Spooky Action at a Distance: The No-Communication Theorem

Pre­vi­ously in se­ries: Bell’s The­o­rem: No EPR “Real­ity”

When you have a pair of en­tan­gled par­ti­cles, such as op­po­sitely po­larized pho­tons, one par­ti­cle seems to some­how “know” the re­sult of dis­tant mea­sure­ments on the other par­ti­cle. If you mea­sure pho­ton A to be po­larized at 0°, pho­ton B some­how im­me­di­ately knows that it should have the op­po­site po­lariza­tion of 90°.

Ein­stein fa­mously called this “spukhafte Fern­wirkung” or “spooky ac­tion at a dis­tance”. Ein­stein didn’t know about de­co­her­ence, so it seemed spooky to him.

Though, to be fair, Ein­stein knew perfectly well that the uni­verse couldn’t re­ally be “spooky”. It was a then-pop­u­lar in­ter­pre­ta­tion of QM that Ein­stein was call­ing “spooky”, not the uni­verse it­self.

Let us first con­sider how en­tan­gled par­ti­cles look, if you don’t know about de­co­her­ence—the rea­son why Ein­stein called it “spooky”:

Sup­pose we’ve got op­po­sitely po­larized pho­tons A and B, and you’re about to mea­sure B in the 20° ba­sis. Your prob­a­bil­ity of see­ing B trans­mit­ted by the filter (or ab­sorbed) is 50%.

But wait! Be­fore you mea­sure B, I sud­denly mea­sure A in the 0° ba­sis, and the A pho­ton is trans­mit­ted! Now, ap­par­ently, the prob­a­bil­ity that you’ll see B trans­mit­ted is 11.6%. Some­thing has changed! And even if the pho­tons are light-years away, spacelike sep­a­rated, the change still oc­curs.

You might try to re­ply:

“No, noth­ing has changed—mea­sur­ing the A pho­ton has told you some­thing about the B pho­ton, you have gained knowl­edge, you have car­ried out an in­fer­ence about a dis­tant ob­ject, but no phys­i­cal in­fluence trav­els faster-than-light.

“Sup­pose I put two in­dex cards into an en­velope, one marked ‘+’ and one marked ‘-’. Now I give one en­velope to you, and one en­velope to a friend of yours, and you get in a space­ship and travel a few light-years away from each other, and then you open your en­velope and see ‘+’. At once you know that your friend is hold­ing the en­velope marked ‘-’, but this doesn’t mean the en­velope’s con­tent has changed faster than the speed of light.

“You are com­mit­ting a Mind Pro­jec­tion Fal­lacy; the en­velope’s con­tent is con­stant, only your lo­cal be­liefs about dis­tant refer­ents change.”

Bell’s The­o­rem, cov­ered yes­ter­day, shows that this re­ply fails. It is not pos­si­ble that each pho­ton has an un­known but fixed in­di­vi­d­ual ten­dency to be po­larized a par­tic­u­lar way. (Think of how un­likely it would seem, a pri­ori, for this to be some­thing any ex­per­i­ment could tell you!)

Ein­stein didn’t know about Bell’s The­o­rem, but the the­ory he was crit­i­ciz­ing did not say that there were hid­den vari­ables; it said that the prob­a­bil­ities changed di­rectly.

But then how fast does this in­fluence travel? And what if you mea­sure the en­tan­gled par­ti­cles in such a fash­ion that, in their in­di­vi­d­ual refer­ence frames, each mea­sure­ment takes place be­fore the other?

Th­ese ex­per­i­ments have been done. If you think there is an in­fluence trav­el­ing, it trav­els at least six mil­lion times as fast as light (in the refer­ence frame of the Swiss Alps). Nor is the in­fluence fazed if each mea­sure­ment takes place “first” within its own refer­ence frame.

So why can’t you use this mys­te­ri­ous in­fluence to send sig­nals faster than light?

Here’s some­thing that, as a kid, I couldn’t get any­one to ex­plain to me: “Why can’t you sig­nal us­ing an en­tan­gled pair of pho­tons that both start out po­larized up-down? By mea­sur­ing A in a di­ag­o­nal ba­sis, you de­stroy the up-down po­lariza­tion of both pho­tons. Then by mea­sur­ing B in the up-down/​left-right ba­sis, you can with 50% prob­a­bil­ity de­tect the fact that a mea­sure­ment has taken place, if B turns out to be left-right po­larized.”

