Anomalous Weak Values are Caused by Disturbance

Abstract

In combination with post-selection, weak measurements can lead to surprising results known as anomalous weak values. These lie outside the bounds of the spectrum of the relevant observable, as in the canonical example of measuring the spin of an electron (along some axis) to be 100. We argue that the disturbance caused by the weak measurement, while small, is sufficient to significantly affect the measurement result, and that this is the most reasonable explanation of anomalous weak values.

Appendix

1.1 A. State Update and Disturbance

For an ideal classical measurement, it is usually assumed that the state of the system after interacting with the apparatus is the same as it was before the interaction. In contrast, the presence of measurement disturbance (or back-action) then means that this condition is not satisfied, i.e. that the meter somehow causes the system to change state.

The situation for quantum measurements is more subtle. Since it is the system that will have our interest, it is helpful to abstract away the ‘inner workings’ of the measurement procedure. As in the main text, let us concentrate on the von Neumann model. We thus consider an apparatus that takes a quantum system as input and produces a number x and a quantum system as output. It can be completely characterized by two pieces of data. One is the probability $P(x|\psi )$ of result x given the input $|\psi \rangle$. In our specific case $P(x|\psi ) = P_\lambda (x|\psi )$ is given by Eq. (3). The second piece of data is the state of the output $|\chi _{\psi ,x}\rangle$ conditioned on a specific result x,

$$\begin{aligned} |\chi _{\psi ,x}^\lambda \rangle := P_\lambda (x|\psi )^{-1/2}\langle x|e^{-i\lambda {{\hat{A}}}{{\hat{p}}}}|\psi \rangle | I\rangle . \end{aligned}$$

(25)

In analogy with the classical case, we want to define quantum measurement disturbance as the change of the state of the system caused by the interaction with the meter (compare with e.g. Refs. [6, 18]). The question is how to compare the state before and after interaction.

One approach would be to look at how close $|\chi _{\psi ,x}^\lambda \rangle$ is to $|\psi \rangle$. Expanding (25) for small $\lambda$ we find

$$\begin{aligned} |\chi _{\psi ,x}^\lambda \rangle = \left( 1+\lambda [{{\hat{A}}}-\langle {{\hat{A}}}\rangle _\psi ]\frac{x}{2} + O(\lambda ^2)\right) |\psi \rangle . \end{aligned}$$

(26)

From this point of view the conclusion would thus be that the the back-action is of order $\lambda$ (unless $|\psi \rangle$ happens to be an eigenstate of ${{\hat{A}}}$), in conflict with DI.

Seemingly, there is just one way to avoid this conclusion, which is to understand a pure quantum states as only expressing partial knowledge about the system (similarly to how a probability distribution expresses partial knowledge about a classical random variable). Then it could be argued that the change $|\psi \rangle \rightarrow |\chi _{\psi ,x}^\lambda \rangle$ is, at least partly, a form of Bayesian update (or conditioning). That is, the claim would be that some of the difference between $|\chi _{\psi ,x}^\lambda \rangle$ and $|\psi \rangle$ reflects the fact that we learn something about the system by learning x, and should thus update our believes about the quantum system. From this point of view we can reformulate DI as

Disturbance Insignificance’ (DI’): Any disturbance caused by the measurement process only shows up in the higher order [i.e. $O(\lambda ^2)$] correction to the conditional state (26). Hence the first order correction must be understood as purely due to some form of Bayesian update.

An interesting discussion of quantum Bayesian conditioning can be found in Section V of Ref. [18].

Let us mention that it is not possible to interpret the update $|\psi \rangle \rightarrow |\chi _{\psi ,x}^\lambda \rangle$ purely as Bayesian conditioning. That is, the update necessarily introduces some amount of disturbance. Indeed, if $|\psi \rangle \rightarrow |\chi _{\psi ,x}^\lambda \rangle$ were a kind of Bayesian update, then marginalizing over x (i.e. ‘unlearning’ the measurement result) should give back the original state, but we actually find¹⁰

$$\begin{aligned} \rho _\psi ^\lambda := \int P_\lambda (x|\psi ) |\chi _{\psi ,x}^\lambda \rangle \langle \chi _{\psi ,x}^\lambda |\text{d}x = \int G(x') e^{-i\lambda {{\hat{A}}}x'/2}|\psi \rangle \langle \psi |e^{i\lambda {{\hat{A}}}x'/2} \text{d}x' \end{aligned}$$

