Weak values in a classical theory with an epistemic restriction

Here, we review the framework of quantum measurement, wherein an observable is coupled to a measurement device followed by a projective measurement of the measurement device's position. With this framework, we can describe both strong (projective) as well as arbitrarily weak measurements.

We describe the measurement device by a one-dimensional quantum system with canonical position observable $\hat{Q}$ and momentum observable $\hat{P}$ satisfying $[\hat{Q},\hat{P}]={\rm{i}}{\rm{\hslash }}$. In what follows, we choose units such that ${\rm{\hslash }}=1$. We denote position eigenstates by $| Q\rangle$ satisfying $\hat{Q}| Q\rangle =Q| Q\rangle$.

Consider the initial state of the wavefunction of the measurement device to be a pure gaussian state with mean position 0. The most general form of such a state is

$$\begin{eqnarray}&&| \Phi \rangle =\int \phi (Q)| Q\rangle \;{\rm{d}}Q,\end{eqnarray} \tag{ 2 }$$

where

$$\begin{eqnarray}&&\phi (Q)\propto \mathrm{exp}\left(\displaystyle \frac{({\rm{i}}\Omega -1){Q}^{2}}{4{\Delta }_{Q}^{2}}+{\rm{i}}Q{\mu }_{P}\right),\end{eqnarray} \tag{ 3 }$$

up to an irrelevant normalization constant. Here, ${\Delta }_{Q}$ is the standard deviation of the position of the device, ${\mu }_{P}$ is the mean momentum of the measurement device and Ω is the covariance of the device

$$\begin{eqnarray}&&\Omega =\langle \hat{P}\hat{Q}+\hat{Q}\hat{P}\rangle -2\langle \hat{P}\rangle \langle \hat{Q}\rangle .\end{eqnarray} \tag{ 4 }$$

As the measurement device is in a pure gaussian state, it saturates the uncertainty principle and hence the uncertainty in its momentum is

$$\begin{eqnarray}&&{\Delta }_{P}^{2}=\displaystyle \frac{1+{\Omega }^{2}}{4{\Delta }_{Q}^{2}}.\end{eqnarray} \tag{ 5 }$$

Such a device can be used to measure a quantity given by the particle observable $\hat{A}$ by coupling the particle and device via the interaction Hamiltonian

$$\begin{eqnarray}&&{\hat{H}}_{\mathrm{int}}=\chi (t)\hat{P}\otimes \hat{A},\end{eqnarray} \tag{ 6 }$$

where $\hat{P}$ acts on the measurement device and $\hat{A}$ acts on the particle.

An impulsive measurement is performed over a time interval $[0,{t}_{0}]$ such that

$$\begin{eqnarray}&&{\displaystyle \int }_{0}^{{t}_{0}}\chi (t)\;{\rm{d}}t=g,\end{eqnarray} \tag{ 7 }$$

where g is the effective interaction strength. Consider an initial state of the particle given as $| \Psi \rangle ={\sum }_{j}{\alpha }_{j}| {a}_{j}\rangle$, where $| {a}_{j}\rangle$ is an eigenstate of $\hat{A}$ with eigenvalue a_j. After the interaction, the state of the device and particle will be

$$\begin{eqnarray}&&{{\rm{e}}}^{-{\rm{i}}g\hat{P}\hat{A}}| \Phi \rangle | \Psi \rangle =\displaystyle \sum _{j}{\alpha }_{j}\left(\int \phi (Q-{{ga}}_{j})| Q\rangle {\rm{d}}Q\right)| {a}_{j}\rangle .\end{eqnarray} \tag{ 8 }$$

Consider the case where the initial uncertainty in the position of the pointer ${\Delta }_{Q}$ is zero, and thus $| \Phi \rangle$ is a position eigenstate with eigenvalue 0. In this case,

$$\begin{eqnarray}&&\underset{{\Delta }_{Q}\to 0}{\mathrm{lim}}{{\rm{e}}}^{-{\rm{i}}g\hat{P}\hat{A}}| \Phi \rangle | \Psi \rangle =\displaystyle \sum _{j}{\alpha }_{j}| Q={{ga}}_{j}\rangle | {a}_{j}\rangle .\end{eqnarray} \tag{ 9 }$$

After the interaction, the measurement device's position is maximally entangled with the eigenstates of $\hat{A}$ of the particle. A projective measurement of the position of the device pointer perfectly resolves the eigenvalue of $\hat{A}$, and collapses the state of the particle into an eigenstate of $\hat{A}$. This measurement, then, corresponds to a strong, projective measurement of $\hat{A}$ on the particle.

The weak measurement limit is the opposite limit of this strong measurement, and aims to reduce the disturbance to an arbitrarily small amount at the expense of a correspondingly small information gain. Within the measurement model described above, we introduce two different ways in which a measurement can be made weak. First, each particle can be coupled arbitrarily weakly to a measuring device by using a vanishingly small interaction strength. Alternatively, a weak measurement can also be obtained by using an initial state of the measurement device with ${\mu }_{P}=0$ and ${\Delta }_{P}\to 0$, which would imply ${\Delta }_{Q}\to \infty$ from (5). (While these two limits lead to identical measurement statistics within quantum theory, we explore each of them separately, as they will correspond to different processes in the context of the epistemically-restricted theory of classical mechanics explored in the next section.) In both of these limits, the disturbance caused by measurement, as well as the amount of information gained, are both very small. If this measurement is repeated on a large number of particles, each prepared in the same initial state $| \Psi \rangle$, it is possible to measure the average value of an observable A of the ensemble of particles with arbitrary accuracy as the number of particles becomes large.

