arXiv:1401.1786v1 [cond-mat.stat-mech] 25 Nov 2013 eatoso u-ytmwt t environment. its in- with the sub-system macroscopic from a arises of of that teractions behavior decoherence the classical explores the objects and collapse tion world macroscopic the phenomena. of related closely nature all ma- classical are density the Neumann von and state the trix, mixed problem, application the measurement of the so-called emergence following The eigenstate operator. single an superposition of a a to from states system to quantum of reference a the in of ‘collapse’ of related word transition the closely collapse the of course the use of traditional as is the This here to to function. wave referred or is system states of sition paper. are this from questions in Both distribution addressed does principles? probability how first and statistical mechanical quantum states, the quantum derive of mixture one a wave superposition to quantum function underlying one the from of get one experience classical quantum the pure time. to a a en- akin at in state an is each as which systems, problem of state, the mixture of or aspects semble statistical the presses inaernee eoeetb h neato ihthe with interaction the by func- decoherent wave rendered sub-system are original super- tion the the (envi- of is, components states That quantum posed einselection). pure or of selection, function mixture wave ronmental a the into causes collapse sub-system to the to reser- or coupled environment voir the that says approach the sentially ia ii fpr ttsi h ujc fogigde- ongoing for of resuscitated subject the periodically mortem is post is states Schr¨odinger’s cat The pure bate; clas- al- time. of observed its a the limit to at with passes sical state system states of one quantum superposition in a lowed whereby be world ever classical process the only precise in can whereas system in states, a simultaneously of exist may superposition system a quantum a that is ics em ftevnNuandniymatrix. density Neumann in von formulation the conventional of the terms in disappears problem unu ttsia ehnc.I eoeec,Wv Funct Wave Decoherence, I. Mechanics. Statistical Quantum n rmsn praht h rbeso aefunc- wave of problems the to approach promising One superpo- a from states of mixture a of appearance The does how arise: naturally questions fundamental Two ehp h otpzln seto unu mechan- quantum of aspect puzzling most the Perhaps nqatmsaitclmcais h superposition the mechanics, statistical quantum In rbe,adt h lsia aueo h arsoi worl macroscopic the decoh of for nature mechanism classical n the einselection that to the function and wave to problem, system results total present conserve the a the of for entanglement law fo conservation the mechanism the and The from results average. it quantum mix the general: a for into form collapse trace states Neumann superposition the that shown also examination. h rbblt prtri eie rmfis rnilsfo principles first from derived is operator probability The .INTRODUCTION I. Dtd 1Ot,21,[email protected]) 2013, Oct., 31 (Dated: emn est Matrix Density Neumann 1–4 hsex- This 5–12 hlAttard Phil Es- emn est arxeeg rmtefis principles first Both the derivation. from su- states. emerge a pure matrix density von from of the Neumann collapse and mixture operator to a probability form. Maxwell-Boltzmann system to the entangled the states causes an of that has perposition this necessarily is system It function total is wave It the the law conservation reservoir). of the a to with due sub- that energy a shown exchange (i.e. can system equilibrium that canonical system a of average cal the A. with Appendix differences in given and is similarities approach the the and present of of decoherence summary discussion to brief a approach A einselection device. conventional measuring or reservoir c:rpeetn h ytmi h ai corresponding basis the im- operator, in is some system axiom to the This mechan- representing quantum of ics: justification. interpretation standard a the by for plied B Appendix See rtri simply is erator otoa otepoaiiyo h ytmbigi h mi- the in being system the crostate of probability the to portional O erator of weight in- the to to proportional need microstate. this each factor would then additional value weight, an expectation equal clude have the not for did expression microstates the If unu ubradlbl h nrymcotts and macrostates, energy g the labels and number quantum that rymcottscnb ae obe to taken be can microstates ergy repre- energy the in system the sentation. of microstates the labels nrp fteeeg irsae is microstates energy the of entropy k 1 = h rsn ae eie oml o h statisti- the for formula a derives paper present The ti xoai that axiomatic is It osdra sltdsbsse ihHmloinop- Hamiltonian with sub-system isolated an Consider rmteeulwih odto,tewih fteen- the of weight the condition, weight equal the From | ζ kg O H| , ,ecagal aibe(uha energy) as (such variable exchangeable d, ˆ ueo ttsgvn h ovninlvon conventional the giving states of ture 2 i H h product the , neulbimqatmsse.I is It system. quantum equilibrium an r N , . . . , ζ ˆ r discussed. are d ζ rne oteqatmmeasurement quantum the to erence, kg E kg O l irsae fa sltdsystem isolated an of microstates all n rhnra nryegnucin such eigenfunctions energy orthonormal and i csaiyflos h eeac of relevance The follows. ecessarily . 2,3 h olpei on ob quite be to found is collapse the r = k ihti,teepcainvleo h op- the of value expectation the this, With E aestedgnrt nrystates; energy degenerate the labels O k I DERIVATION II. | aeeulweight. equal have ( ζ | ψ ψ kg E ψ = ) i o olpe n h von the and Collapse, ion i kg O Here . = ψ kg P O P ∗ kg kg sitrrtda en pro- being as interpreted is k ψ ψ kg O kg O stepicpeenergy principle the is ∗ | ψ ζ S kg kg O O kg E w i O with , = kg E k / k P n the and 1 = B ln kg O ˆ w ψ | kg ζ E kg O kg O ∗ ψ 0, = i ζ (1) kg O kg E = . 2