It’s par­tic­u­larly an­noy­ing that no­body gave me an an­swer, be­cause the an­swer turns out to be sim­ple: If both pho­tons have definite po­lariza­tions, they aren’t en­tan­gled. There are just two differ­ent pho­tons that both hap­pen to be po­larized up-down. Mea­sur­ing one pho­ton doesn’t even change your ex­pec­ta­tions about the other.

En­tan­gle­ment is not an ex­tra prop­erty that you can just stick onto oth­er­wise nor­mal par­ti­cles! It is a break­down of quan­tum in­de­pen­dence. In clas­si­cal prob­a­bil­ity the­ory, if you know two facts, there is no longer any log­i­cal de­pen­dence left be­tween them. Like­wise in quan­tum me­chan­ics, two par­ti­cles each with a definite state must have a fac­tor­iz­able am­pli­tude dis­tri­bu­tion.

Or as old-style quan­tum the­ory put it: En­tan­gle­ment re­quires su­per­po­si­tion, which im­plies un­cer­tainty. When you mea­sure an en­tan­gled par­ti­cle, you are not able to force your mea­sure­ment re­sult to take any par­tic­u­lar value. So, over on the B end, if they do not know what you mea­sured on A, their prob­a­bil­is­tic ex­pec­ta­tion is always the same as be­fore. (So it was once said).

But in old-style quan­tum the­ory, there was in­deed a real and in­stan­ta­neous change in the other par­ti­cle’s statis­tics which took place as the re­sult of your own mea­sure­ment. It had to be a real change, by Bell’s The­o­rem and by the in­visi­bly as­sumed unique­ness of both out­comes.

Even though the old the­ory in­voked a non-lo­cal in­fluence, you could never use this in­fluence to sig­nal or com­mu­ni­cate with any­one. This was called the “no-sig­nal­ing con­di­tion” or the “no-com­mu­ni­ca­tion the­o­rem”.

Still, on then-cur­rent as­sump­tions, they couldn’t ac­tu­ally call it the “no in­fluence of any kind what­so­ever the­o­rem”. So Ein­stein cor­rectly la­beled the old the­ory as “spooky”.

In de­co­her­ent terms, the im­pos­si­bil­ity of sig­nal­ing is much eas­ier to un­der­stand: When you mea­sure A, one ver­sion of you sees the pho­ton trans­mit­ted and an­other sees the pho­ton ab­sorbed. If you see the pho­ton ab­sorbed, you have not learned any new em­piri­cal fact; you have merely dis­cov­ered which ver­sion of your­self “you” hap­pen to be. From the per­spec­tive at B, your “dis­cov­ery” is not even the­o­ret­i­cally a fact they can learn; they know that both ver­sions of you ex­ist. When B fi­nally com­mu­ni­cates with you, they “dis­cover” which world they them­selves are in, but that’s all. The statis­tics at B re­ally haven’t changed—the to­tal Born prob­a­bil­ity of mea­sur­ing ei­ther po­lariza­tion is still just 50%!

A com­mon defense of the old the­ory was that Spe­cial Rel­a­tivity was not vi­o­lated, be­cause no “in­for­ma­tion” was trans­mit­ted, be­cause the su­per­lu­mi­nal in­fluence was always “ran­dom”. As some Hans de Vries fel­low points out, in­for­ma­tion the­ory says that “ran­dom” data is the most ex­pen­sive kind of data you can trans­mit. Nor is “ran­dom” in­for­ma­tion always use­less: If you and I gen­er­ate a mil­lion en­tan­gled par­ti­cles, we can later mea­sure them to ob­tain a shared key for use in cryp­tog­ra­phy—a highly use­ful form of in­for­ma­tion which, by Bell’s The­o­rem, could not have already been there be­fore mea­sur­ing.

But wait a minute. De­co­her­ence also lets you gen­er­ate the shared key. Does de­co­her­ence re­ally not vi­o­late the spirit of Spe­cial Rel­a­tivity?

De­co­her­ence doesn’t al­low “sig­nal­ing” or “com­mu­ni­ca­tion”, but it al­lows you to gen­er­ate a highly use­ful shared key ap­par­ently out of nowhere. Does de­co­her­ence re­ally have any ad­van­tage over the old-style the­ory on this one? Or are both the­o­ries equally obey­ing Spe­cial Rel­a­tivity in prac­tice, and equally vi­o­lat­ing the spirit?

A first re­ply might be: “The shared key is not ‘ran­dom’. Both you and your friend gen­er­ate all pos­si­ble shared keys, and this is a de­ter­minis­tic and lo­cal fact; the cor­re­la­tion only shows up when you meet.”

