Continuous measurements in a composite quantum system and possible exchange of information between its parts

0.15 MB·English

by  E. D. Vol

Continuous measurements in a composite quantum system and possible exchange of
information between its parts.
E. D. Vol
B.Verkin Institute for Low Temperature Physics and Engineering National
Academy of Sciences of Ukraine, Lenin av. 47 Kharkov 61103, Ukraine
We study an influence of the continuous measurement in a composite quantum system C on the
evolution of the states of its parts. It is shown that the character of the evolution (decoherence or
recoherence)dependsonthetypeofthemeasuredquantityandontheinitialstateofthesystem. A
numberofconditionsunderwhichthestatesofthesubsystemsofCdecohereduringthemeasuring
process are established. We propose a model of the composite system and specify the observable
themeasurementofwhich mayresultin therecoherenceofthestateofoneofthesubsystemsofC.
Intheframework ofthismodelwefindtheoptimalregimefortheexchangeofinformation between
5
the parts of C during the measurement. The main characteristics of such a process are computed.
0
Weproposeaschemeofdetectionoftherecoherenceunderthemeasurement inaconcretephysical
0
experiment.
2

n PACSnumbers: 03.65.-w,03.65.Ta
a
J

2 The main goal of the paper is to point out a possibil- concluding part of the paper we propose the scheme for
1 ity of the recoherence of the state of one of the parts of the observationof the recoherence in a concrete physical
a composite quantum system under the measurement of experiment.
1
physicalquantitiesofacertaintype. Anotherwords,the Let us go to the details.
v
decoherenceofthe stateofthe compositesystem,caused During last years the point of view, formulated
8
5 by the interaction with the environment, can be accom- most precisely by Zurek [1], becomes more and more
0 panied, under certain conditions, with the recoherence, widespreadin the quantum physics community. Accord-
1 or purification, of a mixed state of one of its parts. ing to it, the behavior of a quantum system becomes
0 The paper is organized as follows. First, we review moreclassicalduetoitsinteractionwiththemacroscopic
5
shortlysomebackgroundknowledgeofthetheoryofopen environment. Asarule,thisinteractionresultsinthede-
0
quantumsystems(OQS)andofthe theoryofcontinuous coherence of the state of the quantum system, i.e., the
/
h quantum measurements (CQM) used in the paper. The transformation of initially pure states into mixed ones.
p main object under study, the composite system C that Thus, the decoherence suppresses the possibility of in-
-
t consists of non-interacting parts, is introduced, and the terference of quantum states and changes the initial pic-
n
evolution of the states of the subsystems of C during ture of propagating probability waves to the usual sta-
a
the continuous measurement is considered. A number of tistical description. The idea of the decoherence is basic
u
q conditions under which the measurement of C results in alsoforthetheoryofcontinuousquantummeasurements.
: the decoherence of states of all its parts are formulated. The important conceptual advantage of this theory is
v
Then,weconsideramoreinterestingsituation,whenthe the possibility to consider the measurement as the spe-
i
X measurement in the composite system may result in the cific process of interactionof two systems: the measured
r recoherenceof the state of one of the subsystems. In the (quantum) system and the measuring (device) system.
