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A. and Dressel J. ˆ A A ˆ ˆ | 1 claimed [1] ψ oi can it so , oacon- a to i Dtd uy3,2018) 30, July (Dated: i of weak A ˆ (1) A in ˆ ; ehnc 3] hc a rmtdrcn td into study recent of phenomenon prompted mathematical has the wave which classical such [33], in intrinsically occur in mechanics also not can arises been but is mechanical, effect has detector quantum amplification wave-function it continuous the particular, a how In that [28–32]. noted measurements appeared amplification also other have and for [12–27] amplification proposals the several as of derivation (1) original extensions of the theoretical parts of Several imaginary and parameters. real amplification the both pre- remarkable using to am- sensitively cision, [6–11] The to phenomena of successfully variety 5]. a used measure [4, been systems since optical has plification in re- followed detector soon amplified sponse such of confirmation experimental evolution. the to contributing parameters small determination other sensitive of the for allowed potentially which a eandqiecnrvril ic ti generally of is spectrum the it to since constrained not controversial: and complex quite remained has 35]). [34, bevbewti h otx fape n post-selected and the pre- to a of assigned context be counter- the can within disturbance-free observable that a average as conditioned value factual to weak tempting it the make interpret measurements, operationally conditioned appearance weak in approximate its with together values, weak of in- average an the into through identity value h can the expectation one of behavior an sertion anomalous ob- decompose the its still upon in despite loosely interpretation that somewhat primary servation rested Its has literature interpreted? the be it should a h aefr sdcmoigacasclexpectation classical a decomposing value as form same the has values ψ fe hoeia lrfiain ftedrvto n[3], in derivation the of clarifications theoretical After ocpuly oee,tewa au xrsin(1) expression value weak the however, Conceptually, i rbblt ln h ntr o generated flow unitary the along probability | A ˆ httemaueetdsubnefo a from disturbance measurement the that | E eaeo htosral ntelmtof limit the in observable that of verage ψ lcin sn h unu operations quantum the using elections ,wihalw h hfsfraGaussian a for shifts the allows which e, E to h eeaie ekvlea the as value weak generalized the of rt a tde rvd sdrc information direct is provide does it hat eosral prtr pcfial,we Specifically, operator. observable he ( i t ntrso igecmlxweak complex single a of terms in gth etet fqbtmaueet and measurements qubit of reatments i X ( r ohse,NwYr 42,USA 14627, York New Rochester, er, rlzdwa au osntprovide not does value weak eralized X al,wihcni etitdsense restricted a in can which vable, | peain ervsttestandard the revisit we rpretation, = i | mgnr at ftewa value weak the of parts imaginary r ftegnrlzdwa value weak generalized the of art noa vrg of average an into ) i = ) P f P |h f ψ P f | ( ψ f i | i| i ) 2 E ( h ( ψ X f | | odtoe expectation conditioned ,f i, A ˆ superoscillations | ψ .Ti observation, This ). i i / h ψ f | ψ i i ,which ), A ˆ how , (e.g. 2 ensemble even when it is not strictly measured [36–39]. However, we are still left with a mystery: what is the Supporting this point of view is the fact that when the significance of the imaginary part of (1) that appears real part of (1) is bounded by the eigenvalue range of Aˆ, in the von Neumann measurement, and how does it re- it agrees with the classical conditioned expectation value late to the operator Aˆ? We can find a partial answer to for the observable [37]. Moreover, even when the real this question in existing literature (e.g. [12, 37, 41, 42]) part is outside the normal eigenvalue range, it still obeys that has associated the appearance of the imaginary part a self-consistent logic [40] and seems to indicate oddly of (1) in the response of the detector with the intrinsic sensible information regarding the operator Aˆ. As such, disturbance, or back-action, of the measurement process. it has been used quite successfully to analyze and inter- For example, regarding continuous von Neumann detec- pret many quantum-mechanical paradoxes both theoreti- tors Aharonov and Botero [37, p.8] note that “the imag- cally and experimentally, such as tunneling time [41–44], inary part of the complex weak value can be interpreted vacuum Cherenkov radiation [45], cavity QED correla- as a ‘bias function’ for the posterior sampling point [of tions [46], double-slit complementarity [47, 48], superlu- the detector].” Furthermore, they note that “the weak minal group velocities [49], the N-box paradox [50, 51], value of an observable Aˆ is tied to the role of Aˆ as a phase singularities [52], Hardy’s paradox [53–56], pho- generator for infinitesimal unitary transformations” [37, ton arrival time [57], Bohmian trajectories [58–61], and p.11]. Similarly, while discussing measurements of tun- Leggett-Garg inequality violations [62–64]. neling time Steinberg [41] states that the imaginary part Arguably more important for its status as a quantity is a “measure of the back-action on the particle due to pertaining to the measurement of Aˆ, however, is the fact the measurement interaction itself” and that the detector that the real part of (1) appears as a stable weak limit shift corresponding to the imaginary part “is sensitive to point for conditioned measurements even when the detec- the details of the measurement apparatus (in particular, tor is not a von Neumann-coupled continuous wave that to the initial uncertainty in momentum), unlike the [shift can experience superoscillatory interference (e.g. [63– corresponding to the real part].” 67]). As a result, we can infer that at least the real part In this paper, we will augment these observations in of (1) must have some operational significance specifi- the literature by providing a precise operational inter- cally pertaining to the measurement of Aˆ that extends pretation of the following generalized expression for the beyond the scope of the original derivation. This observa- imaginary part of (1), tion prompted our Letter [68] showing that a principled Tr(Pˆf ( i[Aˆ, ρˆi])) treatment of a general conditioned average of an observ- ImAw = − , (3) able can in fact converge in the weak measurement limit 2Tr(Pˆf ρˆi) to a generalized expression for the real part of (1), where [Aˆ, ρˆ ]= Aˆρˆ ρˆ Aˆ is the commutator between Aˆ i i − i Tr(Pˆ Aˆ, ρˆ ) and the initial state. We will see that the imaginary part ReA = f { i} , (2) w ˆ of the weak value does not pertain to the measurement of 2Tr(Pf ρˆi) Aˆ as an observable. Instead, we will interpret it as half the logarithmic directional derivative of the post-selection ˆ ˆ ˆ where A, ρˆi = Aρˆi + ρˆiA is the anti-commutator be- probability along the flow generated by the unitary action { } tween the observable operator and an arbitrary initial of the operator Aˆ. As such, it provides an explicit mea- state ρˆi represented by a density operator, and where sure for the idealized disturbance that the coupling to Aˆ ˆ Pf is an arbitrary post-selection represented by an ele- would have induced upon the initial state in the limit that ment from a positive operator-valued measure (POVM). the detector was not measured, which resembles the sug- The general conditioned average converges to (2) pro- gestion by Steinberg [41]; however, we shall see that the vided that the manner in which Aˆ is measured satis- measurement of the detector can strongly alter the state fies reasonable sufficiency conditions [69, 70] that ensure evolution away from that ideal. The explicit commutator that the disturbance intrinsic to the measurement pro- in (3) also indicates that the imaginary part of the weak cess does not persist in the weak limit. value involves the operator Aˆ in its role as a generator for It is in this precise restricted sense that we can oper- unitary transformations as suggested by Aharonov and ationally interpret the real part of the weak value (2) as Botero [37], in contrast to the real part (2) that involves an idealized conditioned average of Aˆ in the limit of zero the operator Aˆ in its role as a measurable observable. measurement disturbance. Since it is also the only ap- To make it clear how the generalized weak value ex- parent limiting value of the general conditioned average pressions (2) and (3) and their interpretations arise that no longer depends on how the measurement of Aˆ within a traditional von Neumann detector, we will pro- is being made, it is also distinguished as a measurement vide an exact treatment of a von Neumann measurement context-independent conditioned average. These observa- using the formalism of quantum operations (e.g. [74–76]). tions provide strong justification for the treatment of the In addition to augmenting existing derivations in the lit- real part of the weak value (2) as a form of value assign- erature that are concerned largely with understanding ment [36–39, 71, 72] for the observable Aˆ that depends the detector response (e.g. [12, 13, 15, 18–21, 23–27]), only upon the preparation and post-selection [73]. our exact approach serves to connect the standard treat- 3 ment of weak values to our more general contextual val- t [0,T ]. The interaction is also assumed to be impulsive ues analysis that produces the real part [68–70] more with∈ respect to the natural evolution of the initial joint explicitly. We also provide several examples that spe- state ρˆ of the system and detector; i.e., the interaction cialize our exact solution to typically investigated cases: Hamiltonian (4) acts as the total Hamiltonian during the a particular momentum weak value, an arbitrary qubit entire interaction time interval. observable measurement, and a Gaussian detector. As Solving the Schr¨odinger equation, a consequence, we will show that the Gaussian detec- i~∂ Uˆ = Hˆ Uˆ , (5) tor is notable since it induces measurement disturbance t I that purely decoheres the system state into the eigen- with the initial condition Uˆ0 = 1ˆ produces a unitary basis of Aˆ in the Lindblad sense with increasing mea- operator, surement strength. Surprisingly, the pure decoherence ˆ g ˆ allows the shifts in a Gaussian detector to be completely UT = exp A pˆ , (6) i~ ⊗ parametrized by a single complex weak value to all orders T   in the coupling strength, which allows those shifts to be g = dt g(t), (7) completely understood using our interpretations of that Z0 weak value. that describes the full interaction over the time interval The paper is organized as follows. In II we analyze the T . The constant g acts as an effective coupling parameter § von Neumann measurement procedure in detail, starting for the impulsive interaction. If the interaction is weakly with the traditional unconditioned analysis in II A, fol- coupled then g is sufficiently small so that Uˆ 1ˆ and § T lowed by an operational analysis of the unconditioned the effect of the interaction will be approximately≈ negli- case in II B 1 and the conditioned case in II B 2. After gible; however, we will make no assumptions about the § § obtaining the exact solution for the von Neumann detec- weakness of the coupling a priori. tor response, we consider the weak measurement regime The unitary interaction (6) will entangle the system to linear order in the coupling strength in III, which clar- with the detector so that performing a direct measure- § ifies the origins and interpretations of the expressions (2) ment on the detector will lead to an indirect measure- and (3). We discuss the time-symmetric picture in IV ment being performed on the system. Specifically, we § for completeness. After a brief Bohmian mechanics ex- note that the position operator xˆ of the detector satisfies ample in V A that helps to illustrate our interpretation the canonical commutation relation [xˆ, pˆ] = i~1ˆ , and § d of the weak value, we provide the complete solutions for thus will evolve in the Heisenberg picture of the interac- a qubit observable in VB and a Gaussian detector in tion according to, V C. Finally, we present§ our conclusions in VI. § § (1ˆ xˆ) = Uˆ † (1ˆ xˆ)Uˆ , (8) s ⊗ T T s ⊗ T = 1ˆs xˆ + gAˆ 1ˆd. II. VON NEUMANN MEASUREMENT ⊗ ⊗ As a result, measuring the mean of the detector position after the interaction x = Tr((1ˆ xˆ) ρˆ) will produce, The traditional approach for obtaining a complex weak h iT s ⊗ T value [1] for a system observable is to post-select a weak x T = x 0 + g A 0. (9) Gaussian von Neumann measurement [2]. The real and h i h i h i Hence, the mean of the detector position will be shifted imaginary parts of the complex weak value then appear from its initial mean by the mean of the system observ- as scaled shifts in the conditioned expectations of conju- able Aˆ in the initial reduced system state, linearly scaled gate detector observables to linear order in the coupling by the coupling strength g. For this reason we say that strength. To clarify how these shifts occur and how the directly measuring the average of the detector position xˆ weak value can be interpreted, we shall solve the von results in an indirect measurement of the average of the Neumann measurement model exactly in the presence of system observable Aˆ. post-selection. The detector momentum pˆ, on the other hand, does not evolve in the Heisenberg picture since [Uˆ , 1ˆ pˆ]= T s ⊗ A. Traditional Analysis 0. Hence, we expect that measuring the average detector momentum will provide no information about the system observable Aˆ. A von Neumann measurement [1, 2] unitarily couples As discussed in the introduction, however, when one an operator Aˆ on a system Hilbert space to a momen- Hs conditions such a von Neumann measurement of the de- tum operator pˆ on a continuous detector Hilbert space tector upon the outcome of a second measurement made via a time-dependent interaction Hamiltonian of the Hd only upon the system, then the conditioned average of form, both the position and the momentum of the detector can experience a shift. To see why this is so, we will find it Hˆ I (t)= g(t)Aˆ pˆ. (4) ⊗ useful to switch to the language of quantum operations The interaction profile g(t) is assumed to be a function (e.g. [74–76]) in order to dissect the measurement in that is only nonzero over some interaction time interval more detail. 4

