## Violation of Heisenberg's Uncertainty Principle 155 155

mbone writes

*"A very interesting paper (PDF) has just hit the streets (or, at least,**Physics Review Letters*) about the Heisenberg uncertainty relationship as it was originally formulated about measurements. The researchers find that they can exceed the uncertainty limit in measurements (although the uncertainty limit in quantum states is still followed, so the foundations of quantum mechanics still appear to be sound.) This is really an attack on quantum entanglement (the correlations imposed between two related particles), and so may have immediate applications in cracking quantum cryptography systems. It may also be easier to read quantum communications without being detected than people originally thought."
## Magic (Score:2, Insightful)

## Argh science journalism. (Score:5, Insightful)

This article is horrible.

"The Heisenberg uncertainty principle is in part an embodiment of the idea that in the quantum world, the mere act of observing an event changes it."

That's not the Heisenberg uncertainty principle. That's just the observer effect, and it's not something peculiar to quantum mechanics. You want to measure the temperature of a system, so you stick a thermometer in there. Okay, the mercury in the thermometer absorbs a bit of heat from the system, providing you with a temperature measurement at the same time it changes the temperature of the system. If you want to measure the parameters of a particle, you stick a bubble chamber in the way, and as the particle flies through the chamber it smacks into hydrogen molecules, showing you what it's doing but also taking a different path than it would have if none of those hydrogen molecules were in the way. Big fat hairy deal.

The HUP doesn't just say that you can't simultaneously measure the position and momentum of a particle, it says that a particle *does not simultaneously possess* a well-defined position and momentum. If the particle's doing something in a system and is interacting in such a way that you can define its position to arbitrary precision, then it *does not have* a well-defined momentum for you to measure, and vice versa. Position and momentum are what are called quantum conjugate variables, and the HUP says that when you have a pair of those variables, then the product of their uncertainties is greater than or equal to a constant. There is *no state* in which that particle is even *allowed* to exist in which it possesses both a well-defined position and well-defined momentum.

A signal processing analogy, for any analog people. A particle's wavefunction carries information about its position and its momentum. Where the wave exists is where the particle actually is, and the wavelength is the particle's momentum. Take a particle whose momentum you know to the utmost precision, and graph that. Range of momentums on the x axis, probability of the particle having that momentum on the y axis. You'll get a graph that looks like a Dirac function, a value of 0 everywhere except for a single spike corresponding to the particle momentum, area under the curve of 1.

Now switch domains, change from the momentum to the position domain, this is mathmatically the same thing as changing from a time domain to a frequency domain, which means you can use your old friend the Fourier Transform.

What do you get when you do an FT of a Dirac function? You get a constant value everywhere, from -infinity to +infinity. If you know exactly where that particle is, you have no idea *where* it is, and it's not because you disturbed it in measuring it, it's because *it* has no idea where it is, a well-defined position does not exist; since the uncertainty in the momentum measurement approaches zero than the uncertainty in the position measurement has to approach infinity so that the product of those uncertainties remains greater than a constant.

The "you change the system by measuring it" is an analogy, and it's one that Heisenberg himself used to explain the HUP, but *that is not what it says*. The HUP is not a statement about the process of measuring things, it is a statement about the nature of the universe, and finding a way to improve a measuring system to reduce the disturbance it creates in the system it's measuring has nothing to do with the HUP.

## Re:Argh science journalism. (Score:4, Insightful)

## Re:Magic (Score:5, Insightful)

That is a good description of classical entanglement - what, in this context, would be called a hidden variable theory (the cards have a certain face value, even if you can't see them).

Let's see if I can expand this analogy. Suppose you had two decks of cards, each with only two cards - say the king of hearts and the king of spades. Off-stage, I shuffle them, so that there is either one deck of 2 hearts, and one of two spades, or one deck of both, and another of both. Say that the chances of either shuffle are the same.

Now, repeat your experiment, except you and your friend only get to pull 1 card each, each from your own deck. Classically, the chances are

- 50%, you pull from 1 spade and 1 heart

- 25%, you pull from 2 spades

- 25%, you pull from 2 hearts.

And, of course, ditto for your friend.

Now, if you pull a spade, then the classical chances are

2/3 the other card is a heart

1/3 the other card is a spade

and the classical chances for your friend are thus

2/3 he has a spade and a heart

1/3 he has 2 hearts

so his (classical) chances on his card are

2/3 he pulls a heart

1/3 he pulls a spade.

(If you pull a spade, you CANNOT have two hearts, while he can.)

So, if you pull a Spade, you can tell your friend he is likely to have a heart. Do this a lot of times, and you should be correct 2/3 of the time. The cards are indeed entangled, but classically. Experimental error (maybe you can't always see your cards well) will lower this, but (for a long enough term average) cannot raise this.

In Quantum Mechanics, however, you can get correlations that you cannot get in classical physics, i.e., greater than 2/3 in this case. That is the essence of Bell's Theorem - you have correlations that you just can't "get there from here," classically. This is a consequence of having a complex amplitude. Again, it's not just having a correlation, it's that you can get correlations you just can't classically.

I saw a lecture from Dick Feynman once where he showed that you could explain all of this by allowing for negative probabilities for intermediate results, and that this was mathematically the same as the normal (i.e., complex) formulation of QM. (Since you cannot actually measure the intermediate results, you never actually measure a negative probability.) In some ways, I find that helps to grasp the weirdness. YMMV.