A quasi-probability for the arrival time problem with links to backflow and the Leggett-Garg inequalities

# A quasi-probability for the arrival time problem with links to backflow and the Leggett-Garg inequalities

J.J.Halliwell    H.Beck    B.K.B.Lee    S.O’Brien Blackett Laboratory
Imperial College
London SW7 2BZ
UK
###### Abstract

The arrival time problem for the free particle in one dimension may be formulated as the problem of determining a joint probability for the particle being found on opposite sides of the -axis at two different times. We explore this problem using a two-time quasi-probability linear in the projection operators, a natural counterpart of the corresponding classical problem. We show that it can be measured either indirectly, by measuring its moments in different experiments, or directly, in a single experiment using a pair of sequential measurements in which the first measurement is weak (or more generally, ambiguous). We argue that when positive, it corresponds to a measurement-independent arrival time probability. For small time intervals it coincides approximately with the time-averaged current, in agreement with semiclassical expectations. The quasi-probability can be negative and we exhibit a number of situations in which this is the case. We interpret these situations as the presence of “quantumness”, in which the arrival time probability is not properly defined in a measurement-independent manner. Backflow states, in which the current flows in the direction opposite to the momentum, are shown to provide an interesting class of examples such situations. We also show that the quasi-probability is closely linked to a set of two-time Leggett-Garg inequalities, which test for macroscopic realism.

## I Introduction

The arrival time problem in quantum mechanics has been the subject of many papers over the years Allcock (); TQM (); TQM2 (); rev1 (); rev2 (). It continues to be an interesting problem since it is a simple example of a question easily addressed in classical mechanics whilst quantum mechanics provides no unique answer. Most simply stated, in one-dimensional quantum mechanics, it is the question of determing the probability that an incoming wave packet crosses the origin during a given time interval.

There are many approaches to this problem. One of the earliest and most-studied appraoches involves the construction of an arrival time operator AB () by quantizing the classical result (for an incoming classical particle with momentum and position ). This typically leads to arrival time operators which are not self-adjoint Pauli (), although self-adjoint variants have been proposed DM (); MI (); HELM (). Even when self-adjoint, such operators are not obviously connected to a particular measurement scheme, since there is no obvious way of creating a physically realizable coupling between a measuring device and any of the proposed arrival time operators. (See however, Ref.HELM ()).

In the present paper we will focus on arrival times in one-dimensional quantum mechanics defined using sequential measurements onto the positive or negative -axis. Suppose we consider a system prepared in state at and use measurements at a set of closely spaced times to determine its behaviour. Introducing projection operators and onto the positive and negative -axis, the state

 |Ψ123⋯n⟩=P+(tn)P−(tn−1)⋯P−(t2)P−(t1) |Ψ⟩, (1)

represents the amplitude for the history in which the particle lies in at times and is in at , where denotes the projection operator in the Heisenberg picture. For sufficiently close spacing of the times, this object is then a plausible candidate for the amplitude for the particle to make a left-right crossing of the origin, for the first time, during the time interval . The probability for the crossing is then the norm of this state.

Note that in an expression of the form Eq.(1), one would expect the Zeno effect Zeno () to come into play for sufficiently frequent measurements. This is indeed the case – it becomes significant when the time interval between projectors is smaller than , where is the energy scale of the incoming packet HaYe1 (); HaYe5 ().

The probability derived from Eq.(1) is a special case of the standard quantum-mechanical formula for a set of sequential measurements at times,

 p(s1,s2,⋯sn)=Tr(Psn(tn)Psn−1(tn−1)⋯Ps1(t1)ρPs1(t1)⋯Psn−1(tn−1)), (2)

where we have introduced the convenient notation in which the projectors and are written

 Ps=12(1+s^Q), (3)

where and (and the introduction of the dichomotic variable makes notational contact with the related papers Refs.HalLG1 (); HalLG2 (); HalLG3 (); HalLG4 ()). We use the Heisenberg picture in which , denotes the trace, and we have taken a general mixed initial state .