the total probability density for detecting the position x,

p(x) = Tr ( (ρˆ )) = Tr (Eˆ ρˆ ), (13) s Mx i s x i Eˆ = Mˆ †Mˆ = ψ Uˆ † (1ˆ x x )Uˆ ψ , (14) x x x h | T s ⊗ | ih | T | i where Tr ( ) is the partial trace over the system Hilbert s · space. The probability operator Eˆx is a positive system operator that encodes the probability of measuring a par- FIG. 1. (color online) Schematic for a von Neumann mea- ticular detector position x, and can also be written in surement. An initially prepared system state ρˆi and detector terms of the initial detector position wave-function as state ψ ψ become entangled with the von Neumann uni- ˆ ˆ 2 | ih | Ex = ψ(x gA) . To conserve probability it satisfies tary interaction UˆT (6) over a time interval T . Measuring | − | the condition, dx Eˆ = 1ˆ , making the operators Eˆ a a particular detector position x after the interaction updates x s x the detector state to x x and also updates the system state positive operator-valued measure (POVM) on the system | ih | R to x(ρˆi), where x (11) is an effective measurement op- space. erationM that encodesM the entanglement with and subsequent Consequently, averaging the position of the detector measurement of the detector. will effectively average a system observable with the ini- tial system state,

∞ B. Quantum Operations x T = dxxp(x) =Trs(Oˆρˆi), (15) h i −∞ Z ∞ 1. Unconditioned Measurement Oˆ = dx xEˆ = ψ Uˆ † (1ˆ xˆ)Uˆ ψ , x h | T s ⊗ T | i Z−∞ As before, we will assume an impulsive interaction in = x 01ˆs + gAˆ, what follows so that any natural time evolution in the h i joint system and detector state will be negligible on the where we see the Heisenberg evolved position operator time scale of the measurement. (For considerations of (8) naturally appear. the detector dynamics, see [13].) We will also assume for Since the probability operators Eˆx are diagonal in the simplicity of discussion that the initial joint state of the basis of Aˆ, then the effective system operator Oˆ will also system and detector before the interaction is a product be diagonal in the same basis. Hence, by modifying the state and that the detector state is pure, values that we assign to the position measurements, we can arrange an indirect measurement of any system ob- ρˆ = ρˆi ψ ψ , (10) servable spanned by Eˆ in the basis of Aˆ, including Aˆ ⊗ | ih | x itself, { } though we will be able to relax this assumption in our ∞ final results. Conceptually, this assumption states that a x x 0 Aˆ = dx − h i Eˆx, (16) typical detector will be initially well-calibrated and un- −∞ g correlated with the unknown system state that is being Z   probed via the interaction. The chosen set of values (x x 0)/g are contextual values −h i Evolving the initial state with the interaction unitary for Aˆ, which can be thought of as a generalized spectrum ˆ ˆ UˆT (6) will entangle the system with the detector. Hence, that relates A to the specific POVM Ex associated subsequently measuring a particular detector position with the measurement context [68–70].{ } They are {Mx} will be equivalent to performing an operation x upon not the only values that we could assign to the position the reduced system state, as illustrated in FigureM 1, measurement in order to obtain the equality (16), but they are arguably the simplest to obtain and compute, as 1ˆ ˆ ˆ † ˆ ˆ † well as the most frequently used in the literature. It is in x(ρˆi) = Trd(( s x x )UT ρˆUT )= MxρˆiMx, M ⊗ | ih | (11) this precise sense that we can say that the von Neumann coupling leads to an indirect measurement of the average ˆ ˆ Mx = x UT ψ . (12) ˆ h | | i of A in the absence of post-selection. The measurement of Aˆ comes at a cost, however, since where Trd( ) is the partial trace over the detector Hilbert the system state is necessarily disturbed by the opera- · ˆ space, and Mx is the Kraus operator associated with tions in order to obtain the probability operators Eˆ . Mx x the operation x. Furthermore, since x ψ = ψ(x) is The state may even be disturbed more than is strictly re- M h | i the initial detector position wave-function we find Mˆ x = quired to make the measurement of Aˆ, which can be seen da exp( ga∂x)ψ(x) a a = da ψ(x ga) a a , or, by rewriting the measurement operators in polar form, − | ih | − | ih | ˆ ˆ ˆ 1/2 more compactly, Mˆ x = ψ(x gAˆ). Mx = Ux Ex , with the positive root of the probabil- R − R | |ˆ 1/2 If we do not perform a subsequent post-selection on ity operator Ex and an additional unitary operator | | the system state, then we trace out the system to find Uˆ . This decomposition implies that splits into an x Mx 5 effective composition of two distinct operations,

(ρˆ )= ( (ρˆ )), (17a) Mx i Ux Ex i (ρˆ )= Eˆ 1/2ρˆ Eˆ 1/2, (17b) Ex i | x| i| x| (ρˆ′ )= Uˆ ρˆ′ Uˆ †. (17c) Ux i x i x

We can interpret the operation x that involves only the roots of the probability operatorEEˆ 1/2 as the pure mea- | x| surement operation producing Eˆx. That is, it represents the minimum necessary disturbance that one must make FIG. 2. (color online) Schematic for a sequence of two indi- rect measurements. After the von Neumann interaction and to the initial state in order to extract a measurable prob- measurement of x illustrated in Figure 1 that produces the ability. The second operation unitarily disturbs the x effective measurement operation x upon the initial system U ˆ initial state, but does not contribute to Ex. Since only state, a second detector interactsM impulsively with the sys- ˆ ˆ Ex can be used to infer information about A through the tem for a time interval T2 T . The second detector is then identity (16), we conclude that the disturbance from measured to have a particular− outcome f, which updates the Ux is superfluous. system state to f ( x(ρˆi)), where f is another measure- P M P To identify the condition for eliminating x, we can ment operation. Taking the trace of the final system state rewrite the Kraus operator (12) using theU polar form will then produce the joint probability densities (24). of the initial detector position wave-function ψ(x) = exp(iψ (x))ψ (x), s r and that both detector probability operators can be ob- tained as marginals of a Wigner quasi-probability opera- Mˆ = exp(iψ (x gAˆ))ψ (x gAˆ). (18) x s − r − tor on the system Hilbert space, ∞ 1 2ipy/~ † The phase factor becomes the unitary operator Uˆx = Wˆ = dy e Mˆ Mˆ , (23a) x,p π~ x+y x−y exp(iψs(x gAˆ)) for x, while the magnitude becomes −∞ − U ∞Z Eˆ 1/2 Aˆ the required positive root x = ψr(x g ) for x. Eˆ = dp Wˆ , (23b) Hence, to eliminate the superfluous| | operation− fromE a x x,p x Z−∞ von Neumann measurement with coupling HamiltonianU ∞ ˆ ˆ (4), one must use a purely real initial detector wave- Fp = dx Wx,p. (23c) function in position. Z−∞ For contrast, measuring only a particular detector mo- In the absence of interaction, then the Wigner quasi- mentum p will be equivalent to performing a different probability operator reduces to the Wigner quasi- operation p upon the reduced system state, probability distribution W (x, p) for the initial detector N g=0 state, Wˆ x,p W (x, p)1ˆs. (ρˆ ) = Tr ((1ˆ p p )Uˆ ρˆUˆ † )= Nˆ ρˆ Nˆ †, (19) −−→ Np i d s ⊗ | ih | T T p i p ˆ ˆ gp ˆ Np = p UT ψ = exp A p ψ . (20) 2. Conditioned Measurement h | | i i~ h | i   The Kraus operator Nˆp has a purely unitary factor con- To post-select the system, an experimenter must per- taining Aˆ that will disturb the system, regardless of the form a second measurement after the von Neumann mea- form of the initial momentum wave-function p ψ . More- surement and filter the two-measurement event space over, the probability operator associated withh | i the mo- based on the outcomes for the second measurement. In mentum measurement has the form, other words, the experimenter keeps only those pairs of outcomes for which the second outcome satisfies some † 2 constraint. The remaining measurement pairs can then Fˆp = Nˆ Nˆp = p ψ 1ˆs, (21) p |h | i| be averaged to produce conditioned averages of the first measurement. which can only be used to measure the identity 1ˆ . s If we represent the second measurement as a set of For completeness we also briefly note that the conju- Pˆ ˆ ˆ probability operators f indexed by some parameter gate Kraus operators Mx and Np are related through a f that can be derived{ analogously} to (14) from a set of Fourier transform, operations as illustrated in Figure 2, then the total {Pf } ∞ joint probability densities for the ordered sequences of 1 −ipx/~ Nˆp = dx e Mˆ x, (22a) measurement outcomes (x, f) and (p,f) will be, √2π~ −∞ Z ∞ p(x, f) = Trs(Pˆf x(ρˆi)) = Trs(Eˆx,f ρˆi), (24a) ˆ 1 ipx/~ ˆ M Mx = dp e Np, (22b) ˆ ˆ √2π~ −∞ p(p,f) = Trs(Pf p(ρˆi)) = Trs(Fp,f ρˆi), (24b) Z N 6 where the joint probability operators, where,

