Kaleidoscopic Optical Schrödinger Cats

Oktay Pashaev & Aygul Koçak, Izmir Institute of Technology

Most mornings I begin my day by looking at two of my favorite web sites -- NASA's Astronomy Picture of the Day (APOD) where you are sure to find some stunning view of our Universe to lift you out of your daily grind and the Cornell/Los Alamos ArXiv which publishes preprints of fresh new science papers in dozens of different specialties, putting anyone with an iPad in daily touch with some of the most brilliant minds on the planet. All this while sipping a cup of exotic coffee from my friends at Boardwalk Beans in New Jersey.

A few days ago, I discovered a paper on the quantum physics arXiv by two mathematical physicists from Izmar, Turkey (formerly known as "Smyrna") entitled "Kaleidoscope of Quantum Coherent States". These two researchers, Oktay Pashaev and Ayguy Koçak, had devised an infinite set of brand new breeds of Schrödinger Cats.

Schrödinger's Cat in bra-ket notation

In quantum mechanics it is commonplace for a system to be in a SUPERPOSITION of states. An (unmeasured) electron's spin, for instance, can simultaneously exist in a spin-up state |UP> and a spin-down state |DOWN>. When measured, however, the electron is always observed to be in one definite spin state. Austrian physicist Erwin Schrödinger, shortly after he invented his famous quantum wave equation, argued that if unmeasured electrons could exist in two states at once, so could cats, and he devised a famous thought experiment in which an unobserved cat could, according to the laws of quantum physics, exist simultaneous as a live cat |ALIVE> and as a dead cat |DEAD>. Schrödinger's famous alive/dead cat conjecture has generated thousands of physics papers on the possible application of quantum superposition to macroscopic objects and numerous jokes, cartoons and T-shirts ("Schrödinger's Cat is a zombie" reads a T-shirt my neighbor Debi gave me for my birthday.).

Schrödinger's cat walks into a bar. And doesn't.

A brief note on notation. When physicists write down their quantum equations, they commonly use the compact and powerful bra-ket notation devised by British physicist Paul Dirac. In Dirac notation, a quantum initial state A is symbolized by a ket symbol |A> and a quantum final state B by a bra symbol <B|. When multiplied together <B|A> represents the probability amplitude that a quantum system A will be measured to have property B. The probability (different from probability amplitude) that A will be measured to have property B is given by the absolute square of the quantity: <B|A>

As a rough example of this kind of physics talk, let the ket |p,p> represent the initial quantum state of two protons. Let transformation T represent the act of accelerating each of these protons to an energy of 6 Gev in CERN's Large Hadron Collider and nudging them into a head-on collision. And let the bra <H,a| represent the final state that contains a Higgs boson and anything else.

Then, in Dirac's concise notation:

<H,a|T|p,p>

represents a number that expresses the probability amplitude of observing a Higgs boson. Square this quantity to get the probability of observing a Higgs boson.

Dirac's simple notation tells you basically what's going on by concealing a ton of detailed math that you really don't want to know about.

So, using Dirac's bra-ket notation, the quantum state of Schrödinger's cat can be simply represented as:

|ALIVE> + |DEAD>

Or, in a more picturesque description, as:

This is the picture one usual gets about Schrödinger's famous cat -- he's both dead PLUS alive.

Quantum mechanics, however, is more complicated than that, and allows for many more existential possibilities for this hapless quantum cat. Quantum mechanical superposition uses COMPLEX NUMBERS (which possess a direction: North, South, East, West,  for instance) as well as a magnitude. (Numbers that possess only magnitude but not direction -- the kind of numbers we use every day -- are called REAL NUMBERS).

Using the extra degrees of freedom provided by complex numbers, the |ALIVE> and |DEAD> states can be "added together" in an infinite number of ways. If we let the direction "East"  represent "+", then the direction "West" will represent "-". Using "West addition" to combine the two cat states we obtain what might be called a MINUS CAT KET.

Schrödinger's MINUS CAT KET, in pictures, might look like this:

In addition to the PLUS CAT state and the MINUS CAT state, the arithmetical freedom provided by complex numbers allows us to imagine NORTH CAT, SOUTH CAT and NNW CAT states. And, in fact, LIVE and DEAD cats may be added together along any conceivable compass direction.

Whether actual cats can be subjected to quantum superposition is still a matter of some controversy, but there does exist a class of macroscopic states of light that can be placed in a variety of quantum superpositions.

Today's physicists probably know more about light than about any other natural phenomenon. Starting with all the natural forms of electromagnetic radiation, we have created both in theory and in practice a large variety of "unnatural" forms of light, some of which were recently invented in this new paper by Pashaev and Koçak.

Pashaev and Koçak begin their work with a familiar quantum state of light |α> called the "Glauber State" after optical physicist Roy Glauber. The Glauber state is a quantum state (also called "coherent state", hence the title of P&K's paper) that most closely approximates a classical state of light, possessing Heisenberg uncertainty and photons (light quanta) which, however, the corresponding classical state of light does not.  The quantity "α" which labels the Glauber state is a complex number. The square of α represents the average number of photons in the Glauber state. And the direction of α (North, South, East or West) represents the location of the Glauber state in a flat space physicists call the "optical phase plane".

The larger the number α, the more photons in the Glauber state |α>. The special case of α =  0 represents no photons whatsoever, or the vacuum state. Many books could be written about the properties of |0>, the quantum vacuum state. "I've got plenty of nothing. And nothing's plenty for me." might well be the theme song of this particular Glauber state, a state that is completely empty of photons.

Prior to the P&K paper, the Optical Schrödinger Cat (OSC) was well known. It consisted of two states from which all other OSCs could be constructed: the PLUS OPTICAL CAT STATE

|plus optical cat state> = |α> + |-α>

and the MINUS OPTICAL CAT STATE :

|minus optical cat state> = |α> - |-α>

The heart of the Schrödinger Cat controversy concerns the question of how big a system can get before it becomes impossible to place it in a quantum superposition. Optical Schrödinger Cats are in a particularly fortunate position to investigate this question because the larger the number of photons in an optical S-Cat state, the "bigger" the state -- and the more it resembles a classical "cat". In the other direction, when α is small (close to 1 photon), the resulting optical states are sometimes referred to as "Schrödinger Kittens".

To construct their "Optical Cat Kaleidoscopes", the two Turks take advantage of the fact that both α and the coefficients multiplying the optical quantum states |α> are complex numbers -- that is, they possess direction as well as magnitude.

The well-known plus and minus optical cats may be considered "cats of order two (C2)." The first new cat in P&K's infinite series of kaleidoscopic cats may be labeled "cats of order three (C3)." Cats of order three are constructed by adding particular cats with different kinds of dead/aliveness along directions that are separated by 120 degrees (similar to the Mercedes emblem). A caricature of the P&K "three cat" might look like this:

Quantum optical trinity cats in Dirac ket notation.

Or, in keeping with the kaleidoscopic metaphor, C3 could look like this:

Three-fold kaleidoscopic optical Schrödinger's Cat (artist's conception)

Pashaev and Koçak go on to show how kaleidoscopic optical Schrödinger Cats of any order can be constructed, depending on which angle you tilt your mathematical mirrors. On its own terms, theirs is a simple but beautiful achievement of pure mathematics. But the authors go further and show how their kaleidoscopic optical cats may someday find a practical use in quantum computing -- each order of cat representing a different number of quantum bits. Thus, if I am not mistaken, the eighth-order cat (octopussy?) can encode eight quantum bits, the same byte size as the ancient Altair computer and many of its successors.