With the concept of weak measurement, we now derive the weak value, with an emphasis on the difference between the two weak measurement methods described above.

The weak value arises in a measurement scenario wherein a weak measurement of an observable A is followed by a strong (projective) measurement of another observable B, together with postselection on a particular outcome labelled by eigenvalue b of the measurement of B. The observable B on which results are postselected need not commute with the variable A weakly measured as illustrated in figure 1. In such a situation, the weak value is defined [5] to be

$$\begin{eqnarray}&&{\langle \hat{A}\rangle }_{W}=\displaystyle \frac{\langle b| \hat{A}| \Psi \rangle }{\langle b| \Psi \rangle },\end{eqnarray} \tag{ 10 }$$

where $\hat{B}| b\rangle =b| b\rangle$. Using $| \Psi \rangle ={\sum }_{j}{\alpha }_{j}| {a}_{j}\rangle$, the real part of this complex number is

$$\begin{eqnarray}&&\mathrm{Re}[{\hat{A}}_{W}]=\displaystyle \frac{{\displaystyle \sum }_{j,l}{\alpha }_{j}{\alpha }_{l}^{*}\langle b| {a}_{j}\rangle \langle {a}_{l}| b\rangle (\displaystyle \frac{{a}_{j}+{a}_{l}}{2})}{{\displaystyle \sum }_{j,l}{\alpha }_{j}{\alpha }_{l}^{*}\langle b| {a}_{j}\rangle \langle {a}_{l}| b\rangle },\end{eqnarray} \tag{ 11 }$$

and its imaginary part is

$$\begin{eqnarray}&&\mathrm{Im}[{\hat{A}}_{W}]=\displaystyle \frac{{\displaystyle \sum }_{j,l}{\alpha }_{j}{\alpha }_{l}^{*}\langle b| {a}_{j}\rangle \langle {a}_{l}| b\rangle (\displaystyle \frac{{a}_{j}-{a}_{l}}{2i})}{{\displaystyle \sum }_{j,l}{\alpha }_{j}{\alpha }_{l}^{*}\langle b| {a}_{j}\rangle \langle {a}_{l}| b\rangle }.\end{eqnarray} \tag{ 12 }$$

Zoom In Zoom Out Reset image size

We will now show explicitly how the real and imaginary parts of the weak value appear as phase-space displacements in the mean position and momentum of the postselected distribution of the measurement devices performing the weak measurements. The unnormalized state of the post selected particles and the devices after weak measurement is

$$\begin{eqnarray}&&(I\otimes | b\rangle \langle b| ){{\rm{e}}}^{-{\rm{i}}g\hat{P}\otimes \hat{A}}| \Phi \rangle | \Psi \rangle .\end{eqnarray} \tag{ 13 }$$

Recall that $| \Phi \rangle$ is a gaussian state of the form of equation (2). Having postselected particle-measurement device pairs for which the particle is in the final state $| b\rangle$, the selected devices are described by the unnormalized state $| \Phi \prime \rangle ={\sum }_{j}{\alpha }_{j}\langle b| {a}_{j}\rangle \int \phi (Q-{{ga}}_{j})| Q\rangle {\rm{d}}Q$. The mean position of this device state after postselection on b, denoted $\langle \hat{Q}{\rangle }_{b}$, is

$$\begin{eqnarray}{\langle \hat{Q}\rangle }_{b} & = & \displaystyle \frac{{\displaystyle \sum }_{j,l}{\alpha }_{j}{\alpha }_{l}^{*}\langle b| {a}_{j}\rangle \langle {a}_{l}| b\rangle \displaystyle \int Q\phi (Q-{{ga}}_{j}){\phi }^{*}(Q-{{ga}}_{l}){\rm{d}}Q}{{\displaystyle \sum }_{j,l}{\alpha }_{j}{\alpha }_{l}^{*}\langle b| {a}_{j}\rangle \langle {a}_{l}| b\rangle \displaystyle \int \phi (Q-{{ga}}_{j}){\phi }^{*}(Q-{{ga}}_{l}){\rm{d}}Q}\\ & = & \displaystyle \frac{{\displaystyle \sum }_{j,l}{\alpha }_{j}{\alpha }_{l}^{*}\langle b| {a}_{j}\rangle \langle {a}_{l}| b\rangle g\left(\frac{({a}_{j}+{a}_{l})}{2}-\frac{{\rm{i}}\Omega ({a}_{j}-{a}_{l})}{2}\right)\mathrm{exp}\left(\frac{-{({a}_{j}-{a}_{l})}^{2}{\Delta }_{P}^{2}{g}^{2}}{2}+{\rm{i}}g({a}_{j}-{a}_{l}){\mu }_{P}\right)}{\displaystyle \sum _{j,l}{\alpha }_{j}{\alpha }_{l}^{*}\langle b| {a}_{j}\rangle \langle {a}_{l}| b\rangle \mathrm{exp}\left(\displaystyle \frac{-{({a}_{j}-{a}_{l})}^{2}{\Delta }_{P}^{2}{g}^{2}}{2}+{\rm{i}}g({a}_{j}-{a}_{l}){\mu }_{P}\right)}.\end{eqnarray} \tag{ 14 }$$

