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.