As a poetic reprieve from so much gratuitous quantum math, this may be a good place to quote British mystic William Blake from a letter to his friend Thomas Butts:

Now I a fourfold vision see

And a fourfold vision is given to me

Tis fourfold in supreme delight

And three-fold in soft Beulah's night

And twofold Always. May God us keep

From Single vision & Newton's sleep.

Some Notes on Quantum Entanglement

Fig. 1: Kim et al experimental realization of path-entangled photons

SOME NOTES ON QUANTUM ENTANGLEMENT

As pop philosopher Robert Anton Wilson was fond of saying: "Quantum mechanics is as queer as a three-legged duck." The crux of the problem is that even physicists don't know how to speak correctly about quantum reality.

We physicists possess a sophisticated and exact quantum theory -- it's never been wrong. We possess delicate and sophisticated instruments, that Bohr and Heisenberg never dreamt of, for measuring happenings in the quantum world. Quantum theory and quantum measurement are in perfect shape. What more could one ask for?

One thing we might reasonably require is a quantum reality -- we could wish for the ability to tell a story about "what's really happening in the world" -- a story that does justice to the radical queerness of the quantum phenomena. And a quantum reality story is precisely what we physicists simply ain't got. Or rather, we have lots of quantum stories, each differing wildly from one another, and none of them quite satisfactory. Wanna stump your neighborhood physicist? Ask him or her "what really happens during a quantum measurement?"

To describe a complicated quantum experiment like the one pictured in Fig. 1 is almost impossible to do without employing some sort of tentative narrative about "what is really going on". So in addition to putting forth theory and experiment, both of which rest on solid ground but which for most of us seem colorless and boring, these notes will necessarily trespass into the dubious quantum reality zone and slip in some talk about "what seems to be really happening". So let the reader beware!

Everything in this world is ultimately made up of quantum systems. And physicists "represent" each quantum system the same way -- by means of a mathematical object they call the "wave function" (or, more generally, the "wave vector"). I say "represent" instead of "describe" because the relationship of the wave function to observation, let alone to reality, is more indirect than a "description".

To every object, a physicist associates a wave function, which he can point to but never write down. For instance, a physicist can say "Wave Function of the Hydrogen atom" but there is no mathematical image that corresponds to this name. Every representation of a quantum system must specify how you intend to measure it. If I intend to measure the momentum of the Hydrogen atom, I can write down the math for its "momentum representation" -- symbolized |H(p)>. If my intent is to measure position, I can write down its "position representation" -- symbolized |H(x)>. But physics provides no mathematical picture of "the thing in itself" -- no mathematical picture of the Hydrogen atom "as it really is" independent of your measurement intent.

[ CORRECTION: a friend reminds me that most simple quantum systems actually do possess "intrinsic qualities" called "eigenstates". For such systems, there exist measurements of particular observables that always give the same answer -- no probabilities here. For the Hydrogen Atom, such eigen-observables include its energy E and its angular momentum J. Independent of your intention to measure it, a Hydrogen atom can be said to actually possess a particular value of E and J. However, in the H-atom's position/momentum space, Nature has neglected to select a favored eigen-representation. In this space, writing down a wave function needs you to choose a measurement intent.] 

(A shorter word for "representation" is "basis". The Hydrogen wave function can be expressed in the momentum basis or the position basis. (The plural of "basis" = "bases"). A beautiful collection of visualizations of the Hydrogen atom's various position representations is Dean Dauger's Atom in a Box.  

 [CORRECTION: In keeping with the above correction, the different Hydrogen atoms in Dauger's collection are each labeled according to their eigen-values of E and J.]

It goes without saying that I can recommend no better guide to the quantum reality question than my own best-selling book on the topic -- recently made available by Doubleday as an e-book.

The numerical value of the wave function represents the "possibility" that a particular intent will be fulfilled. The square of this "possibility" represents the "probability of fulfillment" of some aspect of that intent -- the probability, say, that in the position representation, the photon will be found at position location x.

QUANTUM REALITY

Mother Nature is ready to give us

Anything we're smart enough to ask for

Where "asking' means choosing

Some unique receptive device.

Each kind of asking brings forth

Its own eigen-sack of possibilities

From which -- unpredictably --

She gives out just one, if we're nice.

Intentions, possibilities, probabilities, actualities? Cut to the chase: what's really going on in the quantum world? Today's physicists can't really answer this question.

Or as Albert Einstein once put it: "Who could have guessed that we would know so much and understand so little?"

But on to "quantum entanglement", which founding father Erwin Schrödinger characterized as not one but the feature of quantum theory that most distinguishes it from classical expectations.

The simplest example of quantum entanglement is a pair of path-entangled photons A and B whose wave function |ψ (A, B)>  can be written:

|ψ (A, B)> = S{ |A1>|B1> + |A2>|B2> }             (1)

This wave function represents a pair of photons A and B, each of which can travel along two paths, represented by numbers 1 and 2. The motions of the photons are correlated such that when photon
A takes path A1, then photon B always takes path B1. And when photon A takes path A2, then photon B always takes path B2. (S is just the number 1/(sqrt(2)).

In addition to being correlated, the two particles are entangled. The wave function |ψ (A, B)> represents a SUPERPOSITION of both possibilities. Thus (here comes a "forbidden story") each correlated photon "takes both paths at once" just like Schrödinger's Cat somehow is "both alive and dead". These human-baffling pictures of photons and cats are (probably misleading) attempts to explain a humanly incomprehensible quantum reality that lies beneath the visible world. In this note you will run across several such dubious stories, but this will be your last warning. Trust in mathematics (such as EQ (1)) and in measurement and you will never go wrong, but take all stories (even my own) about "what's really going on" with a grain of salt.

By the way, the wave function EQ (1) is written in the "path representation" or "path basis".

For me there is no more elegant realization of the classic double-slit experiment than that of Yoon-Ho Kim and his buddies at University of Maryland (henceforth Kim et al). The basic message of the double-slit is that if a quantum particle goes through both slits at once, it will produce interference fringes when the beams from the slits are combined at a screen. But if you measure which slit the photon went through, even if that measurement is "non-disturbing", then the interference at the screen will vanish. So sayeth the high priests of quantum theory.

The beauty of the Kim et al experiment is that Kim is able to effectively send one photon (photon A) through both slits, but also is able to measure which slit photon A went through without disturbing it -- simply by measuring its 100% path-correlated partner photon B.

The Kim et al experiment (Fig. 1) exploits a phenomenon exhibited by certain crystals called "photon down conversion" in which one photon of energy E enters the crystal and two photons with energy E/2 come out the other side. Since these two photons (photons A and B) are created in the same place, their paths are correlated. Now shine the initial source of photons thru a double slit onto the down-conversion crystal and two correlated photon pairs are simultaneously possible. These kinds of superposed possibilities are the meat and potatoes of quantum theory. For this situation quantum theory predicts that from these two slits will emerge from time to time, two photons A and B that enjoy the peculiar quantum situation described by EQ (1), that is, these two photons emerge simultaneously from both slits (and so in principle can produce two-slit interference fringes when combined at a screen.  But also, each photon, say A, is accompanied by a second photon, photon B, which in principle is able to measure (without disturbance) thru which slit A passed -- hence destroying any possibility of interference.

