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. Rate This Message: http://www.cryonet.org/cgi-bin/rate.cgi?msg=8608