Weak values from path integrals

We connect the weak measurements framework to the path integral formulation of quantum mechanics. We show how Feynman propagators can in principle be experimentally inferred from weak value measurements. We also obtain expressions for weak values parsing unambiguously the quantum and the classical aspects of weak couplings between a system and a probe. These expressions are shown to be useful in solving current weak-value related controversies concerning the existence of discontinuous trajectories in interferometers and anomalous weak values in the classical limit.

There has been a growing interest in weak measurements -a specific form of quantum nondemolition measurements -over the last decade.Weak measurements were indeed found to be useful in fundamental or technical investigations, involving both experimental and theoretical works.Nevertheless, ever since their inception [1], weak values -the outcomes of weak measurements -have remained controversial.Since weak values are entirely derived from within the standard quantum formalism, the controversies have never concerned the validity of weak values (WV), but their understanding and their properties.For instance, are WV values similar to eigenvalues, or are they akin to expectation values [2,3]?Do anomalous WV represent a specific signature of quantum phenomena or can they be reproduced by classical conditional probabilities [4][5][6][7]?Are WV related to the measured system properties or do they represent arbitrary numbers characterizing the perturbation of the weakly coupled pointer [8,9]?
The path integral formulation of quantum mechanics is strictly equivalent to the standard formalism based on the Schrödinger equation.Though it is often technically more involved, path integrals give a conceptually clearer picture though the natural, built-in connection with quantities defined from classical mechanics (Lagrangian, action, paths).Surprisingly, very few works have employed weak values in a path integral context.Even then, the interest was restricted to the weak measurement of Feynman paths in semiclassical systems [10][11][12], to WV of specific operators [13][14][15], or as a way to probe virtual histories [16].
In this work, we connect the weak measurements framework to the path integral formulation.We will see that path integrals parse the quantum and classical aspects of weak values, thereby clarifying many of the current controversies involving weak measurements.
In particular we will show how the path integral expression accounts for the discontinuous trajectories observed in interferometers and currently wildly debated [17][18][19].We will also see that in the classical limit the weak value is washed out by coarse graining, implying that anomalous pointer values are a specific quantum feature with no classical equivalent [4][5][6].A nice feature arising from the present approach is the possibility to measure propagators through weak measurements.We will show how a point-like weak interaction and postselection lead to a weak value proportional to the propagator.
A weak measurement of a system observable Â is characterized by 4 steps: 1.The system of interest is prepared in the chosen initial state |ψ i , a step known as preselection.
A quantum probe is prepared in a state |φ i , and the initial state is thus (1) 2. The system and the probe are weakly coupled through an interaction Hamiltonian Ĥint .
3. After the interaction, the system evolves until another system observable, say Typically the probe state |φ i is a Gaussian pointer initially centered at the position Q w and Ĥint is of the form where P is the probe momentum, g(t) is a function non-vanishing only in a small interval centered on t w , and f w reflects the short-range character of the interaction that is only non-vanishing in a small region near Q w .Under these conditions it is well-known [1] that |φ f = e −igA w P |φ i , with g = dt g(t ) and the final probe state is the initial Gaussian shifted by g Re(A w ).
In order to determine the evolution of the coupled system-probe problem from the initial state |Ψ i (t i ) = |ψ i |φ i , we need the full Hamiltonian Ĥ = Ĥs + Ĥp + Ĥint where Ĥs and Ĥp are the Hamiltonians for the uncoupled system and probe resp.In terms of the corresponding evolution operators, the system and the probe evolve first independently, Û0 (t, t i ) = Ûs (t, t i ) Ûp (t, t i ).Then, assuming for simplicity that the interaction takes place during the time interval [t w − τ /2, t w + τ /2] (τ is the duration of the interaction), the total evolution operator from t i to the postselection time t f is given as The propagators K ≡ x 2 | Û (t 2 , t 1 ) |x 1 for the uncoupled evolution of the system and probe are given resp.by where as usual [20,21] D[.] implies integration over all paths connecting the initial and final space-time points, and 2 are the classical system and probe Lagrangians resp.
For the coupled evolution Ûint , the propagator becomes non-separable, where is the classical interacting Lagrangian (see Appdx.); f (q, Q w ) sets the range of the interaction (it becomes a Dirac delta function in the limit of a point-like interaction).Q w will be taken here as a parameter specifying the position of the probe, which makes sense for a probe with a negligible kinetic term.Note that A(q) gives the configuration space value of the classical dynamical variable A(q).
