Decoherence

Pre­vi­ously in se­ries: Feyn­man Paths

To un­der­stand the quan­tum pro­cess called “de­co­her­ence”, we first need to look at how the spe­cial case of quan­tum in­de­pen­dence can be de­stroyed—how the evolu­tion of a quan­tum sys­tem can pro­duce en­tan­gle­ment where there was formerly in­de­pen­dence.

Conf6 Quan­tum in­de­pen­dence, as you’ll re­call, is a spe­cial case of am­pli­tude dis­tri­bu­tions that ap­prox­i­mately fac­tor­ize—am­pli­tude dis­tri­bu­tions that can be treated as a product of sub-dis­tri­bu­tions over sub­spaces.

Reluc­tant tourists vis­it­ing quan­tum uni­verses think as if the ab­sence of a rec­t­an­gu­lar plaid pat­tern is some kind of spe­cial ghostly link be­tween par­ti­cles. Hence the un­for­tu­nate term, “quan­tum en­tan­gle­ment”.

The evolu­tion of a quan­tum sys­tem can pro­duce en­tan­gle­ment where there was formerly in­de­pen­dence—turn a rec­t­an­gu­lar plaid pat­tern into some­thing else. Quan­tum in­de­pen­dence, be­ing a spe­cial case, is eas­ily lost.

Entangler Let’s pre­tend for a mo­ment that we’re look­ing at a clas­si­cal sys­tem, which will make it eas­ier to see what kind of phys­i­cal pro­cess leads to en­tan­gle­ment.

At right is a sys­tem in which a pos­i­tively charged light thingy is on a track, far above a nega­tively charged heavy thingy on a track.

At the be­gin­ning, the two thin­gies are far enough apart that they’re not sig­nifi­cantly in­ter­act­ing.

But then we lower the top track, bring­ing the two thin­gies into the range where they can eas­ily at­tract each other. (Op­po­site charges at­tract.)

So the light thingy on top rolls to­ward the heavy thingy on the bot­tom. (And the heavy thingy on the bot­tom rolls a lit­tle to­ward the top thingy, just like an ap­ple at­tracts the Earth as it falls.)

Now switch to the Feyn­man path in­te­gral view. That is, imag­ine the evolu­tion of a quan­tum sys­tem as a sum over all the paths through con­figu­ra­tion space the ini­tial con­di­tions could take.

Sup­pose the bot­tom heavy thingy and the top thingy started out in a state of quan­tum in­de­pen­dence, so that we can view the am­pli­tude dis­tri­bu­tion over the whole sys­tem as the product of a “bot­tom thingy dis­tri­bu­tion” and a “top thingy dis­tri­bu­tion”.

Superposition2 The bot­tom thingy dis­tri­bu­tion starts with bul­ges in three places—which, in the Feyn­man path view, we might think of as three pos­si­ble start­ing con­figu­ra­tions from which am­pli­tude will flow.

When we lower the top track, the light thingy on top is at­tracted to­ward the heavy bot­tom thingy -

- ex­cept that the bot­tom thingy has a sub-dis­tri­bu­tion with three bul­ges in three differ­ent po­si­tions.

So the end re­sult is a joint dis­tri­bu­tion in which there are three bul­ges in the am­pli­tude dis­tri­bu­tion over joint con­figu­ra­tion space, cor­re­spond­ing to three differ­ent joint po­si­tions of the top thingy and bot­tom thingy.

I’ve been try­ing very care­fully to avoid say­ing things like “The bot­tom thingy is in three places at once” or “in each pos­si­bil­ity, the top thingy is at­tracted to wher­ever the bot­tom thingy is”.

Still, you’re prob­a­bly go­ing to vi­su­al­ize it that way, whether I say it or not. To be hon­est, that’s how I drew the di­a­gram—I vi­su­al­ized three pos­si­bil­ities and three re­sult­ing out­comes. Well, that’s just how a hu­man brain tends to vi­su­al­ize a Feyn­man path in­te­gral.

But this doesn’t mean there are ac­tu­ally three pos­si­ble ways the uni­verse could be, etc. That’s just a trick for vi­su­al­iz­ing the path in­te­gral. All the am­pli­tude flows ac­tu­ally hap­pen, they are not pos­si­bil­ities.