A very subtle and complex state of affairs. But at its core a very simple and computable application of elementary quantum theory. Congratulations, Kim et al for devising (and actually carrying out) this lovely experiment.

So what actually happens in this experiment? Does photon A produce interference fringes or not?

The short answer is: No Fringes. But the longer answer is, under certain conditions: Yes, Fringes.

Let's examine the question of what basis to best describe this quantum experiment so as to better understand the quantum facts. EQ (1) is expressed in the path basis and shows that path A is correlated and entangled with path B.

The experiment illustrated in Fig. 1, shows the B photons being measured in the path basis, but the A photons are combined by a converging lens into a state best described in what I will call the "screen basis". I will also introduce a third representation called the "eraser basis". If you expected that this note was going to be a bit of poetic fluff, you will be seriously disappointed. Warning! Raw, uncensored quantum physics ahead!

About the path basis (photon B for instance in Fig. 1): Photon B takes either path B1 or path B2. And this fact can be verified using photon detectors B1 and B2. If B1 clicks, the photon took path B1; If B2 click, the photon took path B2. It's that simple. Remember, Mother Nature on the quantum level is responsive to the questions that you ask. If you go looking for photon paths, Nature will give you photon paths.

Fig. 2: Screen pattern of A photons

What about photon A, that no longer travels in two beams but has been merged by a converging lens onto a photon sensitive screen? Well, the rules of quantum theory state that if you know which path a photon took, it cannot form an interference pattern. We can determine which path A took by looking at the B counters. If B1 clicks then photon A took path A1, giving rise to pattern # 1 of Fig. 2. If B2 clicks then photon A took path A2, giving rise to pattern #2 of Fig. 2. Both these patterns are featureless blurs. Adding them together gives a featureless blur with twice the intensity. No matter how hard you stare at these pixels, you will never see an interference pattern.

One way of understanding this lack of interference is that because of the perfect A-B path correlation,  observation of the B photon path tells us precisely which path the A photon took. If we possess path info about photon A, no interference is possible -- if photon A took one path, it simply could not have gone thru both slits.

This argument depends on the fact that the paths of the B photons carry info about the paths of the A photons. But suppose we alter our experiment by destroying B's path information with a "quantum eraser"? If the eraser leaves us unable to tell which path A took, will we then be able to observe the A photon interfering at the screen? Let's take a look at how a quantum eraser works.

Fig 3: Beam splitter operating as Quantum Eraser

The quantum eraser consists of a half-silvered mirror. Beam B1 comes in, is half reflected and half transmitted and sent into outputs B3 and B4. Likewise beam B2 comes in, is half reflected and half transmitted and is also sent into outputs B3 and B4. The eraser adds together these two input beams in such a manner as to entirely destroy information about which path the input photon took before entering the beam splitter.

The equations for this path info erasure operation are:

|B3> = S {|B1> + |B2>}                EQ (2)
|B4> = S {|B1> - |B2>}

The minus sign in the second equation looks a little out of place, but is absolutely necessary. If that minus sign were not there, more energy would come out of the eraser than went in -- humans could
create endless energy out of mirrors. However the law of conservation of energy is a very strong prejudice in the physics community and seems to be obeyed to the letter by Nature as well. This minus sign is a special case of the Stokes relations for light traveling across interfaces. Sir George Stokes was an Irish physicist from County Sligo, who also made important discoveries in fluid mechanics.

A little algebra gives us the inverse equation to EQ (2)

|B1> = S {|B3> + |B4>}                               EQ (3)
|B2> = S {|B3> - |B4>}

Substituting EQ (3) in EQ (1) we obtain:

|ψ (A, B)> = S{ S(|A1> + |A2>)|B3> + S(|A1> - |A2>)|B4> }                EQ (4)

Defining  |M(A)> = S(|A1> + |A2>)
and |W(A)> = S(|A1> - |A2>)                        EQ (5)

we get for the system wave function:

|ψ (A, B)> = S{ |M(A)>|B3> + |W(A)>|B4> }                  EQ (6)

EQ (6) expresses a perfect entanglement between photon A (whose wave function is represented in the "screen basis") and photon B (whose wave function is expressed in the "eraser basis". The names for the bases are my own inventions but the physics equations are entirely conventional.

THE SCREEN BASIS consists of two wave functions |M(A)> and |W(A)>. In the case of |M(A)>, both of A's photon paths are ADDED TOGETHER before being combined at the screen. In the case of |W(A)>, both of A's photon paths are SUBTRACTED FROM EACH OTHER before being combined at the screen. I chose the symbols M and W to label these two wave functions because the letters M and W are flipped versions of one another, just as the two wave functions are in the interference sense, precisely one another's opposites, as we shall see.

THE ERASER BASIS consists of the two wave functions |B3> and |B4> which emerge from the beam splitter, aka "quantum eraser". These two wave functions by themselves contain no information about "which path" the B photon took from the source, but taken together the two wave functions could in principle still be combined (in an "anti-eraser") which would resurrect the "which path" info of photon B. However if photon B is actually detected after the eraser either by "Bobski" at detector B3 or by "Boris" at detector B4, then B's path information is definitively erased and cannot ever be recovered. A signal from either Boris or Bobski means: "Da! Dat photon's path? Vorry you not. He is erased."

Fig. 4: Quantum Eraser produces 2 sets of fringes.

Does erasing B photon's which-path information (and via their perfect entanglement also erasing A photon's path info as well) now produce interference when both of A's paths converge on the same screen. The answer is Yes. Erasing B's path info produces interference fringes on A's screen.

But B's eraser and A's screen could be light years apart. If B's action can instantly change A's physical situation, then we can call "B" Bob and A "Alice". In the physics literature, Bob and Alice are iconic figures constantly obsessed with exploiting quantum entanglement to exchange superluminal messages between space-like separated locations. And on the surface, it looks as though path-entangled photons can do the job, because erasing Bob's path will produce Alice's fringes. And not erasing Bob's path will make Alice's fringes disappear.

Of course this entangled eraser scheme can't possibly work. Such a scheme would violate Einstein's laws of relativity. But one can't simply invoke relativity to explain away the eraser scheme. Any such alleged FTL signaling proposal must be refuted on its own terms, by using only the laws of quantum mechanics.

The first thing we notice about Alice's fringes is that there is not just ONE SET OF FRINGES produced on the screen but TWO SETS OF FRINGES. When Bobski announces that he has erased photon B's path, all of photon A's screen pixels that correlate with Bobski's message form a fringe pattern Z. When Boris announces that he has erased photon B's path, all of photon A's screen pixels that correlate with Boris's message form a second fringe pattern "Anti-Z". As shown in Fig. 4, the Z fringes and the anti-Z fringes exactly cancel -- the peaks of one set of fringes fitting exactly into the valleys of the other set of fringes. So Alice's fringes appear when triggered by messages either from Bobski or Boris, but no fringes appear when there is no way to tell whether the dot on Alice's screen was correlated with a click at counter B3 or a click at counter B4. Thus, absent a trigger signal (which must be sent at light speed or slower), what happens at Bob's location remains at Bob's location. Conclusion: FTL signaling using this entangled eraser scheme is impossible.

