Dependence of decoherence-assisted classicality on the ways a system is partitioned into subsystems

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by  Oleg Lychkovskiy

Resolution into subsystems, decoherence and thermalization
Oleg Lychkovskiy
Lancaster University, Physics Department, UK, and
Institute for Theoretical and Experimental Physics, Moscow, Russia.
Choosingaspecificwayofresolvingaclosedsystemintopartsisastartingpointforthedecoher-
ence program and for thequantum thermalization program. It is shown that one can always chose
such way of partitioning that thermalization does not occur and decoherence-assisted classicality
does not emerge. Implications of this result are discussed.
I. INTRODUCTION this simple mathematical fact (which we refer to as par-
2
titioning relativity) has profound consequences to quan-
1
0 Consider a large closed quantum system with a state tum physics [12]. In particular, he argued that a notion
2 space andaHamiltonianH.Itisubiquitousinphysics of entanglement is relativized: for any state which is en-
to resoHlve this large closed system in two parts, system tangled with respect to some TPS one can find another
t
c and bath ,1 and to concentrate on the dynamics of TPS in which this state is factorized.
O S, i.e. on thBe evolution of the reduced density matrix The goal of the present paper is to show that an anal-
Sρ (t). Assume both and are macroscopic. Then, ogous conclusion is valid both for decoherence and for
5 S
1 based on the everydaSy experBience, one expects certain thermalization: for every closed quantum system with a
features of the reduced evolution to manifest at certain statespace andaHamiltonianH thereexistsaTPSin
H
] timescales. At a very short time scale τ the state of the which decoherence and thermalization are absent. This
h
system is expectedto decohere. This meansthat ρ (t) will be proven by explicitly constructing such a TPS.
p S S
can be diagonalizedin a basis with each basis vector be- Theambiguityinresolvingaclosedsystemintosubsys-
-
t ing ”quasiclassical” [1], i.e. providing vanishing uncer- tems was previously realizedto be an important concep-
n
a tainty to every observable from a certain ”classical” set tual problem for the decoherence program [13, 14]. The
u (e.g. center-of-masscoordinate,momentum,energyetc). present result sharpens this conclusion to the extreme.
q To derive such type of short-time behavior from realis- Recently Dugi´c and Jekni´c-Dugi´c also considered the
[
tic Hamiltonians is the goal of the decoherence program, implicationsofthe partitioningrelativitytothedecoher-
1 which was essentially founded by Zeh [1, 2] and Zurek ence program [15]. They studied the system of coupled
v [3, 4] (a detailed list of references can be found in the harmonic oscillators (quantum Brownian motion model)
4 book [5]). Decoherence, when proved, accounts for one and came to a striking conclusion that within different
2 aspectofquantum-to-classicaltransition,namely,theap- TPS the decoherence leads to the emergence of qualita-
1
parent absence of Shr¨odinger cat states in our everyday tively different classical worlds. We do not discuss this
4
experience (and does it irrespectively to the chosen in- outstanding claim in the present paper. We only stress
.
0 terpretation of quantum theory). that it is based on the analysis of a specific model, while
1
At a very large time scale T one expects that the sys- our arguments are completely general.
2
tem thermalizes, which means, roughly speaking, that The plan of the rest of the paper is as follows. In Sec.
1
: ρS(t) approaches an equilibrium density matrix which II we discuss the partitioning relativity, i.e. the freedom
v depends only on some coarse-grained properties of the tochoosedifferentTPS.Westartfromanexamplewhich
i
X bathstate,butnotontheinitialstateofthesystem. We motivates consideration of different TPS. Then we show
refer to the effortto justify this intuitive statement from howanarbitrarybasisin generatesasetofTPS.After
r
a thefirstprinciplesofquantumtheoryasaquantumther- that we construct a TPSHgenerated by the eigenbasis of
malization program. In the past decade a considerable H.
progress was made in this direction (see e.g. [6–10]; a Sec. III is devoted to thermalization. First we discuss
profound list of references can be found in [11]). whatpropertiesofthe reduceddensity matrixconstitute
From a formal point of view, resolution into subsys- thermalization. Thenweshowthatwithintheeigenbasis-
tems(partitioning)isdescribedbyatensorproductstruc- induced TPS one of such properties— initial state inde-
ture (TPS) over the state space : pendence of the equilibrium density matrix — is absent.
