# Re-evaluating Your Quantum Upbringing

You may recall being told by your parents, as you were growing up, **outmoded ideas or outright misconceptions** about quantum mechanics. Examples may have included: the uncertainty principle is due to momentum imparted by photons as you measure a particle’s position; in any given experiment you can observe wave or particle properties but not both; the wave function is a mathematical tool and not part of objective reality; you cannot ask what a particle is doing between measurements; you cannot simultaneously determine position and momentum; the phase of a wave function is not observable; you cannot discuss reality separately from what you choose to measure; you cannot ask what is really there when no one is looking; ** “just shut up and calculate”**.

Maybe it was while you were in college rather than while you were growing up. And perhaps it was your quantum mechanics (QM) instructor rather than your parents. Nonetheless, much of what has been written and taught about QM since its inception has misled students, teachers, researchers, and the general public about the implications it has for reality and observation.

The above notions, and many other bits of nonsensical interpretational issues are being clarified and sometimes overturned by talented theorists and experimentalists. These explorers continue to peel back the curtains to see what is really going on behind the cloak of quantum weirdness. The techniques of *weak measurements* have become vital tools in this quest. The article I wrote a couple weeks ago, *Discontinuous Trajectories in Quantum Mechanics*, was an example of weak measurements. Today, I discuss weak measurements used to reconstruct particle trajectories in a double-slit experiment.

# Imagining Weak Measurements

Weak measurements were initially proposed by Yakir Aharonov, David Albert, and Lev Vaidman about twenty five years ago. The idea is basically this: you prepare particles in some particular initial state (pre-selection) and later detect some of them in a particular final state (post-selection). It may be that only a small subset of your particles end up in the particular final state that you make the selection on, but that is ok, you are going to repeat the experiment many, many times.

You want to know what your particles are doing between these initial and final states; how they get from point A to point B, for example. So you need to do some measurements. However, that would lead to apparent collapse of the state vector into a particular eigenstate, essentially re-setting the experiment. If you measure the particle’s position somewhere along its trajectory, the momentum becomes uncertain and uncorrelated with any initial momentum it may have had.

The key insight that these gentlemen had was to make the disturbance from this intermediate measurement as small as necessary, so as not to disturb the wavefunction too much. When measuring the position of a particle in a weak measurement, the velocity does not become random. However, the uncertainty in the position measurement is large. So, the second trick is to average over a very large number of trials. This leads to precise information about the wavefunction itself.

# Weak Measurements in Action

In a Physics World article, *In Praise of Weakness**, *Aephraim Steinberg and his colleagues discussed their use of weak measurements to map particle trajectories in a double-slit experiment. Their article is* *also available here. The green 3D plot below shows where a quantum particle is most likely to be found as it passes through the double-slit apparatus while behaving as a wave. The black lines on top of the green 3D surface are the average paths that the particles take through the experiment, as reconstructed from weak measurements.

From Steinberg, et al:

“…*it is striking that the average result of such a measurement will yield exactly what common sense would have suggested*.

*What we are arguing – and this admittedly is a controversial point – is that weak measurements provide the clearest operational definition for quantities such as “the average velocity of the electrons that are going to arrive at x = 1”. It is very tempting to say that this value, this hypothetical measurement result, is describing something that’s “really out there”, whether or not a measurement is performed. We should stress: this is for now only a temptation, albeit a tantalizing one. The question of what the “reality” behind a quantum state is – if such a question is even fair game for physics – remains a huge open problem.”*

Aephraim Steinberg spoke about quantum mechanics and weak measurements in the Perimeter Institute Public Lecture Series. The video of his talk: *In Praise of Weakness Public Lecture.*

# Exploring Weak Measurements Further

Dressel, et al, provide a review of the mathematics and applications of weak measurements in their recent paper: *Understanding Quantum Weak Values: Basics and Applications. *They discuss three different types of experimental applications that are revolutionizing our ability to study and manipulate quantum systems using weak measurements: (1) amplifying a signal, enabling the sensitive estimation of unknown evolution parameters, such as beam detection, phase shifts, frequency shifts, time shifts, temperature shifts, etc.; (2) measuring the real and imaginary parts of a complex-valued parameter, enabling new methods for reconstructing the quantum state, including relative phase of the complex value; (3) finding conditioned averages of generalized observable eigenvalues, providing a window into non-classical features of a quantum mechanical system.

For additional discussion of the theory and mathematical background for directly measuring the wavefunction of a quantum system, it is worthwhile to read *Direct Measurement of the Quantum Wavefunction* and *Direct measurement of general quantum states using weak measurement*.

So where will all this lead? This is still, very much, an evolving field of study. In an area as unintuitive as quantum physics, you cannot just take one or two experimental results and assume you understand what is going on. Perhaps the wavefunction is not just a mathematical tool, but rather something that is real and can be directly measured. Perhaps these experiments will clarify the relationship between quantum and classical behaviors. Perhaps these experiments will help reduce the confusion and misunderstanding concerning the meaning of measurement and observation in quantum mechanics. The insights gained from weak measurements will certainly lead to a deeper conceptual understanding of the quantum realm.

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