Weak Values
Introduction
In the two state vector formalism, a weak value is equal to the transition amplitude between a prepared initial state and a postselection state [1]. We focus in this study on the weak value and the “weak variance” of the momentum operator \(\hat{p}\) postselected by position \(\left\langle x\right| \),
$$\begin{aligned} {_{x}\hat{p}_{\psi }} =\frac{\left\langle x\right| \hat{p} \left| \psi \right\rangle }{\left\langle x|\psi \right\rangle }\; ,\quad {_{x}\hat{p}_{\psi }} \in \mathbb {C}. \end{aligned}$$
A number of the following results should generalise to any pair of canonically noncommuting variables, although a general definition of the weak variance is not the objective of this study. The weak value \({_{x}\hat{p}_{\psi }} \in \mathbb {C}\) can be split into real and imaginary parts [2] as
$$\begin{aligned} {_{x}\hat{p}_{\psi }} = \frac{\left\langle x\right| \hat{p} \left| \psi \right\rangle }{\left\langle x|\psi \right\rangle }=\hbar \partial _x \left( \arg \psi \right) -i\left( \frac{\hbar }{2}\frac{\partial _{x}\left| \psi \right| ^{2}}{\left| \psi \right| ^{2}}\right) . \end{aligned}$$
(1)
One might naively attempt to define a weak variance as the second central moment out of weak values \({_{x}\hat{p}^2_{\psi }} - {_{x}\hat{p}_{\psi }}^2\). However, the fact that weak values are complex requires one to take real parts of this naive construction—and it makes a difference whether one squares the second term before or after taking real parts: \(\mathrm {Re}({_{x}\hat{p}_{\psi }}^2) \ne (\mathrm {Re}\,{_{x}\hat{p}_{\psi }})^2\). This ambiguity motivates the search for a more ‘fundamental’ approach to these statistical quantities, which this first section will attempt to describe.
This study is organised as follows. We begin by reviewing the relation between the real parts of weak values and the phase space formalism of quantum mechanics. The quasistatistical interpretation of this formalism suggests a natural and unique definition for the “weak variance” in terms of the Wigner function. In Sect. 2, the embedding of configuration space formalisms into the phase space formalism then allows the weak variance to be related to the thermodynamics of the Madelung fluid, to the de Broglie–Bohm quantum potential \(Q\), and to an experimentally measurable combination \(\mathrm {Re}({_{x}\hat{p}^2_{\psi }}-{_{x}\hat{p}_{\psi }}^2)\) of weak values.
In Sect. 3 we discuss the titular problem, that the negativity of the Wigner function can result in a negative weak variance. The weak variance therefore does not define a standard deviation of weak measurements from the weak value. The sign of the weak variance is shown to be related to the local extrema of the probability amplitude. The relation between weak variances and determinism paves the way for a detailed study of the classical limit in Sect. 4. We derive the classical limit from a variational principle involving the average stochastic contribution to the total momentum variance, and demonstrate the existence of a semiclassical limit where particle motion occurs, on average, classically.
The Importance of the Measurement Apparatus
It is important to note [3] that weak values are not measured independently of an interaction Hamiltonian or a meter system. The real part of a weak value is well known to be the conditional average one would measure in the ideal limit of zero disturbance form the measuring apparatus [4–6]. The imaginary part of a weak measurement is unrelated to observables, it arises from the disturbance due to coupling with a measurement apparatus [4]. As such, it is not uniquely defined and not intrinsic to the measured system [3].
Since only the real part informs us about our operator as an observable, we opt to describe the variance of only the real part of the weak value in this study. However, the imaginary part in (1) will emerge naturally during our investigation of weak variances for the real part, without reference to a specific interaction Hamiltonian or meter system. In principle the ‘ideal’ decomposition (1) would be measured as [3, 7]
$$\begin{aligned} {_{x}\hat{p}_{\psi }} = \mathrm {Re}\left( \frac{\left\langle x\right| \hat{p} \left| \psi \right\rangle }{\left\langle x|\psi \right\rangle }\right) + \eta \; \mathrm {Im}\left( \frac{\left\langle x\right| \hat{p} \left| \psi \right\rangle }{\left\langle x|\psi \right\rangle }\right) , \end{aligned}$$
(2)
such that the theoretically natural imaginary part in (1) is arbitrarily scaled in experiments by adjusting the value of \(\eta \). This parameter depends on meter states and observables, as well as the interaction strength with the apparatus [3].