But this just re­veals a deeper prob­lem. The counter-ob­jec­tion would be: “The mea­sure­ment that you perform over at A, splits both A and B into two parts, two wor­lds, which guaran­tees that you’ll meet the right ver­sion of your friend when you re­unite. That is non-lo­cal physics—some­thing you do at A, makes the world at B split into two parts. This is spooky ac­tion at a dis­tance, and it too vi­o­lates the spirit of Spe­cial Rel­a­tivity. Tu quoque!”

And in­deed, if you look at our quan­tum calcu­la­tions, they are writ­ten in terms of joint con­figu­ra­tions. Which, on re­flec­tion, doesn’t seem all that lo­cal!

But wait—what ex­actly does the no-com­mu­ni­ca­tion the­o­rem say? Why is it true? Per­haps, if we knew, this would bring en­light­en­ment.

Here is where it starts get­ting com­pli­cated. I my­self don’t fully un­der­stand the no-com­mu­ni­ca­tion the­o­rem—there are some parts I think I can see at a glance, and other parts I don’t. So I will only be able to ex­plain some of it, and I may have got­ten it wrong, in which case I pray to some physi­cist to cor­rect me (or at least tell me where I got it wrong).

When we did the calcu­la­tions for en­tan­gled po­larized pho­tons, with A’s po­lariza­tion mea­sured us­ing a 30° filter, we calcu­lated that the ini­tial state

√(1/​2) * ( [ A=(1 ; 0) ∧ B=(0 ; 1) ] - [ A=(0 ; 1) ∧ B=(1; 0) ] )

would be de­co­hered into a blob for

( -(√3)/​2 * √(1/​2) * [ A=(-(√3)/​2 ; 12) ∧ B=(0 ; 1) ] )
- ( 12 * √(1/​2) * [ A=(-(√3)/​2 ; 12) ∧ B=(1; 0) ] )

and sym­met­ri­cally (though we didn’t do this calcu­la­tion) an­other blob for

( 12 * √(1/​2) * [ A=(1/​2 ; (√3)/​2) ∧ B=(0 ; 1) ] )
- ( (√3)/​2 * √(1/​2) * [ A=(1/​2 ; (√3)/​2) ∧ B=(1; 0) ] )

Th­ese two blobs to­gether add up, lin­early, to the ini­tial state, as one would ex­pect. So what changed? At all?

What changed is that the fi­nal re­sult at A, for the first blob, is re­ally more like:

(Sen­sor-A-reads-“ABSORBED”) * (Ex­per­i­menter-A-sees-“ABSORBED”) *
{ ( -(√3)/​2 * √(1/​2) * [ A=(-(√3)/​2 ; 12) ∧ B=(0 ; 1) ] )
-( 12 * √(1/​2) * [ A=(-(√3)/​2 ; 12) ∧ B=(1; 0) ] ) }

and cor­re­spond­ingly with the TRANSMITTED blob.

What changed is that one blob in con­figu­ra­tion space, was de­co­hered into two dis­tantly sep­a­rated blobs that can’t in­ter­act any more.

As we saw from the Heisen­berg “Uncer­tainty Prin­ci­ple”, de­co­her­ence is a visi­ble, ex­per­i­men­tally de­tectable effect. That’s why we have to shield quan­tum com­put­ers from de­co­her­ence. So couldn’t the de­co­her­ing mea­sure­ment at A, have de­tectable con­se­quences for B?

But think about how B sees the ini­tial state:

√(1/​2) * ( [ A=(1 ; 0) ∧ B=(0 ; 1) ] - [ A=(0 ; 1) ∧ B=(1; 0) ] )

From B’s per­spec­tive, this state is already “not all that co­her­ent”, be­cause no mat­ter what B does, it can’t make the A=(1 ; 0) and A=(0 ; 1) con­figu­ra­tions cross paths. There’s already a sort of de­co­her­ence here—a sep­a­ra­tion that B can’t elimi­nate by any lo­cal ac­tion at B.

And as we’ve ear­lier glimpsed, the ba­sis in which you write the ini­tial state is ar­bi­trary. When you write out the state, it has pretty much the same form in the 30° mea­sur­ing ba­sis as in the 0° mea­sur­ing ba­sis.

In fact, there’s noth­ing pre­vent­ing you from writ­ing out the ini­tial state with A in the 30° ba­sis and B in the 0° ba­sis, so long as your num­bers add up.