H
InthelastsubsectionwediscusstheEigenstateThermal-
= . (1)
H S⊗B ization Hypothesis (ETH) [16–18] and demonstrate that
Zanardi pointed out that any state space with a non- it can not be valid within the considered TPS.
H
prime dimension supports infinitely many TPS and that In Sec. IV we turn to the decoherence program. First
we specify the notion of the decoherence-assisted emer-
gent classicality within the context of the present pa-
1 ThroughoutthepaperwerefertoSassimplya”system”without per. Then we demonstrate how to construct a TPS
aprefix”sub-”,whileHisreferredtoasa”closedsystem”. Bath withinwhichthedecoherence-assistedclassicalityfailsto
Bisoftenreferredtoas”environment”. emerge. This new TPS shares some common features
2
with the eigenbasis-induced TPS, in particular, it also B. Constructing a tensor product structure
precludes thermalization. In Sec. IVC we introduce
an Eigenstate Decoherence Hypothesis in a close anal- Forsimplicity,weconsiderafinite-dimensionalHilbert
ogy with the Eigenstate Thermalization Hypothesis and space .Its dimensiondis consideredto be anon-prime
discuss its robustness against partitioning relativity. numbeHr: d=mn, n>m 1. Centralfor our discussion
InthelastSec. Vtheresultsaresummarizedandtheir is the following ≫
implications are discussed.
Lemma. Consider anarbitrarybasis in withdim =
H H
mn, m 2, n 2.Enumerateitbyadoubleindex(i,j):
≥ ≥
II. PARTITIONING RELATIVITY
Ψ , i=1,2,...,n; j =1,2,...,m.
i,j
{ }
A. Example: partitioning relativity in XX spin This enumeration generates a bipartition = of
H S ⊗B
chain the space in a tensor product of two spaces and ,
H S B
dim = m, dim = n, such that every basis state Ψ
i,j
S B
is a product state with respect to this bipartition,
Let us start from a simple but enlightening example.
Considerachainofspins1/2withthe XX Hamiltonian:
Ψ =ϕ χ , ϕ , χ , (4)
i,j i j i j
⊗ ∈S ∈B
N 1 N
H = 1 − (σxσx +σyσy )+ h σz (2) and, moreover, {ϕi} and {χj} are bases in S and B cor-
4 n n+1 n n+1 2 n respondingly.
nX=1 nX=1
Note that all the bases discussed throughout the paper
AfterJordan-WignerandBogolyubovtransformationap-
are orthonormal.
plied to operators σ this model appears to be a free-
n± The lemma essentially states that every basis in the
fermion one [19],
state space of a closed quantum system generates a
H
TPS (in fact, combinatorially many different TPS). The
1
H = ε (c+c ). (3) statement of the lemma is almost self-evident: one for-
Xp p p p− 2 mally introduces Hilbert spaces S and B with orthonor-
mal bases ϕ and χ correspondingly,takes a tensor
i j
{ } { }
An operator c+ creates a fermion mode with quasi- product , maps the basis ϕi χj to the basis
p S ⊗ B { ⊗ }
momentum p (admitting N discrete values) and energy Ψi,j of and thus introduces the identification (4).