We shall therefore discuss the imaginary part as if it were intrinsic to the system, absorbing all dependence on the measurement protocol into the parameter \(\eta \). This has the disadvantage of illusorily disconnecting these results from the experimental setups that give rise to them, which comes hand in hand with the hidden advantage of being a more universal treatment than that derived from any specific experimental setup. As such, the weak variance we define is related to the stochastic nature of quantum theory; it is not the experimental variance of any apparatus (although it may manifest itself as such in a measurement setting). Not without irony, this apparatus-agnostic treatment will suggest multiple possible experimental setups to study the weak variance (Sect. 2.3).
Weak Values as Conditional Expectations
In the quasistatistical interpretation [8–10] of the phase space formalism [11–14], the canonical observables \(x,p\in \mathbb {R}\) are treated as random variables. Since these observables are incompatible, no joint probability distribution can reasonably be given for them—the phase space distributions are understood to be bivariate pseudoprobability distributions. For example, the Wigner function \(\mathcal {W}\left( x,p\right) \) is not positive definite. For a review of extended probability we refer the reader to [10].
Our choice of the Wigner function to define the weak variance is motivated by this quasidistribution’s special status amongst physically equivalent representations of the phase space formalism (which correspond to different choices of operator ordering, \([\hat{x},\hat{p}]\ne 0\)). Indeed, only the Wigner function produces the correct marginals and observable averages by direct integration [13]; other quasidistributions require explicit convolutions over noncommutativity terms. As such, the Wigner function is uniquely singled out by the quasistatistical outlook, where such noncommutativity terms would interfere with the interpretation of observable averages as moments of the pseudodistribution.
The real part of a weak value is a conditional average [5, 6, 15]. Hence, to obtain weak values of momentum, one must consider not the momentum \(p\), but instead the mean value of \(p\) at a given position, for a statistical ensemble. One may consequently express the experimental protocol of postselection with conditional random variables: the quantity \(p|x\) (\(p\) conditioned by \(x\)) is also a random variable. To find the distribution of \(p|x\), we need to consider the conditional distribution \(\mathcal {W}\left( p|x\right) \). The Wigner function has a marginal distribution
$$\begin{aligned} \mathcal {W}\left( x\right) =\int \mathcal {W}\left( x,p\right) dp=\left| \psi (x)\right| ^{2}, \end{aligned}$$
(3)
and so the conditional distribution for the random variable \(p|x\) is simply
$$\begin{aligned} \mathcal {W}\left( p|x\right) =\frac{\mathcal {W}\left( x,p\right) }{\mathcal {W}\left( x\right) }. \end{aligned}$$
(4)
These observables \(p\), \(x\) are incompatible: in what sense can we condition one on the other if they are not meaningful simultaneously? A random variable is defined by its distribution. Individual realisations of \(x\) and \(p\) are meaningful positions and momenta only because their distributions (the marginals of the Wigner quasidistribution) are proper distributions. The random variable \(p|x\) is defined by the conditional quasidistribution \(\mathcal {W}\left( p|x\right) \), which from (4) is clearly not nonnegative. Hence individual realisations of \(p|x\) may occur with negative probability. This follows the same logic as stating that joint realisations \((x,p)\) may occur with negative probability as realisations of the full Wigner function.
However, the fact that individual realisations of \(p|x\) can occur with negative probabilities does not preclude the use of this conditional variable as an intermediate step in a calculation of a physical quantity [10]. Furthermore, the mean or the variance of \(p|x\) need not be confined to intermediate steps in this way, since they are not random variables but instead contribute to the description of the physical state of an ensemble (i.e. they describe the Wigner function as a whole rather than the outcome of a measurement).
The mean value of \(p|x\),
$$\begin{aligned} \mathbb {E}_{p}\left( p|x\right) \equiv \tilde{p}, \end{aligned}$$
(5)
can be thought of as a function of \(x\) [9]. This mean value \(\tilde{p}\) will in Sect. 1.2.1 be shown to equal the real part of the weak value \(_{x}\hat{p}_{\psi }\). One may then express the conditional mean momentum in terms of \(\mathcal {W}\) as the first conditional moment [8], i.e. the first moment of the conditional distribution:
$$\begin{aligned} \tilde{p}= & {} \int p\mathcal {W}\left( p|x\right) \; dp\end{aligned}$$
(6)
$$\begin{aligned}= & {} \frac{1}{\mathcal {W}\left( x\right) }\int \; p\mathcal {W}\left( x,p\right) \; dp. \end{aligned}$$
(7)
It is possible to formulate complex weak values of arbitrary observables directly in the phase space formalism using cross-Wigner functions [16], but as noted earlier our focus is currently limited to the variance of the real part of position-postselected weak momentum measurements.
