The significance of the imaginary part of the weak value

# The significance of the imaginary part of the weak value

J. Dressel    A. N. Jordan Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA
July 27, 2019
###### Abstract

Unlike the real part of the generalized weak value of an observable, which can in a restricted sense be operationally interpreted as an idealized conditioned average of that observable in the limit of zero measurement disturbance, the imaginary part of the generalized weak value does not provide information pertaining to the observable being measured. What it does provide is direct information about how the initial state would be unitarily disturbed by the observable operator. Specifically, we provide an operational interpretation for the imaginary part of the generalized weak value as the logarithmic directional derivative of the post-selection probability along the unitary flow generated by the action of the observable operator. To obtain this interpretation, we revisit the standard von Neumann measurement protocol for obtaining the real and imaginary parts of the weak value and solve it exactly for arbitrary initial states and post-selections using the quantum operations formalism, which allows us to understand in detail how each part of the generalized weak value arises in the linear response regime. We also provide exact treatments of qubit measurements and Gaussian detectors as illustrative special cases, and show that the measurement disturbance from a Gaussian detector is purely decohering in the Lindblad sense, which allows the shifts for a Gaussian detector to be completely understood for any coupling strength in terms of a single complex weak value that involves the decohered initial state.

###### pacs:
03.65.Ta,03.65.Ca,03.67.-a

## I Introduction

In their seminal Letter, Aharonov et al. (1988) claimed that they could consistently assign a particular value to an observable that was being weakly measured in a pre- and post-selected ensemble. To illustrate their technique, they weakly coupled an observable to a continuous detector with an initial Gaussian wave-function. Normally, such a weak von Neumann coupling von Neumann (1932) would approximately shift the mean of the Gaussian detector wave-function by the expectation value of in the initial state , which would effectively measure ; however, they showed that by post-selecting a final state after the weak coupling, the mean of the Gaussian detector wave-function could be made to approximately shift by a complex quantity that they dubbed the weak value of the observable,

 Aw =⟨ψf|^A|ψi⟩⟨ψf|ψi⟩. (1)

Notably, the weak value expression is not constrained to the eigenvalue range for the observable , so it can become arbitrarily large for nearly orthogonal pre- and post-selections.

This complex shift in the mean of the Gaussian detector wave-function was only approximate under weak von Neumann coupling and not directly observable, so its significance was not overtly clear; however, the Letter Aharonov et al. (1988) also showed that both the real and imaginary parts of (1) could be operationally obtained from the linear response of the detector under separate conjugate observable measurements. The practical benefit of this observation was that one could amplify the response of the detector by making a clever choice of post-selection, which potentially allowed for the sensitive determination of other small parameters contributing to the evolution.

After theoretical clarifications of the derivation in Duck et al. (1989), experimental confirmation of such amplified detector response soon followed in optical systems Ritchie et al. (1991); Parks et al. (1998). The amplification has since been used successfully to sensitively measure a variety of phenomena Hosten and Kwiat (2008); Dixon et al. (2009); Starling et al. (2010a, b); Turner et al. (2011); Hogan et al. (2011) to remarkable precision, using both the real and imaginary parts of (1) as amplification parameters. Several theoretical extensions of the original derivation of the amplification Jozsa (2007); Di Lorenzo and Egues (2008); Starling et al. (2009); Geszti (2010); Shikano and Hosoya (2010); Cho et al. (2010); Wu and Li (2011); Haapasalo et al. (2011); Parks and Gray (2011); Zhu et al. (2011); Shikano and Hosoya (2011); Kofman et al. (2011); Nakamura et al. (2011); Koike and Tanaka (2011); Di Lorenzo (2011); Pan and Matzkin (2011) and several proposals for other amplification measurements have also appeared Romito et al. (2008); Brunner and Simon (2010); Zilberberg et al. (2011); Kagami et al. (2011); Li et al. (2011). In particular, it has been noted that how the amplification effect arises in such a continuous wave-function detector is not intrinsically quantum mechanical, but can also occur in classical wave mechanics Howell et al. (2010), which has prompted recent study into the mathematical phenomenon of superoscillations (e.g. Aharonov et al. (2011); Berry and Shukla (2011)).