The mean momentum of the device after the weak measurement and postselection on b is

$$\begin{eqnarray}{\langle \hat{P}\rangle }_{b} & = & \displaystyle \frac{{\displaystyle \sum }_{j,l}{\alpha }_{j}{\alpha }_{l}^{*}\langle b| {a}_{j}\rangle \langle {a}_{l}| b\rangle \displaystyle \int P\tilde{\phi }(P,{a}_{j}){\tilde{\phi }}^{*}(P,{a}_{l}){\rm{d}}P}{{\displaystyle \sum }_{j,l}{\alpha }_{j}{\alpha }_{l}^{*}\langle b| {a}_{j}\rangle \langle {a}_{l}| b\rangle \displaystyle \int \tilde{\phi }(P,{a}_{j}){\tilde{\phi }}^{*}(P,{a}_{l}){\rm{d}}P}\\ & = & \displaystyle \frac{{\displaystyle \sum }_{j,l}-{\rm{i}}g{\Delta }_{P}^{2}{\alpha }_{j}{\alpha }_{l}^{*}\langle b| {a}_{j}\rangle \langle {a}_{l}| b\rangle ({a}_{j}-{a}_{l}){{\rm{e}}}^{\displaystyle \frac{-{({a}_{j}-{a}_{l})}^{2}{\Delta }_{P}^{2}{g}^{2}}{2}}}{\displaystyle \sum _{j,l}{\alpha }_{j}{\alpha }_{l}^{*}\langle b| {a}_{j}\rangle \langle {a}_{l}| b\rangle {{\rm{e}}}^{\displaystyle \frac{-{({a}_{j}-{a}_{l})}^{2}{\Delta }_{P}^{2}{g}^{2}}{2}}},\end{eqnarray} \tag{ 15 }$$

where $\tilde{\phi }(P,{a}_{i})=\mathrm{exp}({P}^{2}{\Delta }_{P}^{2}(1+{\rm{i}}\Omega )/4-{\rm{i}}{{Pga}}_{i})$ is the Fourier transform of $\phi (Q-{{ga}}_{i})$ up to a normalization constant.

2.3.1. Small uncertainty ${\Delta }_{P}$

We now consider these expressions using the first method to obtain weak measurements, wherein the initial position of the measurement device becomes highly uncertain. In the limit of ${\Delta }_{P}\to 0$, we also have $\Omega \to 0$. Consider the case where the mean momentum of the device, ${\mu }_{P}$, is also set to zero. Using equation (14), the mean position of the device is

$$\begin{eqnarray}{\langle \hat{Q}\rangle }_{b} & = & \displaystyle \frac{{\displaystyle \sum }_{j,l}{\alpha }_{j}{\alpha }_{l}^{*}\langle b| {a}_{j}\rangle \langle {a}_{l}| b\rangle g\frac{({a}_{j}+{a}_{l})}{2}}{\displaystyle \sum _{j,l}{\alpha }_{j}{\alpha }_{l}^{*}\langle b| {a}_{j}\rangle \langle {a}_{l}| b\rangle }\\ & = & g\mathrm{Re}[\langle {\hat{A}}_{W}\rangle ]+g\Omega \mathrm{Im}[\langle {\hat{A}}_{W}\rangle ],\end{eqnarray} \tag{ 16 }$$

where we have ignored terms of order ${g}^{3}{\Delta }_{P}^{2}$ and higher as a result of taking ${\Delta }_{P}$ to be small. In the limit ${\Delta }_{P}\to 0$, the covariance $\Omega \to 0$ and this shift becomes

$$\begin{eqnarray}&&\underset{{\Delta }_{P}\to 0}{\mathrm{lim}}{\langle \hat{Q}\rangle }_{b}=g\mathrm{Re}[\langle {\hat{A}}_{W}\rangle ].\end{eqnarray} \tag{ 17 }$$

From equation (15), the mean momentum of the device is

$$\begin{eqnarray}{\langle \hat{P}\rangle }_{b} & = & \displaystyle \frac{\displaystyle \sum _{j,l}-{\rm{i}}g{\Delta }_{P}^{2}{\alpha }_{j}{\alpha }_{l}^{*}\langle b| {a}_{j}\rangle \langle {a}_{l}| b\rangle ({a}_{j}-{a}_{l})}{\displaystyle \sum _{j,l}{\alpha }_{j}{\alpha }_{l}^{*}\langle b| {a}_{j}\rangle \langle {a}_{l}| b\rangle }\\ & = & 2g{\Delta }_{P}^{2}\mathrm{Im}[\langle {\hat{A}}_{W}\rangle ]\end{eqnarray} \tag{ 18 }$$

which in the limit becomes

$$\begin{eqnarray}&&\underset{{\Delta }_{P}\to 0}{\mathrm{lim}}{\langle \hat{P}\rangle }_{b}=0.\end{eqnarray} \tag{ 19 }$$

In this limit, the momentum of the device remains unchanged, and there is no disturbance to the state of the particle. There is, however, a shift in the mean position of the device proportional to the real part of the weak value $\langle {\hat{A}}_{W}\rangle$. Because this limit implies ${\Delta }_{Q}\to \infty$, it is a shift in a uniform distribution and hence not physically resolvable.