r s where kB is Boltzmann’s constant. Note that the same such that Ej1 = Etot − Ei1 . Assuming that the coeffi- s r value is used irrespective of the energy of the isolated sys- cient matrix is dyadic, cig,jh = cigcjh, then for all j 6= j1 r tem and whether or not this is equal to Ek. According one must have cjh = 0. But this would mean that there to the set theoretic formulation of probability for statis- is only one value of the reservoir energy, and hence that 13,14 tical mechanics, the weight of an energy macrostate Ei1 is the only possible value of the sub-system energy. E E is therefore wk = g wkg = Nk and the entropy of an Since this contradicts the fundamental definition of the energy macrostateP is just the logarithm of the number statistical system, namely that the sub-system can ex- E of microstates that it contains, Sk = kB ln Nk. This change energy with the reservoir, one concludes that the is Boltzmann’s original formula. Finally, soon will be coefficient matrix cannot be dyadic and that the wave invoked the thermodynamic definition of temperature, state of the total system must be entangled.15 namely13,14 Using the entangled form of the total wave function, and the vanishing of the coefficients for terms that vi- 1 ∂S(E) = . (2) olate energy conservation, the expectation value of an T ∂E arbitrary operator on the sub-system is

Now allow the sub-system s to exchange energy with 1 (s) (r) ∗ O(ψ ) = c ′ ′ ′ ′ c a reservoir r such that the total system is isolated and tot Z′ i g ,j h ig,jh ig,iX′g′ jh,jX′h′ therefore Etot = Es + Er is constant. The Hilbert space s,E ˆ s,E r,E r,E is Htot = Hs ⊗Hr. If it is possible to factor the wave state × hζi′ g′ |O|ζig ihζj′ h′ |ζjh i of the total system as 1 (s) (r) ∗ s,E = ′ ci′g′,jhcig,jhOi′g′,ig Z ′ ′ |ψtoti = |ψsi |ψri , (3) ig,iXg Xjh 1 (s) (r) ∗ s,E = c ′ c O ′ . (7) then the wave state is said to be a separable state, or a Z′ ig ,jih ig,ji h ig ,ig product state. Here and below the direct product symbol i,g,gX′ Xh on the right hand side is suppressed. Conversely, with ′ s r Here the norm is Z ≡ hψtot|ψtoti, ji is defined such that {|ζi i} a basis for the sub-system and {|ζj i} a basis for r s Eji = Etot − Ei , and the energy conservation condition, the reservoir, the most general wave state of the total Eq. (6), has been used. system is One sees how energy conservation and entanglement

s r have collapsed the original four sums over independent |ψtoti = cij |ζi i|ζj i. (4) principle energy states to a single sum over principle en- Xi,j ergy states. Specifically the principle energy states of s r the sub-system have at this stage collapsed, whilst the If the coefficient matrix is dyadic, cij = ci cj , then the (s) s s degenerate energy states of the sub-system remain in su- wave state is separable with |ψsi = i ci |ζi i, and perposition form. (r) r r |ψri = j cj |ζj i. Alternatively, if the coefficientsP can’t An expression for the reservoir entropy will be required be writtenP in dyadic form, then the wave state is insepa- for the two derivations of the statistical average that fol- 2,3 rable. This is called an entangled state. low. One has the axiomatic result given above that the Whether or not the total system can be written as reservoir entropy for a reservoir energy macrostate is the a product state or is necessarily an entangled state is logarithm of the number of degenerate states, the main difference between the present analysis and the 5–12 r r r r (r) conventional decoherence einselection approach. (See S (Ej )= kB ln Nj , where Nj ≡ . (8) Appendix A.) Xh In general conservation laws give rise to entangled This must equal the general thermodynamic result that, states. This may be seen explicitly for the present case given a total energy Etot and a sub-system energy Es, of energy exchange, with Etot = Es + Er fixed. Using the the reservoir entropy is13,14 energy eigenfunctions described above as a basis for the Es sub-system and the reservoir, the most general expansion Sr(Er)= Sr(Etot − Es) = const. − . (9) of the total wave function is T By definition, the sub-system has been assumed to be s,E r,E |ψtoti = cig,jh|ζig i|ζjh i. (5) infinitely smaller than the reservoir and consequently the ig,jhX Taylor expansion can be terminated at the first term. Before giving the formal result for the statistical aver- In view of energy conservation one has, age of the expectation value, an equivalent but perhaps more physical argument is first given. One can impose c = 0 if Er + Es 6= E . (6) ig,jh j i tot the condition on the coefficients in the entangled expan- sion that It is not possible to satisfy this if cig,jh is dyadic. s r The proof is by contradiction. Suppose that for i1, 1, Ei + Ej = Etot s |cig,jh(Etot)| = s r (10) Ei1 < Etot and hence there exists a corresponding j1  0, Ei + Ej 6= Etot. 3

