Bell’s Theorem: No EPR “Reality”

Pre­vi­ously in se­ries: En­tan­gled Photons

(Note: So that this post can be read by peo­ple who haven’t fol­lowed the whole se­ries, I shall tem­porar­ily adopt some more stan­dard and less ac­cu­rate terms; for ex­am­ple, talk­ing about “many wor­lds” in­stead of “de­co­her­ent blobs of am­pli­tude”.)

The leg­endary Bayesian, E. T. Jaynes, be­gan his life as a physi­cist. In some of his writ­ings, you can find Jaynes railing against the idea that, be­cause we have not yet found any way to pre­dict quan­tum out­comes, they must be “truly ran­dom” or “in­her­ently ran­dom”.

Sure, to­day you don’t know how to pre­dict quan­tum mea­sure­ments. But how do you know, asks Jaynes, that you won’t find a way to pre­dict the pro­cess to­mor­row? How can any mere ex­per­i­ments tell us that we’ll never be able to pre­dict some­thing—that it is “in­her­ently un­know­able” or “truly ran­dom”?

As far I can tell, Jaynes never heard about de­co­her­ence aka Many-Wor­lds, which is a great pity. If you be­longed to a species with a brain like a flat sheet of pa­per that some­times split down its thick­ness, you could rea­son­ably con­clude that you’d never be able to “pre­dict” whether you’d “end up” in the left half or the right half. Yet is this re­ally ig­no­rance? It is a de­ter­minis­tic fact that differ­ent ver­sions of you will ex­pe­rience differ­ent out­comes.

But even if you don’t know about Many-Wor­lds, there’s still an ex­cel­lent re­ply for “Why do you think you’ll never be able to pre­dict what you’ll see when you mea­sure a quan­tum event?” This re­ply is known as Bell’s The­o­rem.

In 1935, Ein­stein, Podolsky, and Rosen once ar­gued roughly as fol­lows:

Sup­pose we have a pair of en­tan­gled par­ti­cles, light-years or at least light-min­utes apart, so that no sig­nal can pos­si­bly travel be­tween them over the times­pan of the ex­per­i­ment. We can sup­pose these are po­larized pho­tons with op­po­site po­lariza­tions.

Po­larized filters block some pho­tons, and ab­sorb oth­ers; this lets us mea­sure a pho­ton’s po­lariza­tion in a given ori­en­ta­tion. En­tan­gled pho­tons (with the right kind of en­tan­gle­ment) are always found to be po­larized in op­po­site di­rec­tions, when you mea­sure them in the same ori­en­ta­tion; if a filter at a cer­tain an­gle passes pho­ton A (trans­mits it) then we know that a filter at the same an­gle will block pho­ton B (ab­sorb it).

Now we mea­sure one of the pho­tons, la­beled A, and find that it is trans­mit­ted by a 0° po­larized filter. Without mea­sur­ing B, we can now pre­dict with cer­tainty that B will be ab­sorbed by a 0° po­larized filter, be­cause A and B always have op­po­site po­lariza­tions when mea­sured in the same ba­sis.

Said EPR:

“If, with­out in any way dis­turb­ing a sys­tem, we can pre­dict with cer­tainty (i.e., with prob­a­bil­ity equal to unity) the value of a phys­i­cal quan­tity, then there ex­ists an el­e­ment of phys­i­cal re­al­ity cor­re­spond­ing to this phys­i­cal quan­tity.”

EPR then as­sumed (cor­rectly!) that noth­ing which hap­pened at A could dis­turb B or ex­ert any in­fluence on B, due to the spacelike sep­a­ra­tions of A and B. We’ll take up the rel­a­tivis­tic view­point again to­mor­row; for now, let’s just note that this as­sump­tion is cor­rect.

If by mea­sur­ing A at 0°, we can pre­dict with cer­tainty whether B will be ab­sorbed or trans­mit­ted at 0°, then ac­cord­ing to EPR this fact must be an “el­e­ment of phys­i­cal re­al­ity” about B. Since mea­sur­ing A can­not in­fluence B in any way, this el­e­ment of re­al­ity must always have been true of B. Like­wise with ev­ery other pos­si­ble po­lariza­tion we could mea­sure—10°, 20°, 50°, any­thing. If we mea­sured A first in the same ba­sis, even light-years away, we could perfectly pre­dict the re­sult for B. So on the EPR as­sump­tions, there must ex­ist some “el­e­ment of re­al­ity” cor­re­spond­ing to whether B will be trans­mit­ted or ab­sorbed, in any ori­en­ta­tion.