ˆ ˆ † ˆ ˆ † Ex,f = MxPf Mx, (25a) (ρˆ ) = Tr ((1ˆ xˆ)Uˆ (ρˆ ψ ψ )Uˆ ), (30a) XT i d s ⊗ T i ⊗ | ih | T ˆ ˆ † ˆ ˆ † Fp,f = Np Pf Np. (25b) (ρˆ ) = Tr ((1ˆ pˆ)Uˆ (ρˆ ψ ψ )Uˆ ), (30b) PT i d s ⊗ T i ⊗ | ih | T are not simple products of the post-selection Pˆf and the are detector averaging operations that affect the system probability operators (14) or (21). Those operators can state before the measurement of the post-selection is per- be recovered, however, by marginalizing over the index f, formed. It is worth noting at this point that we can relax since the post-selection probability operators must satisfy the assumption (10) made about the initial state in the ˆ 1ˆ a POVM condition f Pf = s. exact operational expressions (27) and (30). Similarly, if The joint probabilities (24) will contain information different contextual values are used to average the condi- not only about the firstP measurement and the initial sys- tional probabilities in (29), then corresponding detector tem state, but also about the second measurement and observables with the same spectra will appear in the oper- any disturbance to the initial state that occurred due to ations (30) in place of xˆ or pˆ; for example, averaging the the first measurement. In particular, the joint probabil- values α(x) = (x x 0)/g used in (16) will replace the de- −h i ∞ ity operators (25) can no longer satisfy the identity (16) tector observable xˆ in (30) with αˆ = dx α(x) x x . −∞ | ih | due to the second measurement, so averaging the proba- To better interpret (30), we bring the detector opera- R bilities (24) must reveal more information about the mea- tors inside the unitary operators in (30) using the canon- surement process than can be obtained solely from the ical commutation relations as in (8), operator Aˆ, the initial state ρˆi, and the post-selection ˆ Pˆ . As a poignant example, the unitary disturbance T (ρˆi)= (ρˆi)+ g ( A, ρˆi /2), (31a) f Ux X X E { } in (17) that did not contribute to the operator identity T (ρˆi)= (ρˆi), (31b) (16) will contribute to the joint probability operators, P P ˆ ˆ 1/2 ˆ † ˆ ˆ ˆ 1/2 which splits the T operation into two operations but Ex,f = Ex UxPf Ux Ex . X The total| | probability| for| obtaining the post-selection only changes the form of T . The operation proportional P ˆ outcome f can be obtained by marginalizing over either to g disturbs the symmetrized product A, ρˆi /2 = ˆ ˆ { } x or p in the joint probabilities, (Aρˆi + ρˆiA)/2 of the initial system state with the oper- ˆ ∞ ∞ ator A, while the operations,

