TKO TKOED

Demetrios Kalamidas, creator of the TKO superluminal signaling scheme.

Page 1 of Kalamidas's TKO FTL proposal

Page 5 of Kalamidas's TKO FTL proposal

A few days ago I received 6 or 7 hand-written notes from Demetrios Kalamidas outlining a new faster-than-light (FTL) signaling scheme that he had devised. Three years ago, Kalamidas had proposed (and even published in a major optics journal!) an FTL scheme which was so devilishly clever that it occupied the time of several smart physicists before his scheme (which I irreverently called KISS for Kalamidas's Instant Signaling Scheme) was finally wrestled to the ground and definitively defeated.

Not one to give up so easily, Kalamidas has now come up with another FTL scheme (which I christened TKO, for The Kalamidas Option). "Refute this one, Nick," he challenged.

Well, before I could refute TKO, I had to understand it. So I made a little sketch, which Kalamidas agreed captured the gist of his new scheme.

Kalamidas's TKO Superluminal Signaling Scheme

In the TKO scheme a single photon |1> is divided by a beam splitter into two equal paths |a> and |b> and recombined at ALICE's beam splitter into two other paths |c> and |d>. This simple photon divide-and-recombine scheme is called a Mach-Zehnder (MZ) interferometer which has found numerous uses in the field of optical physics. Before the |b> photon enters her beam recombiner, ALICE has the option to add a phase Q to the |b> beam, altho in the TKO scheme ALICE does not exercise this option.

In BOB's |a> beam is placed a photon up-conversion crystal (symbolized by the blue circle labeled XTL which, with 100% efficiency, converts two incident photons to one photon with twice the energy. This double-energy photon (which Kalamidas calls OMEGA) exits the scene along path |H>.

In the simple MZ configuration the up-conversion crystal XTL is never triggered, since there is never more than one photon |a> in BOB's beam.

But then BOB adds a second pulsed source of light |G> that is timed to strike the XTL at the same time as each of the |a> photons. If |G> were a simple pulsed source of single photons |1>, then this XTL would remove every |a> photon from BOB's beam by transforming |1> + |a> into an OMEGA. No |a> photons would ever be sent to ALICE who would receive only |b> photons. No interference (between photon path |a> and path |b> would ever occur. The resulting situation would be utterly boring.

So instead of letting |G> be a boring source of single photons, Kalamidas makes |G> a more interesting source of "Gray Light" which is a coherent superposition of the zero-photon vacuum state |0> and the single photon state |1>:

|G> = X |0> + Y |1>         EQ 1

where X^2 + Y^2 = 1

So now whenever the Gray Light contains a single photon |1> (which happens with probability Y^2), this photon combines with BOB's photon |a> and is removed from the |B> beam by the up-converting XTL in the form of a doubled-frequency OMEGA photon.

Whenever the Gray Light contains the zero-photon vacuum state |0> which happens with probability X^2, then photon |a> remains unmolested and travels to ALICE's beam combiner where it's mixed with ALICE's photon |b>.

Given this physical setup, how does ALICE send a signal to BOB?  In Kalamidas's scheme, ALICE has two options which I call YES and NO. In the YES option she sets her beam combiner to 50/50 and maximally mixes photons |a> and |b> into her outputs |c> and |d>.

Choosing the NO option, ALICE removes her beam combiner (or equivalently sets its transparency to 100%) so the |a> and |b> photons do not mix. Photon |a> goes directly into counter |c> and photon |b> goes directly into counter |d>.

ALICE's choice amounts to a decision whether to mix photons |a> and |b> (YES) or not to mix the photons (NO). If BOB observes any difference in his results when ALICE switches between YES and NO, then this difference can be used to send a signal faster than light.

If BOB's experience is always the same, then no signaling occurs.

I looked at TKO and came up with an immediate refutation.

Let's suppose that the Gray Light is equally divided into NOTHING (the vacuum state |0>) and SOMETHING (the one-photon state |1>). This means that half the time there is a Gray-Light photon hitting the crystal and half the time there is not.

1. Whenever there is not a Gray Light photon, BOB will see nothing = 50% of the time.

2. Whenever there is a Gray Light photon but no |a> photon, BOB will see 1 photon. This happens 25% of the time since the photon takes path |b> 1/2 the time and Gray light emits a photon 1/2 the time: 1/2 x 1/2 = 1/4 = 25%

3. By the same reasoning whenever there is a Gray Light photon that meets an |a> photon, BOB will see nothing, because the Gray Light photon will be converted into an OMEGA.

