[FOM] From Elena to Kieu through Toby Ord.
Laura Elena Morales Gro.
lemg at math.unam.mx
Mon Apr 12 15:46:11 EDT 2004
On Wed, 7 Apr 2004, Toby Ord wrote resending Kieu's words:
"In the long and critical writings of Elena and Lorenz about the
proposed algorithm, I can only identify the main objection that the
algorithm cannot be realised physically because of (thanks to) the
energy-time uncertainty principle."
Too bad. It is a pity that Kieu could only identify that. In this
objection he is mixing concepts. In one side there is the physical
implausibility of constructing any 'algorithm device' because of the
impossibility of finding forces in nature (or prepare them) to model
any and each (squared!) diophantine hamiltonian, and, on the other hand,
on top of that, there is the uncertainty principle. And the actual, patent,
inevitable, unavoidable need to measure absolutely zero energy value (in
order to have a solution) for the ground state energy. It is, of course,
much easier not to have one. Nobody will ever be able to detect zero
energy!! This is because of (thanks to) the uncertainty principle. This
must of us know without performing any experiment.
>I myself, with obvious vested interests in the problem, would love to
>find out why such proposal cannot be implemented physically.
Just do it. Construct a physical device, not a computer program, and
*measure*. Get a strict zero energy for the energy of whatever ground
state. Do it for one, the simplest case you choose. You'll find out you
cannot measure a strict zero (aside the 'impossibility' of constructing
such a device and reaching a desired ground state), the quantum world
forbids it. You cannot solve ANY equation, not to mention Kieu's procedure
is *not* an algorithm for Hilbert's problem.
>unfortunately, the reason given above by Elena and Lorenz is flawed
Permit me to disagree. The eigenvalues of the Hamiltonian operator are
energies, an elemental statement. Kieu must *measure* energies, cannot
get away with that. Cannot get away with measuring. And measurement of
energy implies uncertainty in time. The more precise the value is wanted,
the more the time required to measure it. Until reaching infinite time,
for a precise energy value, as required to solve an equation.
>because of a serious misunderstanding of the uncertainty principle as
>applied to the algorithm.
Since Kieu is so badly wrong in understanding what his "algorithm" means
and considering that the uncertainty principle can be 'read' differently
when applied to it, better would be he revises his fundamental ideas in
>On the one hand, we can understand the uncertainty principle as
There is only one way to understand it: The time needed to measure
a particular energy value is infinite. In order to have a solution
to any (squared) diophantine equation in particular, one has to measure
zero (not 0 + d0), exactly.
>That is all to the uncertainty principle (which is quite different from
>the other uncertainty principle concerning position and momentum).
Wow!! I can hardly believe this. I insist, there is only one uncertainty
principle, one that can be written in terms of energy and time, OR, the
equivalent, 'changing coordinates', in terms of position and momentum
(there are textbooks in which the uncertainy principle can be seen).
Heisenberg Uncertainty Principle (HUP) can be stated in different ways,
but there is only one principle, for instance, in terms of energy and time
and in terms of momentum and position, as I already said. Let's go into
this last one. Classically, i.e., in our macroscopic world, I can measure
these two quantities to infinite precision (more or less). There is really
no question where something is and what its momentum is.
In the Quantum Mechanical world, the idea that we can measure things
exactly breaks down. Let me state this notion more precisely. Suppose a
particle has momentum p and a position x. In a Quantum Mechanical world, I
would not be able to measure p and x precisely. There is an uncertainty
associated with each measurement, e.g., there is some dp and dx, which I
can never get rid of even in a perfect experiment!!!. This is due to the
fact that whenever I make a measurement, **I must disturb the system**.
(In order for me to know something is there, I must bump into it.) The
size of the uncertainties are not independent, they are related by
dp x dx > h / (2 x pi) = Planck's constant / ( 2 x pi )
The preceding is a statement of The Heisenberg Uncertainty Principle. So,
for example, if I measure x exactly, the uncertainty in p, dp, must be
infinite in order to keep the product constant. This uncertainty leads to
many strange things. For example, in a Quantum Mechanical world, I cannot
predict where a particle will be with 100 % certainty. I can only speak in
terms of probabilities. For example, I can say that an atom will be at
some location with a 99 % probability, but there will be a 1 % probability
it will be somewhere else (in fact, there will be a small but finite
probabilty that it will be found across the Universe). This is strange.
We do not know if this indeterminism is actually the way the Universe
works because the theory of Quantum Mechanics is probably incomplete. That
is, we do not know if the Universe actually behaves in a probabilistic
manner (there are many possible paths a particle can follow and the
observed path is chosen probabilistically) or if the Universe is
deterministic in the sense that I can predict the path a particle will
follow with 100 % certainty. Moreover, a consequence of the Quantum
Mechanical nature of the world, is that particles can appear in places
where they have no right to be (from an ordinary, common sense [classical]
point of view)! Today, even the greatest physicists admit to bafflement at
Heisenberg's mathematical non sequiturs and leaps of logic. "I have tried
several times to read [one of his early papers]," confesses the Nobel
laureate Steven Weinberg, "and although I think I understand quantum
mechanics, I have never understood Heisenberg's motivations for the
mathematical steps ..." No wonder why it is so difficult to understand
On the other side, HUP in quantum mechanics and atomic physics, dealing
with the energies of photons, is written in terms of frequencies, hf,
where h is Planck's constant. So a measurement of the energy corresponds
to a measurement of the frequency, and that, as can be seen, takes time.