2.3.2. Weak coupling g

Consider now the second method for obtaining weak measurements, where the coupling strength g is small and the mean momentum of the device, ${\mu }_{P}$, is also set to zero. In the limit of $g\to 0$, there is no disturbance to the system, however, in this limit, there is also no shift in the average position and momentum of the post selected devices. Hence, we then calculate the mean position and momentum of the measurement device after postselection to leading order in g. The mean position of the device after postselection is

$$\begin{eqnarray}{\langle \hat{Q}\rangle }_{b} & = & \displaystyle \frac{{\displaystyle \sum }_{j,l}{\alpha }_{j}{\alpha }_{l}^{*}\langle b| {a}_{j}\rangle \langle {a}_{l}| b\rangle g[\frac{({a}_{j}+{a}_{l})}{2}-\frac{{\rm{i}}\Omega ({a}_{j}-{a}_{l})}{2}]}{\displaystyle \sum _{j,l}{\alpha }_{j}{\alpha }_{l}^{*}\langle b| {a}_{j}\rangle \langle {a}_{l}| b\rangle }\\ & = & g\mathrm{Re}[\langle {\hat{A}}_{W}\rangle ]+g\Omega \mathrm{Im}[\langle {\hat{A}}_{W}\rangle ],\end{eqnarray} \tag{ 20 }$$

and the mean momentum of the device after postselection is

$$\begin{eqnarray}&&{\langle \hat{P}\rangle }_{b}=\displaystyle \frac{{\displaystyle \sum }_{j,l}-{\rm{i}}g{\Delta }_{P}^{2}{\alpha }_{j}{\alpha }_{l}^{*}\langle b| {a}_{j}\rangle \langle {a}_{l}| b\rangle ({a}_{j}-{a}_{l})}{\displaystyle \sum _{j,l}{\alpha }_{j}{\alpha }_{l}^{*}\langle b| {a}_{j}\rangle \langle {a}_{l}| b\rangle }\end{eqnarray} \tag{ 21 }$$

$$\begin{eqnarray}&&=\;2g{\Delta }_{P}^{2}\mathrm{Im}[\langle {\hat{A}}_{W}\rangle ].\end{eqnarray} \tag{ 22 }$$

(In [16], the expression for ${\langle \hat{Q}\rangle }_{b}$ is written as ${\langle \hat{Q}\rangle }_{b}=g\mathrm{Re}[{\hat{A}}_{W}]+{gm}\frac{{\rm{d}}{\Delta }_{Q}}{{\rm{d}}t}\mathrm{Im}[{\hat{A}}_{W}]$, where m is the mass of the measurement device. To relate the expression of equation (20) with this one, observe that the covariance of the device can be written as mass times the change in variance of position of the device propagating under a free Hamiltonian. Note however that, at the time of interaction, the Hamiltonian is not that of a free particle.)

2.3.3. Shifts in the postselected distributions of the measurement device

Here, we analyse the weak value in the special case where the particle is a one-dimensional canonical quantum system prepared in a state with a gaussian wavefunction. It is this special case which will be directly comparable to the results presented in the next section, within a theory of classical mechanics with an epistemic restriction. Let $\hat{q}$ and $\hat{p}$ be the position and momentum observables for the particle, satisfying $[\hat{q},\hat{p}]={\rm{i}}{\rm{\hslash }}$. Let the initial quantum state of the particle be

$$\begin{eqnarray}&&| \Psi \rangle \propto \int \mathrm{exp}\left(-\displaystyle \frac{{(q-{\mu }_{q})}^{2}}{4{\sigma }^{2}}+{\rm{i}}{\mu }_{p}q\right)| q\rangle {\rm{d}}q.\end{eqnarray} \tag{ 23 }$$

This is a gaussian wavefunction with position mean ${\mu }_{q}$ and variance ${\sigma }^{2}$, and momentum mean ${\mu }_{p}$ and variance $\frac{1}{4{\sigma }^{2}}$. While this state has been chosen to have zero convariance, we emphasize that our results are completely general (see note below). The particle and the measuring device are coupled under the Hamiltonian in equation (6), where the observable $\hat{A}$ we are measuring is one of the form

$$\begin{eqnarray}&&\hat{A}=\mathrm{cos}{\theta }_{A}\hat{q}+\mathrm{sin}{\theta }_{A}\hat{p}.\end{eqnarray} \tag{ 24 }$$

We then postselect using a projective measurement of the observable $\hat{B}=\mathrm{cos}{\theta }_{B}\hat{q}+\mathrm{sin}{\theta }_{B}\hat{p}$, with eigenstate

$$\begin{eqnarray}&&\langle b| =\int \sqrt{\displaystyle \frac{1}{2\pi {\rm{i}}\;\mathrm{sin}\;{\theta }_{B}}}{{\rm{e}}}^{-{\rm{i}}({bq}/\mathrm{sin}{\theta }_{B}-\mathrm{cot}{\theta }_{B}{q}^{2}/2)}\langle q| {\rm{d}}q.\end{eqnarray} \tag{ 25 }$$