{ } H
ε .
p
Whatarethepossiblewaystopartitiontheclosedsys-
C. Eigenbasis-induced tensor product structure
tem under consideration into a two-level system and a
2N 1-level bath? Evidently, one can regard any spin in
−
the chainas a system and other N 1 spins as a bath Let us apply the above lemma to the eigenbasis
. This gives N possiSble choices of p−artitioning. How- El , l = 1,2,...,d of the Hamiltonian H.2 To do this
BevertheseN choicesbynomeansexhaustallpossibilities. {o|ne ih}as to introduce a double index (i,j) instead of a
Anothersetofpossiblebipartitionsemergeswhenonere- single index l, which amounts to arranging basis states
gardsagivenfermionmodeasasystemandotherfermion in a table:
modes as a bath. In fact, there are infinitely many such
χ χ . . . χ
sets of bipartitions generatedby varioustransformations | 1i | 2i | ni
applied to a set of operators σ . This is what we call
n± ϕ E E . . . E
partitioning relativity. | 1i | 1i | 2i | ni
Differentwaysofpartitioningleadtodrasticallydiffer-
ϕ E E . . . E
ent behavior of corresponding reduced states. If a single | 2i | n+1i | n+2i | 2ni
spin is regarded as , its reduced density matrix relaxes
S ... ... ... . . . ...
to an equilibrium one up to finite-size effects, at least
for a certain class of initial conditions [20] (in case when
ϕ E E . . . E
dimS issmalldecoherenceandthermalizationtimescales | mi | n(m−1)+1i | m(n−1)+2i | nmi
(5)
arusuallyofthesameorder,τ T,thereforeoneshould
∼
not separate a relaxation process into decoherence and
thermalization[21]). Onthe contrary,if a single fermion
mode is regarded as , its reduced density matrix does
not evolve with timeSat all, since c+pcp is an integral of 2 sWyseteamlwa|Eysliu,s|eEiD,jiiraacndbr|aE˜-kliet(sneoetabteiloonws)fotor satvaotiedstohfetchoencfulossioedn
motion. Properties of entanglement in the XX model between states and eigenenergies. On the other hand, we often
also drastically depend on how the closed system is re- omit bras and kets for the states ϕ and χ of the subsystems to
solved into parts [22]. simplifynotations.
3
Thistabledefinesamapfromthesetofeigenstates E Whatkindoflong-timebehaviorofρ (t)doweexpect
l S
{| i}
to the set of product states ϕ χ . Combining these from our everyday experience with subsystems? We ex-
i j
{ ⊗ }
basisvectorsinsuperpositionsallowsto extendthis map pect that ρ approaches an equilibrium density matrix
S
to a map between the original state space and the ofsomespecial(e.g. Boltzmann-Gibbs)form. Aswasar-
H
product of state spaces of two subsystems and gued in [10], on closer examination one expects that the
S ⊗B S
, ϕ beingabasisin and χ —in .Forexample, system exhibits four distinct properties, which we refer
i j
B { } S { } B
the following identifications are in order: to as thermalization properties:
E =ϕ χ , 1. Equilibration. System is said to equilibrate if
2 1 2
| i ⊗ S
E + E =ϕ (χ +χ ), (6) ρ (t) approaches a time-averaged density matrix
2 n 1 2 n S
| i | i ⊗
|E2i+|E2ni=ϕ1⊗χ2+ϕ2⊗χn. ρS and stays close to it most of the time. Defined
inthiswayequilibrationdoesnotimplyneitherany
In what follows we often enumerate the vectors of the
specialformofρ ,northeindependenceofρ from
eigenbasis by a double index explicitly: S S
initial conditions.
E E =ϕ χ . (7)
| i,ji≡| (i−1)n+ji i⊗ j 2. Bath initial state independence (Bath ISI). This
We refer to the TPS defined by (5) as TPS-1. meansthatρS doesnotdependontheexactinitial
Note that the eigenvalues E do not have to be ar- microstateofthebath. Ratherρ dependsonsome
l S
ranged in an ascending order. Different orderings of the macroscopiccharacteristicsofthestateofthebath.
sequence of eigenvelues induce different TPS. The primeexampleofsuchcharacteristicis theen-
A very special feature of TPS-1 which will show up in ergy of the bath (which is related to the bath tem-
the nextsectionis that (1)everyHamiltonianeigenstate peraturewhenbathitselfisin(quasi-)equilibrium).