But if no one has mea­sured A, quan­tum the­ory does not pre­dict with cer­tainty whether B will be trans­mit­ted or ab­sorbed. (At least that was how it seemed in 1935.) There­fore, EPR said, there are el­e­ments of re­al­ity that ex­ist but are not men­tioned in quan­tum the­ory:

“We are thus forced to con­clude that the quan­tum-me­chan­i­cal de­scrip­tion of phys­i­cal re­al­ity given by wave func­tions is not com­plete.”

This is an­other ex­cel­lent ex­am­ple of how seem­ingly im­pec­ca­ble philos­o­phy can fail in the face of ex­per­i­men­tal ev­i­dence, thanks to a wrong as­sump­tion so deep you didn’t even re­al­ize it was an as­sump­tion.

EPR cor­rectly as­sumed Spe­cial Rel­a­tivity, and then in­cor­rectly as­sumed that there was only one ver­sion of you who saw A do only one thing. They as­sumed that the cer­tain pre­dic­tion about what you would hear from B, de­scribed the only out­come that hap­pened at B.

In real life, if you mea­sure A and your friend mea­sures B, differ­ent ver­sions of you and your friend ob­tain both pos­si­ble out­comes. When you com­pare notes, the two of you always find the po­lariza­tions are op­po­site. This does not vi­o­late Spe­cial Rel­a­tivity even in spirit, but the rea­son why not is the topic of to­mor­row’s post, not to­day’s.

To­day’s post is about how, in 1964, Bel­l­dandy John S. Bell ir­re­vo­ca­bly shot down EPR’s origi­nal ar­gu­ment. Not by point­ing out the flaw in the EPR as­sump­tions—Many-Wor­lds was not then widely known—but by de­scribing an ex­per­i­ment that dis­proved them!

It is ex­per­i­men­tally im­pos­si­ble for there to be a phys­i­cal de­scrip­tion of the en­tan­gled pho­tons, which speci­fies a sin­gle fixed out­come of any po­lariza­tion mea­sure­ment in­di­vi­d­u­ally performed on A or B.

This is Bell’s The­o­rem, which rules out all “lo­cal hid­den vari­able” in­ter­pre­ta­tions of quan­tum me­chan­ics. It’s ac­tu­ally not all that com­pli­cated, as quan­tum physics goes!

We be­gin with a pair of en­tan­gled pho­tons, which we’ll name A and B. When mea­sured in the same ba­sis, you find that the pho­tons always have op­po­site po­lariza­tion—one is trans­mit­ted, one is ab­sorbed. As for the first pho­ton you mea­sure, the prob­a­bil­ity of trans­mis­sion or ab­sorp­tion seems to be 50-50.

What if you mea­sure with po­larized filters set at differ­ent an­gles?

Sup­pose that I mea­sure A with a filter set at 0°, and find that A was trans­mit­ted. In gen­eral, if you then mea­sure B at an an­gle θ to my ba­sis, quan­tum the­ory says the prob­a­bil­ity (of my hear­ing that) you also saw B trans­mit­ted, equals sin2 θ. E.g. if your filter was at an an­gle of 30° to my filter, and I saw my pho­ton trans­mit­ted, then there’s a 25% prob­a­bil­ity that you see your pho­ton trans­mit­ted.

(Why? See “De­co­her­ence as Pro­jec­tion”. Some quick san­ity checks: sin(0°) = 0, so if we mea­sure at the same an­gles, the calcu­lated prob­a­bil­ity is 0—we never mea­sure at the same an­gle and see both pho­tons trans­mit­ted. Similarly, sin(90°) = 1; if I see A trans­mit­ted, and you mea­sure at an or­thog­o­nal an­gle, I will always hear that you saw B trans­mit­ted. sin(45°) = √(1/​2), so if you mea­sure in a di­ag­o­nal ba­sis, the prob­a­bil­ity is 5050 for the pho­ton to be trans­mit­ted or ab­sorbed.)

Oh, and the ini­tial prob­a­bil­ity of my see­ing A trans­mit­ted is always 12. So the joint prob­a­bil­ity of see­ing both pho­tons trans­mit­ted is 12 * sin2 θ. 12 prob­a­bil­ity of my see­ing A trans­mit­ted, times sin2 θ prob­a­bil­ity that you then see B trans­mit­ted.

