[FOM] From Elena to Kieu through Toby Ord.

[Laura Elena Morales Gro.]

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
quantum physics.
  
>On the one hand, we can understand the uncertainty principle as 
>follows.  
  
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 
them.
  
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 
>algorithm.) 
  
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 
>physical implementation.  
  
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 
>possible

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.  
 
Best wishes,
LE