Conceptually, however, the weak value expression (1) has remained quite controversial: since it is generally complex and not constrained to the spectrum of , how should it be interpreted? Its primary interpretation in the literature has rested somewhat loosely upon the observation that despite its anomalous behavior one can still decompose an expectation value through the insertion of the identity into an average of weak values, , which has the same form as decomposing a classical expectation value into an average of conditioned expectation values . This observation, together with its approximate appearance operationally in weak conditioned measurements, make it tempting to interpret the weak value as a disturbance-free counter-factual conditioned average that can be assigned to the observable within the context of a pre- and post-selected ensemble even when it is not strictly measured Aharonov and Vaidman (1990); Aharonov and Botero (2005); Tollaksen (2007); Hosoya and Koga (2011).

Supporting this point of view is the fact that when the real part of (1) is bounded by the eigenvalue range of , it agrees with the classical conditioned expectation value for the observable Aharonov and Botero (2005). Moreover, even when the real part is outside the normal eigenvalue range, it still obeys a self-consistent logic Aharonov and Vaidman (2008) and seems to indicate oddly sensible information regarding the operator . As such, it has been used quite successfully to analyze and interpret many quantum-mechanical paradoxes both theoretically and experimentally, such as tunneling time Steinberg (1995a, b); Aharonov et al. (2002a, 2003), vacuum Cherenkov radiation Rohrlich and Aharonov (2002), cavity QED correlations Wiseman (2002), double-slit complementarity Wiseman (2003); Mir et al. (2007), superluminal group velocities Brunner et al. (2004), the N-box paradox Aharonov and Vaidman (1991); Resch et al. (2004), phase singularities Solli et al. (2004), Hardy’s paradox Aharonov et al. (2002b); Lundeen and Steinberg (2009); Yokota et al. (2009); Hosoya and Shikano (2010), photon arrival time Wang et al. (2006), Bohmian trajectories Leavens (2005); Wiseman (2007); Kocsis et al. (2011); Hiley (2011), and Leggett-Garg inequality violations Williams and Jordan (2008); Goggin et al. (2011); Dressel et al. (2011a).

Arguably more important for its status as a quantity pertaining to the measurement of , however, is the fact that the real part of (1) appears as a stable weak limit point for conditioned measurements even when the detector is not a von Neumann-coupled continuous wave that can experience superoscillatory interference (e.g. Pryde et al. (2005); Goggin et al. (2011); Dressel et al. (2011a); Iinuma et al. (2011); Dressel et al. (2011b)). As a result, we can infer that at least the real part of (1) must have some operational significance specifically pertaining to the measurement of that extends beyond the scope of the original derivation. This observation prompted our Letter Dressel et al. (2010) showing that a principled treatment of a general conditioned average of an observable can in fact converge in the weak measurement limit to a generalized expression for the real part of (1),

 ReAw =Tr(^Pf{^A,^ρi})2Tr(^Pf^ρi), (2)

where is the anti-commutator between the observable operator and an arbitrary initial state represented by a density operator, and where is an arbitrary post-selection represented by an element from a positive operator-valued measure (POVM). The general conditioned average converges to (2) provided that the manner in which is measured satisfies reasonable sufficiency conditions Dressel and Jordan (2012, 2011) that ensure that the disturbance intrinsic to the measurement process does not persist in the weak limit.

It is in this precise restricted sense that we can operationally interpret the real part of the weak value (2) as an idealized conditioned average of in the limit of zero measurement disturbance. Since it is also the only apparent limiting value of the general conditioned average that no longer depends on how the measurement of is being made, it is also distinguished as a measurement context-independent conditioned average. These observations provide strong justification for the treatment of the real part of the weak value (2) as a form of value assignment Aharonov and Vaidman (1990); Mermin (1993); Spekkens (2005); Aharonov and Botero (2005); Tollaksen (2007); Hosoya and Koga (2011) for the observable that depends only upon the preparation and post-selection 111Note that such a value assignment does not violate the Bell-Kochen-Specker theorem Mermin (1993); Spekkens (2005); Tollaksen (2007) since (2) does not generally obey the product rule, ..