And now you and I perform three statis­ti­cal ex­per­i­ments, with large sam­ple sizes:

(1) First, I mea­sure A at 0° and you mea­sure B at 20°. The pho­ton is trans­mit­ted through both filters on 12 sin2 (20°) = 5.8% of the oc­ca­sions.

(2) Next, I mea­sure A at 20° and you mea­sure B at 40°. When we com­pare notes, we again dis­cover that we both saw our pho­tons pass through our filters, on 12 sin2 (40° − 20°) = 5.8% of the oc­ca­sions.

(3) Fi­nally, I mea­sure A at 0° and you mea­sure B at 40°. Now the pho­ton passes both filters on 12 sin2 (40°) = 20.7% of oc­ca­sions.

Or to say it a bit more com­pactly:

  1. A trans­mit­ted 0°, B trans­mit­ted 20°: 5.8%

  2. A trans­mit­ted 20°, B trans­mit­ted 40°: 5.8%

  3. A trans­mit­ted 0°, B trans­mit­ted 40°: 20.7%

What’s wrong with this pic­ture?

Noth­ing, in real life. But on EPR as­sump­tions, it’s im­pos­si­ble.

On EPR as­sump­tions, there’s a fixed lo­cal ten­dency for any in­di­vi­d­ual pho­ton to be trans­mit­ted or ab­sorbed by a po­larizer of any given ori­en­ta­tion, in­de­pen­dent of any mea­sure­ments performed light-years away, as the sin­gle unique out­come.

Con­sider ex­per­i­ment (2). We mea­sure A at 20° and B at 40°, com­pare notes, and find we both saw our pho­tons trans­mit­ted. Now, A was trans­mit­ted at 20°, so if you had mea­sured B at 20°, B would cer­tainly have been ab­sorbed—if you mea­sure in the same ba­sis you must find op­po­site po­lariza­tions.

That is: If A had the fixed ten­dency to be trans­mit­ted at 20°, then B must have had a fixed ten­dency to be ab­sorbed at 20°. If this rule were vi­o­lated, you could have mea­sured both pho­tons in the 20° ba­sis, and found that both pho­tons had the same po­lariza­tion. Given the way that en­tan­gled pho­tons are ac­tu­ally pro­duced, this would vi­o­late con­ser­va­tion of an­gu­lar mo­men­tum.

So (un­der EPR as­sump­tions) what we learn from ex­per­i­ment (2) can be equiv­a­lently phrased as: “B was a kind of pho­ton that was trans­mit­ted by a 40° filter and would have been ab­sorbed by the 20° filter.” Un­der EPR as­sump­tions this is log­i­cally equiv­a­lent to the ac­tual re­sult of ex­per­i­ment (2).

Now let’s look again at the per­centages:

  1. B is a kind of pho­ton that was trans­mit­ted at 20°, and would not have been trans­mit­ted at 0°: 5.8%

  2. B is a kind of pho­ton that was trans­mit­ted at 40°, and would not have been trans­mit­ted at 20°: 5.8%

  3. B is a kind of pho­ton that was trans­mit­ted at 40°, and would not have been trans­mit­ted at 0°: 20.7%

If you want to try and see the prob­lem on your own, you can stare at the three ex­per­i­men­tal re­sults for a while...

(Spoilers ahead.)

Con­sider a pho­ton pair that gives us a pos­i­tive re­sult in ex­per­i­ment (3). On EPR as­sump­tions, we now know that the B pho­ton was in­her­ently a type that would have been ab­sorbed at 0°, and was in fact trans­mit­ted at 40°. (And con­versely, if the B pho­ton is of this type, ex­per­i­ment (3) will always give us a pos­i­tive re­sult.)

Now take a B pho­ton from a pos­i­tive ex­per­i­ment (3), and ask: “If in­stead we had mea­sured B at 20°, would it have been trans­mit­ted, or ab­sorbed?” Again by EPR’s as­sump­tions, there must be a definite an­swer to this ques­tion. We could have mea­sured A in the 20° ba­sis, and then had cer­tainty of what would hap­pen at B, with­out dis­turb­ing B. So there must be an “el­e­ment of re­al­ity” for B’s po­lariza­tion at 20°.

