# From Schrödinger’s cat to quantum computers (I)

From Schrödinger’s cat to entangled cats

If you are also fans of The Big Bang Theory, you will be aware of Penny and Sheldon’s discussions about Schrödinger’s cat. Penny wants to know whether she should hook up with Leonard, and Sheldon tells her that, in 1935, Erwin Schrödinger designed a mental experiment in which a cat was put inside a closed box with a vial of poison which can be opened at random times. You may not know whether the cat is alive or dead until you open it. Penny thinks that the lesson is that she should try, and only then she will know. But, really, Sheldon only wanted to get rid of her. The question, as all human interaction, was completely irrelevant to him.

I have a board. If you like boards, this is my board.

Although I love the series, the explanation about Schrödinger’s cat is lame. You put a cat in a box and a vial of poison. There’s 50% chance that the vial opens and the cat dies. According to our intuition, the real state of the system is one of them: alive-cat or dead-cat. Since we don’t know which one it is, we represent our knowledge with probabilities:

But that’s not quantum mechanics! Quantum mechanics is far more weird, and tell us that the cat may be alive and dead at the same time. We represent it this way:

(That notation, $\left|X\right>$, is called a “ket”… yes, we physicists are very fond of funny notations.) If, while in that state, you open the box, the cat is forced to choose. With 50% probabilities, it becomes an alive-cat, and with 50% a dead-cat. But then, how is it different from before!?  Because the “alive-and-dead-cat” is a new “catty state” that we may represent this way:

and which has different properties.

Well, with cats this doesn’t really work. We tried, but they move a lot, and miaow, and scratch. We better try with atoms, which are far more peaceful. Most atoms behave like small magnets, and their magnetization can be thought of as a small arrow, called “spin”, pointing in any direction, something like this:

We have our (cat-like) atom in a closed box, with its little arrow pointing in any direction. With cats, you may ask: “are you alive or dead?”, and it gives you an answer. With atoms you may ask, for example: “is your little-arrow (spin) pointing up or down?” Of course, not only for the vertical direction. You just pick up any direction and ask, but let’s say that the vertical direction is clear enough. So, you may have the atom in state $\left|\uparrow\right>$, and the answer will be “up”, or $\left|\downarrow\right>$ and the answer will be “down”. But what happens if you mix them? You can have the state $\left|\uparrow\right> + \left|\downarrow\right>$. Then when you ask “Is your little-arrow pointing up or down?”, the atom chooses $\left|\uparrow\right>$ with 50% probability and $\left|\downarrow\right>$ with 50%.

(By the way, if someone is thinking of erotic analogies, flash news: we physicists have already thought of all of them.)

But I told you that $\left|\uparrow\right> + \left|\downarrow\right>$ is more than just 50% up and 50% down. Let’s change direction. Now, instead of asking about up or down, we ask “is your little-arrow (spin) pointing rightwards or leftwards?” The atom answers “rightwards” with certainty. 100% probability!! So… that was the point!! It answered randomly when asked about up or down, because it was pointing to the right!! Is there any way to prepare the atom so it points always leftwards? Yeah, we write $\left|\uparrow\right> - \left|\downarrow\right>$. And the same happens if you ask the other way round: if you have $\left|\uparrow\right>$ and ask “are you pointing left or right?”, it will answer randomly. Wrong question, random answer.

But let’s come back to cats. We can go beyond Schrödinger and put two killer cats in the same box. They hate each other, and only one will survive. The quantum state can be written as

but… it could be the other way round! It might be

Classically, we would have 50% of each, But, in quantum mechanics, we can have the state

I put both signs because both are possible. It depends on the cat breed, I think.

But, I insist, that’s hard to do with cats. Do it at your own risk. With atoms, it’s a whole different story. We may prepare atoms such that their little-arrows (spins) point, for sure, in opposite directions. Let’s say that they are in the state

$\left| \uparrow\downarrow \right> - \left | \downarrow\uparrow \right>$

This state suffers from what we call entanglement. And very weird things happen to it. That was studied by Einstein and some of his buddies, called Podolsky and Rosen, in 1935 (also) (yeah, good year), when they showed that we could do the following. Take the box containing both atoms and split it in half, making sure that a single atom stays in each half-box. Now, take one of the boxes very far away. When you ask one of the atoms “are you pointing up or down?”, you don’t know what the answer is going to be, because it chooses randomly. If we do it with cats, you don’t know if the box you’ve kept contains the dead or the alive cat. But let’s assume that you get the answer “up”. Then we know what will the other atom reply when asked whether its arrow  points up or down. It will say “down”.

The surprise comes when you ask the atom: “is your arrow pointing left or right?” Its answer will also come randomly, 50% right and 50% left. I am not going to justify that, just believe me. But if you ask the same question to the other atom, no matter how far away it is, its answer will be the opposite!!! You may ask about any direction, and both atoms will give you opposite answers. The question that we may ponder is, of course… how does the second atom know what was measured on the first? Apparently, entangled atoms hold a bond that, like good loves and good hates, survives distance.

This is the first part of a lecture I delivered in the street in Madrid, as a part of the “Uni en la calle” program to protest the budget cuts in education and science in Spain, on March 9, 2013. I have delivered it also at Manuela Malasaña high school. Thanks to you all, guys!