Grade inflation and college investment incentives

Here is Raphael Boleslavsky and Christopher Cotton discussing their model of grade deflation in selective undergraduate programs:

Grade inflation is widely viewed as detrimental, compromising the quality of education and reducing the information content of student transcripts for employers. This column argues that there may be benefits to allowing grade inflation when universities’ investment decisions are taken into account. With grade inflation, student transcripts convey less information, so employers rely less on transcripts and more on universities’ reputations. This incentivises universities to make costly investments to improve the quality of their education and the average ability of their graduates. [Link. h/t Ben Kuhn.]

I’ve only read the column rather than the full paper, but it sounds like their model simply posits that “schools can undertake costly investments to improve the quality of education that they provide, increasing the average ability of graduates”.

But if you believe folks like Bryan Caplan, then you think colleges add very little value. (Even if you think the best schools do add more value than worse schools, it doesn’t at all follow that this can be increased in a positive-sum way by additional investment. It could be that all the value-added is from being around other smart students, who can only be drawn away from other schools.) Under Boleslavsky and Cotton’s model, schools are only incentivized to increase the quality of their exiting graduates, and this seems much easier to accomplish by doing better advertising to prospective students than by actually investing more in the students that matriculate.

Princeton took significant steps to curb grade inflation, with some success. However, they now look to be relaxing the only part of the policy that had teeth.… [continue reading]

How to think about Quantum Mechanics—Part 2: Vacuum fluctuations

[Other parts in this series: 1,2,3,4,5,6,7,8.]

Although it is possible to use the term “vacuum fluctuations” in a consistent manner, referring to well-defined phenomena, people are usually way too sloppy. Most physicists never think clearly about quantum measurements, so the term is widely misunderstood and should be avoided if possible. Maybe the most dangerous result of this is the confident, unexplained use of this term by experienced physicists talking to students; it has the awful effect of giving these student the impression that their inevitable confusion is normal and not indicative of deep misunderstanding“Professor, where do the wiggles in the cosmic microwave background come from?” “Quantum fluctuations”. “Oh, um…OK.” (Yudkowsky has usefully called this a “curiosity-stopper”, although I’m sure there’s another term for this used by philosophers of science.)a  .

Here is everything you need to know:

  1. A measurement is specified by a basis, not by an observable. (If you demand to think in terms of observables, just replace “measurement basis” with “eigenbasis of the measured observable” in everything that follows.)
  2. Real-life processes amplify microscopic phenomena to macroscopic scales all the time, thereby effectively performing a quantum measurement. (This includes inducing the implied wave-function collapse). These do not need to involve a physicist in a lab, but the basis being measured must be an orthogonal one.W. H. Zurek, Phys. Rev. A 76, 052110 (2007). [arXiv:quant-ph/0703160]b  
  3. “Quantum fluctuations” are when any measurement (whether involving a human or not) is made in a basis which doesn’t commute with the initial state of the system.
  4. A “vacuum fluctuation” is when the ground state of a system is measured in a basis that does not include the ground state; it’s merely a special case of a quantum fluctuation.
[continue reading]

Risk aversion of class-action lawyers

The two sides in the potentially massive class-action lawsuit by silicon-valley engineers against Google, Apple, and other big tech companies reached an agreement, but that settlement was rejected by the judge. New York Times:

After the plaintiffs’ lawyers took their 25 percent cut, the settlement would have given about $4,000 to every member of the class.

Judge Koh said that she believed the case was stronger than that, and that the plaintiffs’ lawyers were taking the easy way out by settling. The evidence against the defendants was compelling, she said.

(Original court order.)

I would like to be able to explain this by understanding the economic/sociological motivations of the lawyers. People often complain about a huge chunk of the money going to the class-action lawyers who are too eager to settle, but the traditional argument is that a fixed percentage structure (rather than an hourly or flat rate) gives the lawyers the proper incentive to pursue the interests of the class by tying their compensation directly to the legal award. So this should lead to maximizing the award to the plaintiffs.

My best guess, doubtlessly considered by many others, is this: Lawyers, like most people, are risk adverse for sufficiently large amounts of money. (They would rather have $10 million for sure than a 50% chance at $50 million.) On the other hand, the legal award will be distributed over many more plaintiffs. Since it will be much smaller per person, the plaintiffs are significantly less risk adverse. So the lawyers settle even though it’s not in the best interests of the plaintiffs.

