The Misleading Equivalence of Decoherenceand Branching

Guido Bacciagaluppi

Lausanne, 3 May 2014(slides revised 25 May 2014)

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Introduction

Ken Wharton’s short story ‘Aloha’ (in Analog ScienceFiction and Fact, June 2003) is a love story between twopeople with opposite time arrows meeting around the mid-point of a universe with both past and future low-entropyboundary conditions.

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The funny thing about free will, Felix thought, was that italways maintained its own illusion.

Aloha (for that was what he had named her) had implied thatthey had met several times in her past—in Felix’s future—sonow he knew something about the choices he would soon make.In a universe where free will reigned supreme, it would be asimple matter to create a paradox. Felix merely had to choosenot to see Aloha again, and the universe would be inconsistent.

He laughed out loud at the idea. As if he were more powerfulthan the universe.

No, paradox-prevention had turned out to be a major under-pinning of reality, the lynchpin to explaining why quantummechanics worked the way that it did. He couldn’t force aparadox, no matter how he tried.

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We are familiar with (maybe not quite as) odd thingsthrough the time-reversal and time-recurrence objectionsin the foundations of statistical mechanics.

I suggest we should be worrying about the same kind ofproblems also in the context of decoherence.

4

Standard models of decoherence take unentangled statesto extremely highly entangled states, and thereby veryeffectively diagonalise the reduced state of the system inthe eigenbasis of the decohering variables.

Schematically,

|ψ〉 ⊗ |E0〉 →∑i

ci(t)|ψi〉 ⊗ |Ei(t)〉

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We may imagine (and in finite dimensions we know wehave) recurrence after a suitably long time:

|ψ〉 ⊗ |E0〉 →∑i

ci(t)|ψi〉 ⊗ |Ei(t)〉 → |ψ〉 ⊗ |E0〉

What is happening in the very long time in-between? Andgiven that (except when we artificially isolate a system)decoherence goes on all the time, should we not expectthat whatever is in fact happening is a fairly generic kindof behaviour?

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Here is a possible intuition: the system is in a dynamicallystationary state of entanglement with the environment, inthat the interaction involves as many particles gettingentangled with the system as getting disentangled fromit.

We shall give a very elementary look at this possibility,from two points of view:

• decoherence and branching• dynamical consistency

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Decoherence and branching

What seems to tell against this possibility: decoherenceinduces branching of the quantum state, but we cannothave branching in both directions simultaneously, or onlytrivially so.

And even if it makes sense, we can see that the universalwave function is non-trivially branching all the time.

(Or rather, the evidence is naturally interpreted in termsof constant branching, at least if we adopt a no-collapseapproach to quantum mechanics – especially in Everett.)

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We shall need a few definitions.

A history is a time-ordered sequence of (Heisenberg-picture)projectors:

Pi1(t1), Pi2(t2), . . . , Pin(tn)

The associated history operator is:

Cα(n) := Pin(tn) . . . Pi1(t1)

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If for all k the Pik(tk) are mutually orthogonal and sumto the identity we have a history space.

If we form sums of the projections in {Cα(n)}, we obtaina coarse-graining of the history space.

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Take a quantum state |Ψ〉, and define the decoherencefunctional

D(Cα(n), Cβ(n)) := Tr(Cα(n)|Ψ〉〈Ψ|C

∗β(n)

)The positive number D(Cα(n), Cα(n)) is called the weightof the history Cα(n).

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Weak decoherence condition: for distinct histories,

ReD(Cα(n), Cβ(n)) = 0

Weak decoherence is equivalent to the disappearance ofinterference terms between histories, i. e. we may sumweights when coarse-graining (the weights ‘behave likeprobabilities’).

[As Diósi points out, weak decoherence does not respect composition:the real part of the decoherence functional of a product is not theproduct of the real part of the decoherence functionals of the factors!]

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Decoherence condition: for distinct histories,

D(Cα(n), Cβ(n)) = 0Decoherence is equivalent to orthogonality of the vectorsCα(m)(tm)|Ψ〉, thus to existence of mutually orthogonalprojections Rα(m)(tm) summing to identity, with

Rα(m)(tm)|Ψ〉 = Cα(m)(tm)|Ψ〉 (1)

(‘permanent records’).

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In general, these projections might just be given by

Rα(m)(tm) = Cα(m)(tm)|Ψ〉〈Ψ|C∗α(m)(tm)

(‘generalised records’).

