Thanks Nick. What would Santa do without you in his workshop? ;-)
Remember I have been stressing the relevance of Glauber coherent states.
They are obviously distinguishably non-orthogonal & over-complete.
Congratulations again on your clever FTL-signaling scheme.
I am busy constructing (on my white board) your thought experiment
using my own notation.
First: I hope you do not mind the acronym I have chosen for this project = KISS
KISS = Kalamidas's Instant Signaling Scheme.
Second: It has become conventional to imagine these signals sent between Alice and Bob.
So everything on left side should be labeled "A" and on the right side "B".
Since A and B photons are delivered into two (entangled) modes, I have chosen to label these modes
U and D (for Up and Down). In this labeling convention the basic entangled state vector |ES> becomes
|ES> = |1>(AU)|0>(AD)|1>(BU) |0>(BD) + |0>(AU)|1>(AD)|0>(BU)|1>(BD)
or (dropping the subscripts)
|ES> = |1>|0>|1>|0> + |0>|1>|0>|1>
which is essentially your (unnormalized) EQ 1.
Also it is conventional for beam-splitter modes to be labeled 1, 2, 3, 4
where 1 and 2 are inputs and 3 and 4 are outputs.
So for my thought experiment I will label the 4 modes of Bob's two beam splitters U and D
as |U1>, |U2>, |U3>, |U4> and |D1>, |D2>, |D3> and |D4> with a similar convention for the
50/50 beamsplitter encountered by Alice's photons.
I like your clever use of coherent states to muddle the which-way question. But instead of inputting coherent states
at Bob's beamsplitters U and D, I will be inputting the coherent XYZ states |BU> and |BD>
where |BU> = x|0> + y|1> + z|2>
and |BD> has a similar definition.
These are truncated coherent states sufficient to produce the ambiguities you claim will lead to coincidence-less, Bob-controllable
interference in Alice's 50/50 beamsplitter and are easier to calculate than the infinite sums of real coherent states.
Thanks for the opportunity to return to the algebra of few photons on an asymmetric beam splitter. And for the chance to reformulate
your clever KISS experiment in terms that make sense to me.
I am always looking for (high quality) work to do.
And your KISS proposal is both of high quality and within my modest abilities for calculating quantum outcomes.
If this paper proves correct in the lab, it vindicates my struggle since 1960 or so that MIT Physics Professor David Kaiser has recorded for history in his book "How the Hippies Saved Physics." This will be a science-technology revolution worth billions if not trillions of dollars for visionary venture capitalists.
"Proposal for a feasible quantum-optical experiment to test the validity of the no-signaling theorem
Demetrios A. Kalamidas
Received November 29, 2012; accepted January 17, 2013;
posted January 24, 2013 (Doc. ID 180742)
Motivated by a proposal from [Phys. Scr. T76, 57 (1998)] for superluminal signaling and inspired by an experiment
from [Phys. Rev. Lett. 67, 318 (1991)] showing interference effects within multiparticle entanglement without
coincidence detection, we propose a feasible quantum-optical experiment that purports to manifest the capacity
for superluminal transfer of information between distant parties." © 2013 Optical Society of America
OCIS codes: 270.4180, 270.5290, 270.5565, 270.5585.