I’m working on a project/presentation on whether scalable quantum computers are possible. A quantum circuit can be simplified as application of a unitary matrix to an initial state of qubits. Unitary matrices represent reversible basis shifts, which means that the computation must be shielded from irreversible decoherence events (or subject to quantum error correction to the extent possible) until the purposeful measurement of qubits at the end of the computation.
The word “reversibility” has come up a lot in my reading. Essentially, the idea is that physical laws seem, for the most part, to be the same whether time is run forward or backward. For example, if you were shown a video of a planet orbiting some distant star, you would not be able to tell whether the video was being played forward or in reverse. Yet we experience time to move in a particular direction (namely, the future). This has led to a centuries-long debate about the “arrow of time”: whether physical laws are reversible or whether there is actually some direction built into the fabric of the physical world.
It’s time to nip it in the bud: the physical world is not time-reversible.
As an example of a typical argument for classical reversibility, imagine dropping a porcelain teapot on a wooden floor. Of course, it will irreparably break into probably hundreds of pieces. “In principle,” they say, “if you know the positions and trajectories of all those pieces, you can then apply forces that will completely reverse the process, causing the pieces to recombine to the original teapot.”
But that’s crap. We already know, thanks to the Heisenberg Uncertainty Principle (“HUP”), that the pieces don’t have positions and momenta to infinite precision. That alone is enough to guarantee that any attempt to apply time-reversed forces to the pieces will, thanks to chaos, fail to result in a perfect recombination of the pieces. (One of my favorite papers discusses how even “gargantuan” black holes become chaotic over time, thanks to HUP.) This problem is only compounded by the fact that any measurement of the positions and/or momenta of the pieces will inevitably change their trajectories very slightly also.
So quantum mechanics guarantees that the classical world is not and cannot be time-reversible. But I’ve recently realized that the notion of time-reversibility in quantum mechanics is not only false… it’s actually a contradiction. In Section F of this post, I had already realized and pointed out that there is something logically contradictory about the notion of Schrodinger’s Cat (“SC”) or Wigner’s Friend (“WF”). (I copied the most relevant section of that post below.)
The idea is simple. To actually create SC, which is a macroscopic superposition state, the cat (and its health) has to correlate to a vial of poison (and whether it is broken), which has to correlate to some quantum event. These correlations are colloquially called “measurements.” But to prove (or experimentally show) that the cat is in a macroscopic superposition state, you have to do an interference experiment that undoes the correlations. In other words, to show that the measurements are reversible (as assumed by the universality of QM), you have to reverse the measurements to the extent that there is no evidence anywhere in the universe (including the cat’s own clock) that the measurements happened.
Remember, scientific inquiry depends on evidence. We start by assuming that SC is created in some experiment. But then the only way to show that SC is created is… to show that it was not created. The very evidence we scientifically rely upon to assert that SC exists must not exist. Proving SC exists requires proving that it does not exist. This is gibberish. (David Deutsch tried to explain away this problem in this paper but failed. Igor Salom correctly pointed out in this paper that any attempt to correlate the happening of a measurement inside the otherwise “isolated” SC container will inevitably correlate to the result of that measurement, in which case the measurement event will be irreversible.)
Whether discussing WF, SC, quantum computers, etc., if the evolution of a quantum mechanical system from time t1 to t2 is actually reversible at t2, then that must mean there is no evidence at t2 of its evolution. And if you actually reverse the system to how it was at t1, then there can be no evidence of (and thus no scientific fact or meaning about) its having evolved or done anything from t1 to t2. There can be no evidence anywhere, including as “experienced” by the system itself, because even by its own internal clock, there was no evolution to t2. For a reversible system that is actually reversed, there just is no scientific fact about its having had any evolution. And for a reversible system that is actually measured, so that information exists in the universe about its state (correlations, etc.), then that system is no longer reversible.