(27)

which is mixed (unless $|\psi \rangle$ is an eigenstate of ${{\hat{A}}}$), and thus not equal to the initial state $|\psi \rangle \langle \psi |$. It is a general result that any quantum measurement must be disturbing in this sense [6, 18].¹¹

The fact that the difference between $\rho _\psi ^\lambda$ and $|\psi \rangle \langle \psi |$ is solely due to back-action is used in Sect. 2 to quantify the amount of disturbance. Indeed, the difference between the overall post-selection probability $P_\lambda (\varphi |\psi )$ [see Eq. (10)] and its unperturbed value $|\langle \varphi |\psi \rangle |^2$ can be written as

$$\begin{aligned} P_\lambda (\varphi |\psi ) - |\langle \varphi |\psi \rangle |^2 = \left\langle \rho _\psi ^\lambda - |\psi \rangle \langle \psi | \right\rangle _\varphi . \end{aligned}$$

(28)

1.2 B. Disturbance Insignificance in the Literature

We claim in the introduction that a number of works in the literature apparently conclude or postulate that the disturbance of weak measurements can be neglected. The subtle nature of the subject matter means that there is a substantial risk of misunderstandings. With that in mind we will quote some relevant passages from these papers. After the quotes we make a couple of brief remarks.

• (Vaidman, 1996, p. 899) [23]: “I propose to consider the standard measuring procedure [...] in which we weaken the interaction in such a way that the state of the quantum system is not changed significantly during the interaction.”

• (Resch, Lundeen & Steinberg, 2004, p. 125) [20]: “In particular, [the weak measurement strategy] makes it possible to contemplate the behavior of a system defined both by state preparation and by a later post-selection, without significant disturbance of the system in the intervening period.”

• (Tollaksen, 2007, abstract) [22]: “When measurements are performed which do not disturb the pre- and post-selection (i.e. weak measurements) [...]”

• (Tollaksen, 2007, p. 9063) [22]: “[...] we have now shown that when considered as a limiting process, the disturbance goes to zero more quickly than the shift in the measuring device, which means for a large enough ensemble, information (e.g. the expectation value) can be obtained even though not even a single particle is disturbed.”

• (Hofmann, 2010, abstract) [14]: “[...] weak measurements have negligible back action [...]”

• (Hofmann, 2010, p. 2) [14]: “For very small measurement strengths $\epsilon$, the effects of the quantum state on the measurement probabilities is linear in $\epsilon$ while the measurement back action is quadratic in $\epsilon$. It is therefore possible to realize quantum state tomography with negligible back action.”

• (Dressel & Jordan, 2012, p. 7) [11]: “[the real part of the weak value] can be interpreted as an idealized limit point for the average of ${{\hat{A}}}$ in the initial state ${\hat{\rho }}_i$ that has been conditioned on the postselection ${\hat{P}}_f$ without any appreciable intermediate measurement disturbance.”

• (Bednorz, Franke & Belzig, 2013, abstract) [4]: “We show that it is possible to define general weak measurements, which are noninvasive: the disturbance becomes negligible as the measurement strength goes to zero.”

• (Bednorz, Franke & Belzig, 2013, p. 9) [4]: “[...] weak measurements (both classical and quantum) are noninvasive in a stronger sense: their disturbance vanishes as $g^2$ regardless of the type of measurements before/after.”

• (Vaidman et al., 2017, p. 2) [24]: “The change in the other systems [e.g. a meter] should be large enough to be seen, but the back action on the system should be small enough, such that the change in the two-state vector describing the system can be neglected. Since we allow an unlimited ensemble of experiments with identical pre- and postselection, the required limits are achievable.”

• (Cohen, 2017, p. 1263) [7]: “[...] for $\epsilon \ll 1$, which is indeed justified in the weak measurement regime, then the fidelity¹² is 1 up to $O(\epsilon ^2)$. Therefore, by definition, the state has been negligibly disturbed, and this is now a precise claim.”

• (Cohen, 2017, p. 1264) [7]: “[...] weak measurements are non-invasive, that is, the probability of evolving the initial state to an orthogonal state through a weak measurement decreases like $g^2$. This property significantly limits the amount of backaction [...]”