The weak value of $\hat{A}$ for this postselection is

$$\begin{eqnarray}{\langle \hat{A}\rangle }_{W} & = & \mathrm{cos}{\theta }_{A}\displaystyle \frac{\langle b| \hat{q}| \Psi \rangle }{\langle b| \Psi \rangle }+\mathrm{sin}{\theta }_{A}\displaystyle \frac{\langle b| \hat{p}| \Psi \rangle }{\langle b| \Psi \rangle }\\ & = & \displaystyle \frac{4{\sigma }^{4}\mathrm{cos}{\theta }_{B}(b\mathrm{cos}{\theta }_{A}-{\mu }_{p}\mathrm{sin}(\Delta \theta ))}{4{\sigma }^{4}{\mathrm{cos}}^{2}{\theta }_{B}+{\mathrm{sin}}^{2}{\theta }_{B}}\\ & & +\displaystyle \frac{\mathrm{sin}{\theta }_{B}({\mu }_{q}\mathrm{sin}(\Delta \theta )+b\mathrm{sin}{\theta }_{A})}{4{\sigma }^{4}{\mathrm{cos}}^{2}{\theta }_{B}+{\mathrm{sin}}^{2}{\theta }_{B}}\\ & & +{\rm{i}}\displaystyle \frac{2{\sigma }^{2}(-b+{\mu }_{q}\mathrm{cos}{\theta }_{B}+{\mu }_{p}\mathrm{sin}{\theta }_{B})\mathrm{sin}(\Delta \theta )}{4{\sigma }^{4}{\mathrm{cos}}^{2}{\theta }_{B}+{\mathrm{sin}}^{2}{\theta }_{B}},\end{eqnarray} \tag{ 26 }$$

where $\Delta \theta ={\theta }_{B}-{\theta }_{A}$. We note that this expression depends linearly on ${\mu }_{q}$, ${\mu }_{p}$, and b, and while the real part of this expression has the same form as expected from Bayes' rule, the imaginary components of ${\langle \hat{A}\rangle }_{W}$ is also clearly identified.

Note: in equation (23), we consider a gaussian state with zero covariance for simplicity. However, this choice is equivalent to using a gaussian state with nonzero covariance simply by changing the quadratures of of both weak and strong measurement appropriately, that is, changing ${\theta }_{A}$ and ${\theta }_{B}$. Hence our analysis holds for general gaussian states.

In this section, we will treat the measurement procedure outlined in section 2.1 within ERL theory. That is, we will interact the particles and measuring devices using a classical interaction and Hamilton's equations of motion. However, we will require that all initial Liouville distributions describing these systems obey the epistemic restriction (27). This restriction will lead us to a concept of weak measurement within ERL theory, including a tradeoff between information gain and disturbance much like quantum theory. By following the structure of the quantum derivation in section 2.1, we introduce a general measurement model within ERL theory and then investigate the situations under which the disturbance to the particles is minimized.

We model both the particle to be measured and the measurement device as one-dimensional canonical systems, with $q$ and $p$ the position and momentum coordinates of the particle, and $Q$ and $P$ the position and momentum coordinates of the measurement device. The particle and the measurement device are coupled via the Hamiltonian

$$\begin{eqnarray}&&H=\chi (t){PA},\end{eqnarray} \tag{ 30 }$$

where the observable we are measuring is of the form

$$\begin{eqnarray}&&A=\mathrm{cos}{\theta }_{A}q+\mathrm{sin}{\theta }_{A}p.\end{eqnarray} \tag{ 31 }$$

The distribution of the particle and the measurement device changes according the the classical Hamiltonian equations as a result of this interaction, following

$$\begin{eqnarray}&&\displaystyle \frac{{\rm{d}}q}{{\rm{d}}t}=\displaystyle \frac{\partial H}{\partial p},\quad \displaystyle \frac{{\rm{d}}p}{{\rm{d}}t}=-\displaystyle \frac{\partial H}{\partial q}.\end{eqnarray} \tag{ 32 }$$

After the measurement, the position and momentum of the particle are

$$\begin{eqnarray}&&q={q}_{i}+g\mathrm{sin}{\theta }_{A}{P}_{i},\end{eqnarray} \tag{ 33 }$$

$$\begin{eqnarray}&&p={p}_{i}-g\mathrm{cos}{\theta }_{A}{P}_{i},\end{eqnarray} \tag{ 34 }$$

where ${p}_{i}$ and ${q}_{i}$ are the initial momentum and position of the particle being measured respectively and ${P}_{i}$ is the initial momentum of the measuring device.

The position and momentum of the measurement device after this interaction are

$$\begin{eqnarray}&&Q={Q}_{i}+g(\mathrm{cos}{\theta }_{A}{q}_{i}+\mathrm{sin}{\theta }_{A}{p}_{i}),\end{eqnarray} \tag{ 35 }$$

$$\begin{eqnarray}&&P={P}_{i}.\end{eqnarray} \tag{ 36 }$$

Hence the change in the position of the device gives the measurement outcome. Note there is no change in the momentum of each device due to the measurement interaction, which suggests that any change to the mean momentum of the ensemble of devices, must be statistical bias.