p(f)= dx p(x, f)= dp p(p,f), (26) † −∞ −∞ (ρˆ ) = Tr (Uˆ (ρˆ xˆ, ψ ψ /2)Uˆ ), (32a) Z Z X i d T i ⊗{ | ih |} T ˆ † = Trs(Pf (ρˆi)), (ρˆ ) = Tr (Uˆ (ρˆ pˆ, ψ ψ /2)Uˆ ), (32b) E P i d T i ⊗{ | ih |} T (ρˆ ) = Tr (Uˆ (ρˆ ψ ψ )Uˆ † ), (27) E i d T i ⊗ | ih | T disturb the symmetrized products of the initial detector where the operation is the total non-selective measure- state with the detector operators. E The form of the equations (31) clearly illustrates how ment that has been performed on ρˆi. Since is not the identity operation, it represents the total disturbanceE in- the post-selection will affect the measurement. If the ˆ 1ˆ trinsic to the measurement process. It includes unitary post-selection is the identity operator, Pf = s, then the ˆ evolution of the reduced system state due to the inter- unitary operators UT causing the total disturbance of action Hamiltonian (4), as well as decoherence stemming the initial state will cancel through the cyclic property of from entanglement with the measured detector. the total trace in (29), leaving the averages in the initial By experimentally filtering the event pairs to keep only states that were previously obtained, a particular outcome f of the second measurement, an x = x + g A , (33a) experimenter can obtain the conditional probabilities, h iT h i0 h i0 p = p . (33b) h iT h i0 p(x, f) Trs(Pˆf x(ρˆi)) p(x f)= = M , (28a) In this sense, commuting the detector operators xˆ and | p(f) Tr (Pˆ (ρˆ )) s f E i pˆ in (30) through the unitary operators to arrive at (31) p(p,f) Trs(Pˆf p(ρˆi)) is equivalent to evolving them in the Heisenberg picture p(p f)= = M , (28b) | p(f) Trs(Pˆf (ρˆi)) back from the time of measurement T to the initial time E 0 in order to compare them with the initial states. How- which can then be averaged to find the exact conditioned ever, the presence of the post-selection operator Pˆf will averages for the detector position and momentum, now generally spoil the cancelation of the unitary oper- ators that is implicit in the Heisenberg picture, leading ∞ ˆ Trs(Pf T (ρˆi)) to corrections from the disturbance between the pre- and f x T = dxxp(x f)= X , (29a) h i −∞ | Tr (Pˆ (ρˆ )) post-selection. Z s f E i ∞ The symmetrized products in (31) indicate the mea- Trs(Pˆf T (ρˆi)) f p T = dppp(p f)= P , (29b) surement being made on the initial states of the sys- h i −∞ | Tr (Pˆ (ρˆ )) Z s f E i tem and detector, which is then further disturbed by 7 the unitary operators UˆT as a consequence of the cou- The initial detector state plays a critical role in (36) n n+1 pling Hamiltonian (4). The post-selection both condi- by determining the various moments, p 0, p 0 and n h i h i tions those measurements and reveals the disturbance, p , x /2 0 that appear in the series expansions. No- which corrects each term in (33), yielding the final exact h{tably, if} wei make the initial detector wave-function purely expressions, real so that it minimally disturbs the system state then all moments containing odd powers of pˆ will vanish. We ˆ ˆ ˆ Trs(Pf (ρˆi)) Trs(Pf ( A, ρˆi )) conclude that those moments of the disturbance opera- f x T = X + g E { } , (34a) h i Trs(Pˆf (ρˆi)) 2Trs(Pˆf (ρˆi)) tions are superfluous for obtaining the measurable prob- E E Aˆ ˆ abilities that allow the measurement of , while the mo- Trs(Pf (ρˆi)) p f p T = P . (34b) ments with even powers of ˆ are necessary. h i Trs(Pˆf (ρˆi)) After expanding the corrections (34) to first order in g, E we obtain the linear response of the conditioned detector means due to the interaction, III. THE WEAK VALUE g p, x 0 Trs(Pˆf (adAˆ)(ρˆi)) f x T x 0 + ~ h{ }i (37a) If it were possible to leave the system state undisturbed h i → h i i 2 Trs(Pˆf ρˆi) while still allowing the measurement of Aˆ, then we would Tr (Pˆ Aˆ, ρˆ ) na¨ıvely expect the disturbance to reduce to the iden- + g s f { i} , E ˆ tity operation. Similarly, we would na¨ıvely expect the 2Trs(Pf ρˆi) operations and would reduce to x 0 and p 0 mul- ˆ ˆ X P h i h i g 2 Trs(Pf (adA)(ρˆi)) tiplying the identity operation, respectively. As a result, f p T p 0 + p 0 . (37b) h i → h i i~h i Pˆ ρ the conditioned averages (34) would differ from the un- Trs( f ˆi) conditioned averages (33) solely by the replacement of Measurements for which this linear response is a good the average A with the real part (2) of the complex h i0 approximation are known as weak measurements. generalized weak value expression, After introducing the complex generalized weak value (35), we can write the linear response formulas in a more Trs(Pˆf Aˆρˆi) Aw = . (35) compact form, Trs(Pˆf ρˆi) g p, x x = x + h{ }i0 (2ImA )+ g ReA , (38a) Since this expression depends solely upon the initial state f h iT h i0 ~ 2 w w ρˆi, the post-selection Pˆf , and the operator Aˆ, we are g 2 f p T = p 0 + p 0 (2ImAw), (38b) na¨ıvely tempted to give ReAw an intuitive interpretation h i h i ~h i as the ideal conditioned expectation of Aˆ in a pre- and in terms of not only its real part, but also twice its imag- post-selected state with no intermediate measurement inary part. disturbance. However, it is strictly impossible to remove If the initial detector position wave-function ψ(x) is the disturbance from the measurement while still mak- purely real, so that the measurement is minimally dis- ing the measurement, so we cannot rely on this sort of turbing, then p, x /2 will vanish, leaving only ReA reasoning. We can make a similar interpretation in a re- 0 w in x as weh{ na¨ıvely} i reasoned before. However, the stricted sense, however, by making the coupling strength f T termh proportionali to 2ImA will not vanish in p to g sufficiently small to reduce the disturbance to a minimal w f T linear order in g, making it an element of measurementh i amount that still allows the measurement to be made. disturbance that persists even for minimally disturbing To see how the operations , , and in (27) and weak measurements. (32) depend on the coupling strengthE X g, weP expand them These linear response formulas for the von Neumann perturbatively, measurement have also been obtained and discussed in 1 g n the literature with varying degrees of generality and rigor (ρˆ )= pn (adAˆ)n(ρˆ ), (36a) E i n! i~ h i0 i (e.g. [1, 3, 12, 13, 15, 18–20, 23–27, 40]). However, our