In­deed this is ex­actly what we did do, when we first wrote out the four terms in the two blobs, and didn’t in­clude the sen­sor or ex­per­i­menter.

So when A per­ma­nently de­co­hered the blobs in the 30° ba­sis, from B’s per­spec­tive, this merely solid­ified a de­co­her­ence that B could have viewed as already ex­ist­ing.

Ob­vi­ously, this can’t change the lo­cal evolu­tion at B (he said, wav­ing his hands a bit).

Now this is only a state­ment about a quan­tum mea­sure­ment that just de­co­heres the am­pli­tude for A into parts, with­out A it­self evolv­ing in in­ter­est­ing new di­rec­tions. What if there were many par­ti­cles on the A side, and some­thing hap­pened on the A side that put some of those par­ti­cles into iden­ti­cal con­figu­ra­tions via differ­ent paths?

This is where lin­ear­ity and uni­tar­ity come in. The no-com­mu­ni­ca­tion the­o­rem re­quires both con­di­tions: in gen­eral, vi­o­lat­ing lin­ear­ity or uni­tar­ity gives you faster-than-light sig­nal­ing. (And nu­mer­ous other su­per­pow­ers, such as solv­ing NP-com­plete prob­lems in polyno­mial time, and pos­si­bly Out­come Pumps.)

By lin­ear­ity, we can con­sider parts of the am­pli­tude dis­tri­bu­tion sep­a­rately, and their evolved states will add up to the evolved state of the whole.

Sup­pose that there are many par­ti­cles on the A side, but we count up ev­ery con­figu­ra­tion that cor­re­sponds to some sin­gle fixed state of B—say, B=(0 ; 1) or B=France, what­ever. We’d get a group of com­po­nents which looked like:

(AA=1 ∧ AB=2 ∧ AC=Fred ∧ B=France) +
(AA=2 ∧ AB=1 ∧ AC=Sally ∧ B=France) + …

Lin­ear­ity says that we can de­com­pose the am­pli­tude dis­tri­bu­tion around states of B, and the evolu­tion of the parts will add to the whole.

As­sume that the B side stays fixed. Then this com­po­nent of the dis­tri­bu­tion that we have just iso­lated, will not in­terfere with any other com­po­nents, be­cause other com­po­nents have differ­ent val­ues for B, so they are not iden­ti­cal con­figu­ra­tions.

And uni­tary evolu­tion says that what­ever the mea­sure—the in­te­grated squared mod­u­lus—of this com­po­nent, the to­tal mea­sure is the same af­ter evolu­tion at A, as be­fore.

So as­sum­ing that B stays fixed, then any­thing what­so­ever hap­pen­ing at A, won’t change the mea­sure of the states at B (he said, wav­ing his hands some more).

Nor should it mat­ter whether we con­sider A first, or B first. Any­thing that hap­pens at A, within some com­po­nent of the am­pli­tude dis­tri­bu­tion, only de­pends on the A fac­tor, and only hap­pens to the A fac­tor; like­wise with B; so the fi­nal joint am­pli­tude dis­tri­bu­tion should not de­pend on the or­der in which we con­sider the evolu­tions (and he waved his hands a fi­nal time).

It seems to me that from here it should be easy to show no com­mu­ni­ca­tion con­sid­er­ing the si­mul­ta­neous evolu­tion of A and B. Sadly I can’t quite see the last step of the ar­gu­ment. I’ve spent very lit­tle time do­ing ac­tual quan­tum calcu­la­tions—this is not what I do for a liv­ing—or it would prob­a­bly be ob­vi­ous. Un­less it’s more sub­tle than it ap­pears, but any­way...

Any­way, if I’m not mis­taken—though I’m feel­ing my way here by math­e­mat­i­cal in­tu­ition—the no-com­mu­ni­ca­tion the­o­rem man­i­fests as in­var­i­ant gen­er­al­ized states of en­tan­gle­ment. From B’s per­spec­tive, they are en­tan­gled with some dis­tant en­tity A, and that en­tan­gle­ment has an in­var­i­ant shape that re­mains ex­actly the same no mat­ter what hap­pens at A.

To me, at least, this sug­gests that the ap­par­ent non-lo­cal­ity of quan­tum physics is a mere ar­ti­fact of the rep­re­sen­ta­tion used to de­scribe it.