The formula Eq.(2) gives the probabilities for the set of all possible histories in which the particle may be in or at each of the times . This will permit a quite detailed characterization of the arrival time probability.

In what follows we will focus on the simplest case in which measurements onto the positive and negative -axis are made at just two times, , . The probabilities for the four possible histories of the system are then given by the two-time version of Eq.(2), which we denote,

 p12(s1,s2)=Tr(Ps2(t2)Ps1(t1)ρPs1(t1)). (4)

From this we may obtain the probabilities for, respectively, left-right and right-left crossings, and , and for remaining on the left or on the right, and .

In general, sequential measurements in quantum mechanics have the property that each measurement disturbs any later measurement. To quantify this, consider the probability for a measurement at the second time only in which no earlier measurement was carried out, namely

 p2(s2)=Tr(Ps2(t2)ρ). (5)

If the first measurement in Eq.(4) does not disturb the later measurement, then one would expect that a relation of the form

 ∑s1p12(s1,s2)=p2(s2), (6)

would hold. This is often called a probability sum rule (or, elsewhere, the “no signaling in time” condition KoBr (); Cle ()). However it does not hold in general since we have

 ∑s1p12(s1,s2)=Tr(Ps2(t2)ρM(t1)), (7)

where denotes the measured density operator,

 ρM(t1)=∑s1Ps1(t1)ρPs1(t1). (8)

This does not coincide with Eq.(5) except under very specific conditions, such as an initial density matrix which is diagonal in the appropriate basis.

This feature of the two-time sequential measurement probabilities means that it is difficult to define the arrival time probability in this way. Of course one could still physically measure Eq.(4) and it is still a probability, in the sense that its components all sum to . But the failure to satisfy the sum rule suggests that such a probability formula has in general significant dependence on the measurement procedure employed to determine it and indeed specific models exhibit exactly this property. (It is however, sometimes possible to extract the ideal arrival time distribution proposed by Kijowski Kij () from a particular measurement process DEHM ()).

Furthermore, there is a related question around the issue of “quantumness”. The sum rule Eq.(6) is, as we shall see, a stringent classicality condition since it effectively requires zero interference between different histories. This suggests that the two-time probability formula only gives sensible results in a highly classicalized regime. However, this is on the face of it unusually restrictive, constrasting strongly with the operator approach to the arrival time problem, where there does not appear to be any restrictions on the situations in which arrival time probabilities can be found. Some reasonable questions to ask are therefore as follows: Is it still possible to assign probabilities to two-time histories in a physically sensible and measurement-independent way in the face of non-trivial interference? In what ways does quantumness show itself in the arrival time problem, other than the failure of the sum rule Eq.(6)? How is it measured?

In this paper we will offer interesting possible answers to these questions. The key point is that in attempting to find two-time probabilities for a pair of non-commuting observables (namely, the sign of at two different times), quantum mechanics offers more than one possibility. Different ways correspond to different measurement methods. The first obvious way is via sequential measurements, as described. But there is a second, different sort of approach which is to employ the closely related two-time quasi-probability,

 q(s1,s2)=12Tr((Ps2(t2)Ps1(t1)+Ps1(t1)Ps2(t2))ρ). (9)

Expressions of this type were proposed in Ref.GoPa () and explored in Refs.HalLG1 (); HaYe2 (). This expression has the advantage that it satisfies sum rules of the form Eq.(6), but can be negative, since the projectors at different times do not commute. However, it is clearly a natural analogue of the corresponding classical problem and furthermore, its possible negativity provides exactly the indicator of quantumness that we seek.

The purpose of this paper is to explore the properties of the quasi-probability Eq.(9) as both an indicator of quantumness in the arrival time problem (and beyond) and, when positive, a candidate probability for it.

We start in Section 2 by discussing the general properties of a quasi-probability of the form Eq.(9), for a general dichotomic variable and its relationship to the two-time measurement formula Eq.(4). We will show that it can be positive as long as quantum inteference effects are suitably bounded, a relaxation of the sum rule Eq.(6) (which requires zero interference). We also show that the quasi-probabilty Eq.(9) cannot be smaller than .

