Kalamidas Refuted

The three refuters of Demetrios Kalamidas

Recently Demetrios Kalamidas proposed a clever Bob-to-Alice FTL communication scheme involving a pair of path-entangled photons that passed peer review and was published in an esteemed American optics journal.

DK's scheme involved erasing Bob's which-path information by adding a number-uncertain (weak coherent) state to Bob's paths that produced an essential ambiguity as to which path the photon took. Kalamidas claimed that when this ambiguity was present at Bob's detectors, interference fringes would appear at Alice's distant detectors, without the need for a coincidence signal triggered by Bob's detector outputs. Kalamidas's scheme was not exact but involved two plausible approximations.

Several kinds of refutation were mounted to challenge the Kalamidas claim. Nick Herbert constructed a Kalamidas-inspired ambiguation scheme (2 MB pdf) involving Gray Light (x|0> + y|1>) instead of coherent states |A> which involved NO APPROXIMATIONS. Simple calculation of the EXACT Gray-Light scheme produced no FTL signals.

A different refutation by GianCarlo Ghirardi showed that if all photon operations are unitary, then an EXACT calculation shows that no FTL signaling is possible. However Ghirardi's proof is quite general and does not come to grips with the actual details of the Kalamidas scheme.

Next John Howell at University of Rochester examined a version of the Kalamidas scheme that involves adding a coherent state |A> to a single-photon Fock state |1> to create a displaced Fock state (DFS). Howell showed that an EXACT calculation of this DFS scheme leads to no FTL signaling.

However, Kalamidas's proposal is subtly different from the scheme that Howell refuted. Kalamidas's proposal adds the Fock state |1> and coherent state |A> in reverse order from Howell to produce a photon-added coherent state (PACS) rather than a DFS. Because these operations (photon addition) and (coherent state addition) do not commute, the PACS state is not the same as a DFS state.

So now a question arises: what is the correct output state (DFS or PACS) that emerges from a beam splitter when a coherent state |A> enters at one port and a Fock state |1> enters at the other?

This question was answered by Martin Suda at Austrian Institute of Technology who showed that BOTH OUTPUTS ARE CORRECT. The output of a beam splitter can be correctly described as an entanglement of DFS or of PACS states--a result I have dubbed the Martin Suda Equality. Even tho the relevant Boson operators do not commute, the Stokes relation for beam splitters "magically cancels" out the commutator and leads to this remarkable result.

So Kalamidas is correct in using PACS in his FTL proposal and Howell is also correct in using DCS in his disproof. Kalamidas uses an approximate PACS scheme; Howell's refutation uses an EXACT DCS scheme. Not yet a perfect match between proposal and refutation.

Then Christopher Gerry of Lehman College, CUNY, enters the fray with an EXACT refutation based on PACS states, concluding that "Clearly, if the exact calculation shows no interference, but the approximate calculation does, there is something wrong with the approximate calculation."

That would seem to have settled the matter, Gerry's calculation uses the same PACS states as Kalamidas and shows no FTL signaling. But, at least in my mind, one stumbling block remained: if Kalamidas's approximate calculation is wrong, then where is his mistake? Many of us have searched for this mistake (including myself, Howell, Suda and Gerry) without success.

Now I believe I have found it--not a mistake, but a way of looking at the Kalamidas proposal that exposes its crucial flaw.

Suppose (since none of us can refute it), that Kalamidas's calculation is correct. He has found an approximation scheme that yields true FTL signaling. What this means is that, if you coincidence-trigger on all Bob's terms of the Kalamidas approximation, you will generate interference in Alice's detectors.

But, even assuming Kalamidas's calculation is correct, he still loses the game, because for his scheme to work (since it is only approximate) it necessarily requires coincidence triggering. Because we know from the results of Ghirardi, Howell and Gerry, that the EXACT calculations show no sign of FTL signaling.

Therefore, using his own criterion (no coincidence triggering), the Kalamidas scheme must fail.

Besides the excitement generated by his clever new FTL signaling scheme, and a renewed excuse to delve into the details of Boson calculus, the most notable outcome of the Kalamidas affair seems to have been the discovery of the surprising "Martin Suda Equality".