r N That all the allowed coefficients have the same magni- ji 1 (s) s,E ∗ ′ ′ tude is a statement of the fact that microstates of the = ′′ Oig ,ig dc cig ,jih cig,ji h Z ′ Z total system that have the same total energy have equal i,g,gX hX=1 r weight. Hence all total wave functions ψtot compatible N ji with the energy constraint must have the same weight. 1 (s) s,E 2 = O dc |cig,j h| (See also the more general wave space derivation below.) Z′′ ig,ig Z i Xi,g hX=1 This choice means that the non-zero coefficients are r iαig,jh const. (s) S /kB s,E of the form cig,jh = e , with the α being real. (In = e ji O Z′′ ig,ig essence, this is the so-called EPR state.) Different total Xi,g wave functions correspond to different sets of exponents. s 1 − B s,E = (s)e Ei /k T O If one were to repeat the expectation value with many Z ig,ig different choice of ψtot, the product of coefficients that Xig ∗ ′ appears in Eq. (7), cig ,jihcig,jih, would average to zero ˆ ′ = Tr ℘ˆO . (12) unless g = g. This will be explicitly shown in the statis- n o tical derivation in wave space that follows next. The third equality follows because the terms in which the Using successively the two forms for the reservoir en- integrand is odd vanish. The fourth equality follows be- tropy above, as well as the form for the coefficients, ′ cause all the integrations give the same value irrespective Eq. (10), and setting g = g, the average of the expecta- of the indeces. That this value is infinite is of no concern tion value of the operator, Eq. (7), becomes because it is incorporated into the normalization factor.

r The origin here of the Maxwell-Boltzmann probabil- 1 S /kB s,E ˆ (s) ji O = ′ e Oig,ig ity operator is the same as in the preceding approximate stat Z D E Xig derivation: it comes first from the reservoir entropy due 1 − to the sum over degenerate reservoir energy microstates, = (s)e Ei/kBT Os,E Z ig,ig and then from exploiting the diagonal nature of the en- Xig ergy operator in the energy representation. s,E 1 (s) −Hˆ/kBT s,E Although this expression is identical with the von Neu- = e Oig,i′ g′ Z i′g′,ig mamn expression, no reference to an ensemble of sys- ig,iX′ g′ n o tems is made or implied here. This is also the case in = Tr ℘ˆOˆ , (11) the author’s approach to classical equilibrium13 and non- n o equilibrium14 statistical mechanics.

−1 S/kˆ B −1 −Hˆ/kBT One sees in this derivation that it is the statistical av- with℘ ˆ = Z e = Z e and Z = erage that causes the superposition of the degenerate en- −Hˆ/kBT Tr e . The penultimate equality follows because ergy microstates of the sub-system to collapse into a mix- the energy operator is diagonal in the energy representa- ture. (The collapse of the energy macrostates occurred s,E s,E ˆ s ′ ′ tion, hζi′g′ |H|ζig i = Ei δi,i δg,g . at the level of the expectation value due to the energy The Maxwell-Boltzmann probability operator arises conservation law and the entanglement of the reservoir directly from the sum over the degenerate energy mi- and the sub-system.) Whether one says that this rep- crostates of the reservoir, which gives the exponential resents the wave function collapse, or whether one says of the reservoir entropy for each particular sub-system that it shows only that the superposition states do not energy macrostate. This converts directly to the matrix contribute to the statistical average, is a moot point. In representation of the Maxwell-Boltzmann probability op- the sense that all macroscopic observations and measure- erator in the energy representation. Although the energy ments involve thermodynamic systems and statistical av- representation was used to derive this result, the final ex- erages, either interpretation is compatible with the clas- pression as a trace of the product of the two operators is sical result that a superposition of states is not observed invariant with respect to the representation.1–4 or measured. A more general statistical derivation integrates over In order to see clearly that the von Neumann expres- all possible values of the coefficients that respect en- sion for the statistical average, Eq. (12), really is akin to ergy conservation. Again, a uniform weight density is a classical average over one state at a time rather than a invoked that reflects that fact that all total wave func- quantum average over superposition states, one only has tions with the same total energy have equal weight, to contrast it with the latter, which would be an integral r i dψtot ≡ dc ≡ kg,jh dckg,jh dckg,jh, with the real and over the sub-system wave space of a probability density imaginary partsQ of the coefficient each belonging to the ℘(ψs) and the expectation value O(ψs), real line, ∈ (−∞, ∞). Using the fundamental form for the expectation value, Eq. (7), one obtains for the statis- Oˆ 6= dψs ℘(ψs)O(ψs). (13) tical average D Estat Z