This suggests the following speculative solution for correctly aligning the incentives of the lawyers and the class action plaintiffs: Ensure that the person with the final decision-making power for the plaintiff legal team receives a percentage of the award that is small enough for that person’s utility function to be roughly as linear as the plaintiffs’.… [continue reading]

Lindblad Equation is differential form of CP map

The Master equation in Lindblad form (aka the Lindblad equation) is the most general possible evolution of an open quantum system that is Markovian and time-homogeneous. Markovian means that the way in which the density matrix evolves is determined completely by the current density matrix. This is the assumption that there are no memory effects, i.e. that the environment does not store information about earlier state of the system that can influence the system in the future.Here’s an example of a memory effect: An atom immersed in an electromagnetic field can be in one of two states, excited or ground. If it is in an excited state then, during a time interval, it has a certain probability of decaying to the ground state by emitting a photon. If it is in the ground state then it also has a chance of becoming excited by the ambient field. The situation where the atom is in a space of essentially infinite size would be Markovian, because the emitted photon (which embodies a record of the atom’s previous state of excitement) would travel away from the atom never to interact with it again. It might still become excited because of the ambient field, but its chance of doing so isn’t influenced by its previous state. But if the atom is in a container with reflecting walls, then the photon might be reflected back towards the atom, changing the probability that it becomes excited during a later period.a   Time-homogeneous just means that the rule for stochastically evolving the system from one time to the next is the same for all times.

Given an arbitrary orthonormal basis L_n of the space of operators on the N-dimensional Hilbert space of the system (according to the Hilbert-Schmidt inner product \langle A,B \rangle = \mathrm{Tr}[A^\dagger B]), the Lindblad equation takes the following form:

(1)   \begin{align*} \frac{\mathrm{d}}{\mathrm{d}t} \rho=- i[H,\rho]+\sum_{n,m = 1}^{N^2-1} h_{n,m}\left(L_n\rho L_m^\dagger-\frac{1}{2}\left(\rho L_m^\dagger L_n + L_m^\dagger L_n\rho\right)\right) , \end{align*}

with \hbar=1.… [continue reading]

Potentials and the Aharonov–Bohm effect

[This post was originally “Part 1” of my HTTAQM series. However, it’s old, haphazardly written, and not a good starting point. Therefore, I’ve removed it from that series, which now begins with “Measurements are about bases”. Other parts are here: 1,2,3,4,5,6,7,8. I hope to re-write this post in the future.]

It’s often remarked that the Aharonov–Bohm (AB) effect says something profound about the “reality” of potentials in quantum mechanics. In one version of the relevant experiment, charged particles are made to travel coherently along two alternate paths, such as in a Mach-Zehnder interferometer. At the experimenter’s discretion, an external electromagnetic potential (either vector or scalar) can be applied so that the two paths are at different potentials yet still experience zero magnetic and electric field. The paths are recombined, and the size of the potential difference determines the phase of the interference pattern. The effect is often interpreted as a demonstration that the electromagnetic potential is physically “real”, rather than just a useful mathematical concept.


The magnetic Aharanov-Bohm effect. The wavepacket of an electron approaches from the left and is split coherently over two paths, L and R. The red solenoid in between contains magnetic flux \Phi. The region outside the solenoid has zero field, but there is a non-zero curl to the vector potential as measured along the two paths. The relative phase between the L and R wavepackets is given by \Theta = e \Phi/c \hbar.

However, Vaidman recently pointed out that this is a mistaken interpretation which is an artifact of the semi-classical approximation used to describe the AB effect.… [continue reading]

A dark matter model for decoherence detection

[Added 2015-1-30: The paper is now in print and has appeared in the popular press.]

One criticism I’ve had to address when proselytizing the indisputable charms of using decoherence detection methods to look at low-mass dark matter (DM) is this: I’ve never produced a concrete model that would be tested. My analysis (arXiv:1212.3061) addressed the possibility of using matter interferometry to rule out a large class of dark matter models characterized by a certain range for the DM mass and the nucleon-scattering cross section. However, I never constructed an explicit model as a representative of this class to demonstrate in detail that it was compatible with all existing observational evidence. This is a large and complicated task, and not something I could accomplish on my own.