E is a product form with respect to TPS-1 and, more-
l
| i 3. System initial state independence (System ISI).
over,(2)to constructsuchproductstatesoneneeds only
vectors from a single basis in and a single basis in . This means that ρS does not depend on the ini-
In general, TPS-1 corresponSds to a very unnaturalBbi- tial state of the system
S
partition. However in certain toy models TPS-1 appears
4. Boltzmann-Gibbs form of the equilibrium state:
to be natural, for example, in the paradigmatic central
ρ =Z 1exp( βH˜ ), where H˜ is some effective
spin model [4] with a Hamiltonian S − − S S
Hamiltonian of the system . This property may
S
N be expected if the interaction between the system
H =σz g σz, (8) andthebathisinsomesense”weak”andtheinitial
central n n
nX=1 state of the bath has a small energy uncertainty.
where a central spin 1/2 is regarded as a system and N Thelastthreepropertiesmakesenseonlyifthefirstsone
peripheral spins 1/2 as a bath. holds. The last property makes sense if also the proper-
ties (2) and (3) hold. Note the lack of the symmetry be-
tween the definitions of the bath ISI and the system ISI.
III. THERMALIZATION
This asymmetry arises because the bath is assumed to
be muchlargerthanthe system, so that it canabsorbor
A. Ingredients of thermalization injectanyamountofenergyfromortothesystemandto
completely wipe out any memory about the initial state
Define a time-averagedreduced density matrix: of the system.
Anumber oftheoremswereprovenin[10]whichallow
t
toformulatetheabovelistedthermalizationpropertiesin
ρ lim t 1 ρ (t)dt. (9)
S ≡t − Z S ′ ′ mathematically rigorous terms and to justify the equili-
→∞ 0
brationand the Bath ISI properties under certaincondi-
Itplaysacentralroleinthequantumthermalizationpro-
tions imposed on the spectrum of the total Hamiltonian
gram as it encapsulates the long-time behavior of . In
(but not on the TPS). We are in a position to demon-
S
particular, if ρ (t) equilibrates, the equilibrium density
S strate that within an eigenbasis-induced TPS-1 the sys-
matrixisρS [10]. AswillbeclearfromSec. IV,ρS isalso tem ISI property does not hold.
of substantial importance for the decoherence program.
Throughoutthepaperwesupposeforsimplicitythatthe
spectrum of H is non-degenerate. In this case B. Breakdown of the thermalization
d
ρ = E Ψ(0) 2tr E E , (10) Throughout the paper we consider initial states of a
S Xl=1|h l| i| B| lih l| product form (with respect to a chosen TPS):
where Ψ(0) is an initial state. Ψ(0)=ϕ(0) χ(0), ϕ(0) , χ(0) . (11)
| i∈H ⊗ ∈S ∈B
4
Let us first focus on a very special initial state. Assume and most physically relevant case this functional is just
that the initial state of the bath is described by a basis the total energy: ρ =ρ ( E H E ). ETH is a strong
Seq Seq h l| | li
vector, χ(0) = χ (while the initial state ϕ(0) of the assumption. If the energyuncertainty ofthe initialstate
j
system is still arbitrary). In this case ρ (t) evolves as is small, ETH guarantees Bath ISI and System ISI. If in
S
it were a state of an isolated system with an effective additiontheinteractionbetweenthesystemandthebath
Hamiltonian H E ϕ ϕ : isweak,ETHalsoguaranteestheBoltzmann-Gibbsform
jS ≡ i,j| iih i|
Pi of the equilibrium state. One can easily see, however,
that the ETH fails within TPS-1. Indeed,
ρS(t)=e−iHjSt ϕ(0) ϕ(0)eiHjSt. (12)
| ih |
tr E E = ϕ ϕ , (15)
i,j i,j i i
Evidently, the evolution is unitary and does not display B| ih | | ih |
anyofthethermalizationproperties(italsodoesnotdis- which strongly depends on the exact microstate.
play the decoherence).