However, we are still left with a mystery: what is the significance of the imaginary part of (1) that appears in the von Neumann measurement, and how does it relate to the operator ? We can find a partial answer to this question in existing literature (e.g. Steinberg (1995a, b); Aharonov and Botero (2005); Jozsa (2007)) that has associated the appearance of the imaginary part of (1) in the response of the detector with the intrinsic disturbance, or back-action, of the measurement process. For example, regarding continuous von Neumann detectors Aharonov and Botero (2005, p.8) note that “the imaginary part of the complex weak value can be interpreted as a ‘bias function’ for the posterior sampling point [of the detector].” Furthermore, they note that “the weak value of an observable is tied to the role of as a generator for infinitesimal unitary transformations” (Aharonov and Botero, 2005, p.11). Similarly, while discussing measurements of tunneling time Steinberg (1995a) states that the imaginary part is a “measure of the back-action on the particle due to the measurement interaction itself” and that the detector shift corresponding to the imaginary part “is sensitive to the details of the measurement apparatus (in particular, to the initial uncertainty in momentum), unlike the [shift corresponding to the real part].”

In this paper, we will augment these observations in the literature by providing a precise operational interpretation of the following generalized expression for the imaginary part of (1),

 ImAw =Tr(^Pf(−i[^A,^ρi]))2Tr(^Pf^ρi), (3)

where is the commutator between and the initial state. We will see that the imaginary part of the weak value does not pertain to the measurement of as an observable. Instead, we will interpret it as half the logarithmic directional derivative of the post-selection probability along the flow generated by the unitary action of the operator . As such, it provides an explicit measure for the idealized disturbance that the coupling to would have induced upon the initial state in the limit that the detector was not measured, which resembles the suggestion by Steinberg (1995a); however, we shall see that the measurement of the detector can strongly alter the state evolution away from that ideal. The explicit commutator in (3) also indicates that the imaginary part of the weak value involves the operator in its role as a generator for unitary transformations as suggested by Aharonov and Botero (2005), in contrast to the real part (2) that involves the operator in its role as a measurable observable.

To make it clear how the generalized weak value expressions (2) and (3) and their interpretations arise within a traditional von Neumann detector, we will provide an exact treatment of a von Neumann measurement using the formalism of quantum operations (e.g. Nielsen and Chuang (2000); Breuer and Petruccione (2007); Wiseman and Milburn (2009)). In addition to augmenting existing derivations in the literature that are concerned largely with understanding the detector response (e.g. Jozsa (2007); Di Lorenzo and Egues (2008); Geszti (2010); Wu and Li (2011); Haapasalo et al. (2011); Parks and Gray (2011); Zhu et al. (2011); Kofman et al. (2011); Nakamura et al. (2011); Koike and Tanaka (2011); Di Lorenzo (2011); Pan and Matzkin (2011)), our exact approach serves to connect the standard treatment of weak values to our more general contextual values analysis that produces the real part Dressel et al. (2010); Dressel and Jordan (2012, 2011) more explicitly. We also provide several examples that specialize our exact solution to typically investigated cases: a particular momentum weak value, an arbitrary qubit observable measurement, and a Gaussian detector. As a consequence, we will show that the Gaussian detector is notable since it induces measurement disturbance that purely decoheres the system state into the eigenbasis of in the Lindblad sense with increasing measurement strength. Surprisingly, the pure decoherence allows the shifts in a Gaussian detector to be completely parametrized by a single complex weak value to all orders in the coupling strength, which allows those shifts to be completely understood using our interpretations of that weak value.

The paper is organized as follows. In §II we analyze the von Neumann measurement procedure in detail, starting with the traditional unconditioned analysis in §II.1, followed by an operational analysis of the unconditioned case in §II.2.1 and the conditioned case in §II.2.2. After obtaining the exact solution for the von Neumann detector response, we consider the weak measurement regime to linear order in the coupling strength in §III, which clarifies the origins and interpretations of the expressions (2) and (3). We discuss the time-symmetric picture in §IV for completeness. After a brief Bohmian mechanics example in §V.1 that helps to illustrate our interpretation of the weak value, we provide the complete solutions for a qubit observable in §V.2 and a Gaussian detector in §V.3. Finally, we present our conclusions in §VI.

## Ii von Neumann Measurement

The traditional approach for obtaining a complex weak value Aharonov et al. (1988) for a system observable is to post-select a weak Gaussian von Neumann measurement von Neumann (1932). The real and imaginary parts of the complex weak value then appear as scaled shifts in the conditioned expectations of conjugate detector observables to linear order in the coupling strength. To clarify how these shifts occur and how the weak value can be interpreted, we shall solve the von Neumann measurement model exactly in the presence of post-selection.