Now imag­ine that, in the start­ing state, the bot­tom thingy has an am­pli­tude-fac­tor that is smeared out over the whole bot­tom track; and the top thingy has an am­pli­tude-fac­tor in one place. Then the joint dis­tri­bu­tion over “top thingy, bot­tom thingy” would start out look­ing like the plaid pat­tern at left, and de­velop into the non-plaid pat­tern at right:

Entanglecloud

Here the hori­zon­tal co­or­di­nate cor­re­sponds to the top thingy, and the ver­ti­cal co­or­di­nate cor­re­sponds to the bot­tom thingy. So we start with the top thingy lo­cal­ized and the bot­tom thingy spread out, and then the sys­tem de­vel­ops into a joint dis­tri­bu­tion where the top thingy and the bot­tom thingy are in the same place, but their mu­tual po­si­tion is spread out. Very loosely speak­ing.

So an ini­tially fac­tor­iz­able dis­tri­bu­tion, evolved into an “en­tan­gled sys­tem”—a joint am­pli­tude dis­tri­bu­tion that is not vie­w­able as a product of dis­tinct fac­tors over sub­spaces.

(Im­por­tant side note: You’ll note that, in the di­a­gram above, sys­tem evolu­tion obeyed the sec­ond law of ther­mo­dy­nam­ics, aka Liou­ville’s The­o­rem. Quan­tum evolu­tion con­served the “size of the cloud”, the vol­ume of am­pli­tude, the to­tal amount of grey area in the di­a­gram.

If in­stead we’d started out with a big light-gray square—mean­ing that both par­ti­cles had am­pli­tude-fac­tors widely spread—then the sec­ond law of ther­mo­dy­nam­ics would pro­hibit the com­bined sys­tem from de­vel­op­ing into a tight dark-gray di­ag­o­nal line.

A sys­tem has to start in a low-en­tropy state to de­velop into a state of quan­tum en­tan­gle­ment, as op­posed to just a diffuse cloud of am­pli­tude.

Mu­tual in­for­ma­tion is also ne­gen­tropy, re­mem­ber. Quan­tum am­pli­tudes aren’t in­for­ma­tion per se, but the rule is analo­gous: Am­pli­tude must be highly con­cen­trated to look like a neatly en­tan­gled di­ag­o­nal line, in­stead of just a big diffuse cloud. If you imag­ine am­pli­tude dis­tri­bu­tions as hav­ing a “quan­tum en­tropy”, then an en­tan­gled sys­tem has low quan­tum en­tropy.)

Okay, so now we’re ready to dis­cuss de­co­her­ence.

Multiblobdeco

The sys­tem at left is highly en­tan­gled—it’s got a joint dis­tri­bu­tion that looks some­thing like, “There’s two par­ti­cles, and ei­ther they’re both over here, or they’re both over there.”

Yes, I phrased this as if there were two sep­a­rate pos­si­bil­ities, rather than a sin­gle phys­i­cally real am­pli­tude dis­tri­bu­tion. Se­ri­ously, there’s no good way to use a hu­man brain to talk about quan­tum physics in English.

But if you can just re­mem­ber the gen­eral rule that say­ing “pos­si­bil­ity” is short­hand for “phys­i­cally real blob within the am­pli­tude dis­tri­bu­tion”, then I can de­scribe am­pli­tude dis­tri­bu­tions a lot faster by us­ing the lan­guage of un­cer­tainty. Just re­mem­ber that it is lan­guage. “Either the par­ti­cle is over here, or it’s over there” means a phys­i­cally real am­pli­tude dis­tri­bu­tion with blobs in both places, not that the par­ti­cle is in one of those places but we don’t know which.

Any­way. Deal­ing with highly en­tan­gled sys­tems is of­ten an­noy­ing—for hu­man physi­cists, not for re­al­ity, of course. It’s not just that you’ve got to calcu­late all the pos­si­ble out­comes of the differ­ent pos­si­ble start­ing con­di­tions. (I.e., add up a lot of phys­i­cally real am­pli­tude flows in a Feyn­man path in­te­gral.) The pos­si­ble out­comes may in­terfere with each other. (Which ac­tual pos­si­ble out­comes would never do, but differ­ent blobs in an am­pli­tude dis­tri­bu­tion do.) Like, maybe the two par­ti­cles that are both over here, or both over there, meet twenty other par­ti­cles and do a lit­tle dance, and at the con­clu­sion of the path in­te­gral, many of the fi­nal con­figu­ra­tions have re­ceived am­pli­tude flows from both ini­tial blobs.