It can be argued, however,that the above observation
Tosummarizethissection,wehaveshownhowtocon-
is not of practical significance since χ is a very special,
j struct a tensor product structure within which all (for
almost ”improbable”initial state of the environment. In
special initial states) or some (for generic initial states)
practice we do not ”prepare” environment (since its mi-
ingredients of thermalization are lacking.
crostateislargelyuncontrollable),butratherpickbyran-
dom a state which is restricted only by some global con-
straints (e.g. the total energy should be in a certain IV. DECOHERENCE
narrow window). This idea is advocated and developed
in [10]. Arguably, a more reasonable setup is to consider
A. Decoherence-assisted classicality
a generic initial state of the bath. Following [10], we say
that some property is valid for a generic initial state of
Wave function of a closed system Ψ(t) can be repre-
the bath if it is valid for almost all states of the bath
sented in a Schmidt form,
from some linear subspace (with possible excep-
R
B ⊂B
tions forming a set of an exponentially small in dim R n
B
measure). Ψ(t)= p (t) ϕ (t) χ (t) (16)
i i i
⊗
Itwasshownin[10]thatundercertainreasonablecon- Xi=1p
ditionsequilibrationandbathISIpropertiesarevalidfor
an arbitraryinitial state of the system and a generic ini- where ϕi(t) forma basis in and χi(t) canbe com-
{ } S { }
pleted to a basis in . This corresponds to the diagonal
tial state of the bath.
B
form of the reduced density matrix
Let us show, however, that the equilibrium state
strongly depends on the initial state ϕ(0) of the system
n
within the TPS-1. Indeed, ρ (t)= p (t)1/2 ϕ (t) ϕ (t). (17)
S i i i
| ih |
Xi=1
ρ = ϕ ϕ(0) 2 ϕ ϕ . (13)
S Xi |h i| i| | iih i| States ϕi(t) describe alternatives which an observer in-
side the system is able to perceive (this is true in any
Evidently, one can obtain any set of probabilities interpretation ofSquantum theory). However the major-
|hϕi|ϕ(0)i|2 intheequilibriumreduceddensitymatrixρS ity of states in S are Shr¨odinger-cat-like states which
choosingsuitableinitialstate. Thislackofthesystemini- can not be interpreted classically. What forces ϕ (t) to
i
tial state independence is an immediate consequence of lie in a small quasiclassical domain? The key insight of
the above-mentionedproperty of the Hamiltonian eigen- the decoherence theory is that even if initially ϕ (t) are
i
state factorizability with respect to the TPS-1. Shr¨odinger-cat-likestates,afteraveryshortdecoherence
time τ they become quasiclassical states, as a result of
interaction between the system and the environment (if
C. Eigenstate Thermalization Hypothesis the Hamiltonian is non-trivial enough) [1]. We will re-
fer to this dynamical process as the decoherence-assisted
TheEigenstate Thermalization Hypothesis(ETH)[16– emergence of classicality.
18] is worth mentioning in the present discussion, as it The following subtlety should be emphasized. As-
underlies an important branch of the quantum thermal- sume that for given closed system, total Hamiltonian
izationprogram. ItstatesthatforrealisticHamiltonians andbipartitionthedecoherence-assistedclassicalitydoes
thermalization occurs at the eigenstate level: emerge. Consider two reduced density matrices corre-
sponding to different moments of time (such that the
trB|ElihEl|≃ρSeq, (14) decoherence have already occurred) and/or to different
initial conditions, ρ and ρ˜ . They can be diagonalized
S S
where ρ depends not on the eigenstate itself, but on in the bases ϕ and ϕ˜ . By assumption, all states
Seq { i} { i}
the value of one or few functionals on . In the simplest ϕ and ϕ˜ belong to a quasiclassical domain of the state
i i
H
5
space. However, in general it is not true that bases ϕ B. Failure of classicality
i
{ }
and ϕ˜ coincide or even are close to each other. The
i
{ }
reason is that for sufficiently large dimension of any We do not follow this route in the present paper. As
S
basis represents a very fine-grained set of alternatives, soon as our goal is modest — to demonstrate the fail-
while classicalworldis coarse-grained[2]. It canappear, ure of the decoherence-induced classicality within some
for example, that ϕ˜1 = √12(ϕ1 +ϕ2). Indeed, think of TPS,we insteadidentify aconditionwhichsignifies such
ϕ1 being a state of an alive cat, with a spin of a single a failure.
