the solution (27), we have Hence B w = C w = 1 and A w + D w = −1. As expected the total number of particles traversing the four paths is 1. Moreover, the particle traversed both paths B and C with certainty. These results could have been predicted using property 4. Furthermore, we could have used this fact to arrive at Eq. (26).
This result raises a question regrading the 'past of a quantum particle' [12]. If the particle is understood to be at every place it left a 'weak trace', then we can conclude it traveled through all four paths. However, if we understand the weak value as an effective weak potential [5] then a weak coupling to the particle in B and C will result in the effective H I = +1, while coupling to the particle in A/D will be negative.
This result accords well with the past results of the three box paradox [19], the negative pressure paradox [3], the Cheshire cat [21], Hardy's paradox [22,23] and others, where weak measurements reveal a curious behaviour of pre-and post-selected systems.

The erasure method for the four path paradox
Weak values provide an interesting perspective for the situation described above. However, their physical meaning is lost, if there is no corresponding weak measurement. To perform the desired sequential weak measurement we use the erasure method introduced in [9]. The erasure method uses two basic principles.
1) It is possible undo (erase) the effect of a von Neumann coupling by making an appropriate measurement on the meter.
2) It is possible to make a weak measurement using the following procedure • Coupling S to a meter M s with the standard (strong) von Neumann coupling.
• Coupling M s to a second meter M w using a weak von Neumann coupling.
• Erasing the result on M s .
To show how this is done in our example we define three types of unitary operators CN OT , C ij R(g) and R kl ij (g) as follows: where the state |i corresponds to |+ ,|− and |j corresponds to |0 , |1 . The measurement procedure requires an ancilla and a meter. It works in the following way: First we perform a CN OT on the system and ancilla to measure the system in the Z basis. Next we perform C ij R(g) with (i, j) set to be the inverted sequential operator we want, for example (i, j) = (+, 0) for a measurement of A. This rotates the meter by g around σ x . Finally we erase the first measurement by post-selecting the ancilla in the |+ state. Failing this final step is the same as a unitary operation on the initial state. It is possible to undo this unitary, but the cost is a change of the measurement operator. We will follow the circuit for an arbitrary pre-and post-selection. |ψ = α |0 + β |1 , |φ = γ |+ + δ |− 1. Initial state: 1-System; 2-Ancilla 3-Meter 2. CNOT on 1,2 4. Erasure: We measure σ x on 2 and discard this subsystem. If the result is |+ we ('succeed' and) get the unnormalized state If the result is |− we ('fail' and) get the unnormalized state We will continue the derivation for successful erasure (Eq. 33) and only later return to the failed case.

Post-selection in
We now have four cases corresponding to the different choices of i, j (B) (i, j) = (+, 1) Setting α = γ = cos θ , β = δ = sin θ, and writing R(g) |0 = |g , we have So that given cos 2 θ +cos θ sin θ −sin 2 θ = 0 the pointer in cases B and C will point at g as expected.
Returning to the erasure step, failure would mean the post-erasure state is proportional to Eq. (34). This state corresponds to Eq. (33) with the change β → −β which can be interpreted as a σ x operation on the initial state.
We can 'undo' this operation by applying σ z to the system to get comparing with Eq. (33) we have the same state up to a change in the measurement operator, switching A ↔ C and B ↔ D. For a weak measurement we can apply A, B, C and D simultaneously so that the change in measurement operator can be corrected in post-processing.
Similarly, the above method can be used for performing non-local weak measurements of the peculiar weak values discussed in [24].

IV. CONCLUSIONS
We described three gedanken experiments that showcase the difference between a sequence of measurements and sequential measurements. In the first we ask 'which path did the system go through?', in the second we ask 'did the system go through A/D or through B/C?' Finally, for each path we ask 'did the system go through this path?'. The answers to these three questions do not agree, and moreover, the answer to the last question suggests that the system exhibits paradoxical behavior by deterministically traversing two routes at the same time. It is possible to attribute the strange behavior to the fact that the questions are mutually exclusive, however the results are consistent even when weak measurements are used. Such weak measurements can be applied in an experiment using the 'erasure' method as described in Sec. III D 3.
It is convenient to think of an optical version of the paradox using a double interferometer as in [8]. The advantage of this setup is that the path described by the Z, X measurements corresponds to an actual path along the arms of the interferometer. However, from an experimental point of view an optical test of this experiment may be hard to realize due to the large number of qubits required (i.e. 5 qubits). The mathematical description of our protocol in terms of Pauli operators (rather than optical paths) was deliberately chosen with the view that these experiments are more likely to be performed in other platforms such as NMR [18].