Uncertain on Uncertainty

Over at BBC News, there is an article about a recently published paper (arXiv) by Lee Rozema et al. that could lead to some, ehm, uncertainty about the status of the Heisenberg Uncertainty Principle (HUP).
Before dissecting the BBC article, let’s look at the paper by Rozema et al. The title is “Violation of Heisenberg’s Measurement–Disturbance Relationship by Weak Measurements”. While this title might raise a few eyebrows, the authors make it crystal clear in the opening sentence of the abstract that they didn’t disprove the HUP or some such nonsense. The HUP is a theorem within the standard formulation of quantum mechanics, so finding a violation of that would be equivalent to finding a violation of quantum theory itself! Instead, they look at the so-called measurement–disturbance relationship (MDR), which is a non-rigorous heuristic that is commonly taught to give an intuition for the uncertainty principle.
The HUP is usually stated in the form of the Robertson uncertainty relation, and states that a given quantum state psi cannot (in general) have zero variance with respect to two non-commuting observables. The more modern formulations are stated in a why that is independent of the quantum state; see this nice review by Wehner and Winter for more about these entropic uncertainty relations.
By contrast, the MDR states that the product of the measurement precision and the measurement disturbance (quantified as root-mean-squared deviations between ideal and actual measurement variables) can’t be smaller than Planck’s constant. In 2002, Masanao Ozawa proved that this was inconsistent with standard quantum mechanics, and formulated a corrected version of the MDR that also takes into account the state-dependent variance of the observables. Building on Ozawa’s work, in 2010 Lund and Wiseman proposed an experiment which could measure the relevant quantities using  the so-called weak value.
Rozema et al. implemented the Lund-Wiseman scheme using measurements of complementary observables (X and Z) on the polarization states of a single photon to confirm Ozawa’s result, and to experimentally violate the MDR.  The experiment is very cool, since it crucially relies on entanglement induced between the probe photon and the measurement apparatus.
The bottom line: the uncertainty principle emerges completely unscathed, but the original hand-wavy MDR succumbs to both theoretical and now experimental violations.
Now let’s look at the BBC article. Right from the title and the subtitle, they get it wrong. “Heisenberg uncertainty principle stressed in new test”—no, that’s wrong—“Pioneering experiments have cast doubt on a founding idea…”—also no—the results were consistent with the HUP, and actually corroborated Ozawa’s theory of measurement–disturbance! Then they go on to say that this “could play havoc with ‘uncrackable codes’ of quantum cryptography.” The rest of the article has a few more whoppers, but also some mildly redeeming features; after such a horrible start, though, you might as well quietly leave the pitch. Please science journalists, try to do better next time.

Hardness of NP

In computer science, NP-hard problems are widely believed to be intractable, not because they have been proved so, but on the empirical evidence of no one having found a fast algorithm for any of them in over half a century of trying.  But the concepts of  NP-hardness and NP-completeness are themselves hard for newcomers to understand.   The current American Physical Society piece Unbearable Hardness of Physics makes a common mistake when it takes NP-hard problems to mean problems Not solvable in time Polynomial in the size of their input, rather than those to which all problems solvable in Nondeterministic Polynomial time are efficiently reducible.  Come to think of it, the letters N and P  also breed confusion in other fields, including our own, where  NPT is often taken to stand for Negative Partial Transpose, when it would be more correct to say Nonpositive Partial Transpose, admittedly a tiny imprecision compared to the confusion surrounding what NP means.

What are the odds?

Let’s multiply together a bunch of numbers which are less than one and see how small they get!
If that sounds like fun, then you’ll love this sleek new infographic (of which the above is just the teaser) posted this morning at BoingBoing. The graphic is based on this blog post by Dr. Ali Binazir, who apparently has an AB (same as a BA) from Harvard, an MD from the UC San Diego School of Medicine, and an M.Phil. from Cambridge.
I’ll save you the effort of clicking through: the good doctor estimates the probability of “your existing as you, today”. His estimate consists of (what else?) multiplying a bunch of raw probability estimates together without conditioning! And I’ll give you a hint as to the conclusion: the odds that you exist are basically zero! Astounding.
I should add that it seems he was forced to add a disclaimer that “It’s all an exercise to get you thinking…” and (obliquely) admit that the calculation is bogus at the end of the post, however.
Is there any branch of mathematics which is abused so extravagantly as probability? I think these sorts of abuses are beyond even the most egregious statistical claims, no?

Look Ma, I'm a Financial Journalist!