nucleus in its body pointing up, and ϕ2 being a state Consider two bases in , ϕi and ϕ˜i , related to
S { } { }
of alive cat with a spin of that nucleus pointing down. each other by a discrete Fourier transform (call them
Then ϕ˜1 is a state of the alive cat with a spin of the maximally distant bases):
nucleus pointing in x direction – still a classical state.
Thus we do not expect that ϕi and ϕ˜i coincide, but 1 m−1
rather that each state from t{hes}e two b{ase}s belongs to a ϕ˜i = e−2πi(ki/m) ϕi . (18)
| i √m | i
quasiclassical domain of (cf. [14]). How to define this kX=0
S We argue that the vectors of these two bases can not si-
domain?
multaneously belong to . Indeed, assume that eachϕ
cl i
S
does belong to cl. Although some of the pairs (ϕi,ϕi′)
S
can describe the states which are hardly distinguishable
Ingeneral,thisquestionisnoteasytoanswer. Ausual classically and therefore can form a superposition which
route is to require that the quasiclassical states have is still classical (such as the states of alive cat with two
vanishing quantum uncertainties for a set of observables different directions of a single nuclear spin), a bulk of
used in classical mechanics (center-of-mass coordinate, such pairs should necessarily represent classically distin-
momentum, energy etc). This is not completely satisfy- guishable states (such as states of alive and dead cat),
ing as a priori classical notions are used [14]. Moreover, otherwisetheclassicalrealitywouldbetrivial. Thuseach
this is not directly applicable in our setting, as we allow state ϕ˜ is a superposition of classically distinguishable
i
arbitrarypartitioninginwhichtheseaprioriclassicalob- states, therefore it is highly non-classical.
servables in general are not related to the system (in Let us consider the equilibrium density matrix ρ .
S S
otherwords,correspondingself-adjointoperatorsarenot An evident necessary requirement for the decoherence-
of the product form AS 11B ). One should employ a assistedclassicalitytoemergeisthatρ describesamix-
⊗ S
similarstrategybutavoidusingaprioriclassicalnotions. tureofquasiclassicalstatesforanyinitialconditions. We
This strategy can be sketched as follows. are in a position to demonstrate that, on the contrary,
there exist sucha partitioning inwhich ρ canbe diago-
S
nalineachofthemaximallydistantbases,dependingon
the initial conditions.
One defines a set A of (dimensionless) observ-
α
{ } Assume thatnis even. Considerthe followingbasisin
ables in which are considered to be classical by def-
S :
inition. Then one postulates that the states ϕ
H
∈ S
which provide small uncertainties for all A (i.e. α
ϕA2 ϕ ϕA ϕ 2 < ε with some smallαε) form∀a Ψ = |Ei,ji for j ≤n/2 , (19)
hqu|asiαc|lasis−icahls|ubαse|ti cl . If the decoherence-induced | i,ji (cid:26)|E˜i,ji for j ≥n/2+1
S ⊂S
classicality emerges, in the above-discussed sense, with
where E˜ is an superposition of m eigenstates:
respect to the defined subset cl of quasiclassical states, | i,ji
S
then the whole construction is self-consistent. Note that
m 1
the set Aα should be sufficiently large and non-trivial E˜ = 1 − e2πi(ki/m) E . (20)
{ } i,j i,j
to allow for a non-trivial classical reality (this excludes | i √m | i
kX=0
e.g. the set consisting of one element, identity operator)
but sufficiently coarse-grained to allow the decoherence This basis along with a chosen way of ascribing double
to proceed (this excludes e.g. the complete set of m mu- indices defines a TPS (call it TPS-2):
tually orthogonalprojection operators).