For an ideal measurement, we would require the initial position of the device ${Q}_{i}$ to be known with complete certainty, and this would lead to a perfect correlation between the observable A and the position of the measurement device after interaction. However, due to our epistemic restriction, the uncertainty in the momentum of the device must be infinite in this case, i.e., the observer has no knowledge of the initial momentum of the measurement device ${P}_{i}$. From equations (33) and (34), the position and momentum of the particle are each displaced by an amount proportional to the initial momentum of the device as a result of the interaction. If the initial momentum ${P}_{i}$ of the measurement device is unknown, so is the phase space displacement of the particle. Thus we see how, within the ERL theory, such a measurement that acquires complete information about an observable A of the particle is accompanied by a corresponding unknown disturbance on the state of the particle.

It is in this way that an information gain-disturbance tradeoff appears in the ERL theory, despite its classical ontology. If we removed the epistemic restriction, the state of the particle would still be displaced by an amount proportional to the momentum of the device. However, the observer could in principle know the initial value of the momentum of the device as well as its position and therefore correct for this change. In other words, disturbance due to measurement arises in Newtonian mechanics, and the epistemic restriction just makes the disturbance unrectifiable. Note that, despite the change in the position and momentum of the particles due to the interaction, the expectation value of the observable A has not changed, i.e.,

$$\begin{eqnarray}A(p,P) & = & \mathrm{cos}{\theta }_{A}({q}_{i}-g\mathrm{sin}{\theta }_{A}P)+\mathrm{sin}{\theta }_{A}({p}_{i}+g\mathrm{cos}{\theta }_{A}P)\\ & = & \mathrm{cos}{\theta }_{A}{q}_{i}+\mathrm{sin}{\theta }_{A}{p}_{i}\\ & = & A({p}_{i},{P}_{i}).\end{eqnarray} \tag{ 37 }$$

As in quantum theory, measurements in ERL theory are repeatable, and the disturbance is associated with uncertainty in canonically conjugate observables.

We now introduce weak measurements in the ERL theory, again following by analogy the quantum formalism of section 2.2. From equations (33) and (34), one can again see two methods by which the disturbance due to measurement can be made small: first, by considering small coupling g, and second, by requiring the initial momentum of the measurement device ${P}_{i}$ to be very close to zero. In the second method, to ensure that the momentum of the measurement device has ${P}_{i}=0$ (and not just the mean value of the Liouville distribution), we must require that the variance ${\Delta }_{{P}_{i}}$ is very small as well as the mean value ${\mu }_{{P}_{i}}=0$. Due to the epistemic restriction, the initial uncertainty in position of the measurement device ${\Delta }_{{Q}_{i}}$ must then be very large. Therefore, in the limit of small ${\Delta }_{{P}_{i}}$, we have vanishing knowledge of the initial position of the device and hence any change in this position after the measurement (equation (35)). Thus, in both methods, we obtain weak measurements that yield an arbitrarily small information gain about the system and correspondingly small disturbance.

Within ERL mechanics, we note that these two methods of obtaining weak measurements are physically distinct. In both cases, in the limits $g\to 0$ or ${\Delta }_{P}\to 0$ we have both no disturbance to the system as well as no information gained about the system. There is a difference in the ontology, however. In the limit of $g\to 0$, there is no physical change to the measurement device. In contrast, in the limit of ${\Delta }_{P}\to 0$, there is a shift in the mean position of the measurement device, but our uncertainty about the device's initial position makes the shift undetectable.

With a meaningful notion of weak measurement in ERL theory, we can now consider the appearance of a weak value. Let the initial phase space distribution of the particles being measured be described by a position distribution with mean ${\mu }_{q}$ and variance ${\sigma }^{2}$ and a momentum distribution with mean ${\mu }_{p}$ and variance $\frac{1}{4{\sigma }^{2}}$, i.e., as the Liouville distribution

$$\begin{eqnarray}&&{\rho }_{s}({q}_{i},{p}_{i})=\displaystyle \frac{1}{\pi }\mathrm{exp}\left(\displaystyle \frac{-{({q}_{i}-{\mu }_{q})}^{2}}{2{\sigma }^{2}}-{({p}_{i}-{\mu }_{p})}^{2}2{\sigma }^{2}\right).\end{eqnarray} \tag{ 38 }$$

This distribution is precisely the Wigner function of the initial quantum state $| \Psi \rangle$ of equation (23). The initial phase space distribution of the measurement device is the Wigner function of the initial state $| \Phi \rangle$ in equation (2),

$$\begin{eqnarray}&&{\rho }_{d}({Q}_{i},{P}_{i})=\displaystyle \frac{1}{\pi }\mathrm{exp}\left(-(2{\Delta }_{P}^{2}{Q}_{i}^{2}+2{\Delta }_{Q}^{2}{P}_{i}^{2}-2\Omega {P}_{i}{Q}_{i})\right),\end{eqnarray} \tag{ 39 }$$

where $\Omega =2\langle {P}_{i}{Q}_{i}\rangle -2\langle {P}_{i}\rangle \langle {Q}_{i}\rangle$ form the off diagonal elements of the covariance matrix.

