X-Message-Number: 8608
From: Andre Robatino <>
Subject: Re: CryoNet #8605 - #8607
Date: Thu, 18 Sep 97 11:45:31 EDT

> Message #8607
> Date: Wed, 17 Sep 1997 21:50:16 -0700 (PDT)
> From: John K Clark <>
> Subject: Digital Shakespeare
> 
<snip>

> I #8601  Andre Robatino <>  On Tue, 16 Sep 97 Wrote:
>             
> 
>         >>Me:        

>         >>any set of measurements you make does not correspond to a UNIQUE
>         >>quantum wave function.
> 
>         >One can determine it uniquely up to a phase factor. 
>             
> 
> I'm not sure how much we really disagree in all this, except in our opinion 
> of the importance of the phase factor. It seems pretty important to me, 
> after all, Schrodinger's equation defines a wave in phase space and the 
> number  of dimensions is 3 times the number of particles it works with, 
> hence the computational complexity of making such calculations for even a

> small number of things.  When 2 particles collide so does their quantum wave
> function and when they separate so does the amplitude of their quantum wave  

> function, but their phases do not separate, they remain entangled and produce
> interference effects even if the particles move a billion light years away.
> 
> Unfortunately we can not know the signs of the phase factors because of the  
> nature of complex numbers, measurement just won't tell us. Yes, human beings 
> can and do adopt an arbitrary phase convention that is consistent with 
> experiment, but there is nothing unique about it, other conventions would 
> work just as well. This arbitrary quality is why I maintain that the quantum
> wave function is a calculating device, like longitude and latitude.

  Just to be completely clear about it, when I say a global phase factor I
mean two states that differ only by |psi'> = c|psi>, where c is a complex
_constant_ with |c| = 1.  If the states are expressed as wave functions, then
multiplying one w.f. by a function c(x) depending on x doesn't qualify, even
if |c(x)| = 1 everywhere.  Note that this kind of local phase factor is only
a phase factor with respect to _one particular_ representation, while it makes
sense to talk about a global phase factor regardless of representation.
  I'm _not_ saying that if a system is in an unknown state |psi>, that it's
possible to get it into a known state, without changing the state.  It's
possible to pick a complete set of measurements (not the set of all measure-
ments, just a maximal compatible set) and by doing each of them, get the
system into a known state |psi'>; however, this will in general be different
from |psi>, and it won't be possible to infer what |psi> was, since the act
of measurement is a many-to-one map of states, hence irreversible.  But once
one knows that the system is in state |psi'>, one can make probabilistic
statements that are different from those for any other state (up to global
p.p.).  Here's a simple measurement showing this: for a state |psi>, let
M_{|psi>} be a measurement with 2 possible outcomes, outcome 1 corresponding
to the 1-dimensional subspace H_1 of the Hilbert space generated by |psi>,
and outcome 2 corresponding to its orthogonal complement H_2.  For any state
|psi'>, the probability of outcome 1 for this measurement is |<psi|psi'>|^2,
so for any state c|psi> with |c| = 1, it's 1; for any other state, it's less
than 1.
  Also, the (slight) arbitrariness in the way one represents |psi> is not
unique to QM: in classical mechanics, the coordinates one uses to describe a
system are known only up to an arbitrary translation, rotation, and uniform
velocity.  The phase factor is analogous.

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