We shall focus on the familiar case of environmentally-induced decoherence, with records in the environment, e.g.

Rα(m)(tm) = E1i1

(tm)⊗ E2i2(tm)⊗ . . .⊗ Emim

(tm) (2)

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The (schematic) picture is that at each time tk, the systeminteracts with some degree of freedom in the environmentsuch that

|ψi〉 ⊗ |ek0〉 → |ψi〉 ⊗ |eki 〉 (3)

and each such environmental degree of freedom then evolvesseparately to |eki (tm)〉.

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Each projection (2) thus picks out exactly one componentin the state

|ψ(tm)〉 =∑i

ci(tm)|ψi〉|e1i1(tm)〉|e2i2

(tm)〉 . . . |emim(tm)〉

and is a permanent record in the sense of (1).

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Indeed, if we define a fine-graining of the original historyspace by

Pj1(t1)⊗Rα(1)(t1) . . . , Pjn(tn)⊗Rα(n)(tn) (4)this space decoheres, and for tm ≥ tk the conditionalprobability of

∑α(m)|ik=jk Rα(m)(tm) given Pjk(tk) is 1.

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The space (4) is an example of a history space that isbranching (past-deterministic, backwards deterministic),i. e. any two histories of non-zero weight that coincide atany time tj, coincide also at all previous times ti.

One can prove the branching-decoherence theorem: if ahistory space is branching, then it is decoherent; and if ahistory space is decoherent, then it is a coarse-graining ofa branching history space.

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Note that analogous definitions and results can be derivedif we consider instead the so-called backwards decoherencefunctional

D(C∗α(n), C∗β(n)) = Tr

(C∗α(n)|Ψ〉〈Ψ|Cβ(n)

)Indeed, imagine that at each tk, the system interacts withsome degree of freedom in the environment such that

|ψi〉 ⊗ |fki 〉 ← |ψi〉 ⊗ |fk0 〉 (5)

backwards in time.

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Then the projections

Sα̃(m)(tm) = Fmim

(tm)⊗ Fm+1im+1 (tm)⊗ . . .⊗ Fnin

(tm)

(with α̃(m) the multi-index im, im+1, . . . , in) are recordsin the environment of later events.

It follows that the histories

Pj1(t1)⊗ Sα̃(1)(t1) . . . , Pjn(tn)⊗ Sα̃(n)(tn) (6)

will define a history space that is antibranching (forwarddeterministic, future-deterministic).

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There is also a ‘two-state’ version of the formalism thatuses the time-symmetric decoherence functional

Dsym(α(n), β(n)) = Tr(ρfCα(n)ρiC

∗β(n)

)Note, however, that if both initial and final state are purethen

Dsym(α(n), β(n)) = 〈Ψf |Cα(n)|Ψi〉〈Ψi|Cβ(n)|Ψf〉which will vanish for any distinct histories iff only onehistory has non-zero probability.

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Note also that forwards and backwards decoherence canbe non-trivially satisfied simultaneously.

It further implies that forwards and backwards weightscoincide.

(See GB 2007.)

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Now for the sake of argument assume that (3) and (5) aredynamically consistent, at least during the given interval[t1, tn].

Then our history space

Pi1(t1), Pi2(t2), . . . , Pin(tn)

will indeed decohere both forwards and backwards.

(We shall return below also to the two-state formalism.)

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Further, the history spaces (4) and (6) will be fine-grainingsthat are respectively branching and antibranching, andone would expect that their common refining

Pj1 ⊗Rα(1) ⊗ Sβ̃(1), . . . , Pjn ⊗Rα(n) ⊗ Sβ̃(n)would be in fact both branching (past-deterministic) andantibranching (future-deterministic).

That is, a history space satisfying both (3) and (5) wouldbe a coarse-graining of a deterministic history space.

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It this by itself a fatal objection to the existence of bothrecords of past and of future events?

Not necessarily: branching becomes perspectival.

If one coarse-grains over the records of the future, onerecovers the familiar branching structure of decoherence.

And if one coarse-grains over the records of the past, oneobtains an analogous antibranching structure.

The real question is that of dynamical consistency of (3)and (5).

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Dynamical consistency

We shall look at a simple example. Take n = 2, and letthe state at t−1 be a superposition or mixture of the fourcomponents

|ψ1〉|e10〉|e20〉|f

11 〉|f

21 〉

|ψ1〉|e10〉|e20〉|f

11 〉|f

22 〉

|ψ2〉|e10〉|e20〉|f

12 〉|f

21 〉

|ψ2〉|e10〉|e20〉|f

11 〉|f

22 〉

(no records of past events, but records of future events).