But if B is a kind of pho­ton that would be trans­mit­ted at 20°, then it is a kind of pho­ton that im­plies a pos­i­tive re­sult in ex­per­i­ment (1). And if B is a kind of pho­ton that would be ab­sorbed at 20°, it is a kind of pho­ton that would im­ply a pos­i­tive re­sult in ex­per­i­ment (2).

If B is a kind of pho­ton that is trans­mit­ted at 40° and ab­sorbed at 0°, and it is ei­ther a kind that is ab­sorbed at 20° or a kind that is trans­mit­ted at 20°; then B must be ei­ther a kind that is ab­sorbed at 20° and trans­mit­ted at 40°, or a kind that is trans­mit­ted at 20° and ab­sorbed at 0°.

So, on EPR’s as­sump­tions, it’s re­ally hard to see how the same source can man­u­fac­ture pho­ton pairs that pro­duce 5.8% pos­i­tive re­sults in ex­per­i­ment (1), 5.8% pos­i­tive re­sults in ex­per­i­ment (2), and 20.7% pos­i­tive re­sults in ex­per­i­ment (3). Every pho­ton pair that pro­duces a pos­i­tive re­sult in ex­per­i­ment (3) should also pro­duce a pos­i­tive re­sult in ei­ther (1) or (2).

“Bell’s in­equal­ity” is that any the­ory of hid­den lo­cal vari­ables im­plies (1) + (2) >= (3). The ex­per­i­men­tally ver­ified fact that (1) + (2) < (3) is a “vi­o­la­tion of Bell’s in­equal­ity”. So there are no hid­den lo­cal vari­ables. QED.

And that’s Bell’s The­o­rem. See, that wasn’t so hor­rible, was it?

But what’s ac­tu­ally go­ing on here?

When you mea­sure at A, and your friend mea­sures at B a few light-years away, differ­ent ver­sions of you ob­serve both pos­si­ble out­comes—both pos­si­ble po­lariza­tions for your pho­ton. But the am­pli­tude of the joint world where you both see your pho­tons trans­mit­ted, goes as √(1/​2) * sin θ where θ is the an­gle be­tween your po­lariz­ers. So the squared mod­u­lus of the am­pli­tude (which is how we get prob­a­bil­ities in quan­tum the­ory) goes as 12 sin2 θ, and that’s the prob­a­bil­ity for find­ing mu­tual trans­mis­sion when you meet a few years later and com­pare notes. We’ll talk to­mor­row about why this doesn’t vi­o­late Spe­cial Rel­a­tivity.

Strength­en­ings of Bell’s The­o­rem elimi­nate the need for statis­ti­cal rea­son­ing: You can show that lo­cal hid­den vari­ables are im­pos­si­ble, us­ing only prop­er­ties of in­di­vi­d­ual ex­per­i­ments which are always true given var­i­ous mea­sure­ments. (Google “GHZ state” or “GHZM state”.) Oc­ca­sion­ally you also hear that some­one has pub­lished a strength­ened Bell’s ex­per­i­ment in which the two par­ti­cles were more dis­tantly sep­a­rated, or the par­ti­cles were mea­sured more re­li­ably, but you get the core idea. Bell’s The­o­rem is proven be­yond a rea­son­able doubt. Now the physi­cists are track­ing down un­rea­son­able doubts, and Bell always wins.

I know I some­times speak as if Many-Wor­lds is a set­tled is­sue, which it isn’t aca­dem­i­cally. (If peo­ple are still ar­gu­ing about it, it must not be “set­tled”, right?) But Bell’s The­o­rem it­self is agreed-upon aca­dem­i­cally as an ex­per­i­men­tal truth. Yes, there are peo­ple dis­cussing the­o­ret­i­cally con­ceiv­able loop­holes in the ex­per­i­ments done so far. But I don’t think any­one out there re­ally thinks they’re go­ing to find an ex­per­i­men­tal vi­o­la­tion of Bell’s The­o­rem as soon as they use a more sen­si­tive pho­ton de­tec­tor.

What does Bell’s The­o­rem plus its ex­per­i­men­tal ver­ifi­ca­tion tell us, ex­actly?

My fa­vorite phras­ing is one I en­coun­tered in D. M. Ap­pleby: “Quan­tum me­chan­ics is in­con­sis­tent with the clas­si­cal as­sump­tion that a mea­sure­ment tells us about a prop­erty pre­vi­ously pos­sessed by the sys­tem.”