A von Neumann measurement von Neumann (1932); Aharonov et al. (1988) unitarily couples an operator on a system Hilbert space to a momentum operator on a continuous detector Hilbert space via a time-dependent interaction Hamiltonian of the form,

 ^HI(t) =g(t)^A⊗^p. (4)

The interaction profile is assumed to be a function that is only nonzero over some interaction time interval . The interaction is also assumed to be impulsive with respect to the natural evolution of the initial joint state of the system and detector; i.e., the interaction Hamiltonian (4) acts as the total Hamiltonian during the entire interaction time interval.

Solving the Schrödinger equation,

 iℏ∂t^U =^HI^U, (5)

with the initial condition produces a unitary operator,

 ^UT =exp(giℏ^A⊗^p), (6) g =∫T0dtg(t), (7)

that describes the full interaction over the time interval . The constant acts as an effective coupling parameter for the impulsive interaction. If the interaction is weakly coupled then is sufficiently small so that and the effect of the interaction will be approximately negligible; however, we will make no assumptions about the weakness of the coupling a priori.

The unitary interaction (6) will entangle the system with the detector so that performing a direct measurement on the detector will lead to an indirect measurement being performed on the system. Specifically, we note that the position operator of the detector satisfies the canonical commutation relation , and thus will evolve in the Heisenberg picture of the interaction according to,

 (^1s⊗^x)T =^U†T(^1s⊗^x)^UT, (8) =^1s⊗^x+g^A⊗^1d.

As a result, measuring the mean of the detector position after the interaction will produce,

 ⟨x⟩T =⟨x⟩0+g⟨A⟩0. (9)

Hence, the mean of the detector position will be shifted from its initial mean by the mean of the system observable in the initial reduced system state, linearly scaled by the coupling strength . For this reason we say that directly measuring the average of the detector position results in an indirect measurement of the average of the system observable .

The detector momentum , on the other hand, does not evolve in the Heisenberg picture since . Hence, we expect that measuring the average detector momentum will provide no information about the system observable .

As discussed in the introduction, however, when one conditions such a von Neumann measurement of the detector upon the outcome of a second measurement made only upon the system, then the conditioned average of both the position and the momentum of the detector can experience a shift. To see why this is so, we will find it useful to switch to the language of quantum operations (e.g. Nielsen and Chuang (2000); Breuer and Petruccione (2007); Wiseman and Milburn (2009)) in order to dissect the measurement in more detail.

### ii.2 Quantum Operations

#### ii.2.1 Unconditioned Measurement

As before, we will assume an impulsive interaction in what follows so that any natural time evolution in the joint system and detector state will be negligible on the time scale of the measurement. (For considerations of the detector dynamics, see Di Lorenzo and Egues (2008).) We will also assume for simplicity of discussion that the initial joint state of the system and detector before the interaction is a product state and that the detector state is pure,

 ^ρ=^ρi⊗|ψ⟩⟨ψ|, (10)

though we will be able to relax this assumption in our final results. Conceptually, this assumption states that a typical detector will be initially well-calibrated and uncorrelated with the unknown system state that is being probed via the interaction.

Evolving the initial state with the interaction unitary (6) will entangle the system with the detector. Hence, subsequently measuring a particular detector position will be equivalent to performing an operation upon the reduced system state, as illustrated in Figure 1,

 Mx(^ρi) =Trd((^1s⊗|x⟩⟨x|)^UT^ρ^U†T)=^Mx^ρi^M†x, (11) ^Mx =⟨x|^UT|ψ⟩. (12)

where is the partial trace over the detector Hilbert space, and is the Kraus operator associated with the operation . Furthermore, since is the initial detector position wave-function we find , or, more compactly, .

If we do not perform a subsequent post-selection on the system state, then we trace out the system to find the total probability density for detecting the position ,

 p(x) =Trs(Mx(^ρi))=Trs(^Ex^ρi), (13) ^Ex =^M†x^Mx=⟨ψ|^U†T(^1s⊗|x⟩⟨x|)^UT|ψ⟩, (14)

where is the partial trace over the system Hilbert space. The probability operator is a positive system operator that encodes the probability of measuring a particular detector position , and can also be written in terms of the initial detector position wave-function as . To conserve probability it satisfies the condition, , making the operators a positive operator-valued measure (POVM) on the system space.