In Section 3 we will show that the quasi-probability may be measured experimentally either indirectly, by measuring its moments, or more directly using sequential measurements that are weak or, more generally, ambiguous measurements. In Section 4 we show that the quasi-probability is closely related to the current and to the kinetic energy density in the short time limit, thereby linking to standard results for the arrival time distribution.

Sections 5, 6 and 7 explore the types of states which lead to negative quasi-probability. In Section 5 we explore the relationship between the quasi-probability and quantum backflow – the unusual effect in which a state of positive momenta can have negative current. We find that the component of the quasi-probability is negative for backflow eigenstates. These results also indicate a possible way of measuring quantum backflow. In Section 6 we show how a significantly negative quasi-probability may be obtained using superpositions of gaussians and also that negativity can be obtained from a single gaussian. In Section 7, we write the quasi-probability in terms of the Wigner-Weyl representation (of which the Wigner representation of density matrices is an example). This gives a convenient phase space representation of the quasi-probability which makes it relatively easy to identify what is required of states which make the quasi-probability negative.

As stated above, the quasi-probability is regarded as a candidate probability for the arrival time problem, when non-negative. However, to be a genuine probability it must have a relationship to relative frequencies of certain outcomes in a measurement process. We explain this connection in Section 8.

In Section 9 we show that the condition has the form of a set of two-time Leggett-Garg inequalities, which are tests of macrorealism, a very specific notion of classicality, analogous to local realism in Bell tests LG1 (); ELN (). We will outline how the measurements of the quasi-probabity described in earlier sections may be modified to meet the requirements of a Leggett-Garg test. We summarize and conclude in Section 10.

## Ii General Properties of the Quasi-probability.

In this section we summarize some of the general properties of the quasi-probability Eq.(9). The properties described below concern a general dichomatic variable . Properties relating to the specific form of relating to the arrival time problem, , will be described later.

Because it is linear in both projection operators, the quasi-probability satisfies the relations,

 ∑s1q(s1,s2) = Tr(Ps2(t2)ρ)=p2(s2), (10) ∑s2q(s1,s2) = Tr(Ps1(t1)ρ)=p1(s1). (11)

So unlike the two-time measurement probability Eq.(4), it satisfies the probability sum rules.

Eq.(9) is certainly not the only quasi-probability that matches the single time measurement marginals. One could for example add another term involving the commutator of the two projectors. There is also some similarity with Wigner function constructions for finite dimensional systems Woo (). However, what distinguishes a particular choice of quasi-probability is the choice of measurement method and, as we shall see, the choice. Eq.(9) has a natural link to weak measurements.

Furthermore, the quasi-probability Eq.(9) has a simple relation to the standard quantum-mechanical two-time probability Eq.(4), namely,

 q(s1,s2)=p12(s1,s2)+ReD(s1,s2|−s1,s2), (12)

where the quantity

 D(s1,s2|s′1,s2)=Tr(Ps2(t2)Ps1(t1)ρPs′1(t1)), (13)

is the so-called decoherence functional. Its off-diagonal terms are measures of interference between the two different quantum histories represented by sequential pairs of projectors. (We use here the mathematical language of the decoherent histories approach GH1 (); GH2 (); GH3 (); Gri (); Omn1 (); Hal2 (); Hal3 (); DoK (); Ish (); IshLin () but this is not a decoherent histories analysis of the arrival time problem. Such an analysis was carried out in Ref.HaYe5 ().) When

 ReD(s1,s2|s′1,s2)=0,    for  s1≠s′1, (14)

a condition normally referred to as consistency, there is no interference and we have , and the sum rule Eq.(6) is satisfied exactly. However, noting that is always non-negative, we see from Eq.(12) that will be non-negative if the off-diagonal terms of the decoherence functional are bounded,

 |ReD(s1,s2|−s1,s2)|≤p12(s1,s2). (15)

The requirement that the quasi-probability Eq.(9) is non-negative

 q(s1,s2)≥0, (16)

was named “linear positivity” by Goldstein and Page and is one of the weakest conditions under which probabilities can be assigned to non-commuting variables, subject to agreeing with the expected formulae for commuting projectors and to matching the probabilities for projectors at a single time GoPa (). (See also Ref.HaYe2 () for other weak probability assignment conditions). It is satisfied very easily in numerous models, for suitably chosen ranges of parameters, since it requires only partial suppression of quantum interference, not complete destruction of it.