ˆ 1 This average evidently includes all superposition states O = ′′ dc O(ψtot) of the sub-system. One can show that it is not possible D Estat Z Z 4 to reduce it to the von Neumann expression, Eq. (12), is the problem of the classical world, where macroscopic without explicitly invoking the collapse of the wave func- objects are observed to be in only one state at a time and tion into pure energy quantum states. In contrast, one never in a superposition of states. can show explicitly (see Paper II in this series) that the It is these two issues (rather than quantum statistical average can be written mechanics per se) that have provided the motivation for pursuing the theory of decoherence by einselection. The s ˆ s ˆ s hψ |℘ˆO|ψ i interpretation of them in terms of einselection have been O = dψ s s , (14) 5–12 D Estat Z hψ |ψ i discussed in detail by the advocates of the latter. An alternative interpretation of them based on the present which is really just the continuum analogue of the trace, results is now given. and as such it manifests the collapse of the wave function.

Detecting Schr¨odingers Cat III. CONCLUSION The relevance of the present derivation of quantum The specific motivation for this paper has been to de- statistical mechanics to the measurement problem is that rive from first principle an expression for the probabil- in general a measurement device has to interact with a ity operator and an expression for the statistical average sub-system, generally via the exchange of energy or of of an observable operator in a quantum mechanical sys- some other conserved quantity. As such the measurement tem. Invoking the axiom that all microstates (i.e. states device may be regarded as a reservoir for the sub-system with no further degeneracy) have the same weight, it was that becomes entangled with it via the conservation law. shown that for a canonical equilibrium system (i.e. where The present formalism shows how this causes the wave a sub-system can exchange energy with a reservoir), the function of the sub-system to collapse. probability operator was the Maxwell-Boltzmann oper- ator. In detail, the Maxwell-Boltzmann operator ulti- mately arose from the sum over the degenerate reser- voir energy states for a reservoir energy implied by the Dissecting Schr¨odingers Cat given sub-system energy (i.e. compatible reservoir mi- crostates). Of course the Maxwell-Boltzmann operator The classical nature of the macroscopic world is eluci- has always been used as the von Neumann density oper- dated by addressing several questions that are provoked ator for a canonical system. The difference here is that by the present analysis. Is it necessary actually to ap- the result was derived from first principles, not simply ply an external operator in order to collapse the wave invoked by analogy with the classical result. function and thereby to give a classical macroscopic sys- It was also shown that the statistical average reduced tem? The alternative possibility is that the system spon- to the von Neumann form of the trace of the product taneously collapses (e.g. due to self-interactions). In ei- of the probability operator and the observable operator. ther case, having collapsed, does the system remain in the This form has the interpretation that the system has col- collapsed state between measurements (i.e. is the collapse lapsed into a mixture of quantum states from the un- real and irreversible)? In general quantum systems are derlying superposition of quantum states. In the present not real in the sense that their properties have no ex- paper the collapse occurred in two stages: first the prin- istence independent of measurement. Classical systems ciple energy states collapsed due to the conservation of are the opposite. energy and the consequence entanglement of the wave 1. A system in a mixed state has collapsed and the functions of the sub-system and the reservoir. And sec- statistical average is the same as the classical one (since ond the contribution from the superposition of the sub- the trace becomes an integral over phase space, the prob- system energy microstates (i.e. the distinct states of a ability operator becomes the probability density, and the given energy) vanished upon statistical averaging the to- observable operator becomes the corresponding classical tal entangled wave function over the total Hilbert space. function of phase space). One is therefore justified in That the statistical average of an operator on an open saying that a collapsed system is a classical system. system (i.e. a sub-system that can exchange a conserved 2. It was shown in the text that any measurement on a quantity with a reservoir) collapses from the expectation sub-system able to exchange a conserved variable with a value of a superposition of states to an average over a reservoir revealed a collapsed sub-system. By the above, mixture of states occurs in any statistical system, not just this is a classical system. Since this holds for any mea- the present canonical one, and it has broad implication surement (i.e. any applied operator) one is justified in beyond quantum statistical mechanics itself. Specifically saying that it was not the measurement that caused the there is the measurement problem, wherein the applica- sub-system to collapse but rather that it was the ability tion of an operator on a system in a superposed state to exchange a conserved variable that caused the col- returns a single eigenvalue of the operator and collapses lapse. There are two reasons for this conclusion. First, the system into the corresponding eigenstate. And there the collapse and subsequent classical average occurs for 5 any applied operator. If it depended upon the applica- part of the object is classical, the object as a whole must tion of an operator, then it would be sensitive to the be classical. particular operator that was applied. Second, there is no 4. It follows from this that quantum systems are rather measurement that could ever be made on the sub-system special: they must be prevented from exchanging con- that would yield other than the classical result. The sim- served variables both with their surroundings and also plest picture, then, is that the sub-system’s collapse was within themselves. They must have this property of iso- self-induced (i.e. was due to the ability to exchange con- lation or weak linkage in order to maintain their coherent served quantities), and that this collapse is a real prop- nature. It is not sufficient to isolate a system from its sur- erty of the sub-system that persists between measure- roundings for it to remain quantum; the components of ments. Since no experiment or measurement could ever the system must also be isolated from each other. contradict the results of this picture, by Occam’s razor it Of course just how big an internally interacting, ex- is the only acceptable picture. One must conclude that ternally isolated system can get before it collapses and for a sub-system exchanging with a reservoir, the collapse the time scales for the collapse are quantitative ques- is self-induced, irreversible, and real. tions that depend upon the details of the system and are 3. To address in general the classical nature of a macro- beyond the scope of the present article. In any case one scopic object, one can mentally dissect it into small parts. interesting conclusion from this is that the wave function Each individual part is a sub-system that is able to ex- of the universe as a whole must be in a collapsed state. change conserved variables such as energy with the re- Quantum superposition states are only possible for sub- mainder. Therefore the remainder can be considered a microscopic isolated systems with weak or non-existent reservoir for each part. By the conclusion drawn in the internal interactions. preceding paragraph, each individual part is therefore in Disclaimer. No cats were harmed prior to measure- a collapsed state and is therefore classical. Since each ments performed for this article.