Consequently, averaging the position of the detector will effectively average a system observable with the initial system state,

 ⟨x⟩T =∫∞−∞dxxp(x)=Trs(^O^ρi), (15) ^O =∫∞−∞dxx^Ex=⟨ψ|^U†T(^1s⊗^x)^UT|ψ⟩, =⟨x⟩0^1s+g^A,

where we see the Heisenberg evolved position operator (8) naturally appear.

Since the probability operators are diagonal in the basis of , then the effective system operator will also be diagonal in the same basis. Hence, by modifying the values that we assign to the position measurements, we can arrange an indirect measurement of any system observable spanned by in the basis of , including itself,

 ^A =∫∞−∞dx(x−⟨x⟩0g)^Ex, (16)

The chosen set of values are contextual values for , which can be thought of as a generalized spectrum that relates to the specific POVM associated with the measurement context Dressel et al. (2010); Dressel and Jordan (2012, 2011). They are not the only values that we could assign to the position measurement in order to obtain the equality (16), but they are arguably the simplest to obtain and compute, as well as the most frequently used in the literature. It is in this precise sense that we can say that the von Neumann coupling leads to an indirect measurement of the average of in the absence of post-selection.

The measurement of comes at a cost, however, since the system state is necessarily disturbed by the operations in order to obtain the probability operators . The state may even be disturbed more than is strictly required to make the measurement of , which can be seen by rewriting the measurement operators in polar form, , with the positive root of the probability operator and an additional unitary operator . This decomposition implies that splits into an effective composition of two distinct operations,

 Mx(^ρi) =Ux(Ex(^ρi)), (17a) Ex(^ρi) =|^Ex|1/2^ρi|^Ex|1/2, (17b) Ux(^ρ′i) =^Ux^ρ′i^U†x. (17c)

We can interpret the operation that involves only the roots of the probability operator as the pure measurement operation producing . That is, it represents the minimum necessary disturbance that one must make to the initial state in order to extract a measurable probability. The second operation unitarily disturbs the initial state, but does not contribute to . Since only can be used to infer information about through the identity (16), we conclude that the disturbance from is superfluous.

To identify the condition for eliminating , we can rewrite the Kraus operator (12) using the polar form of the initial detector position wave-function ,

 ^Mx =exp(iψs(x−g^A))ψr(x−g^A). (18)

The phase factor becomes the unitary operator for , while the magnitude becomes the required positive root for . Hence, to eliminate the superfluous operation from a von Neumann measurement with coupling Hamiltonian (4), one must use a purely real initial detector wave-function in position.

For contrast, measuring only a particular detector momentum will be equivalent to performing a different operation upon the reduced system state,

 Np(^ρi) =Trd((^1s⊗|p⟩⟨p|)^UT^ρ^U†T)=^Np^ρi^N†p, (19) ^Np =⟨p|^UT|ψ⟩=exp(gpiℏ^A)⟨p|ψ⟩. (20)

The Kraus operator has a purely unitary factor containing that will disturb the system, regardless of the form of the initial momentum wave-function . Moreover, the probability operator associated with the momentum measurement has the form,

 ^Fp =^N†p^Np=|⟨p|ψ⟩|2^1s, (21)

which can only be used to measure the identity .

For completeness we also briefly note that the conjugate Kraus operators and are related through a Fourier transform,

 ^Np =1√2πℏ∫∞−∞dxe−ipx/ℏ^Mx, (22a) ^Mx =1√2πℏ∫∞−∞dpeipx/ℏ^Np, (22b)

and that both detector probability operators can be obtained as marginals of a Wigner quasi-probability operator on the system Hilbert space,

 ^Wx,p =1πℏ∫∞−∞dye2ipy/ℏ^M†x+y^Mx−y, (23a) ^Ex =∫∞−∞dp^Wx,p, (23b) ^Fp =∫∞−∞dx^Wx,p. (23c)

In the absence of interaction, then the Wigner quasi-probability operator reduces to the Wigner quasi-probability distribution for the initial detector state, .

#### ii.2.2 Conditioned Measurement

To post-select the system, an experimenter must perform a second measurement after the von Neumann measurement and filter the two-measurement event space based on the outcomes for the second measurement. In other words, the experimenter keeps only those pairs of outcomes for which the second outcome satisfies some constraint. The remaining measurement pairs can then be averaged to produce conditioned averages of the first measurement.