But that kind of ex­tra-an­noy­ing en­tan­gle­ment only hap­pens when the blobs in the ini­tial sys­tem are close enough that their evolu­tion­ary paths can slop over into each other. Like, if the par­ti­cles were ei­ther both here, or both there, but here and there were two light-years apart, then any sys­tem evolu­tion tak­ing less than a year, couldn’t have the differ­ent pos­si­ble out­comes over­lap­ping.

Precohered_2 Okay, so let’s talk about three par­ti­cles now.

This di­a­gram shows a blob of am­pli­tude that fac­tors into the product of a 2D sub­space and a 1D sub­space. That is, two en­tan­gled par­ti­cles and one in­de­pen­dent par­ti­cle.

The ver­ti­cal di­men­sion is the one in­de­pen­dent par­ti­cle, the length and breadth are the two en­tan­gled par­ti­cles.

The in­de­pen­dent par­ti­cle is in one definite place—the cloud of am­pli­tude is ver­ti­cally nar­row. The two en­tan­gled par­ti­cles are ei­ther both here, or both there. (Again I’m us­ing that wrong lan­guage of un­cer­tainty, words like “definite” and “ei­ther”, but you see what I mean.)

Now imag­ine that the third in­de­pen­dent par­ti­cle in­ter­acts with the two en­tan­gled par­ti­cles in a sen­si­tive way. Maybe the third par­ti­cle is bal­anced on the top of a hill; and the two en­tan­gled par­ti­cles pass nearby, and at­tract it mag­net­i­cally; and the third par­ti­cle falls off the top of the hill and rolls to the bot­tom, in that par­tic­u­lar di­rec­tion.

Decohered After­ward, the new am­pli­tude dis­tri­bu­tion might look like this. The third par­ti­cle is now en­tan­gled with the other two par­ti­cles. And the am­pli­tude dis­tri­bu­tion as a whole con­sists of two more widely sep­a­rated blobs.

Loosely speak­ing, in the case where the two en­tan­gled par­ti­cles were over here, the third par­ti­cle went this way, and in the case where the two en­tan­gled par­ti­cles were over there, the third par­ti­cle went that way.

So now the fi­nal am­pli­tude dis­tri­bu­tion is fully en­tan­gled—it doesn’t fac­tor into sub­spaces at all.

But the two blobs are more widely sep­a­rated in the con­figu­ra­tion space. Be­fore, each blob of am­pli­tude had two par­ti­cles in differ­ent po­si­tions; now each blob of am­pli­tude has three par­ti­cles in differ­ent po­si­tions.

In­deed, if the third par­ti­cle in­ter­acted in an es­pe­cially sen­si­tive way, like be­ing tipped off a hill and slid­ing down, the new sep­a­ra­tion could be much larger than the old one.

Ac­tu­ally, it isn’t nec­es­sary for a par­ti­cle to get tipped off a hill. It also works if you’ve got twenty par­ti­cles in­ter­act­ing with the first two, and end­ing up en­tan­gled with them. Then the new am­pli­tude dis­tri­bu­tion has got two blobs, each with twenty-two par­ti­cles in differ­ent places. The dis­tance be­tween the two blobs in the joint con­figu­ra­tion space is much greater.

And the greater the dis­tance be­tween blobs, the less likely it is that their am­pli­tude flows will in­ter­sect each other and in­terfere with each other.

That’s de­co­her­ence. De­co­her­ence is the third key to re­cov­er­ing the clas­si­cal hal­lu­ci­na­tion, be­cause it makes the blobs be­have in­de­pen­dently; it lets you treat the whole am­pli­tude dis­tri­bu­tion as a sum of sep­a­rated non-in­terfer­ing blobs.

In­deed, once the blobs have sep­a­rated, the pat­tern within a sin­gle blob may look a lot more plaid and rec­t­an­gu­lar—I tried to show that in the di­a­gram above as well.

Thus, the big headache in quan­tum com­put­ing is pre­vent­ing de­co­her­ence. Quan­tum com­put­ing re­lies on the am­pli­tude dis­tri­bu­tions stay­ing close enough to­gether in con­figu­ra­tion space to in­terfere with each other. And the en­vi­ron­ment con­tains a zillion par­ti­cles just beg­ging to ac­ci­den­tally in­ter­act with your frag­ile qubits, teas­ing apart the pieces of your painstak­ingly sculpted am­pli­tude dis­tri­bu­tion.