6
χ χ ... χ χ ... χ
1 2 n/2 n/2+1 n
| i | i | i | i | i
ϕ E E ... E E˜ ... E˜
1 1,1 1,2 1,n/2 1,n/2+1 1,n
| i | i | i | i | i | i
ϕ E E ... E E˜ ... E˜ (21)
2 2,1 2,2 2,n/2 2,n/2+1 2,n
| i | i | i | i | i | i
... ... ... ... ... ... ...
ϕ E E ... E E˜ ... E˜
m m,1 m,2 m,n/2 m,n/2+1 m,n
| i | i | i | i | i | i
The first n/2 columns of this table coincide with those thecorrespondingproductscannotbeconstructedusing
of the table (5) which defines TPS-1. The state space vectors from a single basis in (one needs two different
S
of the environment can be represented as a direct sum, bases ϕ and ϕ˜ ). Tensor product structures with
i i
{ } { }
= , and beinglinearhullsof χ ,...,χ thesamepropertynaturallyemergeinmoresophisticated
1 2 1 2 1 n/2
B B ⊕B B B { }
and χ ,...,χ correspondingly. The equilibrium versions of a central spin model [23, 24].
n/2+1 n
{ }
density matrix reads
n/2 C. Eigenstate Decoherence Hypothesis
ρS = |hχj|χ(0)i|2 |hϕi|ϕ(0)i|2|ϕiihϕi|
Xj=1 Xi In analogywith the eigenstate thermalizationhypoth-
 
n esis discussed in Sec IIIC, one can put forward an
+ |hχj|χ(0)i|2 |hϕ˜i|ϕ(0)i|2|ϕ˜iihϕ˜i|(,22) Eigenstate Decoherence Hypothesis (EDH). For
j=Xn/2+1 Xi
  realistic Hamiltonians the decoherence-assistedclassical-
ity emerges at eigenstate level, i.e. all states tr E E
where the bases ϕ and ϕ˜ are related by a Fourier l l
{ i} { i} belongs to . B| ih |
transform (18) and thus are maximally distant. If the cl
S
initial state of the environment χ(0) belongs to , the
B1 EDH, if valid, would guaranty that the equilibrium
equilibrium density matrix is diagonalin the basis ϕ ,
{ i} density matrix is diagonal in a quasiclassical basis re-
while if χ(0) ∈ B2, then ρS is diagonal in {ϕ˜i}. This gardless of the initial states of the system and of the en-
means that the decoherence-assisted classicality can not vironment. Howeveritagainappearsthatthishypothesis
emerge. cannot be valid within an arbitrarybipartition. Indeed,
TheabovedemonstratedpropertyofρS isnothingelse it is violated within the TPS-2:
than the lack of the bath initial state independence of
dtheecromhearleiznacteio(nc)o.mTpwaroestuobtshpeacceosrrBe1spaonnddiBn2gopfrothpeerstytatoef trB|Ei,jihEi,j|=(cid:26)||ϕϕ˜iiiihhϕϕ˜ii|| iiff jj ≥≤nn//22,+1, (23)
space of the environment have large dimensions, namely
n/2 each. Therefore we have verified the breakdown of and, as we have argued above, all 2m vectors ϕi , ϕ˜i
| i | i
the bath ISI property of decoherence for generic initial can not belong to cl at the same time.
S
states of the bath. We believe, however, that the eigenstate decoherence
It is clear from eq. (22) that if an initial state of the hypothesis can be valid when the the choice of TPS is
environment has non-vanishing components both in restrictedbysomephysicallymotivatedrequirement(e.g.