n=0 X   derivation has a conceptual advantage in that we see ex- n n 1 g p , x 0 plicitly how the origins of the real and imaginary parts (ρˆ )= h{ }i (adAˆ)n(ρˆ ), (36b) X i n! i~ 2 i of the weak value differ with respect to the measurement n=0 X   of Aˆ. We are therefore in a position to give concrete 1 g n (ρˆ )= pn+1 (adAˆ)n(ρˆ ), (36c) interpretations for each part. P i n! i~ h i0 i n=0 The real part (2) of the weak value ReA stems di-   w X rectly from the part of the conditioned shift of the de- where the operation (adAˆ)( ) = [Aˆ, ] is the left action of tector pointer that corresponds to the measurement of Aˆ · · Aˆ in the adjoint representation of its Lie algebra, which and does not contain any further perturbation induced takes the form of a commutator. That is, (adAˆ) explic- by the measurement coupling that would be indicated itly describes how Aˆ disturbs the initial state due to the by factors of (adAˆ). As a result, it can be interpreted interaction that measures it. as an idealized limit point for the average of Aˆ in the 8 initial state ρˆi that has been conditioned on the post- The logarithmic directional derivative of this post- selection Pˆf without any appreciable intermediate mea- selection probability then produces the factor 2ImAw surement disturbance. To support this point of view, we that appears in (38), have also shown in [68–70] that ReAw appears naturally as such a limit point for minimally disturbing measure- 2ImAw = ∂ε ln pf (ε) ε=0, (41) ments that are not of von Neumann type, provided that those measurements satisfy reasonable sufficiency condi- which is our main result. tions regarding the measurability of Aˆ. In words, the imaginary part of the weak value is half The imaginary part (3) of the weak value ImAw, on the logarithmic directional derivative of the post-selection the other hand, stems directly from the disturbance of probability along the natural unitary flow generated by the measurement and explicitly contains (adAˆ), which Aˆ. It does not provide any information about the mea- is the action of Aˆ as a generator for unitary evolution surement of Aˆ as an observable, but rather indicates due to the specific Hamiltonian (4). The factor 2ImAw an instantaneous exponential rate of change in the post- appears in (38) along with information about the initial selection probability due to disturbance of the initial state detector momentum that is being coupled to Aˆ in the caused by Aˆ in its role as a generator for unitary trans- Hamiltonian (4), as well as factors of ~, in stark contrast formations. Specifically, for small ε we have the approx- to the real part. How then can it be interpreted? imate relation, The significance of 2ImAw becomes more clear once we identity the directional derivative operation that appears pf (ε) pf (0)(1 + (2ImAw)ε). (42) in its numerator, ≈ ˆ For a pure initial state ρˆi = ψi ψi and a projective δA( )= i(adA)( ). (39) | ih | · − · post-selection Pˆ = ψ ψ , the expression (41) simpli- f | f ih f | That is, δA(ρˆi) indicates the rate of change of the initial fies, state ρˆi along a flow in state-space generated by Aˆ. The directional derivative should be familiar from 2ImA = ∂ ln ψ exp( iεAˆ) ψ 2 . (43) w ε |h f | − | ii| ε=0 the Schr¨odinger equation written in the form ∂tρˆ = ˆ ~ Ωˆ [H, ρˆ]/i = δΩ(ρˆ), where the scaled Hamiltonian = Hence, the corrections containing 2ImA w that appear Hˆ /~ is a characteristic frequency operator. The integra- in (38) stem directly from how the specific von Neumann tion of this equation is a unitary operation in exponential Hamiltonian (4) unitarily disturbs the initial system state form ρˆ(t) = exp(tδΩ)(ρˆ(0)) = exp( itΩˆ )ρˆ(0) exp(itΩˆ ) infinitesimally prior to any additional disturbance in- − that specifies a flow in state space, which is a collection duced by the measurement of the detector. Conceptu- of curves that is parametrized both by a time parameter ally, the coupling induces a natural unitary flow of the t and by the initial condition ρˆ(0). Specifying the initial initial system state generated by Aˆ, which for infinitesi- condition ρˆ(0) = ρˆi, picks out the specific curve from the mal g changes the joint probability p(p,f) for a specific flow that contains ρˆi. The directional derivative of the p by the amount (2ImAw)(gp/~), where gp/~ is the in- initial state along that specific curve is then defined in finitesimal parameter ε in (41) that has units inverse to the standard way, ∂tρˆ(t) t=0 = δΩ(ρˆi). Aˆ | . Averaging this correction to the joint probability with The fact that the quantum state space is always a con- the detector observables xˆ or pˆ produces the correction tinuous manifold of states allows such a flow to be de- terms in (38). fined in a similar fashion using any Hermitian operator, such as Aˆ, as a generator. Analogously to time evolu- tion, such a flow has the form of a unitary operation, IV. TIME SYMMETRY ρˆ(ε) = exp(εδA)(ρˆ(0)), where the real parameter ε for the flow has units inverse to Aˆ. Therefore, taking the di- As noted in [40, 77], a quantum system that has been rectional derivative of ρˆi along the specific curve of this pre- and post-selected exhibits time symmetry. We can flow that passes through ρˆi will produce (39). For an explicit example that we will detail in V B, the state- make the time symmetry more apparent in our opera- space of a qubit can be parametrized as§ the continuous tional treatment by introducing the retrodictive state, volume of points inside the unit Bloch sphere; the deriva- ˆ ˆ tive (39) produces the vector field illustrated in Figure 3 ρˆf = Pf /Tr(Pf ), (44) tangent to the flow corresponding to Rabi oscillations of the qubit. associated with the post-selection (see, e.g., [78, 79]) and With this intuition in mind, we define the post- rewriting our main results in the time-reversed retrodic- selection probability for measuring Pˆf given an initial tive picture. state ρˆi(ε) = exp(εδA)(ρˆi) that is changing along the After cancelling normalization factors, the detector re- flow generated by Aˆ, sponse (34) for a system retrodictively prepared in the final state ρˆ that has been conditioned on the pre- ˆ f pf (ε) = Trs(Pf ρˆi(ε)). (40) selection measurement producing the initial system state 9