a case when such a recoherence takes place, one can say The measuring system can be as classical ones as well
about the exchange of information between the subsys- as mesoscopic ones, and, in the framework of the CQM
tems duringthe measuringprocess. Onthe exampleofa theory, the description of the measurement is based on
simplemodelofthecompositesystemwestudytheinflu- the general principles of quantum mechanics and does
enceoftheinitialstatesofthepartsofContheprocessof not involve the ideas of the wave function collapse un-
the exchangeofinformationbetweenthe partsunderthe der the measurement, etc. In the most general form the
measurement of the observable that has only two eigen- CQMtheory has been formulatedinthe papers by Men-
valuesandcommuteswiththeHamiltonianofC.Wefind sky[2]basingonthe methodofrestrictedFeynmanpath
theoptimalregimeoftheinformationexchangeandcom- integrals (the method of quantum corridors). But for
pute the main characteristics of such a process: ∆I , the important class of the problems, connected with the
max
∆S and η, where ∆I is the maximum amount of in- analysis of the behavior of quantum systems under non-
max
formation,receivedbyonesubsystem(receiver),∆S,the selectivemeasurements,anothermoresimplemethodcan
incrementoftheentropyoftheothersubsystem(sender) be used. The method is based on the Lindblad equation
in the same process, and η = ∆I /∆S, the efficiency and yields the same results. Let us remind that at non-
max
coefficient for the information exchange under the mea- selectivemeasurements,in difference with selective ones,
surement. We also consider the special regime of the in- we are interesting not in the result of the measurement,
formationexchangeunder which the energies ofthe sub- but only in the influence of the measuring procedure on
systems do not change under the measurement. In the the state of the measured system. In general case, the
2
Lindbladequation[3],thatdescribestheevolutionofthe increasing of entropy (dS[ρˆ(t)]/dt 0) reflects the irre-
≥
density matrix of an open quantum system (OQS), has versible character of the measurement and witnesses for
the following form: thedecoherenceofthestateofthemeasuredobject. One
cannote that, accordingto Eq. (1), for a certaintype of
dρˆ i γ
= [Hˆ,ρˆ]+ [Rˆρˆ,Rˆ+]+[Rˆ,ρˆRˆ+] (1) interactionofOQSwiththeenvironmenttherecoherence
dt −~ 2 n o ofitsstatemayalsotakeplace. Asthesimplestexample,
(in what follows we use the systems of units, where ~ = we consider a two dimensional OQS with the g and e
| i | i
1). orthonormal basis states. We choose the operator g e
| ih |
The first term in the right hand side of Eq.(1) is con- as the Rˆ operator, and, correspondingly, the e g op-
| ih |
nected with the internal (Hamilton) dynamics of OQS, erator as the Rˆ+ operator. Using Eq. (1) without the
and the second term describes its evolution caused by Hamiltonterm, one can showthat the unique stationary
the interaction with the environment. A concrete form state of the system is the pure state ρˆ = g g . This
st
oftheoperatorsRˆandRˆ+ inEq. (1)isdeterminedbyre- is the attractive state: limt→∞ρˆ(t) = ρˆst =| ihg| g for
| ih |
duction of description of the closed system (”the system any initial state ρˆ(0). Therefore, beginning from a cer-
under investigation” + ”the environment”) with respect tain time, the recoherence of such OQS will take place.
to the environment variables, or directly from the phys- ThisexamplecanbegeneralizedforOQSofanarbitrary
ical reasons. As was shown, for the first time, by Lind- finitedimensionN. Moreover,foranyfixedpurestateof
blad [3], Eq. (1) for ρˆ(t) is the only possible equation thatsystemonecanconstructthecorrespondinginterac-
of the Markov type that satisfies the main postulates of tionwiththeenvironmentthatprovidesevolutionofany
quantum mechanics: 1) the superposition principle (lin- initial state to that pure state.
earity in ρˆ); 2) the conservation of the total probabil-
Letusnowformulatetheproblemofinterest. We con-
ity (tr(ρˆ) = 1); and 3) the condition of non-negativity
sider a composite quantum system C consisting of such
of ρˆ: ψ ρˆ(t)ψ 0 for all t 0. Let us show with
h | | i ≥ ≥ two parts R and S that the interaction between these
the use of Eq. (1) how the interaction of the quan-
parts can be neglected. Let one measures an observable
tum system with the environment (or the form of the Oˆ of the composite system. According to the foregoing
operators Rˆ and Rˆ+) determines the character of evo- stCatement, any state of C decoheres during the measure-
lution of its states, in particular, their possible decoher- ment. But the following question emerges: how does the
ence or recoherence. To do this, we introduce the linear states of R and S evolve under the measurement? Ob-
entropy S[ρˆ(t)] = tr(ρˆ(t) ρˆ2(t)) 1 tr(ρˆ2(t)), the viously, there are two possibilities (alternatives) A1 and
− ≡ −
quantity that characterizesa degree of ”purity” (i.e. the A2 in such a situation.
coherence) of the quantum state with the density ma-
A1. The decoherence of the state of the composite
trix ρˆ(t). One can show that the linear entropy has the
system C is accompanied with the decoherence of both
properties similar to ones of the von Neumann entropy
its parts R and S.