Non-separable propagators are notoriously difficult to handle except when they can be treated perturbatively [21], which is the case here.Standard path integral perturbation techniques (see Ch. 6 of [21]) applied to the system degrees of freedom give K int in terms of the uncoupled propagators and a first order correction in which the system propagates as if it were uncoupled except that each path q(t) is weighed by the perturbative term If the duration τ of the interaction is small relative to the other timescales (as is usually assumed in weak measurements), the time-dependent coupling can be integrated to an effective coupling constant g = tw+τ /2 tw−τ /2 g(t )dt and the uncoupled paths see an effective perturbation gA(q)f (q, Q w )M Q at time t w .Eq. ( 7) becomes (the derivation is given in the Appdx.) Lpdt dqK s (x 2 , t 2 ; q, t w ) From the point of view of the system, the interpretation of Eq. ( 9) is straightforward (see Fig. 1): the transition amplitude from x i to x f involves a sum over the paths directly joining these 2 points in time t f − t i as well as those that interact with the probe in the region determined by f (q, Q w ), hence going from x i to some intermediate point q within this region, and then from this point q to x f .
We now take into account the pre and postselected states and focus on the probe evolution.For convenience we perform the propagation from the initial state involves the direct paths from each x i where ψ is non-vanishing to each x f lying in the support of b f (x).Factorising this term in the full evolution, we get (see Appdx.) where A w is the weak value of Â given by Several remarks are in order.First note that for large M Eq. ( 10) implies a shift of the initial probe state, since the Lagrangian maps each point X 1 to X 1 − gA w .Second, while the denominator in Eq. ( 11) involves all the paths connecting the initial region to the final region, the numerator contains the sole paths connecting the initial and final regions passing through a point q of the interaction region.Each such path is weighed by the classical value of the configuration space function A(q) at that point.Note also that the weak value expressions ( 2) and ( 11) are both the results of asymptotic expansions,but the expansions are not exactly equivalent.For a contact interaction f (q, Q w ) = δ(q − Q w ), it is easy to see that Eqs. ( 2) and ( 11) become identical under the condition A(q) = q| Â |q for q = Q w .
In this case A w is simply given by A(Q w ) × T w /T where T w is the transition amplitude involving the paths connecting x i to x f by going through Q w at t w , while T connects x i to x f through any intermediate point) (see Fig. 1 and Eq. ( 12) below).We then see that: (i) The ratio T w /T can take any complex value, hence if A(Q w ) is bounded, (A w ) can lie beyond these bounds and A w is said to be "anomalous" [29].(ii) If the postselection state is chosen to be the space point |x f , we have if the wavefunctions are known, eg through a previous weak measurement based procedure [22], the propagator K s (x f , Q w ) can be obtained from the measurement of the weak value A w .Both x f and Q w can be varied in order to measure the propagator over the region of interest.
Our weak value expression (11) is useful in cases involving free propagation or in the semiclassical approximation: for large actions stationary phase integration transforms the sum over all arbitrary differentiable paths to a sum containing only the classical paths linking the initial and final points, so that both propagators ( 5) and ( 6) take the form where A k and S k are the amplitude and classical action for the classical path k connecting the initial and endpoints in time t f − t i [20,23]; the phase index µ k counts the number of conjugate points along each trajectory and will be absorbed into the action to simplify the notation.The semiclassical regime, exact for free propagation, remains quantum since the different classical paths still interfere, and A w can be anomalous.Assuming again a point-like interaction at q = Q w , Eq. ( 11) becomes A W and S W label the amplitude and action of a path going through Q w , W runs over all the classical paths connecting x i to Q w and Q w to x f , while J runs over all the classical paths connecting directly x i to x f in time t f − t i .The form of the weak value given by Eq. ( 12).
The first of these issues involves a 3-paths interferometer depicted in Fig. 2. The weakly measured observable is the projector Π j ≡ |x j x j | indicating whether the particle is at point x j .By suitably choosing the preselected and postselected states, the following weak values are obtained [17]: This means that weakly coupling a probe to the system leaves a trace on paths A, B and C, but none on the segments E and F : the particle is seen inside the loop, but not on the entrance and exit paths.A point that has caused some confusion in the literature [18,19] is that in the initial proposal [17] the wavefunction vanished (by destructive interference) on paths E and F , so that an interpretation in terms of vanishing weak values appeared to be flawed.It is possible however to enforce Eq. ( 13) without having destructive interference on E and F .The present path integral approach naturally parses the aspects relevant to a vanishing classical value, to a vanishing superposition, or to a partially propagated state being incompatible with postselection.Indeed, the WV expression ( 12) vanishes (i) if A(Q w ) = 0, which for a projector implies that the particle is not there; (ii) if the term between brackets vanishes, corresponding to destructive interference of the wavefunction at Q w by summing over the different continuous paths W ; (iii) if the integral vanishes, that is the ensemble of points of the preselected state propagated by the sole paths going through Q w up to x f is orthogonal to the postselected state.