By adding up all possibilities we see that 50% of the time BOB sees NOTHING, 25% of the time he sees an OMEGA and 25% of the time he sees ONE PHOTON.

Furthermore this 50/25/25 behavior is completely independent of any action on ALICE's part. Therefore no signaling ever takes place. 

I considered this refutation particularly simple and obvious. So I sent my result to Kalamidas.

"No, no, no, no, Nick! You did not even look at what I have written (his seven pages of hand-inscribed notes). Your calculation is much too simple. IT IGNORES ALL THE PHASES!"

"Phases?"

"Yes, Nick, phases." BOB does not just get either SOMETHING or NOTHING at his detectors, He gets SOMETHING and NOTHING between which there exists a definite phase relationship. And that phase relationship depends on ALICE's choice of YES and NO."

"Phases, Demetri? Phases between SOMETHING and NOTHING?"

"Yes, Nick, phases between SOMETHING and NOTHING. That's what makes Gray Light so special. Gray Light's not a mere incoherent mixture of SOMETHING and NOTHING. Gray Light is a coherent superposition (like Schrodinger's Cat) -- a superposition of two possibilities that are linked by a definite phase relationship, a relationship between two objects that only makes sense in quantum mechanics. "

"Yeah, buddy. I know about phases. Phases are the meat and potatoes of every quantum calculation. But in my way of thinking, phases only exist between actual possibilities of something happening. How can NOTHING possibly possess a phase?"

"It can, it does. And that fact is the secret ingredient of my TKO scheme. Check it out, dude. If you include phases in your calculation for what BOB sees (including the phase of the vacuum state |0>) you'll discover (just like I did) that BOB sees something when ALICE makes her YES choice and BOB sees something different when ALICE makes her NO choice. I don't have to tell you, Nick, that if my result is correct, then FTL signaling is a done deed, hence signaling backwards in time, hence breakdown in causality and hence AN END TO THE WORLD AS WE KNOW IT!"

"Ummph! I gotta sit down and think a bit about whether NOTHING can possess a phase. Let me get back to you, man."

So Nick gets out his optics books and several cups of coffee and generates a little essay called: "Can NOTHING have a phase? And he decides YES. So the Kalamidas TKO proposal must be taken seriously.

Paying attention to vacuum phases, Nick calculates what BOB will see for ALICE's two choices of 1. inserting a beam splitter -- a choice I call YES. and 2. taking out her beam splitter and observing the |a> and |b> photons separately -- a choice I've called NO.

And here are the results: here is what BOB sees when ALICE makes her two choices:

YES ===> [ |G(1)> ] + 1/2 Y [ |1> ] + sY [ |0> ]           EQ 2

NO ===> s [ |G(2)> ] + sX [ |0> ] + sY [ |0> ]               EQ 3

where |G(1)> and |G(2)> are two different kinds of Gray Light given by:

|G(1)> = X |0> + Y/2 |1>  and |G (2)> = X |0> + Y|1>    EQ 4

where the square brackets [ ...  ] indicate a quantity that has "lost its phase" and must be added incoherently. Inside the square bracket, phases still must be taken into account. I have found this unconventional square bracket notation useful in dealing with entangled systems which routinely destroy the phases of entangled sub-systems while preserving the phases of the system as a whole

These results express the quantum amplitudes that appear in BOB's observation channel |B>. To obtain probabilities these amplitudes must be squared. But squaring these raw amplitudes will destroy the phase relations and merely reproduce the results that Nick obtained earlier -- if phases are not important then BOB's results don't depend on ALICE's two choices so no signaling can occur.

But BOB is not restricted to merely passively observing the output of his |B> channel. Instead he has the option to deploy a phase-sensitive detector at |B> that might be able, in principle, to detect the two different forms of Gray Light that appear in EQ 2 and EQ 3. Such a detector might be realized by optical homodyne experiments -- subtle kinds of experiment that have produced such peculiar phenomena as the famous "squeezed vacuum state". Both Demetrios and I begin to look into the homodyne literature for some clue as to how BOB might effectively carry out a phase sensitive measurement.

Our literature search went nowhere. Homodyne experiments seemed designed for tasks far removed from our concerns. At this point Kalamidas and I were stuck. Our search for a REAL MACHINE that could measure the phase between NOTHING and SOMETHING had come up empty handed.