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The state evolves between t−1 and t+1 to the corresponding

superposition or mixture of

|ψ1〉|e11〉|e20〉|f

10 〉|f

21 〉

|ψ1〉|e11〉|e20〉|f

10 〉|f

22 〉

|ψ2〉|e12〉|e20〉|f

10 〉|f

21 〉

|ψ2〉|e12〉|e20〉|f

10 〉|f

22 〉

consistently with (3) and (5).

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Now assume the system evolves freely between t+1 and

t−2 according to some unitary evolution. We obtain thecorresponding superposition or mixture of:

(a|ψ1〉 + b∗|ψ2〉) |e11〉|e20〉|f

10 〉|f

21 〉

(a|ψ1〉 + b∗|ψ2〉) |e11〉|e20〉|f

10 〉|f

22 〉

(b|ψ1〉 − a∗|ψ2〉) |e12〉|e20〉|f

10 〉|f

21 〉

(b|ψ1〉 − a∗|ψ2〉) |e12〉|e20〉|f

10 〉|f

22 〉

(they are orthogonal, so there is no interference even if weassume an initial superposition).

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But now we are in trouble, because |ψi〉 does not alwaysmatch up with |f2i 〉, e. g. the first component has evolvedto

a|ψ1〉|e11〉|e21〉|f

10 〉|f

21 〉 + b

∗|ψ2〉|e11〉|e22〉|f

10 〉|f

21 〉

and in order to preserve unitarity also between t−2 and t+2

this needs to evolve to some

a|ψ1〉|e11〉|e21〉|f

10 〉|f

20 〉 + b

∗|ψ2〉|e11〉|e22〉|f

10 〉|f

2? 〉

which is indeed inconsistent with assuming (5).

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As with Felix in the short story, if the system evolves freelythe universe is inconsistent...

What about paradox prevention?

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There are in fact two (essentially equivalent) methods ofimplementing ‘paradox prevention’.

First, we can postselect for the ‘good’ components, bypostulating there is also a ‘final state’ which is a mixtureof the components

|ψ1〉|e11〉|e21〉|f

10 〉|f

20 〉

|ψ2〉|e11〉|e22〉|f

10 〉|f

20 〉

|ψ1〉|e12〉|e21〉|f

10 〉|f

20 〉

|ψ2〉|e12〉|e22〉|f

10 〉|f

20 〉

(it has to be a mixed state because of (7)).

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The forward-evolving state ρi will develop componentscontaining states orthogonal to |f20 〉, and the backwards-evolving state ρf will develop components containing states

orthogonal to |e10〉, but they will all be assigned weight 0by the symmetrised decoherence functional.

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Alternatively, we can deny there is free evolution betweent+1 and t

−2 , and postulate that each of

|ψ1〉|e11〉|e20〉|f

10 〉|f

21 〉

|ψ1〉|e11〉|e20〉|f

10 〉|f

22 〉

|ψ2〉|e12〉|e20〉|f

10 〉|f

21 〉

|ψ2〉|e12〉|e20〉|f

10 〉|f

22 〉

evolves, respectively,

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to

|ψ1〉|e11〉|e20〉|f

10 〉|f

21 〉

|ψ2〉|e11〉|e20〉|f

10 〉|f

22 〉

|ψ1〉|e12〉|e20〉|f

10 〉|f

21 〉

|ψ2〉|e12〉|e20〉|f

10 〉|f

22 〉

(up to phases).

Note that this can be implemented unitarily.

Both methods yield the same histories (and coefficientscan be chosen to yield the same probabilities.)

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Conclusion

We see that time-symmetric decoherence leads to somesatisfiable but odd constraints on the dynamical evolutionalong histories.

Indeed, the evolution of the system between t+1 and t−2

depends on both the records of the past and the future.

On the other hand, this behaviour might not be obviousto a time-directed observer.

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If some such effect were generically present in decoherenceinteractions, the customary picture of branching throughdecoherence would be misleading.

Branching would be a perspectival effect of coarse-grainingover the (unobserved) records of the future, and therewould be no branching at a more fundamental level.

From an Everettian perspective, there would thus be nosplitting of observers, but only good old uncertainty aboutself-location!

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