Weak Variance as the Second Conditional Cumulant
Through the statistical Wigner–Moyal formalism described above and in [8], the weak value \(\tilde{p}\) takes on the role of a statistical average. However, an average is scientifically worthless unless one can provide an associated spread. It is well established that the variance, i.e. the second central moment of a distribution, is equal to the second cumulant of that distribution. With some effort, the same can be shown of conditional variances and conditional cumulants [17], and extended to those of pseudodistributions like the Wigner function [8].
Derivation
Using the characteristic function associated to \(\mathcal {W}\left( x,p\right) \) and the polar decomposition \(\psi =\sqrt{\rho }e^{iS}\), the conditional cumulants of this distribution can be written for even cumulants (\(n=2,4,\ldots \)) as [8]
$$\begin{aligned} \kappa _{n}|x=\left( \frac{\hbar }{2i}\right) ^{n}\left( \frac{\partial }{\partial x}\right) ^{n}\ln \rho \left( x\right) , \end{aligned}$$
(8)
and for odd cumulants (\(n=1,3,\ldots \)) as
$$\begin{aligned} \kappa _{n}|x=\left( \frac{\hbar }{2i}\right) ^{n-1}\left( \frac{\partial }{\partial x}\right) ^{n}\hbar S\left( x\right) . \end{aligned}$$
(9)
The derivation of this result has been reproduced in full as an Appendix.
By definition, the first two conditional cumulants \(n=1,2\) are also the first two conditional moments, and one finds a simple relation [8],
$$\begin{aligned} \tilde{p}=\frac{\partial \left( \hbar S\right) }{\partial x}, \end{aligned}$$
for the (\(n=1\)) average \(\tilde{p}=\mathbb {E}_{p}\left( p|x\right) \). Simple comparison of this expression with (1) should confirm that \(\tilde{p} = \mathrm {Re}\left( {_{x}\hat{p}_{\psi }}\right) \). The \(n=2\) expression then defines the weak variance of the random variable \(p|x\):
$$\begin{aligned} \mathbb {V}_{p}\left( p|x\right) =-\frac{\hbar ^{2}}{4}\left( \frac{\partial }{\partial x}\right) ^{2}\ln \rho \left( x\right) . \end{aligned}$$
(10)
Dimensional analysis confirms that this quantity has units of \(\left[ \mathrm {momentum}\right] ^{2}\), as expected of a (conditional) momentum variance. Because of the clean splitting between phase-dependent odd cumulants and amplitude-dependent even cumulants, the weak variance does not depend on the wavefunction phase, except indirectly (via the time evolution of the amplitude).
Other Expressions for the Weak Variance
This second conditional cumulant is also expressible directly as the second central conditional moment of the conditional distribution \(\mathcal {W}\left( p|x\right) \):
$$\begin{aligned} \mathbb {V}_{p}\left( p|x\right) =\int \mathcal {W}\left( p|x\right) \left( p|x-\mathbb {E}_{p}\left( p|x\right) \right) ^{2}dp. \end{aligned}$$
(11)
Furthermore, as we shall demonstrate in Sect. 2.1.2, the weak variance also takes the following form, more readily related to weak values:
$$\begin{aligned} \mathbb {V}_{p}\left( p|x\right) = \frac{1}{2}\mathrm {Re}\left( \frac{\left\langle x\right| \hat{p}^2 \left| \psi \right\rangle }{\left\langle x|\psi \right\rangle } -\left( \frac{\left\langle x\right| \hat{p} \left| \psi \right\rangle }{\left\langle x|\psi \right\rangle }\right) ^2\right) . \end{aligned}$$
This last expression shows that despite our use of unphysical conditional random variables \(p|x\) in its derivation, the weak variance is indeed related to the intuitive form \(\langle p^2\rangle - \langle p\rangle ^2\) of a variance. It also shows that the weak variance is measurable in an experimental setting.
The Uncertainty Principle
It is possible to relate the conditional variance \(\mathbb {V}_{p}\left( p|x\right) \) to the unconditional variance \(\mathbb {V}\left( p\right) \) that appears in the uncertainty relations.
