scott_aaronson's picture
David J. Bruton Centennial Professor of Computer Science, University of Texas at Austin; Author, Quantum Computing Since Democritus

In physics, math, and computer science, the state of a system is an encapsulation of all the information you'd ever need to predict what it will do, or at least its probabilities to do one thing versus another, in response to any possible prodding of it. In a sense, then, the state is a system’s “hidden reality,” which determines its behavior beneath surface appearances. But in another sense, there’s nothing hidden about a state—for any part of the state that never mattered for observations could be sliced off with Occam’s Razor, to yield a simpler and better description.

When put that way, the notion of “state” seems obvious. So then why did Einstein, Turing, and others struggle for years with the notion, along the way to some of humankind’s hardest-won intellectual triumphs?

Consider a few puzzles:

  1. To add two numbers, a computer clearly needs an adding unit, with instructions for how to add. But then it also needs instructions for how to interpret the instructions. And it needs instructions for interpreting those instructions, and so on ad infinitum. We conclude that adding numbers is impossible by any finite machine.
  2. According to modern ideas about quantum gravity, space might not be fundamental, but rather emergent from networks of qubits describing degrees of freedom at the Planck scale. I was once asked: if the universe is a network of qubits, then where are these qubits? Isn’t it meaningless, for example, for two qubits to be “neighbors,” if there’s no pre-existing space for the qubits to be neighboring in?
  3. According to special relativity, nothing can travel faster than light. But suppose I flip a coin, write the outcome in two identical envelopes, then put one envelope on earth and the other on Pluto. I open the envelope on earth. Instantaneously, I’ve changed the state of the envelope on Pluto, from “equally likely to say heads or tails” to “definitely heads” or “definitely tails”! (This is normally phrased in terms of quantum entanglement, but as we see, there’s even a puzzle classically.)

The puzzle about the computer is a stand-in for countless debates I’ve gotten into with non-scientist intellectuals. The resolution, I think, is to specify a state for the computer, involving the numbers to be added (encoded, say, in binary), and a finite control unit that moves across the digits adding and carrying, governed by Boolean logic operations, and ultimately by the laws of physics.  It might be asked: what underlies the laws of physics themselves? And whatever the answer, what underlies that? But those are questions for us. In the meantime, the computer works; everything it needs is contained in its state.

This question about the qubits is a cousin of many others: for example, if the universe is expanding, then what’s it expanding into? These aren’t necessarily bad questions. But from a scientific standpoint, one is perfectly justified to respond: “you’re proposing we tack something new onto the state of the world, such as a second space for ‘our’ space to live in or expand into. So would this second space make a difference to observation? If it never would, why not cut it out?”

The question about the envelopes can be resolved by noticing that your decision on earth, to open your envelope or not, doesn’t affect the probability distribution over envelope contents that would be perceived by an observer on Pluto. One can prove a theorem stating that an analogous fact holds even in the quantum case, and even if there’s quantum entanglement between earth and Pluto: nothing you choose to do here changes the local quantum state (the so-called density matrix) over there. This is why, contrary to Einstein’s worries, quantum mechanics is consistent with special relativity.

The central insight here—of equal importance to relativity, quantum mechanics, gauge theory, cryptography, artificial intelligence, and probably 500 other fields—could be summarized as “a difference that makes no difference is not a difference at all.” This slogan might remind some readers of the early 20th-century doctrine of logical positivism, or of Popper’s insistence that a theory that never ventures any falsifiable prediction is unscientific. For our purposes, though, there’s no need to venture into the complicated debates about what exactly the positivists or Popper got right or wrong (or whether positivism is itself positivistic, or falsifiability falsifiable).

It suffices to concentrate on a simpler lesson: that yes, there’s a real world, external to our sense impressions, but we don’t get to dictate from our armchairs what its state consists of. Our job is to craft an ontology around our best scientific theories, rather than the opposite. That is, our conception of “what’s actually out there” always has to be open to revision, both to include new things that we’ve discovered can be distinguished by observation and to exclude things that we’ve realized can’t be.

Some people seem to find it impoverishing to restrict their ontology to the state, to that which suffices to explain observations. But consider the alternatives. Charlatans, racists, and bigots of every persuasion are constantly urging us to look beyond a system’s state to its hidden essences, to make distinctions where none are called for.

Lack of clarity about the notion of “state” is even behind many confusions over free will.  Many people stress the fact that, according to physics, your future choices are “determined” by the current state of the universe.  But this ignores the fact that, regardless of what physics had to say on the subject, the universe’s current state could always be defined in such a way that it secretly determined future choices—and indeed, that’s exactly what so-called hidden-variable interpretations of quantum mechanics, such as Bohmian mechanics, do.  To me, this makes “determination” an almost vacuous concept in these discussions, and actual predictability much more important.

State is my choice for a scientific concept that should be more widely known, because buried inside it, I think, is the whole scientific worldview.