The quasi-probability is very conveniently expanded in terms of its moments,

 q(s1,s2)=14(1+⟨^Q1⟩s1+⟨^Q2⟩s2+C12s1s2), (17)

where the correlation function is given by,

 C12=12⟨^Q1^Q2+^Q2^Q1⟩. (18)

(See Refs.HaYe2 (); Kly () for more on this useful representation). Here, we use the convenient notation and . By contrast the two-time measurement probability, which is always non-negative, has the form

 p12(s1,s2)=14(1+⟨^Q1⟩s1+⟨^Q(1)2⟩s2+C12s1s2). (19)

Here, denotes the average of at time in the context in which an earlier meausrement was made at and is given explicitly by

 ⟨^Q(1)2⟩=⟨^Q2⟩+12⟨[^Q1,^Q2]^Q1⟩ (20)

This extra term on the right-hand side, which clearly vanishes when and commute, is in fact the only difference between and and in particular note that the quasi-probability and the two-time measurement probability have the same correlation function,

 C12=∑s1,s2s1s2p12(s1,s2)=∑s1,s2s1s2q(s1,s2). (21)

as previously noted Fri ().

Finally, one might reasonably ask how negative the quasi-probability may become in quantum theory. This is readily proved using the simply identity,

 q(s1,s2)=18⟨(1+s1^Q1+s2^Q2)2−1⟩, (22)

from which it is easily seen that

 q(s1,s2)≥−18. (23)

## Iii Measurement of the Quasi-probability

There are two obvious methods for determining the quasi-probability experimentally.

### iii.1 Reconstruction from Measurement of the Moments

The first is to simply determine the moments of from a number of different experiments and then assemble the quasi-probability via the moment expansion Eq.(17). This could be done for example, by making use of the sequential measurement probability, which, via the moment expansion Eq.(19) yields and , and then could be determined in a second experiment measuring at time only.

### iii.2 Ambiguous Measurements

A more direct alternative method is to use ambiguous measurements Amb (); Ema (), which include weak measurements weak0 () as a special case. We follow the convenient formulation given by Emary Ema ().

An ambiguous measurement relates to situations in which there is some sort of intrinsic noise in the system so that instead of determining a definite value , the measurement leads to a value , with conditional probability (which is assumed to be known). Unlike conventional measurements, a repeated measurement may yield a different answer each time, constrained only by the conditional probability. In general may run over a range greater than that of , but here we assume that both take just two values. For a single time measurement, the desired probability of , which we denote , may be inferred indirectly from the ambiguous measurement probability , using

 ~p(s)=∑αdsαp(α), (24)

where we have introduced the inverse , which satisfies

 ∑αdsαcαs′=δss′. (25)

(It is the left inverse in the more general case where and run over different ranges). Ambiguous measurements in quantum mechanics are described by POVM’s of the form

 Fα=∑scαsPs, (26)

for which the probability is . The inferred probability for is then readily seen to be , the expected result.

Ambiguous measurements yield more subtle results when used as part of a sequence. We consider a situation in which there are two measurements at times , , in which the first measurement is ambiguous. We first introduce the operator , where , so

 Mα=∑s√cαsPs. (27)

The two time joint probability of an ambiguous measurement obtaining a value followed by a projective measurement obtaining a value is

 p(α,s2)=Tr(Ps2(t)Mα(t1)ρMα(t1)). (28)

The inferred joint probability is then

 ~p(s1,s2) = ∑αds1α p(α,s2) (29) = ∑α∑s,s′ds1α√cαs√cαs′Tr(Ps2(t2)Ps(t1)ρPs′(t1)).