1 von Neumann, J. (1927), G¨ottinger Nachrichten 1, 245. Appendix A: Environmental Selection von Neumann, J. (1932), Mathematische Grundlagen der Quantenmechanik, (Springer, Berlin). 5–12 2 Environmental selection or einselection refers to Messiah, A. (1961), Quantum Mechanics, (North-Holland, Amsterdam, Vols I and II). the purification of a sub-system from a superposed state 3 Merzbacher, E. (1970), Quantum Mechanics, (Wiley, New to a mixture of pure quantum states by its interaction York, 2nd edn). with the reservoir. It is based on the notion that in a 4 Bogulbov, N. N. and Bogulbov, N. N. (1982), Introduc- Hilbert space of large dimension, say with M degrees of tion to Quantum Statistical Mechanics, (World Scientific, freedom, two normalized wave functions chosen at ran- Singapore). dom are almost certainly orthogonal. One expects that 5 Zurek, W. H. (1982), Phys. Rev. D 26, 1862. on average their relative scalar product would go like 6 Joos, E. and Zeh, H. D. (1985), Z. Phys. B 59, 223. 7 Zurek, W. H. (1991), Phys. Today 44, 36. Also hψ|φi∼ cM −1/2 ≈ δ(ψ − φ). (A1) arXiv:quant-ph/0306072. 8 Zurek, W. H. and Paz, J. P. (1994), Phys. Rev. Lett. 72, The stochastic variable c is order unity and is on average 2508. 9 zero. Of course if ψ = φ, this would equal unity. Zeh, H. D. (2001), The Physical Basis of the Direction of Time (Springer, Berlin, 4th ed.). To see how einselection works in detail, suppose that in 10 Zurek, W. H. (2003), Rev. Mod. Phys. 75, 715. some orthonormal basis one has superposed wave func- 11 Zurek, W. H. (2004), arXiv:quant-ph/0405161. tions of the isolated sub-system, 12 Schlosshauer, M. arXiv:quant-ph/0312059v4. 13 Attard, P. (2002), Thermodynamics and Statistical Me- |ψi = an |ζni , and |φi = bn |ζni . (A2) chanics: Equilibrium by Entropy Maximisation, (Academic Xn Xn Press, London). 14 Attard, P. (2012), Non-Equilibrium Thermodynamics and The scalar product of these is Statistical Mechanics: Foundations and Applications, (Ox- ford University Press, Oxford). ∗ ∗ 15 hψ|φi = a bn hζm|ζni = a bn, (A3) An alternative interpretation of this result is that in those m n Xm,n Xn sub-systems in which the energy macrostates are so spaced as to preclude energy exchange with a reservoir, entangle- and the expectation value of some observable is ment and the consequent collapse of the wave function may not occur. ˆ ∗ O(ψ) ≡ hψ| O |ψi = amanOmn, (A4) Xm,n