Fig. 5: Computing the observed pattern of photons on Alice's screen.

To see how this fringe and anti-fringe business works, we calculate the intensity of the patterns at a single location on Alice's screen. Fig. 5 illustrates where the two beams M(A) and W(A) impinge in the screen. These two beams have been slightly shifted for clarity, In reality they would be exactly superposed.

The wave function |M> represents the possibility for a photon to hit the screen. Squaring the wave function we obtain the probability for a photon to hit the screen. This probability is proportional to the INTENSITY of the light produced by the |M> beam at a particular location x on the screen, illustrated by the vertical line in Fig. 5.

If we represent possibilities A1 and A2 as amplitudes a1 and a2 times phases exp i(θ1) and exp i(θ2), we obtain for the probability (intensity) at location x for the wave function |M>

                               Intensity (M)  = C + D cos θ                 EQ (9)

where C = (a1 x a1+ a2 x a2), D = (a1 x a2) and θ is the phase difference at location x between |A1> and |A2>


Carrying out this same calculation for the wave function |W> at location x, we obtain:

                                 Intensity (W) = C - D cos θ                 EQ (10)

C is the intensity at x and D represents the magnitude of the interference fringes at x. Here you can explicitly see that the interferences terms at location x cancel when both terms are summed. Since x is an arbitrary position on the screen, if the interference term cancels at x, then the interference terms cancel everywhere. 

We see here precisely in what sense the two "screen basis" wave functions |M> and |W> can be regarded as "opposites". Each of these wave functions produces interference fringes on Alice's screen. But these two sets of interference fringes exactly cancel one another. As far as interference fringes go, |M> and |W> by themselves produce precisely opposite effects.

Path-entangled systems possess a variety of "magical" qualities which I will only mention in passing. There is, for instance, the so-called "delayed-choice" quantum eraser which involves invoking Bob's quantum eraser long after Alice's photon pattern has been indelibly recorded. Hence apparently after-the-fact producing matched sets of mutually canceling Alice fringes that correlate with Bobski's and Boris's detector events.

But wait, there's more. The path eraser pictured in Fig. 3 represents only one out of an infinite array of possible ways of erasing Bob's which-path information. By changing the position of Bob's movable mirror, Bob can add a phase angle φ tο photon path B2. Now when we trigger on Bobski's and Boris's detector outputs we get two complementary patterns of Alice fringes as before, but these fringes have moved to a new location on Alice's screen -- a new location that depends on Bob's choice of the phase angle φ.

Thus altho there are no fringes visible in Alice's total pattern of photons on her screen, Bob can, by changing the phase angle φ, seemingly shift the location of Alice's two mutually canceling fringe patterns. And Bob can carry out this invisible Alice fringe pattern shifting in the "delayed-choice" mode, that is, long after Alice has already recorded every pixel of her pattern "in stone". Given the peculiar behavior of this simplest of all quantum entanglements, it is hard not to imagine that "something" must be being transmitted faster-than-light (or even backwards-in-time) from Bob to Alice. Something must be really being sent (and really fast too) in these experiments. But physicists can't really say what that "something" might be.

We have introduced here three different photon representations, the path basis P, the screen basis S and the eraser basis E. Applying these three bases separately to photon A or photon B, we could obtain nine different experiments on photon entanglement. Since P(A)P(B) is the same as P(B)P(A), the number of unique entanglement experiments reduces to six, namely (PP), (PE), (PA), (EE), (ES) and (SS). We have already considered most of these six experiments, but not the two (SS) and (EE) where both Bob and Alice combine their photons either on screens (SS) or using erasers (EE).

Both (SS) and (EE) have such similar behaviors that I will just consider the (SS) case for which the wave function (in the screen - screen basis) looks like this :

|ψ (A, B)> = S{ |M(A)>|M(B)> + |W(A)>|W(B)> }    EQ (11)

We see from EQ (11) that Alice's interference pattern Z is perfectly correlated with Bob's interference pattern Z. Likewise Alice's anti-interference pattern anti-Z is perfectly correlated with Bob's anti-Z pattern. Thus on both Alice and Bob's screens appear two complementary fringe patterns exactly as illustrated in Fig. 4. The two interference patterns exactly cancel. And again no FTL signaling is possible between Alice and Bob in the (SS) basis.

[ ADDED 5/14: we have seen, that depending on your measurement intention, a simple path-entangled photon pair can be written as a Path-Screen, an Eraser- Screen or a Screen-Screen entanglement -- each a different but equally valid way of envisioning the same quantum wave function. In fact, by adding a phase φ to one side of either the Eraser basis or to the Screen basis, one can generate a continuous infinity of different bases, each of which takes a different experimental perspective on the very simple path-entangled photon pair represented by EQ (1).]

If a photon takes two paths to a single location it can produce interference fringes, If which-path information is available these fringes will vanish. Because we can entangle two photons, it seems that we can distantly decide whether path information for photon A exists or not, depending on what we do with its entangled partner photon B. Distant which-path erasure seems on the face of it to be a viable road to achieving FTL signaling using entangled photons. As these examples show, erasing Bob's which-path information does indeed lead (instantly?) to the appearance of fringes on Alice's screen.

But these Alice fringes always appear in pairs -- as a set Z of fringes and a set anti-Z of complementary fringes which exactly cancel out: with the total result that no fringes appear on Alice's site when Bob chooses to erase his which-path information.

Is this fringe/anti-fringe behavior a general feature of quantum erasure schemes? Or is it merely an accidental feature of this particular method of path erasure?

Recently Demetrios Kalamidas came up with an ingenious FTL signaling scheme that uses a radically new method of which-path erasure. Instead of merely erasing photon paths, the Kalamidas scheme makes the paths "ambiguous" by mixing each path with a weak coherent state. A coherent state has the property that its photon number is uncertain. The result of this mixing is to create a situation in which when you measure a photon in a particular path, you can never be sure whether that path was "full" and you are measuring the "real" entangled photon. Or whether that path was "empty" and you are measuring a "fake photon" originating from the coherent state.

Claiming that if Bob's uses his clever new means of path erasure, uncompensated fringes will be produced on Alice's screen, Kalamidas published his FTL scheme in a well-regarded optics journal. His scheme was immediately refuted in general terms by a number of different physicists in a number of different ways, but it took several months of work before Martin Suda and Nick Herbert were able to finally demonstrate exactly where Kalamidas went off the rails. It turns out that even in the Kalamidas case, Bob's path erasure produces complementary sets of fringes which totally cancel out all interference at Alice's screen.

Even though his clever FTL signaling scheme was eventually refuted, the Kalamidas scheme led to intense discussions of the subtle details of few-photon states and coherent states. And contributed certainly to my knowledge of such topics and perhaps added to the physics community's store of few-photon lore. Thank you, Demetrios.

I would also like to thank Jack Sarfatti for prodding me to reconsider the physics of eraser-based FTL communication schemes.

Despite its enormous successes, quantum theory has left physicists with two big mysteries, both of which involve quantum reality -- the ability to tell a convincing story about "what's really happening in the world" that does justice to quantum theory and the quantum facts.