Noting that the terms in the trace on the right-and side yield the probability , and the terms yield the decoherence functional Eq.(13), we have

 ~p(s1,s2)=p12(s1,s2)+∑s≠s′(∑αds1α√cαs√cαs′) D(s,s2|s′,s2). (30)

To proceed further we need to make a specific choice of POVMs. The following pair of POVMs are convenient for our purposes:

 F+ = (1+ε)2P++(1−ε)2P−, (31) F− = (1−ε)2P++(1+ε)2P−, (32)

where . These clearly correspond to projective measurements as and to weak measurements as . The matrices corresponding to the coefficients and are then,

 c=12(1+ε1−ε1−ε1+ε), d=12ε(1+ε−1+ε−1+ε1+ε). (33)

The coefficient in Eq.(30) is now readily evaluated with the result,

 ~pε(s1,s2)=p12(s1,s2)+√1−ε2 ReD(s1,s2|−s1,s2). (34)

Using the relation Eq.(12), this is now readily rewritten in the convenient form,

 ~pε(s1,s2)=(1−√1−ε2) p12(s1,s2)+√1−ε2 q(s1,s2). (35)

This simple and appealing result shows how the inferred probability from ambiguous measurements mediates continuously from the usual projective measurement result for unambiguous measurements when , to the quasi-probability in the limit of a very weak measurement. Hence the desired quasi-probability is therefore obtained either by a weak measurement, or more generally by using an ambiguous measurement of arbitrary strength together with a projective measurement to determine . Alternatively, one could use two ambiguous measurements of different strengths and use Eq.(35) to determine both and in terms of and .

These results show quite generally that the quasi-probability Eq.(9) for any dichotomic variable arises very naturally in a weak or ambiguous measurement scheme, not just in the context of the arrival time problem. Weak measurements in the specific context of the arrival time problem were previously considered in Ref.RuKa (), with essential agreement with our results. (See also Ref.MaRe () for connections between the quasi-probability and weak measurements).

Specific experiments implementing both of the above approaches could be designed using simple modifications of existing protocols designed to measure arrival times. An example is described in Ref.MBDE (), which involves an incoming packet moving along the -axis interrupting a laser beam in the -direction.

## Iv Relation to Standard Arrival Time Distribution Results

In general the probability for a free particle crossing the origin during interval has the form

 p(t1,t2)=∫t2t1dt Π(t) (36)

for some arrival time distribution , for which a number of different candidates have been suggested. Most approaches yield the quantum-mechanical current in the semiclassical limit,

 Π(t) = −iℏ2m[ψ∗(0,t)ψ′(0,t)−ψ(0,t)ψ′∗(0,t)], (37) = ⟨ψ|^J(t)|ψ⟩,

where the dash denotes spatial derivative and is the current operator,

 ^J=12m(^pδ(^x)+δ(^x)^p). (38)

Note for future reference that

 ^J(t)=ddtθ(^x(t)). (39)

For situations in which the wave function vanishes at , the expected result is the kinetic energy density,

 Π(t) = N⟨ψ|^pδ(^x(t))^p|ψ⟩ (40) = Nℏ2∣∣ψ′(0,t)∣∣2

for some normalization factor , which is typically model-dependent kin1 (); kin2 (); kin3 (). We expect our quasi-probability to make some link to these standard results, at least in the small time limit, and this we now show.

The quasiprobability can be written as

 q(−,+)=Re⟨Ψ|P+(τ)P−|ψ⟩, (41)

where denotes the real part and we have chosen and . Since

 P+(τ)P− = (P+(τ)−P+)P− (42) = ∫τ0dt^J(t)P−,

where we have used Eq.(39), we have

 q(−,+)=∫τ0dt Re⟨ψ∣^J(t)P−∣ψ⟩. (43)

To evaluate the matrix element we note that

 ⟨ψ∣eiℏHt^Je−iℏHtP−∣ψ⟩ = 12m(⟨ψ∣eiℏHt^p∣0⟩⟨0∣e−iℏHtP−∣ψ⟩ (44) + ⟨ψ∣eiℏHt∣0⟩⟨0∣^pe−iℏHtP−∣ψ⟩).