And you can’t just mag­i­cally make the pieces of the scat­tered am­pli­tude dis­tri­bu­tion jump back to­gether—these are blobs in the joint con­figu­ra­tion, re­mem­ber. You’d have to put the en­vi­ron­men­tal par­ti­cles in the same places, too.

(Sounds pretty ir­re­versible, doesn’t it? Like try­ing to un­scram­ble an egg? Well, that’s a very good anal­ogy, in fact.

This is why I em­pha­sized ear­lier that en­tan­gle­ment hap­pens start­ing from a con­di­tion of low en­tropy. De­co­her­ence is ir­re­versible be­cause it is an es­sen­tially ther­mo­dy­namic pro­cess.

It is a fun­da­men­tal prin­ci­ple of the uni­verse—as far as we can tell—that if you “run the film back­ward” all the fun­da­men­tal laws are still obeyed. If you take a movie of an egg fal­ling onto the floor and smash­ing, and then play the film back­ward and see a smashed egg leap­ing off the floor and into a neat shell, you will not see the known laws of physics vi­o­lated in any par­tic­u­lar. All the molecules will just hap­pen to bump into each other in just the right way to make the egg leap off the floor and re­assem­ble. It’s not im­pos­si­ble, just un­be­liev­ably im­prob­a­ble.

Like­wise with a smashed am­pli­tude dis­tri­bu­tion sud­denly as­sem­bling many dis­tantly scat­tered blobs into mu­tual co­her­ence—it’s not im­pos­si­ble, just ex­tremely im­prob­a­ble that many dis­tant start­ing po­si­tions would end up send­ing am­pli­tude flows to nearby fi­nal lo­ca­tions. You are far more likely to see the re­verse.

Ac­tu­ally, in ad­di­tion to run­ning the film back­ward, you’ve got to turn all the pos­i­tive charges to nega­tive, and re­verse left and right (or some other sin­gle di­men­sion—es­sen­tially you have to turn the uni­verse into its mir­ror image).

This is known as CPT sym­me­try, for Charge, Par­ity, and Time.

CPT sym­me­try ap­pears to be a re­ally, re­ally, re­ally deep prin­ci­ple of the uni­verse. Try­ing to vi­o­late CPT sym­me­try doesn’t sound quite as awful to a mod­ern physi­cist as try­ing to throw a base­ball so hard it trav­els faster than light. But it’s al­most that awful. I’m told that Gen­eral Rel­a­tivity Quan­tum Field The­ory re­quires CPT sym­me­try, for one thing.

So the fact that de­co­her­ence looks like a one-way pro­cess, but is only ther­mo­dy­nam­i­cally ir­re­versible rather than fun­da­men­tally asym­met­ri­cal, is a very im­por­tant point. It means quan­tum physics obeys CPT sym­me­try.

It is a uni­ver­sal rule in physics—ac­cord­ing to our best cur­rent knowl­edge—that ev­ery ap­par­ently ir­re­versible pro­cess is a spe­cial case of the sec­ond law of ther­mo­dy­nam­ics, not the re­sult of time-asym­met­ric fun­da­men­tal laws.)

To sum up:

De­co­her­ence is a ther­mo­dy­namic pro­cess of ever-in­creas­ing quan­tum en­tan­gle­ment, which, through an amaz­ing sleight of hand, mas­quer­ades as in­creas­ing quan­tum in­de­pen­dence: De­co­her­ent blobs don’t in­terfere with each other, and within a sin­gle blob but not the to­tal dis­tri­bu­tion, the blob is more fac­tor­iz­able into sub­spaces.

Thus, de­co­her­ence is the third key to re­cov­er­ing the clas­si­cal hal­lu­ci­na­tion. De­co­her­ence lets a hu­man physi­cist think about one blob at a time, with­out wor­ry­ing about how blobs in­terfere with each other; and the blobs them­selves, con­sid­ered as iso­lated in­di­vi­d­u­als, are less in­ter­nally en­tan­gled, hence eas­ier to un­der­stand. This is a fine thing if you want to pre­tend the uni­verse is clas­si­cal, but not so good if you want to fac­tor a mil­lion-digit num­ber be­fore the Sun burns out.

Part of The Quan­tum Physics Sequence

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