Quantum Mystery # 1: What does the quantum wave function really represent? What is really happening in the world before any measurements are made?

Quantum Mystery # 2: What really happens during a quantum measurement? How does a quantum possibility decide to turn into an actuality?

To these fundamental questions, the fact of quantum entanglement adds a third:

Quantum Mystery # 3: What (if anything) is actually exchanged between two distant entangled quantum systems? When will physicists get smart enough to be able to tell their kids a believable story about what's really going on between Alice and Bob?

Wave function (in the position basis) for an excited Hydrogen Atom: from Dean Dauger's Atom in a Box.

The Quantum Olympics

Selection of molecules which show quantum interference in matter-wave interferometers. (Graphic by Sandra Eibenberger.)

At the beginning of the 20th century, one of the biggest problems in physics was to understand the interaction between matter and light. Today we possess an impressively broad and detailed knowledge of matter-light interactions expressed in the language of quantum theory.

In 1900, it was generally believed that light was made of waves and that matter was made of particles. This belief was shattered when Albert Einstein (better known for his relativity theories) showed that light in some situations acted like a particle. Einstein's particles of light were christened "photons". Later in the century, Louis de Broglie, the French prince who became a physicist, proposed that particles should possess wave properties. And de Broglie was able to calculate the supposed wavelength of the electron (the lightest of the known particles). He submitted this proposal for his PhD thesis at the Sorbonne. His professors were prepared to reject his thesis on the grounds of preposterousness. But through the intervention of Einstein the prince was awarded his degree which was crowned a few years later by a Nobel Prize when some American physicists at Bell Labs measured the wavelength of the electron which was precisely the value that de Broglie had predicted using his "preposterous" theory.

De Broglie's wave theory of matter predicts that every piece of matter possesses frequency, wavelength and "phase" (whatever these quantities might mean). Not only electrons, but protons, cats, bicycles and you yourself possess wave properties. The catch is that the more massive a particle gets, the smaller the particle's de Broglie wavelength. Hence it becomes more difficult to experimentally demonstrate a particle's wave properties as its mass gets larger.

Enter the Quantum Olympics. Open only to experimental quantum physics. What is the biggest (most massive) particle whose wave properties you can demonstrate in the laboratory?

The electron was the first to show its waviness, later the neutron -- almost 2000 times more massive than the electron -- was shown to be a wave. In the 1990s several large atoms such as Helium, Iodine and Sodium vapors were shown to possess wave properties. And in 1999, someone in Vienna succeeded in diffracting a buckyball -- a soccer-ball-shaped molecule consisting of 60 Carbon atoms.

A Talbot Carpet demonstrating near-field interference from multiple slits

Recently, spectacularly impressive records have been set in the Quantum Olympics. Using a novel matter-wave detection technique developed by John Clauser (of Bell's Theorem fame) called Talbot-Lau interferometry, experimenters from Vienna, Basel and Duisburg-Essen have demonstrated high-contrast quantum interference for a remarkable assortment of complex and increasingly massive molecules culminating with the current winner of Olympic Gold -- molecule "m" shown above.

The 2015 Olympic champ is a "functionalized porphyrin" with atomic formula:

C(284) H(190) F(320) N(4) S(12)

Congratulations to the assortment of clever physicists who showed that this assortment of increasingly massive particles behave like waves as well as like particles.

As admirable as these experiments might seem to the ordinary person, they seem even more remarkable, even impossible to the average physicist. These experiments seem impossible on the face of it because wave interference is a very delicate affair, requiring stability and coherence over large times and distances (compared to the sizes of these atoms). Although it might be possible to observe interference with atoms with very little structure, it should be impossible to do so for buckyballs and especially impossible for the grotesquely complicated molecules pictured above.

The reason that such experiments should be impossible is that these complex molecules are not rigid objects but possess hundreds of degrees of rotational, vibrational and conformational freedom. They are turning, vibrating, bending in hundreds of different ways. Certainly the waves associated with such a busy, buzzing, bendable object could never be moving coherently long enough to form a clean high-contrast Talbot Carpet such as the figure above in green. So goes the conventional wisdom.

But the conventional wisdom is wrong.

It can be shown (by quantum calculations) that as long as the internal motion of the molecule (no matter how grotesque this motion) is UNCORRELATED with the external trajectory of the molecule, then this internal motion will not destroy the coherence of the external motion. Hence these delicate experiments can even be carried out at room temperature when the internal motion of the molecule is as complicated as Times Square on New Year's Eve. However as the temperature is raised and the internal motion becomes hot enough to emit photons, photons that can perturb the molecule's external motion, then coherence is lost and the molecule's wave properties become impossible to detect.

This intrinsic decoupling of internal motions from the external motions of a complex object reminds me of a similarly engaging problem in theoretical physics: How do cats always manage to land on their feet when dropped?

It would seem impossible for a cat to turn over in midair because of conservation of angular momentum. And whatever could the cat push against to begin its spin? Like the busy, buzzing, bendable molecules, a cat's internal motion is completely decoupled from the trajectory of its center of mass. Yet it turns (as Galileo might have said). The cat turns. And lands on its feet. All without violating a single law of physics. Clever cat.

The Falling Cat Problem (from an illustration in the journal Nature 1894)

Quantum Vampire Effect

Bela Lugosi as Dracula

Quantum mechanics is full of subtle and unusual processes that challenge our common-sense understanding of the world, for instance simultaneous particle/wave behavior, quantum non-locality, quantum entanglement and instant quantum teleportation. Now, thanks to four Russian researchers in Moscow and Calgary, Canada, a new example of quantum weirdness has been added to the list, the peculiar phenomenon called Quantum Vampire Effect (QVE).

Ilya Fedorov, Alex Ulanov, Yury Kurochkin and Alex Lvovsky announced the discovery of the Quantum Vampire Effect in a recent ArXiv post. Here I will attempt a brief description of the four Russians' discovery.

Quantum Vampire Effect: Removing a photon from part of a state removes the photon from the entire state.

The picture above illustrates an INTERFEROMETER, one of the physicist's most sensitive measuring devices. It works like this. A quantum state ψ at the left is split into two parts by beam splitter BS1 and then later recombined at beam splitter BS2. If the two paths suffer no change, the state ψ re-emerges intact. However if one of the two paths is perturbed by even the most minuscule change, reconstruction fails and some of the state ψ will end up in detector A.

Possible changes that might hinder the perfect reconstruction of ψ include changes in the density of the air, vibrations of the mirrors caused by a car passing outside or by people talking in the room. The INTERFEROMETER is a very very sensitive device.

So now we take this change-sensitive INTERFEROMETER and remove one photon from one of its paths by means of a weakly-transmitting mirror and a single-photon detector (see above). What do you think will happen? The answer to this question is the basis of the Quantum Vampire Effect.

What happens if we remove one photon from one of the beams is NOTHING. Well not exactly nothing, but almost nothing. The original state is preserved at the final detector; sensitive-change-detector A does not click. But the final state is missing one photon.