The Law of Total Variance
One may readily verify that the law of total expectation,
$$\begin{aligned} \mathbb {E}\left( p\right) =\mathbb {E}_{x}\left( \mathbb {E}_{p}\left( p|x\right) \right) =\mathbb {E}_{x}\left( \tilde{p}\right) , \end{aligned}$$
(12)
reads in our quantum setting
$$\begin{aligned} \left\langle \psi \right| \hat{p}\left| \psi \right\rangle =\int \left| \psi \right| ^{2}\tilde{p}dx. \end{aligned}$$
(13)
In the same way that one derives the law of total expectation, one may derive the Law of Total Variance:
$$\begin{aligned} \mathbb {V}\left( p\right)= & {} \mathbb {E}_{x}\left( \mathbb {V}_{p}\left( p|x\right) \right) +\mathbb {V}_{x}\left( \mathbb {E}_{p}\left( p|x\right) \right) \nonumber \\= & {} \mathbb {E}_{x}\left( \mathbb {V}_{p}\left( p|x\right) \right) +\mathbb {V}_{x}\left( \tilde{p}\right) , \end{aligned}$$
(14)
where \(\mathbb {V}_{x}\left( \tilde{p}\right) \) is the variance of the weak value \(\tilde{p}\) as a function of \(x\), and where \(\mathbb {E}_{x}\left( \mathbb {V}\left( p|x\right) \right) \) is the average value of the weak variance (as a function of \(x\)). \(\mathbb {V}_{x}\left( \tilde{p}\right) \) can be simply given by the marginal variance
$$\begin{aligned} \mathbb {V}_{x}\left( \tilde{p}\right) =\int \mathcal {W}\left( x\right) \left( \tilde{p}-\mathbb {E}_{x}\left( \tilde{p}\right) \right) ^{2}dx, \end{aligned}$$
(15)
where the law of iterated expectation (13) gives \(\mathbb {E}_{x}\left( \tilde{p}\right) =\left\langle \psi \right| \hat{p}\left| \psi \right\rangle \). Hence, \(\mathbb {V}_{x}\left( \tilde{p}\right) \) measures the standard deviation of the weak value \(\tilde{p}\) from the usual expectation value \(\left\langle \hat{p} \right\rangle \) as the postselection position \(x\) changes. Particularly, it does not measure the variance of \(p|x\) (or of any other random variable): the variable \(\tilde{p}\) in the variance \(\mathbb {V}_{x}\) is deterministically defined by (15). The nonweak variance \(\mathbb {V}_{x}\left( \tilde{p}\right) \) contributes to the total variance of \(\hat{p}\) of the uncertainty principle, but it is not itself a manifestation of quantum randomness.
By contrast the conditional variance,
$$\begin{aligned} \mathbb {V}_{p}\left( p|x\right) =\int \mathcal {W}\left( p|x\right) \left( p|x-\mathbb {E}_{p}\left( p|x\right) \right) ^{2}dp, \end{aligned}$$
is truly the variance of a random variable: it says how far from the mean \(\mathbb {E}_{p}\left( p|x\right) = \tilde{p}\) one may expect to find the random variable \(p|x\). Unlike the total variance or the variance of \(\tilde{p}\), this “scedastic function” depends on the postselection \(x\). It is in this sense that we identify it as the weak variance.
These two parts of the total variance allow interesting limits where their contributions are negligible one with respect to the other. For example, in stationary states there is no variability in \(\tilde{p}(x)\) by construction, and the total variance of the state is supported entirely by the weak variance. A more detailed discussion of these limits, and their relevance to the classical limit, is postponed until the relevant properties of the weak variance \(\mathbb {V}_{p}\left( p|x\right) \) are introduced (Sect. 4.1).
Apparatus versus Fundamental Weak Variances
This connection between the weak variance and the momentum variance reconfirms that the weak variance is central to understand the predictions of QM. Oddly, we find that the weak variance is related to the fundamental momentum variance \(\mathbb {V}(p)\)
of a state (as described in the Robertson uncertainty relations) rather than the momentum change \(\Delta p\) when a state is disturbed (as originally described by Heisenberg).
Since \(\mathbb {V}_{p}\left( p|x\right) \) is given by differentiation of the wavefunction amplitude (10), it depends only on the values of the wavefunction amplitude in the neighbourhood of the postselection point \(x\). This choice of postselection point is not apparatus-dependent, although issues such as pointer variance for the postselection variable will certainly affect the analysis of any experimental study.
The weak variance is further distinguished from an ‘apparatus quantity’ by the fact that it is not an ‘observable’, in the sense that it does not correspond to a Hermitean operator (cf Sect. 2.1.3) or equivalently (in the presentation adopted above) to a real random variable. Instead, as a cumulant of the Wigner function, it describes the state of the system, irrespective of how it is measured.