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 t_{1} to t_{2}
is actually reversible at t_{2}, then that must mean there is no evidence
at t_{2} of its evolution. And
if you actually reverse the system to how it was at t_{1}, then there can
be no evidence of (and thus no scientific fact or meaning about) its having
evolved or done anything from t_{1} to t_{2}. 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 t_{2}. 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 t_{2} if it is set up at earlier time t_{1}
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 t_{2} to its earlier state Ψ(t_{1}), then
there can never be scientific evidence that it was ever in state Ψ(t_{2}). 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(t_{1}), you could presumably let them evolve to state Ψ(t_{2}),
and then measure all of them except one, which you would then reverse to state Ψ(t_{1}). If the measured systems yield statistics that
are consistent with the Born rule applied to state Ψ(t_{2}), then you
might logically infer that the system you reversed actually “was” in state Ψ(t_{2})
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 t*_{1} of
a radioactive sample correlates to the integrity of a glass vial of poison gas,
and the vial’s integrity correlates at time t_{2} 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 t_{1},
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 t_{1} (to the cat, the external
observer, or anyone else) about the first correlation event nor a fact at t_{2} 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.