ρˆi has the form, has been recently discussed by Leavens [58], Wiseman [59], and Hiley [61], where the operator Aˆ = pˆ being ∗ ∗ ˆ Trs( (ρˆf )ρˆi) Trs( (ρˆf ), A ρˆi) measured is the momentum operator of the system par- i x T = X∗ + g {E ∗ } , (45a) h i Trs( (ρˆf )ρˆi) 2Trs( (ρˆf )ρˆi) ticle. Since the wave-number operator kˆ = pˆ/~ gener- E ∗ E − Trs( (ρˆf )ρˆi) ates a flow that is parametrized by the position x, then i p T = P∗ , (45b) we expect from the discussion surrounding (41) that the h i Trs( (ρˆf )ρˆi) E imaginary part of a momentum weak value will give in- where the retrodictive operations ∗, ∗, and ∗ are the formation about how the post-selection probability will adjoints of the predictive operationsE inX (27) andP (32), change along changes in position. If we restrict our initial system state to be a pure state ∗ ˆ † ˆ (ρˆf )= ψ U ρˆf UT ψ , (46a) ρˆ = φ φ , and post-select the measurement of the mo- E h | T | i i † | ih | ∗ ˆ ˆ mentum on a particular position Pˆf = x x , then the (ρˆf )= ψ xˆ, UT ρˆf UT /2 ψ , (46b) X h |{ } | i detector will have the linear response relations| ih | (38) with ∗(ρˆ )= ψ pˆ, Uˆ † ρˆ Uˆ /2 ψ . (46c) P f h |{ T f T } | i the complex weak value given by, Notably, the symmetric product with Aˆ that appears x pˆ φ i~∂ φ(x) p = h | | i = − x . (50) in the detector response (45) involves the retrodictive w x φ φ(x) state that has been evolved back to the initial time due h | i to the non-selective measurement operation ∗. Hence, We can split this value naturally into its real and imagi- in both pictures the measurement of Aˆ is beingE made nary parts by considering the polar decomposition of the with respect to the same initial time. initial system state φ(x)= r(x)exp(iS(x)), After expanding the retrodictive operations perturba- tively as in (36), pw = ~∂xS(x) i~∂x ln r(x). (51) − n ∗ 1 g ∗ The real part of the weak value Re p = ~∂ S(x) is (ρˆ )= pn (ad Aˆ)n(ρˆ ), (47a) w x E f n! i~ h i0 f the phase gradient, or Bohmian momentum for the ini- n=0 X   tial state, which we can now interpret operationally as the 1 g n pn, x ∗(ρˆ )= h{ }i0 (ad∗Aˆ)n(ρˆ ), (47b) average momentum conditioned on the subsequent mea- X f n! i~ 2 f n=0 surement of a particular x in the ideal limit of no mea- X   1 g n surement disturbance. This connection between the real ∗(ρˆ )= pn+1 (ad∗Aˆ)n(ρˆ ), (47c) P f n! i~ h i0 f part of a weak value and the Bohmian momentum that n=0 X   was pointed out in [58, 59] has recently allowed Koc- sis et al. [60] to experimentally reconstruct the averaged where (ad∗Aˆ)( )= (adAˆ)( ) = [ , Aˆ] is the right action ˆ · − · · Bohmian trajectories in an optical two-slit interference of A, then the linear response of the detector (38) can be experiment using such a von Neumann measurement. written in terms of the retrodictive forms of the real and The imaginary part of the weak value, Im pw = imaginary parts of the complex weak value, ~∂x ln r(x), on the other hand, is the logarithmic gradi- − 2 ˆ ent of the root of the probability density ρ(x)= φ(x) = Trs( ρˆf , A ρˆi) 2 | | ReAw = { } , (48) r (x) for the particle at the point x. Written in the form, 2Trs(ρˆf ρˆi) ~ Tr (( i[ρˆ , Aˆ])ρˆ ) 2Im pw = ∂x ln ρ(x), (52) 2ImA = s − f i , (49) − w Tr (ρˆ ρˆ ) s f i it describes the instantanous exponential rate of posi- = ∂ε ln Trs(exp( εδA)(ρˆf )ρˆi) , tional change of the probability density with respect to − ε=0 the particular post-selection point x, as expected. This which should be compared with (2), (3), and (41). quantity, scaled by an inverse mass 1/m, was introduced We see that the imaginary part of the weak value can under the name “osmotic velocity” in the context of a also be interpreted as half the logarithmic directional stochastic interpretation of quantum mechanics devel- derivative of the pre-selection probability as the retro- oped by Nelson [80], where it produced a diffusion term dictive state changes in the opposite direction along the in the stochastic equations of motion for a classical point ˆ flow generated by A. particle with diffusion coefficient ~/2m. Nelson’s inter- pretation was carefully contrasted with a stochastic inter- pretation for the Bohmian pilot wave by Bohm and Hiley V. EXAMPLES [81], and the connection of the osmotic velocity with a weak value was recently emphasized by Hiley [61]. A. Bohmian Mechanics Hence, the imaginary part of the momentum weak value does not provide information about a measurement To make the preceding abstract discussion of the weak of the momentum in the initial state. Instead, it indicates value more concrete, let us consider a special case that the logarithmic directional derivative of the probability 10 density for measuring x along the flow generated by pˆ. has the exact form, The scaled derivative ~∂x appears since pˆ = ~kˆ and − − c(g)r1 s(g)r2 kˆ generates flow along the position x. (ρˆi)= ρˆi + − σˆ1 (56a) E 2 c(g)r + s(g)r + 2 1 σˆ , 2 2 ∞ n 2n ( 1) 2Ag 2n c(g)= − p 0, (56b) B. Qubit Observable (2n)! ~ h i n=1   X∞ ( 1)n+1 2Ag 2n−1 s(g)= − p2n−1 . (56c) To make the full von Neumann measurement process (2n 1)! ~ h i0 n=1 more concrete, let us also consider a simple example X −   where Aˆ operates on the two-dimensional Hilbert space The correction term can be interpreted as a Rabi oscilla- of a qubit. (See also [12, 13, 15, 18–21, 23–27].) We can tion of the qubit that has been perturbed by the coupling in such a case simplify the perturbative expansions (36) to the detector. Indeed, if the detector operator pˆ were using the following identities, replaced with a constant p, then the interaction Hamilto- nian (4) would constitute an evolution term for the qubit that would induce Rabi oscillations around the σˆ axis of ˆ 3 A = Aσˆ3, (53a) the Bloch sphere, which would be the natural flow in state space generated by the action of Aˆ. With the substitu- 1 n n ρˆi = 1ˆs + rkσˆk , (53b) tion pˆ p then p p , so c(g) cos(2gAp/~) 1 2 → h i0 → → − k ! and s(g) sin(2gAp/~), which restores the unperturbed X → [σˆj , σˆk]=2iǫjklσˆl, (53c) Rabi oscillations. Similarly, we find that the averaging operations for σˆ , σˆ =2δ 1ˆ , (53d) { j k} jk s the detector position and momentum (32) have the exact forms, 3 where σˆk k=1 are the usual Pauli operators, the com- cx(g)r1 sx(g)r2 { } 3 (ρˆi)= x 0ρˆi + − σˆ1 (57a) ponents of the initial system state rk k=1 are real and X h i 2 satisfy the inequality 0 r2 { 1,} ǫ is the com- ≤ k k ≤ jkl cx(g)r2 + sx(g)r1 pletely antisymmetric Levi-Civita pseudotensor, and δjk + σˆ2, P 2 is the Kronecker delta. We have defined σˆ3 to be diago- ∞ n 2n 2n Aˆ ( 1) 2Ag p , x 0 nal in the eigenbasis of and have rescaled the spectrum cx(g)= − h{ }i , (57b) Aˆ (2n)! ~ 2 of for simplicity to zero out its maximally mixed mean n=1   ˆ1ˆ X Trs(A s/2) = 0. As a result, A 0 = Ar3. ∞ n+1 2n−1 2n−1 h i ( 1) 2Ag p , x 0 s (g)= − h{ }i , It follows that for positive integer n the repeated ac- x (2n 1)! ~ 2 tions of Aˆ on the various qubit operators have the forms, n=1 −   X (57c) and, n (adAˆ) (1ˆs)=0, (54a) cp(g)r1 sp(g)r2 (ρˆi)= p 0ρˆi + − σˆ1 (58a) 2n−1 2n−1 P h i 2 (adAˆ) (σˆ1)= iσˆ2(2A) , (54b) cp(g)r2 + sp(g)r1 ˆ 2n 2n + σˆ2, (adA) (σˆ1)= σˆ1(2A) , (54c) 2 ˆ 2n−1 2n−1 ∞ n 2n (adA) (σˆ2)= iσˆ1(2A) , (54d) ( 1) 2Ag 2n+1 − cp(g)= − p 0, (58b) ˆ 2n 2n (2n)! ~ h i (adA) (σˆ2)= σˆ2(2A) , (54e) n=1   X∞ − (adAˆ)n(σˆ )=0, (54f) ( 1)n+1 2Ag 2n 1 3 s (g)= − p2n . (58c) p (2n 1)! ~ h i0 n=1 X −   which collectively imply that, These operations differ from only in how the various moments of the initial detectorE distribution weight the se- i ries for the Rabi oscillation. In particular, given the sub- ˆ 2n−1 2n−1 n n (adA) (ρˆi)= (2A) (r1σˆ2 r2σˆ1) , (55a) stitutions pˆ p and xˆ x, then p , x /2 0 p x 2 − n+1 → n+1 → h{ } i ~→ 1 and p 0 p , so cx(g) x (cos(2gAp/ ) 1), (adAˆ)2n(ρˆ )= (2A)2n (r σˆ + r σˆ ) , (55b) s (g)h xi sin(2→ gAp/~), c (g) → p (cos(2gAp/~) − 1), i 2 1 1 2 2 x → p → − and sp(g) p sin(2gAp/~). Therefore, if the detec- tor remained→ uncorrelated with the system the averag- and hence that the nonselective measurement operation ing operations would reduce to (ρˆ ) x (ρˆ ) and X i → E i 11