But then came the crucial breakthrough. We both realized this: "We don't got to show you no steenking measuring device". At this early stage the TKO proposal is only a thought experiment, which meant that Demetrios and I had unrestricted access to the vast warehouses of the ACME thought experiment Super Store. The fabled ACME warehouse contains all conceivable measuring devices provided only that they don't violate the laws of physics. The ACME shelves, for instance, are empty of perpetual motion devices and quantum-state Xerox machines. Who supplied that box-on-a-spring which could weigh a single photon, that Einstein used in his famous debate with Bohr? ACME, of course. Or its European equivalent.

Before we raid the ACME shelves, let's take a closer look at BOB's two results. On the surface his YES and NO results look completely different, with the exception of the last zero-photon event sY [ |0> ] which occurs only when an OMEGA is created. This OMEGA term is common to both of ALICE's choices. On the other hand the fact that BOB's two remaining terms seem distinctly different (the same result Kalamidas obtained on page 5 of his hand-written manuscript) gives us hope that, equipped with a 100% sensitive phase-discriminating device, BOB might be able to detect a difference between ALICE's YES and ALICE's NO choices. Hence, given an appropriate device from the ACME store, the TKO proposal might actually work as an FTL signaling machine. Such was our optimistic expectation.

So this is what I ordered from ACME -- a device (called GL (MAX) that splits reality into two orthogonal kinds of Gray Light which I call |S> and |D>:

|S> = s ( |0> + |1> )     And |D> = s ( |0> - |1> )      EQ 5

  

where s = 1/SQRT (2)

The detector GL (MAX) is maximally sensitive to the phase angle between NOTHING |0> and SOMETHING |1>. If this phase angle is positive, the photon ends up in detector |S>. If this phase angle is negative, the photon ends up in detector |D>. In the general case where the phase angle (and amplitude) can be anything, the photon has a definite (and calculatable) probability of ending up either in detector |S> or detector |D>. How the detector GL (MAX) might be physically realized is not our concern. If there were a way to make tons of money from this kind of photon phase detection, a detector of the type GL (MAX) would soon be realized.

Lacking a plausible real way to measure photon phases, Kalamidas and I resort to the ACME thought-experiment warehouse. The price is certainly right: this "ACME Miracle Detector" costs absolutely nothing.

The first thing to notice about the ACME Miracle Detector is that BOB's basis states NOTHING |0> and SOMETHING |1> can be conveniently expressed in terms of AMD states |S> and |D> as:

|0> = s ( |S> + |D> )      |1> = s ( |S> - |D> )                EQ 6

These two expressions will be especially useful for expressing EQ 2 and EQ 3 in terms of phase-sensitive quantum states |S> and |D>. And also useful for calculating the probabilities of the responses of our two orthogonal miracle-detector results <S|S> and <D|D>

Expressing EQ 2 and EQ 3 in terms of the miracle detector bases |S> and |D>, we easily obtain:

YES => s { [ (X + 1/2 Y) |S> + (X - 1/2 Y) |D>]  + 1/2 Y [ |S> - |D> ]}
+ 1/2 Y [|S> - |D>]

NO => s^2 { [(X +Y) |S> + (X-Y) |D>] + X [ |S> + |D>]}
+ 1/2 Y [ |S> - |D> ]

EQ 7 & EQ 8

where the square brackets [... ] indicate no external phase -- inside the brackets, amplitudes do coherently combine, but each bracketed quantity as a whole must be added incoherently to each of its bracketed fellows.

EQ 7 & EQ 8 represent the quantum amplitudes at BOB's |S> and |D> phase-sensitive detectors for each of ALICE's choices.

To determine the quantum probabilities at BOB's |S> and |D> phase-sensitive detectors, we calculate the absolute squares of EQ 7 & EQ 8.

YES PROB ==> 1/2 {(X^2 + XY + Y^2) <S|S>
+ (X^2 - XY + Y^2) <D|D> }

NO PROB ==> 1/2 {(X^2 + XY + Y^2) <S|S>
 + (X^2 - XY + Y^2) <D|D> }

EQ 9 & EQ 10

The final result is that both of these probabilities are exactly the same for all values of the Gray Light parameters X and Y. Thus what happens at BOB's |B> channel, even if BOB is able to deploy perfect miracle phase-sensitive detectors from ACME,  is completely independent of ALICE's actions. ALICE can send no signal, superluminal or otherwise, to BOB.  The exact equality of EQ 9 and EQ 10 means that the TKO proposal totally fails. This result was initially obtained using unconventional square bracket notation, but Kalamidas has independently reached the same conclusion using a standard density matrix calculation.

Although, in common with all previous FTL schemes, the TKO proposal ultimately failed, its detailed refutation led me to places I'd never been before. Highly rewarding was the journey. Thanks much, Demetrios, for taking me along on your trip.

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