References [4, 7, 14, 22] all conclude that the disturbance is of second order in the interaction strength using variants of the argument discussed, and found unconvincing, in Sect. 2. The arguments of Ref. [11] are not discussed in this paper, but in Appendix D we address some related ideas by the same authors. It is not clear (to this author) what definition is being referred to in the first quote from Ref. [7].

1.3 C. Collective Weak Measurements

In the usual protocol for measuring the weak value the variance of x is large compared to the shift $\lambda {{\,\mathrm{Re}\,}}{{\hat{A}}}_w$. We thus have to repeat the experiment many times to get a good estimate. Another possibility is to let a single meter interact with a large number N of identically prepared systems. [3] In this case both the uncertainty of the (single) measurement result and the back-action on each system can, in principle, be made as small as one would like. It could thus seem as if this would exclude the possibility of the kind of statistical effects we have discussed so far. An analysis along the lines of Sect. 2, however, shows that the presence of significant disturbance, if anything, is more evident.

We again consider the interaction to be of von Neumann type,

$$\begin{aligned} |\psi \rangle ^N | I\rangle \rightarrow e^{-i\lambda {{\bar{A}}}{{\hat{p}}}}|\psi \rangle ^N | I\rangle , \end{aligned}$$

(29)

but now ${{\bar{A}}}$ is the averaged observable

$$\begin{aligned} {{\bar{A}}}:= N^{-1}\sum _{n=1}^N {{\hat{A}}}_n, \end{aligned}$$

(30)

and we abbreviate

$$\begin{aligned} |\psi \rangle ^N := |\psi \rangle _1|\psi \rangle _2\cdots |\psi \rangle _N . \end{aligned}$$

(31)

The conditional distribution of x is

$$\begin{aligned} P_\lambda (x|\varphi ^N,\psi ^N) = P_\lambda (\varphi ^N|\psi ^N)^{-1}\left| \langle x | \left( \langle \varphi |e^{-i\lambda N^{-1}{{\hat{A}}}{{\hat{p}}}}|\psi \rangle \right) ^{N}| I\rangle \right| ^2 . \end{aligned}$$

(32)

If we now keep $\lambda$ fixed (not necessarily small), but send $N\rightarrow \infty$, we can expand¹³

$$\begin{aligned} \left( \langle \varphi |e^{-i\lambda N^{-1}{{\hat{A}}}{{\hat{p}}}}|\psi \rangle \right) ^{N}&= (\langle \varphi |\psi \rangle )^N[1-i\lambda N^{-1} {{\hat{A}}}_w {{\hat{p}}}+ O(N^{-2})]^N\nonumber \\&= (\langle \varphi |\psi \rangle )^N [e^{-i\lambda {{\hat{A}}}_w{{\hat{p}}}} + O(N^{-1})], \end{aligned}$$

(33)

leading to

$$\begin{aligned} P(x|\varphi ^N,\psi ^N)&= P_\lambda (\varphi ^N|\psi ^N)^{-1}|\langle \varphi |\psi \rangle |^{2N} (2\pi )^{-1/2}\left| e^{-(x-\lambda {{\hat{A}}}_w)^2/4}+ O(N^{-1})\right| ^2\nonumber \\&= G(x-\lambda {{\,\mathrm{Re}\,}}{{\hat{A}}}_w)+ O(N^{-1}) . \end{aligned}$$

(34)

We again find a shift proportional to the weak value, but now without any condition that the constant of proportionality $\lambda$ is small.

Turning to the question of back-action, we first note that the strength of the coupling to each individual system is $N^{-1}\lambda$, which is certainly small in the limit we are considering. However, a more relevant quantity in connection with the conditional distribution of x is the overall post-selection probability

$$\begin{aligned} P_\lambda (\varphi ^N|\psi ^N) = |\langle \varphi |\psi \rangle |^{2N} \left[ e^{\lambda ^2({{\,\mathrm{Im}\,}}{{\hat{A}}}_w)^2/2}+O(N^{-1})\right] , \end{aligned}$$

(35)

which is not in general close to the unperturbed value of $|\langle \varphi |\psi \rangle |^{2N}$. We see that, when looking at all N systems as a whole, the disturbance from the interaction (29) is simply large! It thus seems difficult to argue that it nevertheless should be considered insignificant.