(ρˆi) p (ρˆi), which are the decoupled intial detec- H L ∆Σ3 Ρi torP means→ scalingE the Rabi-oscillating qubit state. r2 Since we have assumed that Aˆ does not have a compo- nent proportional to the identity, the symmetric product Aˆ, ρˆi /2 = Ar31ˆs/2 = A 0(1ˆs/2) for a qubit will act effectively{ } as an inner producth i that extracts the part of the initial state proportional to Aˆ. Therefore, the cor- rection to A 0 in f x T that appears in (34a) has the simple form,h i h i

Tr (Pˆ ( Aˆ, ρˆ /2)) g A r1 g s f E { i} = h i0 . (59) Tr (Pˆ (ρˆ )) p˜(f) s f E i where the conditioning factor, p˜(f) = 2Tr (ρˆ (ρˆ )), (60) s f E i = 2Trs(ρˆf ρˆi) + (c(g)r s(g)r )Tr (ρˆ σ ) 1 − 2 s f 1 + (c(g)r2 + s(g)r1)Trs(ρˆf σ2), FIG. 3. (color online) The projection onto the plane r3 = 0 of the qubit Bloch sphere, showing the vector field δσ (ρˆi)= is (2/Trs(Pˆf )) times the total probability of obtaining the 3 r2σˆ 1 + r1σˆ2 for arbitrary initial states ρˆi = (1ˆ + r1σˆ1 + post-selection. We have expressedp ˜(f) more compactly − r2σˆ2 + r3σˆ 3)/2. The curves of the flow through this vector in terms of the retrodictive state (44) to show how the ˆ field are the Rabi oscillations around the r3 axis that are deviations from the initial state that are induced by A generated by the unitary action of σˆ3. The quantity 2ImAw become effectively averaged by the post-selection state. (63) is the logarithmic rate of change of the post-selection In the absence of post-selection, the retrodictive state will probability (41) along this vector field. be maximally mixed ρˆf = 1ˆs/2andp ˜(f) 1, recovering the unconditioned average A . → h i0 The correction to the detector mean position x 0 in (34a) can be expressed in a similar way, h i initial state, conditioned by the disturbance-free overlap between the predictive and retrodictive states. The imag- Trs(Pˆf (ρˆi)) 1 inary part, on the other hand, contains δA(ρˆi), which is X = 2 x 0Trs(ρˆf ρˆi) (61) a tangent vector field on the Bloch sphere—illustrated in Trs(Pˆf (ρˆi)) p˜(f) h i E  Figure 3—that corresponds to an infinitesimal portion of + (cx(g)r1 sx(g)r2)Trs(ρˆf σˆ1) ˆ − the Rabi oscillation being generated by A. This tangent vector field contains only the components r and r from + (cx(g)r2 + sx(g)r1)Trs(ρˆf σˆ2) , 1 2 bases orthogonal to Aˆ in the initial state ρˆi, so 2ImAw  as can the correction to the detector mean momentum contains only the retrodictive averages of corrections to orthogonal Aˆ p 0 in (34b), bases to , and thus contains no information h i about the measurement of Aˆ as an observable. As dis- Trs(Pˆf (ρˆi)) 1 cussed in (41), 2ImAw is the logarithmic rate of change P = 2 p 0Trs(ρˆf ρˆi) (62) of the post-selection probability along the vector field Trs(Pˆf (ρˆi)) p˜(f) h i E  δA(ρˆi). Scaling it by a small factor with units inverse + (c (g)r s (g)r )Tr (ρˆ σˆ ) p 1 − p 2 s f 1 to A will produce a probability correction to linear or- der. In the absence of post-selection, then ρˆ 1ˆ /2, + (cp(g)r2 + sp(g)r1)Trs(ρˆf σˆ2) . f → s ReAw A 0, and ImAw 0.  → h i → Expanding (34) using (59), (61), and (62) to linear order in g, we find the linear response (38) in terms of the real and imaginary parts of the qubit weak value,

A 0 C. Gaussian Detector ReAw = h i , (63a) 2Trs(ρˆf ρˆi) Trs(ρˆf δA(ρˆi)) We can also apply our general results to the traditional 2ImAw = , (63b) Trs(ρˆf ρˆi) case when the initial detector state in (10) is a zero-mean δ (ρˆ )= A( r σˆ + r σˆ ). (63c) Gaussian in position, A i − 2 1 1 2 As expected, the real part contains information re- ˆ x ψ = (2πσ2)−1/4 exp( x2/4σ2), (64) garding the measurement of A as an observable in the h | i − 12

Then the measurement operators for position detection can see more clearly by rewriting the non-selective mea- (12) have the initial Gaussian form shifted by gAˆ, surement operation in (70) as,