Which is ex­actly right: Mea­sure­ment de­co­heres your blob of am­pli­tude (world), split­ting it into sev­eral non­in­ter­act­ing blobs (wor­lds). This cre­ates new in­dex­i­cal un­cer­tainty—un­cer­tainty about which of sev­eral ver­sions of your­self you are. Learn­ing which ver­sion you are, does not tell you a pre­vi­ously un­known prop­erty that was always pos­sessed by the sys­tem. And which spe­cific blobs (wor­lds) are cre­ated, de­pends on the phys­i­cal mea­sur­ing pro­cess.

It’s some­times said that Bell’s The­o­rem rules out “lo­cal re­al­ism”. Tread cau­tiously when you hear some­one ar­gu­ing against “re­al­ism”. As for lo­cal­ity, it is, if any­thing, far bet­ter un­der­stood than this whole “re­al­ity” busi­ness: If life is but a dream, it is a dream that obeys Spe­cial Rel­a­tivity.

It is just one par­tic­u­lar sort of lo­cal­ity, and just one par­tic­u­lar no­tion of which things are “real” in the sense of pre­vi­ously uniquely de­ter­mined, which Bell’s The­o­rem says can­not si­mul­ta­neously be true.

In par­tic­u­lar, de­co­her­ent quan­tum me­chan­ics is lo­cal, and Bell’s The­o­rem gives us no rea­son to be­lieve it is not real. (It may or may not be the ul­ti­mate truth, but quan­tum me­chan­ics is cer­tainly more real than the clas­si­cal hal­lu­ci­na­tion of lit­tle billiard balls bop­ping around.)

Does Bell’s The­o­rem pre­vent us from re­gard­ing the quan­tum de­scrip­tion as a state of par­tial knowl­edge about some­thing more deeply real?

At the very least, Bell’s The­o­rem pre­vents us from in­ter­pret­ing quan­tum am­pli­tudes as prob­a­bil­ity in the ob­vi­ous way. You can­not point at a sin­gle con­figu­ra­tion, with prob­a­bil­ity pro­por­tional to the squared mod­u­lus, and say, “This is what the uni­verse looked like all along.”

In fact, you can­not pick any lo­cally speci­fied de­scrip­tion what­so­ever of unique out­comes for quan­tum ex­per­i­ments, and say, “This is what we have par­tial in­for­ma­tion about.”

So it cer­tainly isn’t easy to rein­ter­pret the quan­tum wave­func­tion as an un­cer­tain be­lief. You can’t do it the ob­vi­ous way. And I haven’t heard of any non-ob­vi­ous in­ter­pre­ta­tion of the quan­tum de­scrip­tion as par­tial in­for­ma­tion.

Fur­ther­more, as I men­tioned pre­vi­ously, it is re­ally odd to find your­self differ­en­ti­at­ing a de­gree of un­cer­tain an­ti­ci­pa­tion to get phys­i­cal re­sults—the way we have to differ­en­ti­ate the quan­tum wave­func­tion to find out how it evolves. That’s not what prob­a­bil­ities are for.

Thus I try to em­pha­size that quan­tum am­pli­tudes are not pos­si­bil­ities, or prob­a­bil­ities, or de­grees of un­cer­tain be­lief, or ex­pres­sions of ig­no­rance, or any other species of epistemic crea­tures. Wave­func­tions are not states of mind. It would be a very bad sign to have a fun­da­men­tal physics that op­er­ated over states of mind; we know from look­ing at brains that minds are made of parts.

In con­clu­sion, al­though Ein­stein, Podolsky, and Rosen pre­sented a pic­ture of the world that was dis­proven ex­per­i­men­tally, I would still re­gard them as hav­ing won a moral vic­tory: The then-com­mon in­ter­pre­ta­tion of quan­tum me­chan­ics did in­deed have a one per­son mea­sur­ing at A, see­ing a sin­gle out­come, and then mak­ing a cer­tain pre­dic­tion about a unique out­come at B; and this is in­deed in­com­pat­i­ble with rel­a­tivity, and wrong. Though peo­ple are still ar­gu­ing about that.

Part of The Quan­tum Physics Sequence

Next post: “Spooky Ac­tion at a Dis­tance: The No-Com­mu­ni­ca­tion The­o­rem

Pre­vi­ous post: “En­tan­gled Pho­tons