“Yes,” he said at last in rather a strained drawl. “I did have a question. Or rather, what I actually have is an Answer. I wanted to know what the Question was.”
Prak nodded sympathetically, and Arthur relaxed a little.
“It’s… well, it’s a long story,” he said, “but the Question I would like to know is the Ultimate Question of Life, the Universe and Everything. All we know is that the Answer is Forty-Two, which is a little aggravating.”
Prak nodded again.
“Forty-Two,” he said. “Yes, that’s right.”
He paused. Shadows of thought and memory crossed his face like the shadows of clouds crossing the land.
“I’m afraid,” he said at last, “that the Question and the Answer are mutually exclusive. Knowledge of one logically precludes knowledge of the other. It is impossible that both can ever be known about the same universe.”
From Douglas Adams’ Life, The Universe and Everything
Few numbers have such a geek cult following than the number 42, thanks to Douglas Adams’ science-fiction series The Hitchhiker’s Guide to the Galaxy. I guess one of the reasons is that behind the sophisticated humour of Douglas Adams lies an interesting philosophical question. Is there an “ultimate” scientific or philosophical question? If so, what is it? (Perhaps this one?).
Of course, the humour of Douglas Adams sarcastically dismisses the idea of an ultimate question as silly. And in a way it is. But last month, in a meeting at the Perimeter Institute for Theoretical Physics called “Reconstructing Quantum Theory“, Bill Wootters presented the closest I’ve ever seen to a candidate. Bill found a formalism in which quantum mechanics can be represented in a real vector space, as opposed to the usual formulation in terms of complex vectors spaces.
The upshot is that he needs an universal rebit to be able to reconstruct quantum mechanics with that formalism. A rebit is just like a qubit but instead of a superposition of states with complex coefficients, you have one with real coefficients.
As an aside for those who don’t know what a qubit is. A quantum bit it is the quantum extension of the concept of a bit—the unit of information, the amount of information one obtains when finding the answer to a yes/no question. To represent a bit, all you need is one thing in one of two possible states, which are usually denoted ‘1’ or ‘0’. A coin, for example. Heads indicates ‘1’, say, and tails indicates ‘0’. Given a previously agreed code, you can transmit information with a sequence of coins.
In quantum mechanics, for every two possible states of a system (say, ‘1’ and ‘0’), there are an infinity of possible states related to these two states by what are called complex superpositions. Those are mathematical structures that can be represented in the form c1 ‘1’ + c2 ‘2’. Except that in quantum mechanics a state like ‘1’ is represented by the symbol |1>, a notation introduced by Dirac. So for example we could define states like |+> = |1> + |2>, |-> = |1> – |2> or |R> = |1> + i |0>, etc. However, in any complete measurement of a qubit, only two outcomes are possible; a qubit can give you one bit of information, but one bit of information about an infinitude of mutually exclusive (or what Bohr called complementary) questions.
A rebit, or real bit, is an intermediate case where the coefficients c1 and c2 are allowed to be only real numbers.
This rebit is called universal because—and this is where things get interesting—it is in some sense shared by all other systems in the universe, according to the model of Wootters. It interacts nonlocally to all other rebits in the theory. In the model it is also necessary (so as to reconstruct quantum theory) that we are unable to determine the state of this rebit, even though it interacts with everything in the universe. This raises some interesting questions. Is the state of the rebit unknowable in practice or in principle? Wootters showed some interesting models for mechanisms responsible for this epistemic censorship. In one of these, the rebit is randomised by a very rapid rotation of its direction, much faster than we can experimentally detect.
But thinking about it, the rebit itself represents the answer to one question. One binary question, the answer of which is relevant to all systems in the universe. An ultimate universal question.
After a moment of suspense, Bill advanced his hunch: the ultimate question associated to the rebit relates to the direction of time. (Chris Fuchs, who was also attending the meeting, exclaimed disappointedly: “Direction of time? I was thinking of something more like the triumph of good over evil!”) However, this interpretation would seem to pose a problem to his model of a rotating rebit. In which “time” would the rebit be rotating, if it itself “encodes the direction of time”? Furthermore, by the principle of superposition there wouldn’t be just one question, but a continuum of complementary binary questions, the answers of which would be all the orthogonal pairs of real superpositions of |1> and |0>. Would those correspond to other possible pairs of directions for time?
I am very curious to read Bill’s paper about this work to understand his model in more detail. But whatever the question associated to the universal bit is, the answer can’t be 42 after all. It can only be yes or no. Or any real superposition thereof.