∗ assuming ψ is normalized, n anan = 1. P 6

If the sub-system is allowed to interact with the reser- states. This is the result and meaning of einselection: the voir, then the total system Hilbert space is the direct interaction with the reservoir results in a loss of coherence product of the Hilbert spaces of the sub-system and the of the original wave function and transforms it from a reservoir, and it is conventionally assumed that the total superposition of states into a weighted mixture of pure 5–12 wave function may be written |ψis |εir ≡ |ψ,εi. Note quantum states. that in contrast to the analysis in the text that accounted for the conservation laws, this is a product state, not an entangled state. Discussion Suppose that the reservoir is initially in the nor- ′ malized state |ε0ir. After time t , suppose that the ′ The approach taken in the present paper is obvi- t ′ sub-system wave function evolves to |ψis ⇒ |ψ is = ously rather similar to this einselection mechanism for ′ 5–12 n an |ζnis. Also suppose that when the sub-system is in decoherence, in that the collapse of the system ul- Pthe state |ζnis, the initial reservoir wave function evolves timately arises from the interactions between the sub- ′ t ′ system and the reservoir. to |ε0ir ⇒ |εnir. The subscript n occurs because this evo- lution is affected by the state of the sub-system. By the The main difference between the two approaches is quantum linear superposition principle, the total scalar that einselection decoherence takes the sub-system and product evolves to its environment to form a product state, whereas in the text it is shown that they must form an entangled state, hε , ψ|φ, ε i = a∗ b hε , ζ |ζ ,ε i and it is this entanglement and conservation law that 0 0 m n 0 m n 0 specifically leads to the decoherence. A second difference Xm,n ′ is that the einselection expectation value, Eq. (A6) is a t ′∗ ′ ′ ′ ⇒ ambn hεm, ζm|ζn,εni sum over a single index of only the diagonal terms of Xm,n both the operator matrix and the dyadic density matrix ′∗ ′ ′ in an arbitrary basis, whereas the von Neumann trace, = a b c δ δ m n m m,n m,n Eq. (12), is in general a sum over all components of the Xm,n two matrices that only reduces to a single sum of diag- ′∗ ′ ′ = ambmcm onal terms in the operator basis or in the entropy (here Xm energy) basis. ∗ Of course one very nice feature of the einselection ap- = ambm. (A5) Xm proach to decoherence is that it provides a detailed phys- ical picture of the evolution of the collapse of the wave ′ ′ ′ Here hεm|εnir = cmδm,n, because due to their interac- function. Despite the differences in detail, the conclu- tion, the individual evolution operators on each space are sions from the present approach and from the einselection ′ not unitary; the εn may be approximately orthogonal but approach are the same: wave function collapse ultimately they do not necessarily remain normalized, as is reflected arises from the interactions between the sub-system and ′ in the constant cm. However because the time propagator the environment or reservoir. for the total system is unitary, it preservers scalar prod- ′∗ ′ ′ ∗ ucts and so one must have ambmcm = ambm hε0|ε0i = ∗ r ambm. This holds term by term because of the linear su- Appendix B: Uniform Weight Density perposition principle. One sees that the scalar product of wave functions of the sub-system interacting with the Schr¨odinger’s equation for an isolated system is reservoir are unchanged from that of the isolated system. In contrast, the expectation value of an arbitrary ob- i¯h |ψ˙ i = Hˆ |ψi , or i¯hψ˙ = H · ψ, (B1) servable acting only on the sub-system is

ˆ ∗ ˆ in some arbitrary representation for the matrix form. hε0, ψ| O |ψ,ε0i = aman hε0, ζm| O |ζn,ε0i One can construct a trajectory from this, ψ(t|ψ0,t0). Xm,n ′ Associated with each point in wave space is a weight t ′∗ ′ ′ ˆ ′ ⇒ aman hεm, ζm| O |ζn,εni density ω(ψ,t), which when normalized is the probabil- Xm,n ity density ℘(ψ,t). The quantity dψ℘(ψ,t) gives the ′∗ ′ ′ probability of the system being within |dψ| of ψ. Asa = amanOmncmδm,n probability the product is a real non-negative number: Xm,n dψ℘(ψ,t) = |dψ℘(ψ,t)| = |dψ| |℘(ψ,t)|. Since it is al- ∗ = amamOmm. (A6) ways the product that occurs, without loss of generality Xm one may take each to be individually real. The normal- ization is This is different to what was obtained for a sub-system alone. It is the expectation value for a mixture of pure dψ℘(ψ,t) = dψ ℘(ψ,t) =1. (B2) quantum states rather than that of a superposition of Z Z 7