The case for asserting that the particle is not at Q w = E or F when the WV vanishes is unproblematic in case (i): this is independent of postselection and would also be the case classically.(ii) corresponds to the initial proposal [17] in which 2 paths with opposite phases interfere destructively.This is a non-classical effect (since we have A(Q w ) = 0) but it does not necessarily depend on postselection and an interpretation in terms of weak values appears moot: if there is no wavefunction, there is nothing to measure, as remarked in Refs [19].The genuinely interesting case is (iii): the paths do not interfere destructively, but the preselected wavefunction propagated by the sole paths passing through E or F is incompatible with the postselected state.As a result a weakly coupled probe placed there does not detect the system.However if a probe is placed further away at B or C, after these paths have propagated and become spatially separated, the sum over W in Eq. ( 12) is different and the preselected state propagated by these paths is not orthogonal to the postselected state anymore, hence a nonzero WV.The crucial point is that the paths contained in the propagator are continuous, but each probe's motion results from the overlap at t = t f between the preselected state propagated by the paths hitting the probe and the postselected state.
The other current controversy mentioned above concerns the quantumness of anomalous weak values that has been recently questioned as resulting from a mere statistical effect [4][5][6][7].A path integral approach is convenient to obtain heuristically the classical limit of the weakly coupled probe's motion.This involves two well-known steps.The first step, deriving the semiclassical propagator K sc was recalled above.The second one involves retaining only the diagonal terms of the semiclassically propagated density matrix [24].This is grounded on the fact that the non-diagonal terms contain wildly oscillating phases (the classical actions) that vanish when coarse-grained on classically meaningful scales [25].The density matrix are given by Eqs. ( 4) and (1) resp.and Û is expanded through K sc in terms of the semiclassical amplitudes and actions for the particle and probe.Coarse-graining leads to ρ(X, x; X , x ) = J |A s J | 2 |A p J | 2 that is well-known to represent a purely classical objectthe classical paths density in configuration space [20,26].|A J | 2 obeys the classical transport equation and can be obtained by solving the equations of motion for each point of the initial preselected distribution: using the Lagrangian given by Eq. ( 8) each probe position is depicted in black remain unaffected by the interaction with the particle, while the probes depicted in orange see their pointers shift after postselection, see Eq. ( 13)).The Feynman paths of the particle, represented as pencils of trajectories along the arms, are continuous and progagate the interactions with the probes up to D, where superposition with the postselected state yield zero WV at E and F even when the wavefunction on these segments does not vanish (see text for details).
and the mean ∆Q is obtained by averaging over the classical distribution ρ s (q) normalized over configuration space (see Appdx.).Postselection is specified in terms of a domain B f at time t f defined such that the chosen configuration space function b(q, t f ) ∈ B f .The average probe shift with postselection is then ∆Q = g Bw A(q, t w )f (q, Q w ) ρ s (q, t w ) Bw ρ s (q , t w )dq dq.
The integral is taken at t w over the configuration space domain B w such that at t f the condition b(q, t f ) ∈ B f is obeyed; the denominator renormalizes ρ s over the postselected ensemble.In the classical limit, the average probe shift is given by this conditional expectation value that can never be anomalous.Classically, the only way to have an anomalous probe shift would be to replace A(q, t w ) by a different configuration space function or to have the numerator and denominator integrated over different domains, say B f and B f .In both cases this would be the result of a perturbation, due to the detection process in the latter case.
To sum up, we have obtained an expression of weak values from a path integral approach.
We have shown this expression to be useful when semiclassical propagators are involved -in particular for free propagation and well-localized interactions.We have seen that the present approach gives a consistent account of two current controversies involving weak values (the observation of discontinuous trajectories and the quantumness of anomalous weak values).
We have also suggested a method for measuring the Feynman propagator through weak values.The approach introduced in this work will be fruitful to tackle extensions of the weak measurements framework to relativistic and cosmological settings, where quantities analogous to weak values are known to emerge [27].

Figure 1 :
Figure 1: Schematic illustration of the paths involved in the weak value expression when the preand postselected states are well localized, and the propagator is given as a sum over the classical trajectories [see Eq. (12)].For the specific choice of postselection b(x f ) = δ(x − x f ) the weak value resulting from the coupling at q = Q w becomes proportional to the propagator K s (x f , Q w ).The propagator can hence be observed by measuring the weak values.

Figure 2 :
Figure 2: A quantum particle propagating in an interferometer and postselected at D displays discontinuous trajectories when observed with weak measurements (the weakly coupled probes B, is measured through a standard measurement process.Of all the possible eigenstates |b k that can be obtained, a filter selects only the cases for which |b k = |b f , where |b f is known as the postselected state.4.When postselection is successful, the probe is measured.The final state of the probe |φ f has changed relative to |φ i by a shift depending on the weak value of Â.Let Ûs (t 2 , t 1 ) denote the system evolution operator, and let t i , t w and t f represent the preparation, interaction, and postselection times resp.The weak value [1] of Â is then given by