We may then use the wave function notations,

 ⟨0∣e−iℏHt∣ψ⟩ = ψ(0,t), (45) ⟨0∣^pe−iℏHt∣ψ⟩ = −iℏ∂ψ∂x(0,t), (46)

to write this as,

 ⟨ψ∣eiℏHt^Je−iℏHtP−∣ψ⟩ = iℏ2m⟨0∣e−iℏHtP−∣ψ⟩∂ψ∗(0,t)∂x (47) − iℏ2mψ∗(0,t)[∂∂x⟨x∣e−iℏHtP−∣ψ⟩]x=0.

The two remaing matrix elements are then evaluated for small by following the methods in Refs.HELM (); Sok (). We thus obtain, at some length, the small-time expansion,

 q(−,+) = 12J(0,0)τ (48) +

The final result connects with the standard formulae given above. The leading order behaviour is essentially the current, Eq.(37), apart from the factor of a half, which is due to the fact that the quasi-probability is for a left-right crossing. Hence there is an agreement with semiclassical notions for short times. For situations in which the wave function vanishes at , we get a result proportional to the kinetic energy density, Eq. (40).

Note that we do not make any contact with the ideal arrival time distribution of Kijowksi Kij (). This is not surprising since the Kijowski distribution is always non-negative, yet here, we are working with a quasi-probability.

## V Relationship to Quantum Backflow

Quantum backflow is the surprising non-classical effect that a free particle in a state consisting entirely of positive momenta can have a negative current BrMe (); Back (). It means, for example, that the probability

 p−(t)=∫0−∞dx |ψ(x,t)|2, (49)

of remaining in the negative -axis can actually increase for periods of time, even though in the long run it decreases, in accordance with classical intuition. The degree of increase can be determined by considering the flux operator

 ∫t2t1^J(t)=P+(t2)−P+(t1), (50)

restricted to the space of states with positive momentum. Following Bracken and Melloy BrMe (), and subsequent authors, one considers the eigenvalue equation

 θ(^p)∫t2t1^J(t) |λ⟩=λ|λ⟩. (51)

It has been shown that the spectrum lies in the range , where is the Bracken-Melloy backflow constant and has been determined numerically to take the approximate value . Furthermore, the eigenstates with negative eigenvalue seem to have a discrete spectrum and those with positive eigenvalues are have a continuous spectrum. Backflow is easily shown to be a consequence of a negative Wigner function Wig (), or equivalently, due to interference effects. To date it has not been measured experimentally, although at least one proposal to do so exists BackE ().

For our purposes, backflow states seem like natural states to try in looking for situation with is negative. We note that the eigenvalue equation Eq.(50) may be written,

 12θ(^p)(^Q2−^Q1)|λ⟩=λ|λ⟩, (52)

and that from Eq.(22), the quasi-probability may be written

 q(−,+) = 18((2λ+1)2−1), (53) = 12λ(λ+1).

For a maximal backflow state, this means . This is clearly small in comparison to the lower bound on of (see Eq.(23)), suggesting that the most negative values of the quasi-probability arise from superpositions of positive and negative momentum states.

These results also suggest that the measurements of the quasi-probability described earlier may also provide a possible experimental test of backflow, if it is possible to prepare non-trivial initial states with non-negative momenta. The results of such an experiment would clearly be interesting. However, such an approach may be unnecessarily complicated in terms of measuring the backflow effect. The quasi-probability requires measurements of the system at two times, whereas the presence of backflow can be determined by measuring the probability Eq.(49), which would clearly be simpler.

## Vi The Quasi-Probability for Gaussian States

To illustrate some of the properties of the quasi-probability we compute it explicitly for initial states consisting of superpositions of gaussian wavepackets. Our first rather simple example is a superposition of four gaussians,

 |ψ⟩=∑s,s′ass′|ϕss′⟩, (54)

where . The four states are chosen so that they are strongly localized in or at and . The states remain in those regions during that time interval and the states cleanly cross the origin. The states are therefore approximately orthogonal and we may take the normalization . The quasi-probability is readily evaluated, since the states are approximate eigenstates of the projector product and we find

 q(s1,s2)≈|as1s2|2, (55)

the expected semiclassical answer.