In other words,
Df.Dt > ~ 1 or, in non-mathematical language:
(time taken to measure f) times (error in f) is about one or greater.
Mutliplying our previous inequality by h on both sides gives us
D(hf).Dt > ~ h
Which means: (uncertainty in energy) times (uncertainty in time) is
greater than about h. Corollary, to measure a particular energy value one
needs an infinite time.
For the present, for our case, to solve a diophantine equation, any
equation, we are concerned with the measurement of *energy*. And so, with
the time for the measurement; an infinite time to get a particular energy
value in analogy to an infinite momentum needed to locate it in a particular,
precise x position.
>Seen in this light, the energy-time uncertainty principle has nothing
>to do with our algorithm,
Incredible. It's amazing; was not this a quantum algorithm??? Quantum
inevitableness implies Uncertainty principle; Heisenberg's one and only
one. Cannot believe this...
>spread/variance of the energy measured. (The measuring time involved
>in the uncertainty principle is NOT the evolution time of our
Nobody is saying (implying) the contrary... No games here.
>obtain E_k = 0 then we can immediately stop the whole thing and declare
>that the Diophantine equation under consideration has a solution.
This is precisely the point. We come to terms. I'm glad. One (Kieu) can
never get (measure) a plain, straight, strict zero, such as the one needed
to assure there is a solution to the D-equation from a real, actual, experiment
due to the uncertainty principle. It will always be better (and much more
comfortable) to stop the whole thing any time! -including before performing
the experiment- and say there is no a solution. Moreover, Kieu already said
we cannot approach (as we all know) the absolute zero energy temperature
but through a limiting process admitting, even inadvertently, when comparing
to his process, the absolute zero energy value cannot be attained. Then, a
solution to the equations cannot be attained.
>thus, I am quite mystified by the remark by Elena that measurement
>would affect our process? In which way?)
Happy to oblige. In any quantum physical procedure one interacts,
disturbs, perturbs, bumps, into the system to find out (measure) about the
system's observables (be them position, momentum, energy,...) And the inherent
uncertainty in every measurement, accumulates.
>In summary, the uncertainty principle cannot be invoked here to rule
>out the possibility of some physical implementation of our algorithm.
I did not say HUP is invoked here to rule out the possibility of *some
physical implementation* (by 'physical implementation' I understand to
construct a physical device) of his algorithm. This is wrong. The physical
implementation, once practised, will speak out the impossiblity of ruling
the "algorithm" by itself. Prepare a physical device for it and measure
what you have to, and then, speak out. Note that physical implementation
does not mean doing numerical calculations. You do not need computers here.
>Up to now, we cannot find any physical principle that prohibits such a
There is no physical principle the prohibits the *physical implementation*
I did not say that. Simply said that there are no forces in nature to cope
with the (any) "algorithm"; one needs to implement the known forces. It is
only the real world that will prohibit *such a physical implementation*
and by this physical implementation I mean: to construct a real, physical
device to model the (squared!) diophantine equation -with whatever
coefficients- (aside measurements of anything). If you think the contrary,
I insist: implement a device for one, the simplest diophantine equation
and you would be improving the obvious. But, unfortunately, not even the
obvious can Kieu improve, not even one case Kieu'll be able to handle.
Because he (nobody) can measure zero energy value. Assuring as well
beforehand, there is no a device, nor will be, (one for each and any DE!
without any physical meaning) to solve Hilbert's problem.
I repeat it, just implement one case. Which does not mean prepare
computer programs to run in whatever sort of comupter. Do physics, not
numerical methods. No one needs computers (of any sort) to prepare
(construct) a physical device and measure whatever he likes to measure
(with the given inaccuracy and uncertainty).
>On the other hand, physical implementation aside, we have indicated
>that a simulation of the algorithm on classical computers is indeed also
Simulating 'algorithms' on any sort of computer is not dealing with the
quantum world. The simulation he talks about, for cases whose solution
is known, is not dealing with physics, only with its equations; the
unsurmountable technical dificulties in implementing (preparing) an
"algorithm" device, are not present in computer programs, not to mention
the uncertainty principle is not playing any role in mathematical numerical
methods ('on classical computers', as Kieu distinguishes). May I insist?, is
not dealing with the uncertainty principle, the unavoidable law in the quantum
world which forbids to measure zero energy value for whatever state or
sets of them (in case there is a solution).
Coming back from a couple of days away from the office, rushed
to reply something. But will retake this issue in due time.
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