If we represent the second measurement as a set of probability operators indexed by some parameter that can be derived analogously to (14) from a set of operations as illustrated in Figure 2, then the total joint probability densities for the ordered sequences of measurement outcomes and will be,

 p(x,f) =Trs(^PfMx(^ρi))=Trs(^Ex,f^ρi), (24a) p(p,f) =Trs(^PfNp(^ρi))=Trs(^Fp,f^ρi), (24b)

where the joint probability operators,

 ^Ex,f =^M†x^Pf^Mx, (25a) ^Fp,f =^N†p^Pf^Np. (25b)

are not simple products of the post-selection and the probability operators (14) or (21). Those operators can be recovered, however, by marginalizing over the index , since the post-selection probability operators must satisfy a POVM condition .

The joint probabilities (24) will contain information not only about the first measurement and the initial system state, but also about the second measurement and any disturbance to the initial state that occurred due to the first measurement. In particular, the joint probability operators (25) can no longer satisfy the identity (16) due to the second measurement, so averaging the probabilities (24) must reveal more information about the measurement process than can be obtained solely from the operator , the initial state , and the post-selection . As a poignant example, the unitary disturbance in (17) that did not contribute to the operator identity (16) will contribute to the joint probability operators, .

The total probability for obtaining the post-selection outcome can be obtained by marginalizing over either or in the joint probabilities,

 p(f) =∫∞−∞dxp(x,f)=∫∞−∞dpp(p,f), (26) =Trs(^PfE(^ρi)), E(^ρi) =Trd(^UT(^ρi⊗|ψ⟩⟨ψ|)^U†T), (27)

where the operation is the total non-selective measurement that has been performed on . Since is not the identity operation, it represents the total disturbance intrinsic to the measurement process. It includes unitary evolution of the reduced system state due to the interaction Hamiltonian (4), as well as decoherence stemming from entanglement with the measured detector.

By experimentally filtering the event pairs to keep only a particular outcome of the second measurement, an experimenter can obtain the conditional probabilities,

 p(x|f) (28a) p(p|f) (28b)

which can then be averaged to find the exact conditioned averages for the detector position and momentum,

 f⟨x⟩T =∫∞−∞dxxp(x|f)=Trs(^PfXT(^ρi))Trs(^PfE(^ρi)), (29a) f⟨p⟩T =∫∞−∞dppp(p|f)=Trs(^PfPT(^ρi))Trs(^PfE(^ρi)), (29b)

where,

 XT(^ρi) =Trd((^1s⊗^x)^UT(^ρi⊗|ψ⟩⟨ψ|)^U†T), (30a) PT(^ρi) =Trd((^1s⊗^p)^UT(^ρi⊗|ψ⟩⟨ψ|)^U†T), (30b)

are detector averaging operations that affect the system state before the measurement of the post-selection is performed. It is worth noting at this point that we can relax the assumption (10) made about the initial state in the exact operational expressions (27) and (30). Similarly, if different contextual values are used to average the conditional probabilities in (29), then corresponding detector observables with the same spectra will appear in the operations (30) in place of or ; for example, averaging the values used in (16) will replace the detector observable in (30) with .

To better interpret (30), we bring the detector operators inside the unitary operators in (30) using the canonical commutation relations as in (8),

 XT(^ρi) =X(^ρi)+gE({^A,^ρi}/2), (31a) PT(^ρi) =P(^ρi), (31b)

which splits the operation into two operations but only changes the form of . The operation proportional to disturbs the symmetrized product of the initial system state with the operator , while the operations,

 X(^ρi) =Trd(^UT(^ρi⊗{^x,|ψ⟩⟨ψ|}/2)^U†T), (32a) P(^ρi) =Trd(^UT(^ρi⊗{^p,|ψ⟩⟨ψ|}/2)^U†T), (32b)

disturb the symmetrized products of the initial detector state with the detector operators.

The form of the equations (31) clearly illustrates how the post-selection will affect the measurement. If the post-selection is the identity operator, , then the unitary operators causing the total disturbance of the initial state will cancel through the cyclic property of the total trace in (29), leaving the averages in the initial states that were previously obtained,

 ⟨x⟩T =⟨x⟩0+g⟨A⟩0, (33a) ⟨p⟩T =⟨p⟩0. (33b)

In this sense, commuting the detector operators and in (30) through the unitary operators to arrive at (31) is equivalent to evolving them in the Heisenberg picture back from the time of measurement to the initial time in order to compare them with the initial states. However, the presence of the post-selection operator will now generally spoil the cancelation of the unitary operators that is implicit in the Heisenberg picture, leading to corrections from the disturbance between the pre- and post-selection.