In this Saturday’s New York Times, in an article titled The Chasm Between Consumers and the Fed, I found the most amazing chart:

Of course I am not a financial journalist, so I have absolutely no understanding of the gigantic amoeba-like-shaded-area in this chart. But it looks very cool and very much like it represents something about which the article has much to say. Sadly, however, the New York Times does not provide the methodology it used in obtaining the amazing fact that six of the points can be grouped together while those other two points are excluded from the party. What astounding mathematical finance model did the Grey Lady use to come up with this plot (I’ll be it involves Ito calculus)?
Frustrated by the lack of transparency, I decided that it would be best if I tried to come up with my own methods and models for obtaining this graph. My first attempt, after scouring the economics literature and using some advance methods (related to integrating over Banach spaces) was the following

As you can see this model seems to pick out the overall rate of return as the defining characteristic. After much great reflection, and reacquainting myself with some obscure results from the theory of hyperbolic partial differential equations and new deep learning techniques from machine learning, I was able to tweak my model a bit and obtained the following

Now this is a beautiful plot, but it clearly does not reproduce the graph from the New York Times. What exactly, was I missing in order to obtain the giant amoeba of correlation?
But then I remembered…I’m not a financial journalist. I’m a physicist. And so, I took a look at the stats notes I took as a physics major at Caltech, quickly plugged in some numbers, and obtained a new, reality based, version of the plot

Well it’s not the New York Time plot. But I like it a lot.

A What Bit?

A correspondent writes to me about a recent article in the APS News describingThe Top Ten Physics Stories of 2008 and notes a very troubling sentence:

Diamond Detectors
Work on the molecular structure of carbon continues to show great promise for quantum computing. This year scientists were able to construct a nano-scale light source that emits a single photon at a time. The team first removed a solitary atom from the carbon’s otherwise regular matrix and then introduced a nitrogen atom nearby. When they excited this crystal with a laser, single polarized photons were emitted from the empty space. These photons could be used to detect very small magnetic forces. Additionally the photons emitted contained two spin states and were able to exist in that state for nearly a millisecond before their wave function collapsed. The emitted photon is essentially a long-lasting qbit which could, with further development, be entangled with other adjacent qbits for uses in quantum computing. Another team at the University of Delft in the Netherlands, working in conjunction with UCSB, was able to detect the spin of a single electron in a diamond environment. At the same time, a group at Harvard was able to locate within a nanometer a single Carbon-11 impurity using its nuclear spin interactions.

Qbit? What’s a qbit? Doh.

A Curmudgeon's and Improv's Guide to Outliers: Chapter 2

Part three in my continuing pedantic slow-as-molasses walk through Outliers: The Story of Success by Malcolm Gladwell.

List of posts here: introduction, ch 1, ch 2.

SPOILER ALERT: Dude, I can’t talk about the book without giving away what the book is about, so if you don’t want the book’s main ideas to be spoiled, don’t continue reading.

IDIOT ALERT: I’m in no way qualified in most of the fields Gladwell will touch on, so please, a grain of salt, before you start complaining about my ignorance. Yes I’m an idiot, please tell me why!

Continue reading “A Curmudgeon's and Improv's Guide to Outliers: Chapter 2”

A Curmudgeon's and Improv's Guide to Outliers: Chapter 1

Moving on to Chapter 1 in my ongoing pedantic plodding through Malcolm Gladwell’s Outliers: The Story of Success. See here for what this is all about. Note that I really am doing this as I read the book (I’m reading it really really slowly), so what I say here may be outdated by the time I get further into the book.
List of posts here: introduction, ch 1.
SPOILER ALERT: Dude, I can’t talk about the book without giving away what the book is about, so if you don’t want the book’s main ideas to be spoiled, don’t continue reading.
IDIOT ALERT: I’m in no way qualified in most of the fields Gladwell will touch on, so please, a grain of salt, before you start complaining about my ignorance. Yes I’m an idiot, please tell me why!
Continue reading “A Curmudgeon's and Improv's Guide to Outliers: Chapter 1”

Nitpicker's Paradiso: We Don't Need no Stinking Scientists and Engineers Edition

Researchers Dispute Notion That America Lacks Scientists and Engineers in the Chronicle of Higher Education is a fine example of how thinking that scientific or engineering degree’s are like technical training degrees will lead you to say all sorts of funny things. Yep, it’s another edition of Nitpicker’s Paradiso.
Continue reading “Nitpicker's Paradiso: We Don't Need no Stinking Scientists and Engineers Edition”