Finally, I want to mention that even for a quantum mechanically reversible system, in order to reverse it, you must have already set up the system to be reversible. For example, if you want an exploding bomb to be reversible, you can’t let the explosion happen and then go hunting for all the fragments to measure their trajectories, etc. Setting aside the classical problems I mentioned earlier (e.g., by measuring the particles you change their positions/momenta), the problem quantum mechanically is that once the happening of the event correlates to some particle that you don’t already have full control over, it’s too late… evidence now exists. If a quantum superposition did exist at an earlier time, it no longer does because it has now, thanks to the decoherence event, irreversibly reduced to a definite state.
This is an error that Scott Aaronson seems to make. Aaronson, one of the most brilliant people ever to discuss the relationship between physics and consciousness (such as in this paper), makes a compelling argument here (also here) that consciousness might be related to irreversible decoherence. However, he seems to think of quantum mechanical reversibility as something that depends on a future event, like whether we take the time to search for all the records of an event and then reverse them. For example, he posits that the irreversible decoherence related to one’s consciousness means that “the records of what you did are now heading toward our de Sitter horizon at the speed of light, and for that reason alone – even if for no others – you can’t put Humpty Dumpty back together again.”
But that’s wrong. The reason you can’t put Humpty Dumpty back together again is not because evidence-carrying photons are streaming away… it’s because the fall of Humpty Dumpty was not set up before his fall to be reversible. So a system described by wave function Ψ(t) can only be reversible at t2 if it is set up at earlier time t1 to be reversible (which means, at least in part, isolating it from decoherence sources). But if you actually do succeed in reversing it at time t2 to its earlier state Ψ(t1), then there can never be scientific evidence that it was ever in state Ψ(t2). Therefore, as a scientific matter, reversibility is a contradiction because the only way to show that a system is reversible is to show that it did not do something that it did.
Of course, assuming you could prepare lots of systems in identical states Ψ(t1), you could presumably let them evolve to state Ψ(t2), and then measure all of them except one, which you would then reverse to state Ψ(t1). If the measured systems yield statistics that are consistent with the Born rule applied to state Ψ(t2), then you might logically infer that the system you reversed actually “was” in state Ψ(t2) at some point. However, there’s a real problem, especially with macroscopic objects, with producing “identical” states, as I discuss here. It is simply not physically possible, “in principle” or not, to make an identical copy of a cat. Therefore, any attempt to scientifically show that SC exists requires showing that it does not exist.
Physical reversibility is a contradiction.
From Section F of this post:
Consider this statement:
Statement Cat: “The measurement at time t1 of a radioactive sample correlates to the integrity of a glass vial of poison gas, and the vial’s integrity correlates at time t2 to the survival of the cat.”
Let’s assume this statement is true; it is a fact; it has meaning. A collapse theory of QM has no problem with it – at time t1, the radioactive sample either does or does not decay, ultimately causing the cat to either live or die. According to U [the "universality" assumption that quantum states always evolve linearly and reversibly], however, this evolution leads to a superposition in which cat state |dead> is correlated to one term and |alive> is correlated to another. Such an interpretation is philosophically baffling, leading countless students and scholars wondering how it might feel to be the cat or, more appropriately, Wigner’s Friend. Yet no matter how baffling it seems, proponents of U simply assert that a SC superposition state is possible because, while technologically difficult, it can be demonstrated with an appropriate interference experiment. However, as I pointed out above, such an experiment will, via the choice of an appropriate measurement basis that can demonstrate interference effects, necessarily reverse the evolution of correlations in the system so that there is no fact at t1 (to the cat, the external observer, or anyone else) about the first correlation event nor a fact at t2 about the second correlation event. In other words, to show that U is true (or, rather, that the QM wave state evolves linearly in systems at least as large as a cat), all that needs to be done is to make the original statement false:
1) Statement Cat is true;
2) U is true;
3) To show U true, Statement Cat must be shown false.
4) Therefore, U cannot be shown true.
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