Observable A is measured using weak measurements as described above. After the weak measurement, the actual position and momentum of each particle and device changes according to equations (33)–(36), and the phase space distributions of the particles and devices become correlated as

$$\begin{eqnarray}{\rho }_{{sd}}^{\prime }(q,p,Q,P) & = & {\rho }_{s}(q-g\mathrm{sin}{\theta }_{A}P,p+g\mathrm{cos}{\theta }_{A}P)\\ & & \times {\rho }_{d}(Q-g(\mathrm{cos}{\theta }_{A}q+\mathrm{sin}{\theta }_{A}p),P).\end{eqnarray} \tag{ 40 }$$

The particles are then measured using a strong measurement of observable $B=\mathrm{cos}{\theta }_{B}q+\mathrm{sin}{\theta }_{B}p$, and postselect on the outcome $\mathrm{cos}{\theta }_{B}q+\mathrm{sin}{\theta }_{B}p=b$. The postselected distribution of the measurement devices is then

$$\begin{eqnarray}&&{\rho }_{d}^{\prime\prime }(Q,P)=\int \int {\rho }_{{sd}}^{\prime }(q,p,Q,P)\delta (B=b)\;{\rm{d}}q\;{\rm{d}}p\end{eqnarray} \tag{ 41 }$$

$$\begin{eqnarray}&&=\int {\rho }_{{sd}}^{\prime }(q,\displaystyle \frac{b-\mathrm{cos}{\theta }_{B}q}{\mathrm{sin}{\theta }_{B}},Q,P)\;{\rm{d}}q.\end{eqnarray} \tag{ 42 }$$

The mean position of the postselected subset is

$$\begin{eqnarray}{\langle Q\rangle }_{b} & = & \displaystyle \int \displaystyle \int Q{\rho }_{d}^{\prime\prime }(Q,P)\;{\rm{d}}P\;{\rm{d}}Q\\ & = & \displaystyle \frac{\alpha (g,{\theta }_{A},{\theta }_{B},{\Delta }_{Q},{\mu }_{p},\sigma ,\Omega ,b)}{\beta (g,{\theta }_{A},{\theta }_{B},{\Delta }_{Q},\sigma ,\Omega )},\end{eqnarray} \tag{ 43 }$$

where α and β are real-valued functions defined as

$$\begin{eqnarray} & & \alpha (g,{\theta }_{A},{\theta }_{B},{\Delta }_{Q},{\mu }_{p},\sigma ,\Omega ,b)={g}^{3}({\mu }_{q}\mathrm{cos}{\theta }_{A}+{\mu }_{p}\mathrm{sin}{\theta }_{A})(1+{\Omega }^{2}){\sigma }^{2}{\mathrm{sin}}^{2}(\Delta \theta )\\ & & +g{\Delta }_{Q}^{2}\left(4{\sigma }^{4}\mathrm{cos}{\theta }_{B}(b\mathrm{cos}{\theta }_{A}-{\mu }_{p}\mathrm{sin}(\Delta \theta ))\right.\\ & & +\mathrm{sin}{\theta }_{B}({\mu }_{q}\mathrm{sin}(\Delta \theta )+b\mathrm{sin}{\theta }_{A})\\ & & +\left.2\Omega {\sigma }^{2}\mathrm{sin}(\Delta \theta )({\mu }_{q}\mathrm{cos}{\theta }_{B}+{\mu }_{p}\mathrm{sin}{\theta }_{B}-b)\right)\end{eqnarray} \tag{ 44 }$$

$$\begin{eqnarray} & & \beta (g,{\theta }_{A},{\theta }_{B},{\Delta }_{Q},\sigma ,\Omega )={g}^{2}(1+{\Omega }^{2}){\sigma }^{2}{\mathrm{sin}}^{2}(\Delta \theta )+{\Delta }_{Q}^{2}(4{\sigma }^{4}{\mathrm{cos}}^{2}{\theta }_{B}+{\mathrm{sin}}^{2}{\theta }_{B}).\end{eqnarray} \tag{ 45 }$$

The mean momentum of the postselected subset is

$$\begin{eqnarray}{\langle P\rangle }_{b} & = & \displaystyle \int \displaystyle \int P{\rho }_{d}^{\prime\prime }(Q,P)\;{\rm{d}}P\;{\rm{d}}Q\\ & = & \displaystyle \frac{g(1+{\Omega }^{2}){\sigma }^{2}({\mu }_{q}\mathrm{cos}{\theta }_{A}+{\mu }_{p}\mathrm{sin}{\theta }_{A}-b)\mathrm{sin}(\Delta \theta )}{\beta (g,{\theta }_{A},{\theta }_{B},{\Delta }_{Q},\sigma ,\Omega )}.\end{eqnarray} \tag{ 46 }$$

3.3.1. Small uncertainty ${\Delta }_{P}$

As with the quantum case, we consider these expressions using the first method to obtain weak measurements, wherein the initial position of the measurement device becomes highly uncertain. We characterize this case by choosing ${\mu }_{p}=0$ and take the limiting case ${\Delta }_{P}\to 0$, which implies $\Omega \to 0$ as well. We find we can express the average shift in the position of the measurement devices upon postselection using the same expression for ${\langle \hat{A}\rangle }_{W}$ as calculated in equation (26), to give

$$\begin{eqnarray}&&{\langle Q\rangle }_{b}=g\mathrm{Re}[{\langle \hat{A}\rangle }_{W}]+g\Omega \mathrm{Im}[\langle {\hat{A}}_{W}\rangle ],\end{eqnarray} \tag{ 47 }$$

where again we have ignored terms of order ${g}^{3}{\Delta }_{P}^{2}$ and higher. In the limit ${\Delta }_{P}\to 0$, the covariance $\Omega \to 0$ and this becomes

$$\begin{eqnarray}&&\underset{{\Delta }_{P}\to 0}{\mathrm{lim}}{\langle \hat{Q}\rangle }_{b}=g\mathrm{Re}[\langle {\hat{A}}_{W}\rangle ].\end{eqnarray} \tag{ 48 }$$

