From my college’s computer-science club, I received a T-shirt that reads:
while(not_dead){
sleep--;
time--;
awesome++;
}
/*There’s a reason we can’t hang out with you…*/
The message is written in Java, a programming language. Even if you’ve never programmed, you likely catch the drift: CS majors are the bees’ knees because, at the expense of sleep and social lives, they code. I disagree with part of said drift: CS majors hung out with me despite being awesome.
The rest of the drift—you have to give some to get some—synopsizes the physics I encountered this fall. To understand tradeoffs, you needn’t study QI. But what trades off with what, according to QI, can surprise us.
The T-shirt haunted me at the University of Nottingham, where researchers are blending QI with Einstein’s theory of relativity. Relativity describes accelerations, gravity, and space-time’s curvature. In other sources, you can read about physicists’ attempts to unify relativity and quantum mechanics, the Romeo and Tybalt of modern physics, into a theory of quantum gravity. In this article, relativity tangos with quantum mechanics in relativistic quantum information (RQI). If I move my quantum computer, RQIers ask, how do I change its information processing? How does space-time’s curvature affect computation? How can motion affect measurements?
Answers to these questions involve tradeoffs.
Nottingham researchers kindly tolerating a seminar by me
For example, acceleration entangles particles. Decades ago, physicists learned that acceleration creates particles. Say you’re gazing into a vacuum—not empty space, but nearly empty space, the lowest-energy system that can exist. Zooming away on a rocket, I accelerate relative to you. From my perspective, more particles than you think—and higher-energy particles—surround us.
Have I created matter? Have I violated the Principle of Conservation of Energy (and Mass)? I created particles in a sense, but at the expense of rocket fuel. You have to give some to get some:
Fuel--; Particles++;
The math that describes my particles relates to the math that describes entanglement.* Entanglement is a relationship between quantum systems. Say you entangle two particles, then separate them. If you measure one, you instantaneously affect the other, even if the other occupies another city.
Say we encode information in quantum particles stored in a box.** Just as you encode messages by writing letters, we write messages in the ink of quantum particles. Say the box zooms off on a rocket. Just as acceleration led me to see particles in a vacuum, acceleration entangles the particles in our box. Since entanglement facilitates computation, you can process information by shaking a box. And performing another few steps.
When an RQIer told me so, she might as well have added that space-time has 106 dimensions and the US would win the World Cup. Then my T-shirt came to mind. To get some, you have to give some. When you give something, you might get something. Giving fuel gets you entanglement. To prove that statement, I need to do and interpret math. Till I have time to,
Fuel--; Entanglement++;
offers intuition.
After cropping up in Nottingham, my T-shirt reared its head (collar?) in physics problem after physics problem. By “consuming entanglement”—forfeiting that ability to affect the particle in another city—you can teleport quantum information.
Entanglement--; Quantum teleportation++;
My research involves tradeoffs between information and energy. As the Hungarian physicist Leó Szilárd showed, you can exchange information for work. Say you learn which half of a box*** a particle occupies, and you trap the particle in that half. Upon freeing the particle—forfeiting your knowledge about its location—you can lift a weight, charge a battery, or otherwise store energy.
Information--; Energy++;
If you expend energy, Rolf Landauer showed, you can gain knowledge.
Energy--; Information++;
No wonder my computer-science friends joked about sleep deprivation. But information can energize. For fuel, I forage in the blending of fields like QI and relativity, and in physical intuitions like those encapsulated in the pseudo-Java above. Much as Szilard’s physics enchants me, I’m glad that the pursuit of physics contradicts his conclusion:
while(not_dead){
Information++;
Energy++;
}
The code includes awesome++ implicitly.
*Bogoliubov transformations, to readers familiar with the term.
**In the fields in a cavity, to readers familiar with the terms.
***Physicists adore boxes, you might have noticed.
With thanks to Ivette Fuentes and the University of Nottingham for their hospitality and for their introduction to RQI.
and 
be positive integers satisfying 
. Then, 
; that is, the numbers 
, then I could claim half the prize. Well, I didn’t quite get there, but I made some progress using knowledge found in these two blog posts: 
and 
, for 
. Moreover, the general case of even exponents can be reduced to the case of 
and 
, for 
primes. Which makes one wonder if the general case has a similar reduction, where two of the three exponents can be assumed equal.
is equivalent to 
. In other words, the triplet 
is a Pythagorean triple (sides of a right triangle), so we must have 
, for some positive integers 
with no common factors (otherwise, our assumption that 
is even. This implies that 
and 
, where 
and 
have no factors in common. Moreover, since 
and 
have no common factors either (why?) Hence, 
, where 
and 
have no factors in common. But, 
, implying that 
, where 
have no common factors.
is even. This implies that 
and 
, where 
, implying that 
, where 
have no common factors.
and 
, then it holds for 
, for arbitrary 
. As it turns out, when 
, Beal’s conjecture becomes Fermat’s Last Theorem, implying that the conjecture holds for all exponents 
, with 
and 
.
, for 
?
