# Why should multiple versions of a weak measurement preclude it from being a measurement of intrinsic properties of some system

I've been trying to understand Stephen Parrott's criticisms of weak measurement, outlined most concisely here: http://arxiv.org/abs/0909.0295

One of his major criticisms is that weak measurement is not unique. I'll try to give a brief outline of his argument here:

We consider a system of interest S, and a system of measurement M, each with an associated hilbert space $\mathbb{H}_S$ and $\mathbb{H}_M$. We start with a separable state state $|s\rangle\otimes |m\rangle \in \mathbb{H}_S\otimes\mathbb{H}_M$ and let it evolve for some short time $\epsilon$ under the action of some hamiltonian $H=A\otimes G$. The resulting state is $|r(\epsilon)\rangle=e^{-i\epsilon H}|s\rangle\otimes |m\rangle=|s\rangle\otimes |m\rangle-i\epsilon A|s\rangle\otimes G|m\rangle+ O(\epsilon^2)$.

If we choose B and G such that $\langle m|B|m\rangle=0$ and $Im(\langle m|BG|m \rangle)=1/2$ then one can calculate that $\lim_{\epsilon ->0}[ \frac{\langle r(\epsilon)| 1 \otimes B |r(\epsilon)\rangle}{\epsilon}] =\langle s|A|s\rangle$.

We then consider a scenario where one repeatedly performs this measurement $\langle r(\epsilon)|1\otimes B|r(\epsilon)\rangle$ followed by a projective postselection measurement $P_f \otimes 1= |f\rangle \langle f| \otimes 1$, throwing away the data from any experiment for which our postselection measurement gives 0. If we take $\epsilon$ to be very small, this procedure can be described by $NE(B|f)=\lim_{\epsilon ->0}[ \frac{\langle r(\epsilon)| P_f \otimes B |r(\epsilon)\rangle}{\epsilon \langle r(\epsilon)| P_f \otimes 1 |r(\epsilon)\rangle}] =Re[\frac{\langle f|A|s\rangle}{\langle f| s\rangle}]$, an expression which Parrott claims the weak measurement literature identifies as the conditional expectation of A in state $|s\rangle$ given succesful postselection to state $|f\rangle$

Parrott then considers a scenario where $\langle m |BG |m \rangle =\rho +1/2 i$, which satisfies our original constraint when $\rho$ is real, but when inserted into the above procedure yields $NE(B|f)=Re[\frac{\langle f|A|s\rangle}{\langle f| s\rangle}]+2\rho Im[\frac{\langle f|A|s\rangle}{\langle f| s\rangle}]$. From this Parrott deduces that the weak measurement is not unique, by virtue of the variability of $\rho$, and that this casts doubt on the weak measurement actually measuring some intrinsic property of system S.

So I suppose my confusion lies in trying to understand what is preventing a proponent of weak measurement from simply adding an additional constraint on$BG$, namely that $Re[\langle m |BG|m\rangle]=0$, this would mean $\rho=0$. Then there would be a unique definiton of weak measurement. Is it not the case that the mathematical construction of the weak measurement is the way it is simply because it results in a convenient expression for $Re[\frac{\langle f|A|s\rangle}{\langle f| s\rangle}]$, and is not this most conveniently expressed when $\rho=0$?

Even more to the point, how is it necessarily the case that because there is more than one way to make your measurement this implies your measurement is not of some intrinsic property of the system? Parrott gives an example of two voltmeters which give different readings, he correctly states that if one expects them to read the same voltage, and they don't, then they must be making measurements of different things, maybe one or both isn't even measuring voltage. This analogy doesn't seem apt in this scenario, since it seems to me that $\rho$ is akin to some kind of calibration, and so in the two scenarios we would expect them to read different things.

I should add that I'm aware of the controversial status of weak measurement, and I hold no allegiance to either camp. I merely wish to more thoroughly understand Parrott's arguments, as I suspect I must be missing something fundamental.

Thanks!

• That's fine. You're adding a "calibration" condition $Re\langle m |BG|m\rangle=0$ which constrains the meter variables $B,G$. Your measurement therefore depends on a particular choice of state/observables of the meter system so you should admit that what you measured was some combined property of the system of interest and the meter system, not just a property of the system of interest. This is not just a formality; the $\rho=0$ choice isn't really preferred in any way and you may construct situations in which the natural value of $\rho$ will be different out of cases where $\rho=0$ is natural – Luboš Motl Mar 26 '13 at 10:03
• Incidentally, to assume that the Hamiltonian has the form $H=A\otimes G$ is utterly unrealistic - it violates locality. The normal Hamiltonian for two decoupled systems would be $H = 1\otimes G+A\otimes 1$ where $G,A$ are the Hamiltonians for the subsystems. The strange assumption for a single tensor-product form of the Hamiltonian already imposes some (unrealistic) correlation between the relative phases in the meter system and those in the system of interest that don't really exist in reality. So $\rho=0$ may look "more natural" to you but that's an illusion. – Luboš Motl Mar 26 '13 at 10:07
• So suppose we conclude that what is measured in weak measurement is some combined property of the system of interest and the meter system. Is Parrott's criticism ultimately that one could never consistently devise a protocol for extracting only the information about the system of interest, whether that protocol be setting $\rho$ to 0, or knowing $\rho$ and doing some mathematical manipulation of the data? If that is the case, I am still not certain about what reasoning leads one to that conclusion. If it is not the case, then isn't weak measurement a valid measurement? – Joel Klassen Mar 26 '13 at 21:00

There are several important points to make. First note that in the definition of the weak value you have an observable $\hat{A}$. So, when you measure weak value it is not just the issue that you measure "some property", it is good for what you are measuring to have some plausible interpretation in connection with the observable $\hat{A}$. Sokolovski illustrates the discussion with a double slit. There the observable $\hat{A}$ has two eigenvalues $1$ or $2$, which indicate the particle passing through slit $1$ or $2$, respectively. If you take the expectation value of the observable $\hat{A}$ in the usual case of projective measurements you will find that the expectation value $\langle\hat{A}\rangle$ is within the interval defined by the minimim and the maximum possible eigenvalue. Because of that you can meaningfully interpret your measurement as saying something about the observable $\hat{A}$; in this case through which slit the particle passed.
Unfortunately, in the weak measurement case, you can get any value for $Re[\frac{\langle f|A|s\rangle}{\langle f| s\rangle}]$ ranging from $-\infty$ to $\infty$; note that the issue is not just real $Re[]$ vs. imaginary $Im[]$, so the "calibration" suggested by you cannot help you much. In the given example for pre-selected and post-selected states Sokolovski obtains a real weak value of $-100$. So, can you tell from that through which slit did the particle pass? Did it pass through slit $-100$, even though there are only two slits? Or to quote Sokolovski:
"A broken speedometer may read $50$ mph each time the car goes at $100$ mph, and might convince the driver, but not the traffic policemen who stops him for speeding. Similarly, the slit number $-100$ may come up in a weak measurement, but cannot be used for any other purpose, such as convincing a potential user that the screen he is about to buy has more than two holes drilled in it."