The final state is missing one photon. This means that taking ONE PHOTON from the PARTIAL STATE (one of the two beams into which the original state was split) is entirely equivalent to taking ONE PHOTON from the ENTIRE STATE. As far as the final result is concerned, the photon-eating device (weakly-transmitting mirror and detector) in partial beam φ1 could just as well have been placed at the beginning of the experiment where it would grab a photon from the whole beam instead of just a part.

In physics slang this photon-grabbing device is the material realization of the "photon annihilation operator" (symbolized by lower case a) which removes one photon from the associated quantum state. If operator a is applied to partial beam φ1 , this operation is written a(φ1). If the full state ψ has one photon removed, this photon removal operation is written a(ψ). The Quantum Vampire Effect amounts to the discovery that under very general conditions:
 

a (φ1) = a (ψ). 

That is, taking a photon from part of the state is exactly equivalent to taking a photon from ALL OF THE STATE.

So what?

Here's what.

If ordinary light absorption worked this way, objects would not cast shadows (hence the Dracula-inspired name). Instead the light as a whole would be dimmed.

The Quantum Vampire Effect does not cast a shadow but reduces the intensity of the light as a whole.

Ordinary Optical Absorption is much more complicated than the simple removal of one photon at a time by physical application of the annihilation operator "a". The realization of "a" is a DIP & CLICK operation which rarely happens in real life but is easy to do in an optics lab. What you do is DIP a mirror into the beam. And if you happen to catch a photon, your detector CLICKS. The Quantum Vampire Effect illustrated above can only be demonstrated if you throw away all ordinary absorptions and just keep the few chance events associated with DIPS & CLICKS.

But however artificial the Vampire Effect might seem, it has enormous philosophical import because it is a concrete example of an action on the part being equivalent TO THE SAME ACTION acting on the whole. One more little puzzle piece in the Great Quantum Mystery.

And perhaps the key to a brand-new super technology. If an action on a part can instantly affect the whole, can one perhaps use the Quantum Vampire Effect to send signals faster-than-light (FTL)?

Since I have been devising impromptu FTL signaling devices for most of my physics career, it was not difficult to see how to exploit the Quantum Vampire Effect to achieve ultra-fast telegraphy.

The trick is this: to start with a quantum state that possesses very few photons, so that the hyper-holistic DIP & CLICK operation results in an enormous change. For this purpose, the best input state one could imagine would be a 2-photon state. Then the DIP & CLICK operation would maximally switch the state (non-locally?) from a state consisting of two photons (symbolized |2>) to a state consisting of just one photon (symbolized |1>).

Accordingly, my new FTL design consists of a 2-photon Diagonally-polarized (D) input state which is split by a polarized beam splitter (PBS) into a Vertically-polarized (V) beam sent to ALICE and a Horizontally-polarized (H) beam sent to BOB.

For starts the initial D beam possesses two photons, which in each pulse are shared between ALICE and BOB. If ALICE detects two photons, BOB detects none. If ALICE detects one photon, BOB gets one too. And so on.

But now ALICE introduces a DIP & CLICK machine into her beam which locally removes one photon from her beam. But according to the Quantum Vampire Effect, Alice's act is NOT ONLY LOCAL BUT GLOBAL. Suddenly instead of sharing a two-photon D state, both ALICE and BOB are sharing a one-photon D state.

And these two kinds of state are easily distinguishable -- mainly by the utter lack of any two-photon counting events.

Is this it? Has the fair Muse of Physics finally delivered Nick his long-sought FTL signaling scheme? Please read on.

VLAD: a proposed FTL signaling scheme based on Quantum Vampire Effect

I decided to call it VLAD (for Vampire-Licit Ansible Device) -- "Ansible" being a famous fictitious FTL signaling device invented by sci-fi writer Ursula Le Guin.

And sure enough it works. The above illustration shows the VLAD scheme, including ALICE'S use of a partially-reflecting mirror to DIP & CLICK single photons out of her beam and record every such events with her "a Detector". ("a", you will recall, stands for the quantum photon annihilation operator).

For the VLAD setup, the Quantum Vampire Effect works as advertised: when nothing is done, the input is a two-photon state. But whenever ALICE snatches a photon from her local beam, the ENTIRE SYSTEM acts as though the input was just one photon all along.

The two equations accompanying the drawing represent the quantum wavefunctions for these two cases. 1. When there are two photons in the starting state, ψ(1) describes the situation; 2. When there is only one photon in the starting state, ψ(2) is the correct description.

VLAD "works" in the sense that BOB could easily distinguish which of these two wavefunctions describes the situation by looking at the pattern of photon counts at his detector. VLAD "works" in the sense that ALICE can select, by deploying her DIP & CLICK device or not, whether the wavefunction shall consist of two photons (ψ(1)) or of only one photon (ψ(2)).

But, alas, VLAD finally fails to work after all, because the output of ALICE'S DIP & CLICK occurs at random. Only if Alice sends BOB a signal (at light speed or slower) whenever her "a detector" clicks, can BOB know for sure that that he is looking at a one-photon state. Absent news of ALICE'S "a detector" click, BOB sees no change whatsoever in his pattern of photon clicks. Even through ALICE'S action has randomly embedded a one-photon pattern in the full photon stream, without a decoding signal from ALICE, BOB cannot extract these special events from the original two-photon situation. So says the mathematics. Once again clever Nature has prevented us from using Her marvelous quantum entanglement to send signals faster than light.

Goodbye to VLAD as an FTL signaling device. Farewell, VLAD. Da svedanya.

And the four Russians conclude their QVE paper thus: "We expect the quantum vampire effect to find applications in quantum information technology...The ability to "steal" a photon without casting a shadow may prove useful for eavesdropping in quantum key distributions as well as developing quantum cloaking devices. We also believe the effect to be of fundamental interest, as quantum action at a distance that is not associated with a local state collapse has not yet been studied."

I wish to thank Doctor Alex Lvovsky for patiently clarifying for me many subtle features of the Quantum Vampire Effect. Without his help this post would have been impossible.

New Father-and-Son Quantum Text Book

Samarkand, Uzbekistan by Richard-Karl Karlovitch Zommer

Samarkand, one of the world's oldest inhabited cities, once prospered as a trading post on the Silk Road between China and Europe. During the Islamic Golden Age (750 AD -- 1258 AD) the city became a famous focus of Arab scholarship in astronomy, medicine and mathematics. In more modern times, there graduated from the State University of Samarkand a physicist Moses Fayngold, who with his son Vadim, also a physicist, has written a new text book on quantum mechanics, intended for advanced undergraduates and beginning graduate students. I found this book rich and unpredictable and, like the romantic Silk Road metropolis, offering something fresh and exotic around every corner.

Why does the world need yet another book about quantum mechanics? This question was raised by the father. "[The father], who by his own admission, used to think of himself as something of an expert in QM, was not initially impressed by the idea, citing a huge number of excellent contemporary presentations of the subject. Gradually, however, as he grew involved in discussing the issues brought up by his younger colleague, he found it hard to explain some of them even to himself. Moreover, to his surprise, in many instances he could not find satisfactory explanations even in those texts he had previously considered to contain authoritative accounts on the subject." (from the Preface).

Unlike most conventional quantum physics texts which merely explain things, this book also focuses on many of the loopholes, exceptions, imperfections, misunderstandings, man traps and pitfalls that exist in this complex field.