In the matrix representation, ψ = {ψn}, n = 1, 2,.... Hence the partial time derivative is r i Since the coefficients are complex, ψn = ψn + iψn, one ∂℘(ψ,t) ∗ can write the infinitesimal volume element as = −ψ˙ · ∂ ∗ ℘(ψ,t) − ψ˙ · ∂ ∗ ℘(ψ,t). (B9) ∂t ψ ψ dψ = dψr dψi ≡ dψr dψi dψr dψi ... (B3) 1 1 2 2 This result is valid for an isolated system that evolves r under Schr¨odinger’s equation. With this the integrations are over the real line, ψn ∈ i Inserting this into the second equality in Eq. (B6), it [−∞, ∞], and ψn ∈ [−∞, ∞]. The compressibility of the trajectory is may be seen that the total time derivative of the prob- ability density vanishes, d℘(ψ,t)/dt = 0. Hence for an ˙ dψ r i isolated system, the probability density is a constant of = ∂ r · ψ˙ + ∂ i · ψ˙ dψ ψ ψ the motion, ∗ = ∂ · ψ˙ + ∂ ∗ · ψ˙ ψ ψ ℘(ψ(t),t)= ℘(ψ0,t0), (B10) 1 1 = Tr H − Tr H i¯h i¯h where ψ(t) ≡ ψ(t|ψ0,t0). = 0. (B4) In an equilibrium system, (i.e. the Hamiltonian opera- tor does not depend upon time), the probability density (The first two equalities are general; the final two hold is not explicitly dependent on time, so that one can write for Schr¨odinger’s equation.) ℘(ψ). In this case the result becomes The compressibility gives the logarithmic rate of change of the volume element.13,14 Hence from the van- ℘(ψ(t)) = ℘(ψ0). (B11) ishing of the compressibility for the adiabatic Schr¨odinger Finally, one may invoke what might be called the weak equation one sees that the volume element is a constant form of the classical ergodic hypothesis, which says that of the motion of the isolated system, a single trajectory passes sufficiently close to all relevant

dψ(t) = dψ0. (B5) points of the state space. This means that any two points in state space, ψ1 and ψ2, with the same norm and energy The total time derivative of the probability density is but otherwise arbitrary, lie on a single trajectory. By the above, they therefore have the same probability density d℘(ψ,t) ∂℘(ψ,t) ˙ r ˙ i = + ψ · ∂ψr ℘(ψ,t)+ ψ · ∂ψi ℘(ψ,t) dt ∂t ℘(ψ2)= ℘(ψ1), (B12) ∂℘(ψ,t) ∗ = + ψ˙ · ∂ ℘(ψ,t)+ ψ˙ · ∂ ∗ ℘(ψ,t) ∂t ψ ψ if E(ψ2)= E(ψ1) and N(ψ2)= N(ψ1). One possible objection to the weak form of the ergodic ∂℘(ψ,t) ˙ = + ∂ψ · ψ℘(ψ,t) hypothesis (that the trajectory visits all points on the ∂t h i energy hypersurface) is that an isolated system always ˙ ∗ + ∂ψ∗ · ψ ℘(ψ,t) . (B6) remains in its initial energy microstate or superposition h i of energy microstates, since these are eigenstates of the The first and second equalities are definitions that hold energy operator. However, if the wave function is repre- in general. The final equality, which uses the vanishing sented in some other basis, then the concept of a trajec- of the compressibility, is valid for an isolated system that tory through wave space ψ(t) is meaningful, (successive evolves via Schr¨odinger’s equation. One can identify from measurements with the same operator do not necessarily ˙ this J℘(ψ,t) ≡ ψ℘(ψ,t) as the probability flux. yield the same value) and the weak form of the ergodic In view of the deterministic nature of Schr¨odinger’s hypothesis is plausible and leads to the conclusion that equation, the evolution of the probability density is given the weight density of wave space is uniform (on a con- by stant energy and magnitude hypersurface). Using this result the weight density of state space ℘(ψ1,t1)= dψ0 ℘(ψ0,t0) δ(ψ1 − ψ(t1|ψ0,t0)) . (B7) Z must be of the form ω(ψ) = ω(E(ψ)). (For simplic- ity the magnitude is neglected since it only trivially af- Using the constancy of the volume element, dψ0 = dψ1, fects the analysis.) The only time one needs to com- setting t1 = t0 + ∆t, and expanding to linear order in pare the weight densities of points with different ener- ∆t → 0, one has gies is when energy exchange with a reservoir occurs. In such a case the total weight density must be of the form ∂℘(ψ1,t0) ℘(ψ1,t0) + ∆t (B8) ω(ψ )= ω(E (ψ ))ω(E −E (ψ )), which has solu- ∂t tot s tot tot s tot 0 tion ω(ψ)= weαE(ψ), with w and α arbitrary constants.