As for the ordinary weak measurement, it is instructive to work out the conditional distribution for a measurement of ${{\hat{x}}}' = 2{{\hat{p}}}$. We find

$$\begin{aligned} P(x'|\varphi ^N,\psi ^N)&= P_\lambda (\varphi ^N|\psi ^N)^{-1}|\langle x' | I\rangle |^2\left| \langle \varphi |e^{-i\lambda N^{-1}{{\hat{A}}}x'/2}|\psi \rangle \right| ^{2N}\nonumber \\&= P_\lambda (\varphi ^N|\psi ^N)^{-1}G(x')|\langle \varphi |\psi \rangle |^{2N} \left[ e^{\lambda {{\,\mathrm{Im}\,}}[{{\hat{A}}}_w]x'}+O(N^{-1})\right] \nonumber \\&= G(x'-\lambda {{\,\mathrm{Im}\,}}A_w)+ O(N^{-1}) \end{aligned}$$

(36)

This shift is entirely due to the back-action, but still of the same order of magnitude as that of x [Eq. (34)]!

We conclude that, also in the case of collective weak measurements, there is no indication that the disturbance of the intermediate measurement is negligible.

1.4 D. Disturbance and the Lindblad Super-Operator

Here we make some remarks on the claims about disturbance in Refs. [8, 12] (some related ideas can be found in Ref. [11]). We have to begin by making some technical definitions. First we write the joint probability as

$$\begin{aligned} P_\lambda (x,\varphi |\psi ) = |\langle \varphi |{{\hat{M}}}_x^\lambda |\psi \rangle |^2, \end{aligned}$$

(37)

where ${{\hat{M}}}_x^\lambda$ is an operator acting on the system Hilbert space. In the von Neumann case we get the equation

$$\begin{aligned} \left| \langle \varphi | \langle x| e^{-i\lambda {{\hat{A}}}{{\hat{p}}}}|\psi \rangle | I\rangle \right| ^2 = |\langle \varphi |{{\hat{M}}}_x^\lambda |\psi \rangle |^2, \end{aligned}$$

(38)

so we can take ${{\hat{M}}}_x^\lambda$ to be given by [Eq. (38) does not uniquely fix ${{\hat{M}}}$, since a phase ${{\hat{M}}}_x\rightarrow e^{if_x}{{\hat{M}}}_x$ drops out]

$$\begin{aligned} {{\hat{M}}}_x^\lambda&= \langle x|e^{-i\lambda {{\hat{A}}}{{\hat{p}}}}| I\rangle \nonumber \\&= \langle x|[1-i\lambda {{\hat{A}}}{{\hat{p}}}-\frac{1}{2} \lambda ^2{{\hat{A}}}^2{{\hat{p}}}^2+O(\lambda ^3)]| I\rangle \nonumber \\&= \sqrt{G(x)}\left[ 1+\lambda \frac{x}{2} {{\hat{A}}}+ \frac{\lambda ^2}{2}\left( \frac{x^2}{4}-\frac{1}{2}\right) {{\hat{A}}}^2+O(\lambda ^3)\right] . \end{aligned}$$

(39)

We will not assume that ${{\hat{M}}}_x^\lambda$ is Hermitian, although it happens to be in this specific example.

The key step of Refs. [8, 12] is to rewrite (37) as

$$\begin{aligned} P_\lambda (x,\varphi |\psi ) = P^w_\lambda (x,\varphi |\psi ) + {\mathcal {E}}_\lambda (x,\varphi |\psi ) , \end{aligned}$$

(40)

with $P^w$ define by ($\{\cdot ,\cdot \}$ denotes the anti-commutator)

$$\begin{aligned} P^w_\lambda (x,\varphi |\psi ) := |\langle \varphi |\psi \rangle |^2 {{\,\mathrm{Re}\,}}[([{{\hat{M}}}_x^\lambda ]^\dagger {{\hat{M}}}_x^\lambda )_w] = \frac{1}{2} \left\langle \{[{{\hat{M}}}_x^\lambda ]^\dagger {{\hat{M}}}_x^\lambda ,|\varphi \rangle \langle \varphi |\}\right\rangle _\psi . \end{aligned}$$

(41)

It follows that the ‘error term’ is given by

$$\begin{aligned} {\mathcal {E}}_\lambda (x,\varphi |\psi ) = \left\langle {\mathcal {L}}[{{\hat{M}}}_x^\lambda ](|\varphi \rangle \langle \varphi |)\right\rangle _\psi , \end{aligned}$$