When you buy a new car, you will find an Owner's Manual in the glove compartment that tells you how to change the oil and how to replace the light bulbs. But if you are handy with tools you will also want to purchase the Mechanic's Manual to learn how to do things that only professionals should attempt. And, in particular, to learn things that YOU SHOULD NOT DO. (Never unscrew part A before releasing part B.)

This new quantum text book is the equivalent of a Mechanic's Manual that makes previous text books seem mere Owner's Manuals.

Most quantum text books tell you how to do things, but I have never run across a text book like Moses and Vadim's which tells you WHAT NOT TO DO. Over and over again in this text, I ran across comments to the effect that "The naive way to do this is B, but B will give you the wrong answer. Here's how to do things right." The authors seem to have anticipated many pitfalls that lie in wait for the quantum neophyte and have posted the appropriate warnings. My guess is that these pitfalls are those into which Moses and Vadim have themselves fallen. Niels Bohr once claimed that the definition of an "expert" in a field is a person who has made all the mistakes in that field. In this unusual book Moses and Vadim give you the advantage of that kind of street-smart expertise.

Their book begins by describing some major phenomena that classical physics could not explain (black-body radiation, photoelectric effect, low-temperature specific heats and atomic spectra), then show how one simple concept--the quantization of energy--could correctly reproduce these results.

Moses and Vadim then describe the origin of Louis DeBroglie's hypothesis--that matter possesses a wave-like nature whose wavelength DeBroglie could calculate. Altho this textbook confines itself to non-relativistic quantum mechanics, I was surprised (one surprise of many) to discover that DeBroglie's calculation was motivated by special relativity which means that his discovery is deeper than necessary and transcends its non-relativistic buddies such as the Schrödinger equation.

Using the DB hypothesis to physically justify energy quantization (similar to the way that resonance modes quantize the notes of stringed instruments), Moses and Vadim then use the Superposition Principle for waves to construct an "embryonic quantum mechanics" from which much more good physics can be derived without yet mentioning the Schrödinger Equation.

This book includes in-depth discussions (always accompanied by Moses and Vadim's dependable pitfall warning signs) of most of the conventional topics in quantum theory including Hilbert space, Dirac notation, angular momentum, scattering theory, band structure, quantum tunneling, density matrices, Kaon and neutrino oscillations, quantum entanglement, CHSH, POVMs, CNOT and XOR gates, the Bloch sphere, Zeno's paradox, Schrödinger's Cat, and much much more.

Moses and Vadim also introduce a novel topic they call "submissive quantum mechanics" in which they show how to manipulate potentials to create customized wave functions never before realized in nature--a useful skill that may prove profitable in the emerging field of nanotechnology.

Again and again while reading this book I got the feeling of a wise adviser at my side. The ratio of explanatory text to equations is large--resulting in a lucidity reminiscent of the classic Feynman Lectures as well as Quantum Theory by David Bohm.

Besides devising the shortest proof of Bell's theorem, Nick Herbert's main claim to physics fame is his FLASH (First Laser-Amplified Superluminal Hookup) proposal which purported to send signals faster-than-light using a "laser-like device" to clone single photons. The FLASH proposal was refuted by Wooters and Zurek who proved that "a single (unknown) photon cannot be cloned", a result which crucially limits what quantum computers can do--for instance, when quantum hard drives or quantum DVDs are built, the no-cloning theorem provides automatic copy protection courtesy of the laws of physics.

Naturally I was curious about how Moses and Vadim would deal with my FLASH proposal in their hyper-informative "Mechanic's Manual" style. In this I was not disappointed.

The authors agree that the W&Z "no perfect cloning of unknown states" proof definitively refutes my FLASH proposal. But what about "imperfect cloning"?, they ask. And what about the cloning of states that are not completely unknown but part of a small prearranged set of known states? Moses and Vadim carefully consider these loopholes (and a few more) to the standard FLASH refutation and definitively decide that FLASH won't work. But in the course of their detailed refutation the reader learns a lot about quantum cloning machines.

This book is a wonderful Mechanic's Manual crammed full of intimate details about the operation of one of the most elegant intellectual sports cars we possess--the theory of non-relativistic quantum mechanics. But in addition to this Mechanic's Manual, I urge you to also purchase an Owner's Manual of your choice, a book that you can use to solve everyday problems in simple ways. (My own favorite Owner's Manual is the classic text by Leonard Schiff from which I learned QM in those bygone days when the world's largest particle accelerator was the Berkeley Bevatron.)

But next to your trusted Owner's Manual, be sure to include this helpful Mechanic's Manual on your book shelf, both to deepen your knowledge of quantum mechanics and to help you avoid some of its more obvious pitfalls.

This book is perfect for those quantum mechanics who know how to fix Volkswagons and now want to go to work on Porsches.

New father-and-son quantum text book

The Kalamidas Experiment

Nick calculating the Kalamidas effect (NKE)

Recently Demetrios Kalamidas, a young New York quantum-optics physicist, proposed an imaginative new superluminal signaling scheme--OKE (see also KISS and Demetrios! the Opera). The Kalamidas experiment proposes for Bob to send instant signals to Alice via the medium of a pair of path-entangled photons (A and B).

When Bob knows (from observation of his photon B) which path Alice's A photon took, then Alice cannot observe two-path interference. But if Bob can erase which-path information then, in principle, Alice can get her path-uncertain photon to interfere with itself.

Switching between these two options (knowledge of Alice's photon's path and erasure of that knowledge) Bob can instantly send a signal to Alice no matter how large the distance that separates them. So goes the argument for FTL signaling via quantum entanglement.

In my analysis of the Kalamidas experiment: NKE (caution: 2-MB pdf download), I consider three possible choices that Bob could make to either obtain or erase which-path info concerning Alice's photon A.

I call these three choices: 1. the Fock Choice; 2. the Frost choice and 3. the Kalamidas choice. The Fock choice preserves which-path info and both the Frost choice and the Kalamidas choice erase which-path info—but in two different ways.

The Frost-choice method for quantum-erasure of which-path info is well-known—scramble Bob's two paths in a beam splitter. This choice does indeed lead to interference of Alice's photons. But this interference is INVISIBLE because it is superposed with an exactly complementary anti-interference pattern, the sum of which produces a completely random signal at Alice's detectors. These two patterns (signal and anti-signal) can however be separated by a coincidence trigger from Bob that tells Alice which of Bob's two detectors fired. If detector B1 went off then Alice sees a signal; if detector B2 went off, then Alice sees an anti-signal.

So (making the Frost choice) Bob's erasure of which-path info does indeed produce interference at Alice's detectors but ALICE'S INTERFERENCE IS ENCRYPTED using a random key that Bob can only send by conventional slower-than-light methods. Hence the Frost choice fails as a superluminal signaling device.

What about the Kalamidas choice?

The gist of the Kalamidas experiment is the novel method he has invented for Bob to quantum-erase which-path info. In his original article: OKE = Original Kalamidas Experiment, Kalamidas demonstrates that his method leads to UNENCRYPTED INTERFERENCE at Alice's detectors. Hence it appears that Demetrios Kalamidas has devised a viable mechanism for sending signals faster-than-light. Furthermore all of the components of the Kalamidas device are available in most modern quantum-optics labs. No exotic processes needed—everything in principle is completely understood.