2 2 1 −(x−gAˆ) /4σ ˆ Mˆ = e , (65) ρˆi(ǫ)= (ρˆi) = exp ǫ [A] (ρˆi), (71a) x (2πσ2)1/4 E L 1   [Aˆ](ρˆ )= Aˆρˆ Aˆ† ρˆ , Aˆ†Aˆ . (71b) while the conjugate measurement operators for momen- L i i − 2{ i } tum detection (20) have the initial Gaussian modified by ˆ a unitary factor containing Aˆ, The operation [A](ρˆi) is the Lindblad operation [75, 76, 82] that producesL decoherence in continuous dynamical 1/4 ˆ 2 2 2 2 systems, with A playing the role of the Lindblad operator ˆ 2σ −p σ /~ gpAˆ/i~ Np = e e . (66) ρˆ Aˆ ρˆ π~2 that decoheres the system. Since ∂ǫ i(ǫ)= [ ]( i(ǫ)),   the Gaussian measurement acts as an effectiveL Lindblad The Wigner quasi-probability operator (23a) corre- evolution that decoheres the system state with increas- ˆ spondingly decouples into a product of Gaussian distri- ing ǫ via the action of A, but does not cause unitary ˆ butions, with only the position shifted by the system op- disturbance along the natural flow of A [83]. erator, The exact expressions for the conditioned Gaussian de- tector means follow from (34) and (70), 2 2 2 2 2 ˆ 1 −(x−gAˆ) /2σ −2p σ /~ Wx,p = e e . (67) ˆ ˆ π~ Trs(Pf ( A, ρˆi )) f x T = g E { } , (72a) h i 2Tr (Pˆ (ρˆ )) Marginalizing the Wigner operator over momentum s f E i 2 and position separately produces the probability oper- g ~ Trs(Pˆf (adAˆ)( (ρˆi))) f p T = E . (72b) ators (14) and (21), h i i~ 4σ2 Tr (Pˆ (ρˆ )) s f E i 2 2 1 −(x−gAˆ) /2σ Eˆx = e , (68a) Surprisingly, the special properties of the Gaussian mo- √2πσ2 ments (69) allow (72) to be written in a form proportional σ 2 2 2 ~2 to the real and imaginary part of a complex weak-value Fˆ = e−2p σ / 1ˆ . (68b) p ~ π s involving the decohered system state (71) to all orders in r the coupling strength g, As anticipated, the probability operator for momentum ˆ ˆ no longer contains any information about the system op- Trs(Pf Aρˆi(ǫ)) ˆ Aw(ǫ)= , (73a) erator A and is proportional to the identity, so measuring Trs(Pˆf ρˆi(ǫ)) the momentum provides zero information about any sys- x = gReA (ǫ), (73b) tem operator not proportional to the identity. f h iT w In the presence of post-selection we can also exactly g ~2 p = (2ImA (ǫ)). (73c) compute the disturbance operations (27), (30a), and f h iT ~ 4σ2 w (30b) using the following identities for the Gaussian de- Following the interpretations outlined in this paper we tector moments, can therefore understand the position shift ReAw(ǫ) to ~ 2n all orders in g as the average of the observable Aˆ in the 2n p 0 = (2n 1)!!, (69a) decohered initial system state ρˆ (ǫ) conditioned on the h i 2σ − i   ˆ 2n−1 post-selection Pf . Similarly, we can understand the fac- p 0 =0, (69b) tor 2ImAw(ǫ) in the momentum shift to all orders in g as nh i p , x /2 0 =0, (69c) the logarithmic directional derivative of the probability of h{ } i ˆ (2n 1)!! 1 post-selecting Pf given the decohered initial system state − = , (69d) ˆ (2n)! 2n n! ρˆi(ǫ) along the unitary flow generated by A. If the measured operator is the qubit operator Aˆ = which hold for positive integer n. We find the simple Aσˆ3 as in (53), then we can further simplify the expres- results, sion (70) using the identities (55) to find,

2 2 1 g 2 1 −(Ag/σ) /2 (ρˆi) = exp (adAˆ) (ρˆi), (70a) (ρˆ )= ρˆ + (e 1) (r σˆ + r σˆ ) , (74) E −2 2σ E i i 2 − 1 1 2 2     1 2 (ρˆi)=0, (70b) = 1ˆ + r σˆ + e−(Ag/σ) /2 (r σˆ + r σˆ ) , X 2 3 3 1 1 2 2 g ~2 (ρˆ )= (adAˆ)( (ρˆ )). (70c)   P i i~ 4σ2 E i which shows how the measurement decoheres the bases orthogonal to Aˆ in the initial state with an increase in the The quantity ǫ = (g/2σ)2 with units inverse to Aˆ2 dimensionless flow parameter (Ag/σ)2 [84]. This deco- emerges as the natural decoherence parameter, which we herence is illustrated in Figure 4. The conditioned means 13

state (ρˆi) is essentially diagonal in the basis of Aˆ, it ΡiHΕL ∆ HΡ HΕLL E Σ3 i will no longer Rabi oscillate, so the directional derivative r2 r2 along the flow generated by Aˆ will be essentially zero. Hence, the probability correction factor represented by 2ImAw(ǫ) vanishes.

r 3 r1

VI. CONCLUSION

We have given an exact treatment of a conditioned von Neumann measurement for an arbitrary initial state and FIG. 4. (color online) Two projections of the Bloch sphere an arbitrary post-selection using the language of quan- showing the pure decoherence of the specific state ρˆi(ǫ) = tum operations. The full form of the conditioned de- exp(ǫ [σˆ3])(ρˆi)=(1ˆ + exp( 2ǫ)√3σˆ 2/2 + σˆ 3/2)/2 due to tector response (34) naturally indicates how the mea- the GaussianL detector (74).− (left) The projection onto the surement disturbance and conditioning from the post- plane r1 = 0 showing the progressive collapse of ρˆi(ǫ) onto selection modify the unconditioned detector response. the r3 axis with increasing ǫ. (right) The projection onto the The corresponding linear response of the detector (38) plane r3 = 0 showing the vector field δσ3 (ρˆi(ǫ)) during the can be parametrized by the generalized complex weak progressive collapse. Notably the quantity 2ImAw(ǫ) (73) is value (35), but the origins of the real and imaginary parts the rate of change of the post-selection probability (41) along differ. this vector field for all ǫ, but not along the purely decohering trajectory that ρˆi(ǫ) actually follows. The real part of the weak value (2) stems directly from the measurement of Aˆ made on the initial system state, and can be interpreted as the idealized zero-disturbance (72) of a Gaussian qubit detector consequently have the conditioned average of the operator Aˆ acting in its role exact form, as an observable. The imaginary part of the weak value A (3), on the other hand, contains no information about the x = g h i0 , (75a) f h iT p˜(f) measurement of Aˆ as an observable, but instead arises 2 from the disturbance due to the von Neumann coupling. g ~ 2A 2 p = e−(Ag/σ) /2 (75b) We interpret it as the logarithmic directional derivative f T ~ 4σ2 p˜(f) h i × (41) of the post-selection probability along the unitary (r1Trs(ρˆf σˆ2) r2Trs(ρˆf σˆ1)), ˆ − flow in state space generated by the operator A in its p˜(f)=1+ r3Trs(ρˆf σˆ3) (75c) role as the element of a Lie algebra. The complex weak 2 value therefore captures both halves of the dual role of + e−(Ag/σ) /2 (r Tr (ρˆ σˆ )+ r Tr (ρˆ σˆ )) , 1 s f 1 2 s f 2 the operator Aˆ in the quantum formalism. to all orders in the coupling strength g. When expanded To illustrate this interpretation, we considered the to linear order in g, (75) reduces to (38) with the real weak value of momentum (51) post-selected on a par- and imaginary parts of the qubit weak value (63), as ticular position. Its real part is the Bohmian momen- expected. tum representing the average momentum conditioned on For contrast, as g becomes large the unconditioned a position detection, while its imaginary part (52) is pro- ˆ measurement of A becomes essentially projective and the portional to the “osmotic velocity” that describes the operation almost completely decoheres the initial state logarithmic derivative of the probability density for mea- E ˆ (74) into the basis of A as the pointer basis, suring the particular position directed along the flow gen- 1 erated by the momentum. (ρˆi) 1ˆs + r3σˆ3 . (76) E ≈ 2 Finally, we applied our exact solution to the useful Hence, in this strong measurement regime,
the condi- special cases of a qubit operator and a Gaussian detec- tioned means (75) approximate, tor. We showed how the natural qubit Rabi oscillations that would be generated by the von Neumann interaction A 0 Hamiltonian (4) become disrupted by the measurement. f x T g h i , (77a) h i ≈ 1+ r3Trs(ρˆf σˆ3) We also showed that the Gaussian detector purely de- ˆ f p T 0. (77b) coheres the initial system state into the basis of A in h i ≈ the Lindblad sense, which allows the exact interpreta- The position shift contains the average of Aˆ in the deco- tion of the position and momentum shifts of the detector hered initial system state (ρˆi), conditioned by the post- in terms of a complex weak value (73) that involves the selection. Moreover, sinceE the decohered initial system decohered system state. 14

ACKNOWLEDGMENTS

This work was supported by the NSF Grant No. DMR- 0844899, and ARO Grant No. W911NF-09-0-01417.

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