= dψ1 ℘(ψ0,t0) δ(ψ1 − ψ(t1|ψ0,t0)) This means that any variation in the weight density of Z wave space with energy of the sub-system will cancel identically with the equal and opposite dependence of = ℘ ψ1 − ∆tψ,t˙ 0   the weight density of the reservoir on the sub-system en- ˙ ˙ ∗ = ℘(ψ1,t0) − ∆tψ · ∂ψ℘(ψ,t0) − ∆tψ · ∂ψ∗ ℘(ψ,t0). ergy. (This is a true statement for each individual point 8 in the wave space of the reservoir. Summing over the turn the Jacobean for the transformation χ ⇒ ψ˜, reservoir degeneracy remains to be carried out, as in the 1/2 text.) Hence without loss of generality one can set α =0 ˜ ∂E(ψ) ∂E(ψ) ˜ ˆ ˆ ˜ 1/2 and the weight density may be set to unity, ∇E = = hψ|HH|ψ i , (B16)  ∂ |ψ˜i ∂ hψ˜| 

ω(ψ)=1. (B13) and for the transformation ψ˜ ⇒ ψ, This result applies to the whole of state space of an ∂N(ψ) ∂N(ψ) 1/2 isolated system, not just to the hypersurface to which |∇N| = = hψ|ψi1/2 . (B17) Schr¨odinger’s trajectory is constrained in any given case.  ∂ |ψ i ∂ hψ |  By design, this is real and positive. The logarithm of this With these the full weight density is gives the internal entropy of the wave states, ω(ψ) = ω(χ|NE) |∇˜ E| |∇N| S(ψ)=0. (B14) hψ˜|HˆH|ˆ ψ˜i1/2 hψ|ψi1/2 ∝ hψ|HˆH|ˆ ψ i1/2 Alternative Derivation = 1. (B18) The interpretation of this is that the weight density is An arguably more satisfactory derivation is as fol- inversely proportional to the speed of the trajectory (i.e. lows. The magnitude of the wave function is redundant large speed means less time per unit volume), and linearly in the sense that expectation values only depend upon proportional to the number of hypersurfaces that pass the normalized wave function. For example, denoting through each volume of wave space (i.e. for fixed spacing the latter as ψ˜ ≡ N(ψ)−1/2ψ, with the magnitude be- between discrete hypersurfaces, ∆ and ∆ , large gra- ˜ ˆ ˜ E N ing N(ψ) ≡ hψ|ψi, the energy is E(ψ) = hψ|H|ψ i. In dients correspond to more hypersurfaces per unit wave addition, since the isolated system is confined to a hy- space volume). persurface of constant magnitude N and energy E, one Just as in the classical case for Hamilton’s equations of can denote a representation of that particular Hilbert motion,13 it is a remarkable consequence of Schr¨odinger’s sub-space by χ. In this notation the normalized wave equation that the speed is identical to the magnitude of ˜ ˜ function is ψ = ψ(E,χ), and the full wave function is the gradient of the hypersurface, so that these two cancel ψ = ψ(N, ψ˜)= ψ(N,E,χ). to give a weight density that is uniform in wave space. Now the weight density on the hypersurface, ω(χ|NE) An objection to the uniform weight density of wave will be derived, and then this will be transformed to the space may be raised, namely that a non-linear transfor- weight density on the full wave space. The derivation mation of wave space would lead to a non-uniform weight is the quantum analogue of that given for classical sta- in the transformed coordinates. However, any such non- tistical mechanics in §5.1.3 of Ref. 13. The axiomatic linear transformation of the wave function would de- starting point is that the fundamental statistical average stroy the linear homogeneous form for the operator equa- is a simple time average. The implication of this is that tions of quantum mechanics. In this sense the form the weight is uniform in time, which is to say that it must of Schr¨odinger’s equation determines uniquely that the be inversely proportional to the speed, weight of wave space should be uniform. Both derivations (the uniformity of the probability ˙ −1 ω(χ|NE) ∝ |ψ| density along a trajectory, and the proportionality of − = h ψ|HˆH|ˆ ψ i 1/2 . (B15) weight to time and gradients) yield the same result, namely that the weight density is uniform in wave space. This is just the time that the system spends in a volume Since the microstates of an operator are just its eigen- O element |dχ|. The proportionality factor is an immaterial functions in wave space, |ζkg i, this result implies that constant. This weight density is now successively trans- these microstates all have equal weight, which is the ax- formed to the full wave space. For this one requires in iomatic starting point taken in the main text.