If you write a 3-di­men­sional vec­tor as “30° west of north, 40° up­ward slope, and 100 me­ters long,” it doesn’t mean that the uni­verse has a ba­sic com­pass grid, or that there’s a global di­rec­tion of up, or that re­al­ity runs on the met­ric sys­tem. It means you chose a con­ve­nient rep­re­sen­ta­tion.

Physics, in­clud­ing quan­tum physics, is rel­a­tivis­ti­cally in­var­i­ant: You can pick any rel­a­tivis­tic frame you like, redo your calcu­la­tions, and always get the same ex­per­i­men­tal pre­dic­tions back out. That we know.

Now it may be that, in the course of do­ing your calcu­la­tions, you find it con­ve­nient to pick some refer­ence frame, any refer­ence frame, and use that in your math. Green­wich Mean Time, say. This doesn’t mean there re­ally is a cen­tral clock, some­where un­der­neath the uni­verse, that op­er­ates on Green­wich Mean Time.

The rep­re­sen­ta­tion we used talked about “joint con­figu­ra­tions” of A and B in which the states of A and B were si­mul­ta­neously speci­fied. This means our rep­re­sen­ta­tion was not rel­a­tivis­tic; the no­tion of “si­mul­tane­ity” is ar­bi­trary. We as­sumed the uni­verse ran on Green­wich Mean Time, in effect.

I don’t know what kind of rep­re­sen­ta­tion would be (1) rel­a­tivis­ti­cally in­var­i­ant, (2) show dis­tant en­tan­gle­ment as in­var­i­ant, (3) di­rectly rep­re­sent space-time lo­cal­ity, and (4) evolve each el­e­ment of the new rep­re­sen­ta­tion in a way that de­pended only on an im­me­di­ate neigh­bor­hood of other el­e­ments.

But that rep­re­sen­ta­tion would prob­a­bly be a lot closer to the Tao.

My sus­pi­cion is that a bet­ter rep­re­sen­ta­tion might take its ba­sic math­e­mat­i­cal ob­jects as lo­cal states of en­tan­gle­ment. I’ve ac­tu­ally sus­pected this ever since I heard about holo­graphic physics and the en­tan­gle­ment en­tropy bound. But that’s just raw spec­u­la­tion, at this point.

How­ever, it is im­por­tant that a fun­da­men­tal rep­re­sen­ta­tion be as lo­cal and as sim­ple as pos­si­ble. This is why e.g. “his­to­ries of the en­tire uni­verse” make poor “fun­da­men­tal” ob­jects, in my hum­ble opinion.

And it’s why I find it sus­pi­cious to have a rep­re­sen­ta­tion for calcu­lat­ing quan­tum physics that talks about a rel­a­tivis­ti­cally ar­bi­trary “joint con­figu­ra­tion” of A and B, when it seems like each lo­cal po­si­tion has an in­var­i­ant “dis­tant en­tan­gle­ment” that suffices to de­ter­mine lo­cal evolu­tion. Shouldn’t we be able to re­fac­tor this rep­re­sen­ta­tion into smaller pieces?

Though ul­ti­mately you do have to re­trieve the phe­nomenon where the ex­per­i­menters meet again, af­ter be­ing sep­a­rated by light-years, and dis­cover that they mea­sured the pho­tons with op­po­site po­lariza­tions. Which is prov­ably not some­thing you can get from in­di­vi­d­ual billiard balls bop­ping around.

I sus­pect that when we get a rep­re­sen­ta­tion of quan­tum me­chan­ics that is lo­cal in ev­ery way that the physics it­self is lo­cal, it will be im­me­di­ately ob­vi­ous—right there in the rep­re­sen­ta­tion—that things only hap­pen in one place at a time.

Hence, no faster-than-light com­mu­ni­ca­tors. (Dam­mit!)

Now of course, all this that I have said—all this won­drous nor­mal­ity—re­lies on the de­co­her­ence view­point.

It re­lies on be­liev­ing that when you mea­sure at A, both pos­si­ble mea­sure­ments for A still ex­ist, and are still en­tan­gled with B in a way that B sees as in­var­i­ant.

All the am­pli­tude in the joint con­figu­ra­tion is un­der­go­ing lin­ear, uni­tary, lo­cal evolu­tion. None of it van­ishes. So the prob­a­bil­ities at B are always the same from a global stand­point, and there is no supralu­mi­nal in­fluence, pe­riod.

If you tried to “in­ter­pret” things any differ­ently… well, the no-com­mu­ni­ca­tion the­o­rem would be­come a lot less ob­vi­ous.

Part of The Quan­tum Physics Sequence

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