(42)

with the Lindblad super-operator¹⁴ defined as

$$\begin{aligned} {\mathcal {L}}[{{\hat{M}}}]({\hat{O}}) = \frac{1}{2} ([{{\hat{M}}}^\dagger ,{\hat{O}}]{{\hat{M}}}+{{\hat{M}}}^\dagger [{\hat{O}},{{\hat{M}}}]) . \end{aligned}$$

(43)

It is now observed that, if ${\mathcal {E}}$ can be neglected in the $\lambda \rightarrow 0$ limit, then $P_\lambda (x,\varphi |\psi ) \approx P^w_\lambda (x,\varphi |\psi )$ and so the conditional expectation value of x is approximately [8, 12]

$$\begin{aligned} \frac{\int xP^w_\lambda (x,\varphi |\psi )\text{d}x}{\int P^w_\lambda (x,\varphi |\psi )\text{d}x} = \frac{|\langle \varphi |\psi \rangle |^2 \int x{{\,\mathrm{Re}\,}}[([{{\hat{M}}}_x^\lambda ]^\dagger {{\hat{M}}}_x^\lambda )_w]\text{d}x}{|\langle \varphi |\psi \rangle |^2} = \lambda {{\,\mathrm{Re}\,}}{{\hat{A}}}_w + O(\lambda ^2). \end{aligned}$$

(44)

Here we have used (in the von Neumann case there are actually no higher order corrections)

$$\begin{aligned} \int x[{{\hat{M}}}_x^\lambda ]^\dagger {{\hat{M}}}_x^\lambda \text{d}x = \lambda {{\hat{A}}}+ O(\lambda ^2) \end{aligned}$$

(45)

and

$$\begin{aligned} \int [{{\hat{M}}}_x^\lambda ]^\dagger {{\hat{M}}}_x^\lambda \text{d}x = {\hat{1}} \end{aligned}$$

(46)

to simplify the numerator and denominator. This argument shows that any weak measurement (i.e. not necessarily of von Neumann type) that can be parametrized by operators ${{\hat{M}}}_x^\lambda$ satisfying (45) and (46) will result in $\lambda {{\,\mathrm{Re}\,}}{{\hat{A}}}_w$ as the conditional expectation value, as long as the error terms can be neglected.¹⁵

The connection with back-action come from the claim [8, 12]¹⁶ which we will call

Generalized Disturbance Insignificance (GDI): If the error term ${\mathcal {E}}_\lambda (x,\varphi |\psi )$ can be neglected¹⁷then the disturbance caused by the intermediate measurement is insignificant.

As the names suggests, DI is a special case of GDI. To see this, we just need to check that the error term can be neglected for weak von Neumann measurements. Inserting (39) into (42) we find¹⁸

$$\begin{aligned} {\mathcal {E}}_\lambda (x,\varphi |\psi ) = |\langle \varphi |\psi \rangle |^2 G(x)\left[ \frac{\lambda ^2}{4} \left( |{{\hat{A}}}_w|^2-{{\,\mathrm{Re}\,}}[({{\hat{A}}}^2)_w]\right) x^2+O(\lambda ^3)\right] \end{aligned}$$

(47)

which is indeed $O(\lambda ^2)$. This in turn means that the paradoxical consequences of adopting DI also follow from adopting GDI. In Ref. [12] it is remarked that

$$\begin{aligned} P_\lambda (\varphi |\psi ) = \int P_\lambda (x,\varphi |\psi )\text{d}x \approx \int P^w_\lambda (x,\varphi |\psi ) \text{d}x = |\langle \varphi |\psi \rangle |^2 \end{aligned}$$

(48)

whenever ${\mathcal {E}}$ can be neglected. This is taken as evidence that “[t]he Lindblad operation indicates disturbance that the intermediate measurement introduces to the measurement sequence.” [12] But, as discussed in Sect. 2, the approximate equality $P_\lambda (\varphi |\psi ) \approx |\langle \varphi |\psi \rangle |^2$ does not imply that the disturbance is insignificant.

To summarize, GDI is, like DI, an additional postulate which does not follow from ordinary quantum mechanics. We do not know of any convincing arguments in favor of GDI, while the arguments against DI works equally well against GDI.