The symmetrized products in (31) indicate the measurement being made on the initial states of the system and detector, which is then further disturbed by the unitary operators as a consequence of the coupling Hamiltonian (4). The post-selection both conditions those measurements and reveals the disturbance, which corrects each term in (33), yielding the final exact expressions,

 f⟨x⟩T =Trs(^PfX(^ρi))Trs(^PfE(^ρi))+gTrs(^PfE({^A,^ρi}))2Trs(^PfE(^ρi)), (34a) f⟨p⟩T =Trs(^PfP(^ρi))Trs(^PfE(^ρi)). (34b)

## Iii The Weak Value

If it were possible to leave the system state undisturbed while still allowing the measurement of , then we would naïvely expect the disturbance to reduce to the identity operation. Similarly, we would naïvely expect the operations and would reduce to and multiplying the identity operation, respectively. As a result, the conditioned averages (34) would differ from the unconditioned averages (33) solely by the replacement of the average with the real part (2) of the complex generalized weak value expression,

 Aw =Trs(^Pf^A^ρi)Trs(^Pf^ρi). (35)

Since this expression depends solely upon the initial state , the post-selection , and the operator , we are naïvely tempted to give an intuitive interpretation as the ideal conditioned expectation of in a pre- and post-selected state with no intermediate measurement disturbance. However, it is strictly impossible to remove the disturbance from the measurement while still making the measurement, so we cannot rely on this sort of reasoning. We can make a similar interpretation in a restricted sense, however, by making the coupling strength sufficiently small to reduce the disturbance to a minimal amount that still allows the measurement to be made.

To see how the operations , , and in (27) and (32) depend on the coupling strength , we expand them perturbatively,

where the operation is the left action of in the adjoint representation of its Lie algebra, which takes the form of a commutator. That is, explicitly describes how disturbs the initial state due to the interaction that measures it.

The initial detector state plays a critical role in (36) by determining the various moments, , and that appear in the series expansions. Notably, if we make the initial detector wave-function purely real so that it minimally disturbs the system state then all moments containing odd powers of will vanish. We conclude that those moments of the disturbance operations are superfluous for obtaining the measurable probabilities that allow the measurement of , while the moments with even powers of are necessary.

After expanding the corrections (34) to first order in , we obtain the linear response of the conditioned detector means due to the interaction,

Measurements for which this linear response is a good approximation are known as weak measurements.

After introducing the complex generalized weak value (35), we can write the linear response formulas in a more compact form,

 f⟨x⟩T =⟨x⟩0+gℏ⟨{p,x}⟩02(2ImAw)+gReAw, (38a) f⟨p⟩T =⟨p⟩0+gℏ⟨p2⟩0(2ImAw), (38b)

in terms of not only its real part, but also twice its imaginary part.

If the initial detector position wave-function is purely real, so that the measurement is minimally disturbing, then will vanish, leaving only in as we naïvely reasoned before. However, the term proportional to will not vanish in to linear order in , making it an element of measurement disturbance that persists even for minimally disturbing weak measurements.

These linear response formulas for the von Neumann measurement have also been obtained and discussed in the literature with varying degrees of generality and rigor (e.g. Aharonov et al. (1988); Duck et al. (1989); Jozsa (2007); Aharonov and Vaidman (2008); Di Lorenzo and Egues (2008); Geszti (2010); Wu and Li (2011); Haapasalo et al. (2011); Parks and Gray (2011); Kofman et al. (2011); Nakamura et al. (2011); Koike and Tanaka (2011); Di Lorenzo (2011); Pan and Matzkin (2011)). However, our derivation has a conceptual advantage in that we see explicitly how the origins of the real and imaginary parts of the weak value differ with respect to the measurement of . We are therefore in a position to give concrete interpretations for each part.

The real part (2) of the weak value stems directly from the part of the conditioned shift of the detector pointer that corresponds to the measurement of and does not contain any further perturbation induced by the measurement coupling that would be indicated by factors of . As a result, it can be interpreted as an idealized limit point for the average of in the initial state that has been conditioned on the post-selection without any appreciable intermediate measurement disturbance. To support this point of view, we have also shown in Dressel et al. (2010); Dressel and Jordan (2012, 2011) that appears naturally as such a limit point for minimally disturbing measurements that are not of von Neumann type, provided that those measurements satisfy reasonable sufficiency conditions regarding the measurability of .