S = tr(ρˆ(t)lnρˆ(t))[4]. But, for the analysis of the
vN
−
evolution of ρˆ(t) in the framework of Eq. (1) the linear A2. In spite of the decoherence of the state of C, the
entropy serves as more convenient measure of the coher- purification(orthe recoherence)ofaninitialstateofone
ence of the state. Let us consider now the important part (for definiteness, R) of the composite system takes
special case of Eq. (1), when the operators Rˆ and Rˆ+ place.
commute with eachother. One can prove (see Appendix
In this paper we are mainly interested in the alterna-
A) that the linear entropy of such OQS increases with
tive A2. Ifthis possibility is realizedone cansaythatan
time dS[ρˆ]/dt 0. This result is valid for any hermitian
information is transferred from S to R during the mea-
≥
operator(Rˆ =Rˆ+ =Oˆ). Inthis casethe Lindbladequa- surement. It is natural to call the subsystem R as the
tion (without the Hamilton term) can be written in the receiver and the subsystem S as the sender. Prior the
form: studyofthepossibilityA2weshouldpointoutanumber
dρˆ γ of conditions that exclude its realization.
= [Oˆ,[Oˆ,ρˆ]]. (2)
dt −2 One can prove (see Appendix B) that the measure-
mentofanadditive observableOˆ =Aˆ ˆ1 +ˆ1 Bˆ
According to the CQM theory (see [2]), it is just the C R⊗ S R⊗ S
results in the decoherence of the state of R and S for
equation that describes the evolution of a quantum sys-
any initial state ρˆ (0). Another case of realization of
temundera continuousnon-selectivemeasurementofan C
observableOˆ. It is necessaryto emphasize the difference the possibility A1 is the measurementof amultiplicative
observable Oˆ =Aˆ Bˆ under condition that the ini-
of the von Neumann and the CQM theory schemes of C R⊗ S
tial states of the subsystems R and S are uncorrelated:
description of the measurement. In the von Neumann
approach the measurement of the observable Oˆ is an in- ρˆC(0)=ρˆR(0)⊗ρˆS(0) (see Appendix B).
stant process that results in a sharp change (collapse) of Now we consider the simplest model of the composite
the state of the system. In the CQM theory the same system C and construct the observable Oˆ the measure-
C
final state is reached for the finite time ∆t γ−1. This ment of which, under certain initial conditions, realizes
0
∼
timeisneglectedinthevonNeumannscheme. Thelawof the case A2. Let the Hamiltonian of C (with noninter-
3
acting parts R and S) has the form Thus, in this model, the measurement of the observ-
able Oˆ (4) results in the simple picture of evolution
C
Hˆ =Hˆ ˆ1 +ˆ1 Hˆ , (3) of the states of the subsystems R and S of the com-
C R S R S
⊗ ⊗
posite system C. If the initial states ρˆ (0) and ρˆ (0)
S R
where Hˆ and Hˆ are the Hamiltonians of the subsys- are specified, the system of equations (7), (8) for ρˆ (t)
R S S
tems R and S. and ρˆ (t) allows to determine the states of the parts S
R
Letusassumenowthatthe subsystemsSandRareof and R at an arbitrary time t. For further consideration
thesamedimension: dimR=dimS =N. Wealsoimply we need to know the relation between the final states
thattheHamiltoniansHˆS andHˆRareunitaryequivalent: ρˆR( ) = limt→∞ρˆR(t), ρˆS( ) = limt→∞ρˆS(t) and the
Hˆ = UˆHˆ Uˆ+, where UˆUˆ+ =Uˆ+Uˆ =ˆ1. We define the initi∞al states ρˆ (0), ρˆ (0). ∞Integrating the system (7),
S R R S
operator Oˆ as (8) and approaching the limit as t we obtain the
C
→ ∞
desired relation in the form of two equations
Oˆ =(Uˆ+ Uˆ)Tˆ, (4)
C
⊗ ρˆ (0)+Uˆ+ρˆ (0)Uˆ
R S
where Tˆ is the hermitian operator that permutes the ρˆR(∞)= 2 , (9)
statesofthesubsystemsRandS.Theactionoftheoper-
atorTˆ onthebasisstatesofCisdefinedbytheequation:
ρˆ (0)+Uˆρˆ (0)Uˆ+
S R
Tˆ i R j S = j R i S. (5) ρˆS(∞)= 2 . (10)
| i ⊗| i | i ⊗| i
Three essential properties of the operator Oˆ follows di- Now we have all that is needed for the study of the pro-
C
rectly from the definitions (4) and (5): a) Oˆ = Oˆ+ = cess of the exchange of information between the parts R
Tˆ(Uˆ Uˆ+); b) Oˆ2 =ˆ1 ; c) [Oˆ ,Hˆ ]=0. C C and S under the measurementof the observableOˆC. We
⊗ C C C C consider this problem in the following formulation. Let
The property a) shows that Oˆ is the hermitian oper-
C at time t = 0 (the starting time of the measurement)
ator and, consequently, Oˆ is the observable. It follows
C the state of the sender ρˆ (0) is known. We will find the
from b) that the eigenvalues of OˆC are equal to 1 or -1. initialstateofthe receiveSrρ˜ˆ (0)thatprovidesthe maxi-
The property c) indicates that the total energy of the R
mumamountofinformation∆I transferredfromStoR
R
compositesystemCconservesunderthemeasurementof
during the measurement. The increment of the amount
Oˆ (while the energies of its parts may not conserve).