I tried hard to find an existing model in the literature that met my requirements, but without luck. So I had to argue (with referees and with others) that this was properly beyond the scope of my work, and that the idea was interesting enough to warrant publication without a model. This ultimately was successful, but it was an uphill battle. Among other things, I pointed out that new experimental concepts can inspire theoretical work, so it is important that they be disseminated.

I’m thrilled to say this paid off in spades. Bateman, McHardy, Merle, Morris, and Ulbricht have posted their new pre-print “On the Existence of Low-Mass Dark Matter and its Direct Detection” (arXiv:1405.5536). Here is the abstract:

Dark Matter (DM) is an elusive form of matter which has been postulated to explain astronomical observations through its gravitational effects on stars and galaxies, gravitational lensing of light around these, and through its imprint on the Cosmic Microwave Background (CMB).

[continue reading]

Diagonal operators in the coherent state basis

I asked a question back in November on Physics.StackExchange, but that didn’t attract any interest from anyone. I started thinking about it again recently and figured out a good solution. The question and answer are explained below.I posted the answer on Physics.SE too since they encourage the answering of one’s own question. How lonely is that?!?a  

Q: Is there a good notion of a “diagonal” operator with respect the overcomplete basis of coherent states?
A: Yes. The operators that are “coherent-state diagonal” are those that have a smooth Glauber–Sudarshan P transform.

The primary motivation for this question is to get a clean mathematical condition for diagonality (presumably with a notion of “approximately diagonal”) for the density matrix of a system of a continuous degree of freedom being decohered. More generally, one might like to know the intuitive sense in which X, P, and X+P are all approximately diagonal in the basis of wavepackets, but RXR^\dagger is not, where R is the unitary operator which maps

(1)   \begin{align*} \vert x \rangle \to (\vert x \rangle + \mathrm{sign}(x) \vert - x \rangle) / \sqrt{2}. \end{align*}

(This operator creates a Schrodinger’s cat state by reflecting about x=0.)

For two different coherent states \vert \alpha \rangle and \vert \beta \rangle, we want to require an approximately diagonal operator A to satisfy \langle \alpha \vert A \vert \beta \rangle \approx 0, but we only want to do this if \langle \alpha \vert \beta \rangle \approx 0. For \langle \alpha \vert \beta \rangle \approx 1, we sensibly expect \langle \alpha \vert A \vert \beta \rangle to be within the eigenspectrum of A.

One might consider the negativity of the Wigner-Weyl transformCase has a pleasingly gentle introduction.b   of the density matrix (i.e. the Wigner phase-space quasi-probability distribution aka the Wigner function) as a sign of quantum coherence, since it is known that coherent superpositions (which are clearly not diagonal in the coherent state basis) have negative oscillations that mark the superposition, and also that these oscillations are destroyed by decoherence.… [continue reading]

Comments on Tegmark’s ‘Consciousness as a State of Matter’

[Edit: Scott Aaronson has posted on his blog with extensive criticism of Integrated Information Theory, which motivated Tegmark’s paper.]

Max Tegmark’s recent paper entitled “Consciousness as a State of Matter” has been making the rounds. See especially Sabine Hossenfelder’s critique on her blog that agrees in several places with what I say below.

Tegmark’s paper didn’t convince me that there’s anything new here with regards to the big questions of consciousness. (In fairness, I haven’t read the work of neuroscientist Giulio Tononi that motivated Tegmark’s claims). However, I was interested in what he has to say about the proper way to define subsystems in a quantum universe (i.e. to “carve reality at its joints”) and how this relates to the quantum-classical transition. There is a sense in which the modern understanding of decoherence simplifies the vague questions “How does (the appearance of) a classical world emerge in a quantum universe? ” to the slightly-less-vague question “what are the preferred subsystems of the universe, and how do they change with time?”. Tegmark describes essentially this as the “quantum factorization problem” on page 3. (My preferred formulation is as the “set-selection problem” by Dowker and Kent. Note that this is a separate problem from the origin of probability in quantum mechanicsThe problem of probability as described by Weinberg: “The difficulty is not that quantum mechanics is probabilistic—that is something we apparently just have to live with. The real difficulty is that it is also deterministic, or more precisely, that it combines a probabilistic interpretation with deterministic dynamics.” HT Steve Hsu.a  .)

Therefore, my comments are going to focus on the “object-level” calculations of the paper, and I won’t have much to say about the implications for consciousness except at the very end.… [continue reading]