To obtain more interesting examples in which the quasi-probability is negative we need gaussians which are significantly chopped by the projections at either time. We take

 |ψ⟩=α−∣ϕ−⟩+α+∣ϕ+⟩, (56)

where are complex coefficients and . We choose the wave packets to be,

 ϕ±(x)=1(2πσ2)14exp(−(x∓L)24σ2∓iℏp0x). (57)

They are therefore peaked at the points and we choose so that they are approximate eigenstates of the projectors at the initial time. (We elaborate on this approximation below). We have chosen the momenta to be equal and opposite, with , so that the wave packets approach each other. This means that after a time , the wave packets meet at the origin and we have,

 ϕ±(x,τ)=1(2π(Δx)2τ)14exp(−x24σ2τ∓iℏp0x−iℏEτ), (58)

where , and

 (Δx)2τ=σ2+ℏ2τ24m2σ2. (59)

The quasiprobability with and can be written,

 q(±,s) =Re⟨ψ∣Ps(τ)P±∣ψ⟩ (60) =Re(|α±|2⟨ϕ±∣Ps(τ)∣ϕ±⟩+α±α∗∓⟨ϕ∓∣Ps(τ)∣ϕ±⟩), (61)

where takes values of . It is easily seen that

 ⟨ϕ±∣Ps(τ)∣ϕ±⟩=∫∞0dx |ϕ±(x,τ)|2=12 (62)

The second term is the overlap integral

 ⟨ϕ∓∣Ps(τ)∣ϕ±⟩=1(2π(Δx)2τ)12∫∞0dxexp(−x22(Δx)2τ∓2isp0x). (63)

These integrals may be evaluated in terms of the imaginary error function Bess () and we find

 ⟨ϕ∓∣Ps(τ)∣ϕ±⟩=12e−y2(1±i erfi(y)) (64)

where . We thus obtain for the quasi-probability,

 q(±,s)=12Re(|α±|2+α±α∗∓e−y2(1∓i erfi(y))), (65)

We will suppose that the parameter is fixed and find values of the coefficients which minimize the quasi-probability.

The complex coefficients can be parameterised as follows

 α+=cosϕeiθ+ α−=sinϕeiθ− (66)

with the condition . Define , then

 q(+,±) =12cos2ϕ+14sin(2ϕ)f±(y,θ), (67) q(−,±) =12sin2ϕ+14sin(2ϕ)f±(y,θ). (68)

where

 f±(y,θ)=e−y2(cosθ±sinθerfi(y)). (69)

From the compound angle formula we easily find that for fixed an angle may be chosen so that has maximum value,

 fm(y)=e−y2√1+erfi(y)2. (70)

Similarly, the minimum of the quasiprobability can be seen most clearly by expressing it in the form,

 q(+,±) =14[1+√1+f2±cos(2ϕ−arctanf±)], (71) q(−,±) =14[1−√1+f2±cos(2ϕ+arctanf±)]. (72)

which may all be minimized to the value for some . Hence we find that the coefficients in the superposition may be chosen so that the minimum value of the quasi-probability for fixed is,

 qmin(y)=14(1−√1+e−2y2(1+erfi(y)2)). (73)

This takes its lowest value of at (which is pretty close to the absolute lower bound on the quasi-probability of ) and increases monotonically with .

There is some tension between choosing a small value of and meeting the requirement necessary for the approximation that the states are approximate eigenstates of the projectors and the value is therefore not feasible. We consider this approximation in more detail. It means that there is an error in the state of the form . This means that to leading order there will be an error in the terms in Eq.(61) involving terms of the form , which have the upper bound,

 |⟨ϕ±∣Ps(τ)P∓∣ϕ±⟩|2 ≤ ⟨ϕ±∣Ps(τ)∣ϕ±⟩⟨ϕ±∣P∓∣ϕ±⟩ (74) = 14 erfc(L√2σ)

using Eq.(62), where is the complementary error function Bess (). By plotting (or simply plugging in some numbers), this error is of order (i.e. half a percent error) at , so it does not have to be that large for a small error.