C of information in the subsystem R for the time of the
Besides that, we note thatthe commutativity of OˆC and measurement can be written in the form:
Hˆ results in that the evolution of the density matrix
C
ρˆC is totally determines by the process of the measure- ∆IR = ∆SR =SR[ρˆR(0)] SR[ρˆR( )]
mWˆentcoofnnOˆeCct.eIdndweietdh, ρlˆet ubsyintthreoduuncietatrhyetdreannssiftoyrmmaattiroinx − =tr(ρˆ2R(∞))−−tr(ρˆ2R(∞0)). (11)
C C
ρˆC =e−iHˆCtWˆCeiHˆCt. As is easily seen, the matrix WˆC Substituting Eq. (9) into Eq. (11) we find the depen-
satisfiesthesameequationasthe matrixρˆC butwithout dence of ∆IR on the initial states of R and S
the Hamilton term:
3 1 1
∆I = tr(ρˆ2(0))+ tr(ρˆ2(0))+ tr(ρˆ (0)Uˆ+ρˆ (0)Uˆ).
dWˆ 1 R −4 R 4 S 2 R S
C = OˆC, OˆC,WˆC . (6) (12)
dt −2h h ii ComputingthemaximumofthequadraticfunctionalEq.
Eq. (6) for Wˆ is the master equation of the CQM the- (12) with respect to ρˆR(0) under the additional restric-
C
tion tr(ρˆ (0))=1 we determine the desired density ma-
ory, written in the dimensionless form. Let us introduce R
now the density matrices ρˆR and ρˆS that describe the trix ρ˜ˆR(0)
states of the parts R and S of the system C and derive
1 2
the equations of their evolution under the measurement ρ˜ˆ (0)= Uˆ+ρˆ (0)Uˆ + ˆ1 (13)
ofOˆ . By definition, ρˆ =tr (Wˆ ) andρˆ =tr (Wˆ ). R 3 S 3N R
C R S C S R C
Takingthetracesofthelefthandsideandtherighthand
and the maximum amount of information ∆I trans-
R
side of Eq. (6) over the states of the subsystem S and
ferred from S to R in the optimal regime for the given
usingthepropertiesoftheoperatorOˆC wefindtheequa- initial state of the part S
tion of evolution for ρˆ :
R
1 1
dρˆR =Uˆ+ρˆ Uˆ ρˆ . (7) ∆IRmax{ρˆS(0)}= 3(cid:20)tr(ρˆ2S(0))− N(cid:21). (14)
S R
dt −
Asfollowsfromtheexpression(14),foranyinitialstateof
Analogously, we obtain the equation for ρˆ :
S the sender (excluding the disordered one ρˆ (0)=ˆ1 /N)
S S
dρˆS =UˆρˆRUˆ+ ρˆS. (8) oinnfeorcmanatfiionnd∆inIitriaeclesitvaetdesdρuˆRri(n0g)tfhorewmheiacshurtehmeeanmtowuinlltboef
dt −
4
positive. The global maximum of the amount of infor- ∆I , ∆S and η remains the same. For instance, to com-
R
mation transferred from S to R is reached for the pure pute∆Ie oneshouldfindthemaximumofthefunctional
R
initial state of S and it is equal to (12) with respect to ρˆ (0) under two additional condi-
R
tions: tr(ρˆ (0)) = 1 and tr(ρˆ (0)Hˆ ) = tr(ρˆ (0)Hˆ ).