The imaginary part (3) of the weak value , on the other hand, stems directly from the disturbance of the measurement and explicitly contains , which is the action of as a generator for unitary evolution due to the specific Hamiltonian (4). The factor appears in (38) along with information about the initial detector momentum that is being coupled to in the Hamiltonian (4), as well as factors of , in stark contrast to the real part. How then can it be interpreted?

The significance of becomes more clear once we identity the directional derivative operation that appears in its numerator,

That is, indicates the rate of change of the initial state along a flow in state-space generated by .

The directional derivative should be familiar from the Schrödinger equation written in the form , where the scaled Hamiltonian is a characteristic frequency operator. The integration of this equation is a unitary operation in exponential form that specifies a flow in state space, which is a collection of curves that is parametrized both by a time parameter and by the initial condition . Specifying the initial condition , picks out the specific curve from the flow that contains . The directional derivative of the initial state along that specific curve is then defined in the standard way, .

The fact that the quantum state space is always a continuous manifold of states allows such a flow to be defined in a similar fashion using any Hermitian operator, such as , as a generator. Analogously to time evolution, such a flow has the form of a unitary operation, , where the real parameter for the flow has units inverse to . Therefore, taking the directional derivative of along the specific curve of this flow that passes through will produce (39). For an explicit example that we will detail in §V.2, the state-space of a qubit can be parametrized as the continuous volume of points inside the unit Bloch sphere; the derivative (39) produces the vector field illustrated in Figure 3 tangent to the flow corresponding to Rabi oscillations of the qubit.

With this intuition in mind, we define the post-selection probability for measuring given an initial state that is changing along the flow generated by ,

 pf(ε) =Trs(^Pf^ρi(ε)). (40)

The logarithmic directional derivative of this post-selection probability then produces the factor that appears in (38),

 2ImAw =∂εlnpf(ε)∣∣ε=0, (41)

which is our main result.

In words, the imaginary part of the weak value is half the logarithmic directional derivative of the post-selection probability along the natural unitary flow generated by . It does not provide any information about the measurement of as an observable, but rather indicates an instantaneous exponential rate of change in the post-selection probability due to disturbance of the initial state caused by in its role as a generator for unitary transformations. Specifically, for small we have the approximate relation,

 pf(ε) ≈pf(0)(1+(2ImAw)ε). (42)

For a pure initial state and a projective post-selection , the expression (41) simplifies,

 2ImAw =∂εln|⟨ψf|exp(−iε^A)|ψi⟩|2∣∣ε=0. (43)

Hence, the corrections containing that appear in (38) stem directly from how the specific von Neumann Hamiltonian (4) unitarily disturbs the initial system state infinitesimally prior to any additional disturbance induced by the measurement of the detector. Conceptually, the coupling induces a natural unitary flow of the initial system state generated by , which for infinitesimal changes the joint probability for a specific by the amount , where is the infinitesimal parameter in (41) that has units inverse to . Averaging this correction to the joint probability with the detector observables or produces the correction terms in (38).

## Iv Time Symmetry

As noted in Aharonov et al. (1964); Aharonov and Vaidman (2008), a quantum system that has been pre- and post-selected exhibits time symmetry. We can make the time symmetry more apparent in our operational treatment by introducing the retrodictive state,

 ^ρf =^Pf/Tr(^Pf), (44)

associated with the post-selection (see, e.g., Pegg et al. (2002); Amri et al. (2011)) and rewriting our main results in the time-reversed retrodictive picture.

After cancelling normalization factors, the detector response (34) for a system retrodictively prepared in the final state that has been conditioned on the pre-selection measurement producing the initial system state has the form,

 i⟨x⟩T =Trs(X∗(^ρf)^ρi)Trs(E∗(^ρf)^ρi)+gTrs({E∗(^ρf),^A}^ρi)2Trs(E∗(^ρf)^ρi), (45a) i⟨p⟩T =Trs(P∗(^ρf)^ρi)Trs(E∗(^ρf)^ρi), (45b)

where the retrodictive operations , , and are the adjoints of the predictive operations in (27) and (32),

 E∗(^ρf) =⟨ψ|^U†T^ρf^UT|ψ⟩, (46a) X∗(^ρf) =⟨ψ|{^x,^U†T^ρf^UT}/2|ψ⟩, (46b) P∗(^ρf)