🚀 go-pugleaf

An analysis of the physical description of Sarfatti's spin-flip mechanism.\
I know this is redundant and obvious to almost anyone who has taken
the usual first year quantum courses, but I just had to...

Abstract: I show that the proposed description of the spin-
flip device can either be local or unitary, but assuming both
leads to a contradiction.  Hence, if the description is the
"correct" quantum mechanical one it is the nonlocal nature of
the spin-flip which gives rise to FTL communication, NOT
quantum entanglement.

Evolution in quantum mechanics is given by unitary
transformations on the hilbert space of vectors associated
with the possible state of the system.  Consider S as such a
unitary operator.  Then, for any vector |psi> in the hilbert
space, the transformation corresponding to the system is
given by
     |chi> = S |psi>
Since S is unitary, its inverse is given by its adjoint
(hermetian conjugate):
      inverse(S) = adjoint(S), adjoint(S) S =I (I is the
identity operator), and the inner product is preserved
<chi|chi> = <psi|adjoint(S) S |psi> = <psi|psi>.
If |psi> is initially normalized (i.e. <psi|psi> =1), then
the final state is normalized (<chi|chi> =1).

On the tensor product of two hilbert spaces, if the evolution
of the two systems is independent, then the transformation
operator factors into the tensor product of transformation
operators which act only on the individual subspaces.  In
this case the hamiltonian for the composite system can be
written as
H = H1 + H2 = h1xi2 +i1xh2,
where x means tensor product, hn (n=1,2) is the hamiltonian
which acts only on system n and in is the identity operator
for the subspace n, then the transformation operator can
always be written as
S = s1xs2,
where sn are unitary transformations on the subspace n.

In the proposed FTL communication scheme, there are two
subsystems, both of which are single photon states spanned by
standard polarizations bases (|hn> and |vn>).

The transition in question is
(|v1>|v2> +|h1>|h2>)/sqrt2 -> (|v1>e^(i phi) +
|h1>)|v2>/sqrt2
It is elementary to show that this is norm preserving in that
the initial state has the same norm as the final state (and
is equal to one) so that probability is conserved.  However,
the two subsystems are (presumably) evolving independently so
that the transformation operator should be factorable:
S = s1xs2.
In addition, subsystem 1 is undisturbed, so that S1 = i1, the
identity operator on subsystem 1.  (System 1 is only to be
measured after the evolution of the composite system, i.e. the action
of the spin-flip.)
S|psi> = S1xS2 |psi> = i1xS2 |psi>
= (|v1>s2|v2> +|h1>s2|h2>)/sqrt2 -> (|v1>e^(i phi) +
|h1>)|v2>/sqrt2.Since

We then have sufficient information to completely construct
S2 since the |v1> and |h1> terms are linearly independent:
<v2|s2|v2> = e^(i phi)
<v2|s2|h2> = 1
<h2|s2|v2> = 0
<h2|s2|h2> = 0

Representing this basis in terms of column vectors
|v2> = 1
       0


|h2> = 0
       1

explicitly constructing the 2 by 2 matrix S2, we have
S2 = e^(i phi)   1
       0         0

adjoint(S2) S2 = 2  0
                 0  0
clearly, S2 is not unitary, nor is it proportional to a
unitary operator.

This is not to say that a unitary operator on the product
space which produces the proposed transition cannot be
constructed.  We have simply shown that such a unitary
operator cannot be local.  The apparent FTL effects are due
not to the entanglement of the initial conditions |psi>, but
are in fact due to the (unphysical) nonlocal description of
the spin-flip device on the entangled state.