For his FTL machine, Kalamidas employs an unusual method for which-path erasure. Bob "ambiguifies"  the number of photons in each of his two paths by mixing each photon (which is normally in a "Fock state" of definite photon number—either zero or one) with a state of uncertain photon number. In Kalamidas's original paper (OKE), he used for this number-uncertain state a truncated coherent state. In Nick's version of the Kalamidas experiment (NKE), I use a state |U> = x|0> + y|1> (which I call "gray light") as my number-uncertain input.

The beam-splitter math for the NKE experiment is simple but tedious—30 terms that must be carefully squared, added together and matched correctly with the right output detectors. The first time I carried out this calculation, I verified Kalamidas's claim: Bob, by his choice of what to measure could seemingly cause: 1. nothing to happen at Alice's detectors or 2. unencrypted interference to happen at Alice's detectors with a very large amplitude (when gray light parameters were maximized) of 25%.

I was happy to see this result. Not because I believed that I had verified FTL signaling. But because I believed that I had created a paradox (the Kalamidas-Herbert paradox?) which would be resolved in some clever way that might teach us something new about the subtleties of few-photon quantum physics.

I sent my results to Demetrios, who scrutinized them with a critical eye, eventually discovering a simple conceptual error that I had missed over and over again. It's easy to overlook your own mistakes—another good reason for peer review in science.

Correcting my mistake I recalculated and obtained Bob-induced interference at Alice's detectors. But this correctly calculated interference was completely encrypted—only visible (like the Frost choice) if Bob sends a coincidence-triggered decryption signal to Alice at slower-than-light speeds.

My conclusion?

(Quoted from NKE): I wish to congratulate Demetrios Kalamidas for coming up with his imaginative new FTL scheme which gave me much pleasure and excitement to analyze. I would also like to thank him for correcting an error in my work which, up until his intervention, seemed to show confirmation of his FTL signaling claim. After his timely input, the present (presumably correct) calculation demonstrates a complete refutation of any FTL effect. However, the Kalamidas scheme of erasing which-path info by mixing Fock light with gray light is clever and may yet find new technical applications in areas other than superluminal communication technology.

Sketch of the original Kalamidas experiment (OKE)

Quantum Teleportation

Alice and Bob make measurement choices concerning entangled photons A and B

One reason why medieval philosophers such as Thomas Aquinas never developed a sophisticated description of the material world might have been that the priorities of thinkers in the Middle Ages were different than our own. The aim of Thomas Aquinas and his colleagues was to discover the nature of God. Hence they treated the world not as a physical object with its own intrinsic laws but as the personal creation of a divine being. The facts of this world, unimportant in themselves, could tell us about the nature of God in much the same way as a painting or sculpture might inform us about the nature of the artist that created it. A bit of this same theological spirit surfaces in Einstein's famous statement: "I want to know the mind of God; all else is details."

Anyone seeking to know the nature of God by studying the physical universe will certainly be fascinated by quantum theory, our deepest and most successful theory of matter. Quantum theory is deeply paradoxical and seems to obey a distinctly non-human logic. One of the most peculiar feature of this theory is the way it seems to effortlessly embody seemingly contradictory aspects in the same phenomena. One of the most elegant examples of quantum theory's union of opposites is the recently discovered fact of quantum teleportation which unites in one system both a faster-than-light transmission of information plus a clear prohibition against humans using this undeniable FTL connection for sending FTL messages.

Quantum theory also embodies the unusual feature that the world we see depends on the questions that we pose. Hence the more sophisticated we become in asking questions of Nature, the more sophisticated will be Her replies.

Quantum teleportation is a special feature of quantum entanglement in which two photons emitted from a special source give up their individual identities and enter a collective state. The collective two-photon state has definite properties but the individual photons do not, until they are actually observed. For example, in the entangled state W(A,B,+), photons A and B will always be observed to have the same polarization; in the state X(A,B, -), photons A and B are always observed to have opposite polarization. These two entangled states W and X are part of a complete set of entangled two-photon states W, X, Y and Z, called the "Bell states" (after Irish physicist John Stewart Bell). Any two-photon state, whether entangled or not, can be expressed as a sum of the 4 Bell states. This fact is essential to the process of teleportation.

Alice obtains unknown photon "?" she wants to send to Bob

Alice acquires a photon "?" with an unknown polarization which she wants to teleport to Bob. This photon IS NOT ENTANGLED with Alice's photon A but Alice employs a clever trick--only possible in quantum theory. Alice expresses the quantum state of photon "?" and photon A as the sum of the four entangled Bell states W, X, Y and Z. She does this sum in such a way that all the entanglements cancel and the total quantum state of "?" and "A" is unentangled.

Alice's move reminds me of a string trick I learned as a kid in which you wrap a loop of string around your fingers in a complicated way so that it looks as though the fingers are entangled in the string. But upon pulling the string the fingers are freed--every loop of string was cancelled by an anti-loop. It's the same with the two photons--every seeming entanglement is cancelled somewhere by an anti-entanglement.

However because Bob's B photon is entangled with Alice's A photon, a kind of quantum magic occur in which the polarization "?" of Alice's unknown photon is transferred to Bob's photon B, although in a somewhat hidden form. To every term W, X, Y, Z in Alice's expression for her two states, there corresponds on Bob's side of things a quantum state that is either identical to "?" or differs from "?" only by a rotation R and/or a phase shift S. (R and S are fixed by the nature of the original AB entanglement and do not depend on "?".)

Given this setup, here's how quantum teleportation works.

Alice asks the question: which Bell state is my system in? This question can have one of four answers W, X, Y or Z. If the answer is W, then Bob's photon has the polarization "?". Teleportation is accomplished.

If the answer is X, Y or Z, the polarization of Bob's photon differs from "?" only by a rotation R, a phase shift S or a combination of both. So for 100% efficient teleportation all Bob has to know is what Alice's result was--W, X, Y or Z--a piece of knowledge that consists of only 2 bits of information. Without these two bits all that Bob sees is a random hash. With these two bits an infinite amount of information can be teleported. (The polarization of a photon can point anywhere on a sphere. The teleported information corresponds then to sending an unknown latitude and longitude on the surface of the Earth to a distant location faster than light. However this information cannot be decoded without the 2-bit key which must be sent by Alice to Bob at light speed or slower.) Thus a large quantity of quantum information can be teleported faster-than-light but this information is unrecognizable in the absence of a 2-bit code which can only be transferred over conventional channels.

Alice sends a 4-bit signal allowing Bob to decode an infinite-bit message

Quantum teleportation was discovered by a six-man team in 1993 and experimentally demonstrated a few years later. Teleportation is a particularly elegant example of quantum theory's subtle union of opposites--in this case the coexistence of a large FTL data transmission with the impossibility of sending signals faster-than-light.

Let's face it. We are only at the beginning of experiencing and appreciating the inhumanly beautiful mysteries of the quantum world.

QUANTUM REALITY

Shall I look at Her
Or shall I not?

Hard, small, separated
If I look;
Soft, spread-out, connected
If I don't.

Hard particle and soft wave: both?
Utterly random and perfectly predictable: both?
Small right-here and spread-out everywhere: both?
Deep connected yet lonely separate?

Honey
Some day You gotta show me
How You do that.