Consider now the possible values of . We have

 y2=p20σ2ℏ2+L24σ2 (75)

where we have inserted the value . It turns out to be convenient to take to be very small, so now . With the above value we have and Eq.(73) then yields the minimum value for the quasi-probability as,

 qmin=−0.05, (76)

with an estimated error of due to the approximations, a factor of smaller. This minimum value is a reasonable fraction of the most negative quasi-probability of . Hence reasonably negative values for the quasi-probability may be obtained with a simple superposition of gaussians.

A final example consists of a single initial gaussian. The quasi-probability can be written in the momentum space representation as,

 q(s1,s2)=∫dp1∫dp2 ~ψ∗(p1)~ψ(p2)fs1s2(p1,p2), (77)

where

 fs1s2(p1,p2)=Re ⟨p1|Ps2(τ)Ps1|p2⟩. (78)

We concentrate on and take an initial gaussian state very strongly concentrated in momentum about a value and with zero average position. The quasi-probability is therefore proportional (up to a postive constant) to,

 f−+(p,p)= Re[√m2πiℏτ∫0−∞dx∫∞0dye−iℏp(x−y)eim(x−y)22ℏτe−ip2τ2mℏ], = τ Re[p2mℏerfc(−p√iτ2mℏ)+1√i2πmℏτe−ip2τ2mℏ], (79)

where is the complementary error function. As shown in Ref.RuKa (), this quantity approaches the expected classical value for large positive , but can have a small amount of negativity for small negative . The timescale for the transition from negativity to postivity as increases from zero is determined by the argument of the error function, and is readily seen to be , where . This is precisely the energy time mentioned in the Introduction, at which quantum-mechanical effects in sequential measurements become important HaYe1 ().

## Vii The Quasi-probability in the Wigner-Weyl Representation

Some useful insights into the properties of the quasi-probability may be obtained using the Wigner-Weyl representation, in which hermitian operators are mapped into real phase space functions, according to the transform,

 WA(X,p)=12πℏ∫∞−∞dξe−ipξ ⟨X+ξ2∣^A∣X−ξ2⟩. (80)

The transform of the density operator yields the well-known Wigner function Wig (). The relevant properties of this transform are conveniently summarized in Ref.Hal4 (). A standard property that we will make use of is the relation,

 Tr(^Aρ)=2πℏ∫dX∫dp WA(X,p) Wρ(X,p). (81)

The quasi-probability Eq.(9) clearly has the form of the left-hand side, so is readily written in Wigner-Weyl form,

 q(s1,s2)=2πℏ∫dX∫dp Ws1s2(X,p) Wρ(X,p), (82)

where is the Wigner-Weyl transform of the operators

 As1s2=12(Ps2(t2)Ps1(t1)+Ps1(t1)Ps2(t2)). (83)

The transform may be carried out, at some length, with the results,

 W±,±(X,p) =14π[θ(±X)+θ(±Xτ)−12+1πSi(2mXτXτ)], (84) W±,∓(X,p) =14π[θ(±X)+θ(∓Xτ)+12−1πSi(−2mXτXτ)], (85)

where for convenience we have set , , and . Also, is the sine integral Bess (), defined by

 Si(u)=∫u0sinttdt. (86)

The behaviour of is best seen by plotting , which is shown in Figure 1. This clearly shows that is a quantum version of the step function , tending exactly to this -function for large , but displaying oscillations around .

This means that in the regime of large , have the approximate form

 W±,±(X,p) =14π[θ(±X)+θ(±Xτ)−θ(−XXτ)], (87) W±,∓(X,p) =14π[θ(±X)+θ(∓Xτ)+θ(XXτ)]. (88)

This is the expected form in the classical regime.

These results show that is non-negative throughout most of the phase space space. The only way it can be negative is in the small regime, as long that the and