This exactly reproduces the quantum mechanical shift in mean position. As our model has a classical ontology, it allows joint probability distributions over all variables, unlike quantum mechanics. As a result, we are able to calculate conditional expectation values of any two observables. In our situation, we can exploit this fact to compare the shift in the average position of the device with the expectation value of A of the particles conditioned on an outcome b of observable B. We find that the real part of the weak value is indeed the same as the conditional expectation value.

In this limit we find that the mean momentum of the devices is

$$\begin{eqnarray}&&{\langle P\rangle }_{b}=2g{\Delta }_{P}^{2}\mathrm{Im}[\langle {\hat{A}}_{W}\rangle ],\end{eqnarray} \tag{ 49 }$$

which in the limit becomes

$$\begin{eqnarray}&&\underset{{\Delta }_{P}\to 0}{\mathrm{lim}}{\langle \hat{P}\rangle }_{b}=0.\end{eqnarray} \tag{ 50 }$$

Again, this exactly reproduces the quantum mechanical expression.

3.3.2. Weak coupling g

Just as in the quantum case, we take the mean momentum ${\mu }_{P}=0$. If we take g to be finite but small enough to ignore higher than first order terms of g, the shift in the average position of the postselected devices is

$$\begin{eqnarray}&&{\langle Q\rangle }_{b}=g\mathrm{Re}[{\langle \hat{A}\rangle }_{W}]+g\Omega \mathrm{Im}[{\langle \hat{A}\rangle }_{W}],\end{eqnarray} \tag{ 51 }$$

and the average momentum shift is

$$\begin{eqnarray}&&{\langle P\rangle }_{b}=2g{\Delta }_{P}^{2}\mathrm{Im}[{\langle \hat{A}\rangle }_{W}].\end{eqnarray} \tag{ 52 }$$

From equation (36), we see that the momentum of the individual measurement devices are not changed due to the weak measurement. However in this method for weak measurements, we do not require ${\Delta }_{P}\to 0$, i.e., there is uncertainty of the initial momentum of the measurement devices. The state of the particles can be shifted in phase space by the momentum of the devices, as shown in (33) and (34), and so initial uncertainty in the momentum of the measurement device leads to an unknown disturbance of the particles. By postselecting on particles that have been perturbed by the momentum of the device, we arrive at a final momentum distribution of the device that is biased.

An example of this effect is illustrated in figure 2, where the position of the particle is weakly measured, followed by postselection on the particle's momentum, that is, ${\theta }_{A}=0$ and ${\theta }_{B}=\frac{\pi }{2}$. The shift in the mean momentum of the measurement devices does not arise as a result of the dynamics of the interaction, but instead from postselection biasing the momentum distribution. If the position distribution of the devices is correlated to the momentum distribution via a nonzero covariance Ω, the position distribution of the devices will also be biased.

Zoom In Zoom Out Reset image size

In summary, we have seen how terms in both the position and momentum shifts that are proportional to the imaginary part of the weak value are primarily the result of postselection biasing the device distributions, not dynamics. This observation allows us to formulate an operational interpretation of the imaginary part of the weak value as a measure of how much postselection will bias the device distribution, given finite disturbance, which we emphasize results from uncertainty in the initial properties of the devices.

As an additional insight that arises from our analysis of the weak value, we now provide a better bound on the range of couplings g for which the standard weak value results will hold. In order to consider g to be small (and therefore to ignore higher order terms), from equations (14), (20), (15), (21), one can see that this approximation is good provided

$$\begin{eqnarray}&&{g}^{2}{\Delta }_{P}^{2}\ll \displaystyle \frac{1}{{({a}_{j}-{a}_{l})}^{2}}\quad \forall j,l,\end{eqnarray} \tag{ 53 }$$

where a_i is an eigenvalue of the measurement operator. This condition is very restrictive and is not useful for variables with a continuous eigenspectrum. However, we can derive a less restrictive upper bound by looking at the approximations made on the classical distributions of the devices. From equations (43) and(46), we can see that in order to ignore higher order terms of g, we require

$$\begin{eqnarray}&&{g}^{2}{\Delta }_{P}^{2}\ll \displaystyle \frac{(4{\sigma }^{4}{\mathrm{cos}}^{2}{\theta }_{B}+{\mathrm{sin}}^{2}{\theta }_{B})}{4{\sigma }^{2}{\mathrm{sin}}^{2}({\theta }_{A}-{\theta }_{B})}.\end{eqnarray} \tag{ 54 }$$

Because ERL theory is operationally equivalent to gaussian quantum mechanics, this upper bound is also true for gaussian quantum states, and could equally well be derived from the quantum formalism. This is a significantly less restrictive upper bound for gaussian states than the one discussed in [29].