1
B involving locality). This will be discussed elsewhere.
and , then the diagonal basis of ρ depends also on
B2 S It is worthstressingthat EDH is a weakerassumption
the initial state ϕ(0) of the system. This situation is
then ETH. For example, it is easy to verify that in a
moreinvolvedasthecoefficients ϕi ϕ(0) and ϕ˜i′ ϕ(0)
h | i h | i centralspin model (8) the ETHfails, in accordancewith
are not independent. We do not consider this case in
the general result of Sec. IIIC, but EDH holds.
detail.
Ifoneconsidersinitialstatesofthe bathfromthe sub-
space only, thenTPS-2coincides withTPS-1. There- To summarize this section, we have shown how to
1
fore thBe breakdown of thermalization established in the construct a tensor product structure within which the
previous section for TPS-1 also occurs for TPS-2. How- decoherense-assistedclassicality fails to emerge.
everthereversereasoningisnotvalid: onecannotextend
the arguments of the present section to prove failure of
decoherense-assisted classicality for TPS-1. This is evi- V. DISCUSSION AND SUMMARY
dentfromcomparingeqs. (13)and(22). The underlying
reasonis thatwhile alleigenvectorsarefactorizablewith Oneoftheimportantdifficultiesbeyondthe quantum-
respect both to TPS-1 and to TPS-2, in the latter case to-classicaltransition can be outlined by the following
7
Question. Due to superposition principle all states of however,thatourargumentsarecompletelyindependent
Hilbert space are equally real. Why in everyday life we from this line of reasoning: they are equally valid in a
observe only tiny fraction of them, but never observe non-relativistic setting.
Shr¨odinger-cat-likestates, which constitute the vast ma-
Let us make a final brief remark on locality require-
jorityof Hilbertspace? Howarethe quasiclassicalstates
ment. At first sight, one could take a straightforward
chosen from a Hilbert space?
approach: toequipquantumtheorywithanotionofspa-
tiallocality(forexampleasisdescribedinthebook[25])
The decoherence program gives the following
andto admitonly local (i.e. confinedinbounded regions
Answer. The only thing the superposition principle of three-dimensional space) subsystems. This prescrip-
guarantees is that every state of Hilbert space can be tion, however, seems to be too restrictive. For example
(in principle) prepared. However as long as we deal itdoesnotcoveracommonsituationinwhichbothther-
with an open system interacting with an environment, malization and decoherence are well established [26]: a
itsstateρ (t)evolvesinsuchawaythatShr¨odinger-cat- particle or a macroscopic object regarded as the system
S
like states decay almost instantly, and we are left with interacts through collisions with a gas of host particles
the quasiclassicalstates robust against decoherence [5]. regarded as the environment. In this case the particle
or the object can explore unbounded regions of space.
However our results motivate another This example illustrates that a straightforward applica-
tionofthe localityrequirementisnotagoodsolution. It
Question. The above answer is true for some tensor
remains an open question how the partitioning ambigu-
product structures (in other words, for some ways of re-
ity should be resolved in the decoherence and quantum
solving a closed system into subsystems) but necessarily
thermalization programs.
not true for other tensor product structures. All tensor
To summarize, the main result of the present paper
productstructuresareequallyreal. Howisa“right”TPS
is as follows. For any large closed quantum system
chosen from the space of all TPS?
a partitioning always exists with respect to which the
Thisquestionisingeneralunanswered[13,14](andvery decoherense-assistedclassicalityfailstoemergeandther-
often even unaddressed) in the decoherence program. malization fails to occur.
It seems that the only conceivable way out is to ad-
mit that not all TPS are ”equally real”: some physical
requirement distinguishing a certain class of acceptable
tensor product structures should be explicitly specified
ACKNOWLEDGEMENTS
in the decoherenceprogram(as wellas inquantum ther-
malization program).
Of course, it is clear from causality arguments based The author is grateful to ERC (grant no. 279738
on the finiteness of the speed of communication that NEDFOQ) for financial support. The partial support
there exists a class of distinguished ”good” partition- from grants NSh-4172.2010.2, RFBR-11-02-00778 and
ings which respect locality in some form. We emphasize, RFBR-10-02-01398is also acknowledged.
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