R R R S S
1 1
∆I = 1 (15) Using the Lagrange multipliers method we obtain, after
max
3(cid:18) − N(cid:19) simplecomputations,theoptimalinitialstateofthepart
R
Another essential characteristics of the information
transfer process is the increment of the entropy of the Uˆ+ρˆ (0)Uˆ 2 2tr(ρˆ (0)Hˆ )
sender ∆S. Let us find this quantity for the optimal ρˆeR(0)= S3 +3Nˆ1R+3 trS(Hˆ2)S HˆR. (20)
regime of the information exchange considered above. R
∆S S[ρˆ ( )] S[ρˆ (0)] Substitutingthisexpressionintoequation(12)weobtain
S S
=≡tr(ρˆ2(0∞)) −tr(ρˆ2( )) the maximum amount of information transferred in the
= 3tr(ρˆ2(0)) 1tr(ρ˜ˆ2(0)) 1trS(ρˆ (0−)Uˆ˜ρˆ (S0)U∞ˆ+)(16) isoenergetic regime of the measurement of OˆC
4 S − 4 R − 2 S R
2
tr(ρˆ (0)Hˆ )
Note that we use Eq. (10) for the derivation of (16). ∆Ie ρˆ (0) = 1tr(ρˆ2(0)) 1 (cid:16) S S (cid:17) 
Substituting the expression (13) for ρ˜ˆ (0) into Eq. (16) R{ S } 3 S − N − tr(Hˆ2) 
R R
we obtain
 (21)
5 1 Notethatthederivationof(21)wasdoneundertheaddi-
∆S[ρˆ (0)]= tr(ρˆ2(0)) . (17)
S 9(cid:20) S − N(cid:21) tionalconditiontr(Hˆ )=0thatfixesthereferencepoint
R
for the energy. This condition does not influence on the
Followingthe approachacceptedin the thermodynamics
generality of the results obtained.
of the informational processes (see [6]) we introduce the
We also present the result for the increment of the
coefficient of efficiency of the information transfer η. By
entropy of the sender S in the isoenergetic regime of the
definition, η ∆IR/∆S. It follows from the law of the measurement of Oˆ :
≡ C
increase of entropy that η 1 for any evolution of the
≤
composite system. Comparing the expressions (14) and 2
tr(ρˆ (0)Hˆ )
(17)wefindthatη =3/5anditdoesnotdependonthe 5 1 S S
0 ∆Se ρˆ (0) = tr(ρˆ2(0)) (cid:16) (cid:17) 
initial state of the sender in the optimal regime of the { S } 9 S − N − tr(Hˆ2)
information exchange.  R 
 
Letusnowdiscussthequestionontherelationbetween (22)
the energy transfer and the information transfer under Comparing the expressions (21) and (22), we find that
the measurement of Oˆ in the model considered. For the coefficient of efficiency of the information transfer in
C
this purpose we determine the evolution of the average such a regime η is also equal to 3/5.
energiesofthesubsystemsRandS:E (t) tr(ρˆ (t)Hˆ ) To illustrate the expression (21) we consider the op-
R R R
≡
andE (t) tr(ρˆ (t)Hˆ ). UsingEqs. (7),(8),weobtain timal process of the measurement for the simplest com-
S S S
the simple≡equations of evolution of E (t) and E (t): posite systemC ofthe dimension4 (dimC =4, dimR=
R S
dimS = 2) without energy exchange between its parts.
dER We use the representation where the Hamiltonian of the
=E E , (18)
dt S − R sender is diagonal: Hˆ = ∆ 1 0 with 2∆ , the
S 0(cid:18)0 1(cid:19) 0
−
distance between the energy levels of the part S. Let at
dE
S
=E E . (19) thestartingtimeofthemeasurementthestateofSisde-
R S
dt −
a c
Note that the unitary equivalence of the Hamiltonians scribed by the density matrix ρˆS(0) = (cid:18)c∗ b(cid:19). Then
IHtRfoallnodwsHfSromwasthuesseedeqfourattihoensdtehriavtattihoen eonfe(r1g8y),of(1t9h)e. tdre(rρˆaScHˆcoSu)n=tin∆g0t(hae−nbo)r,mtra(liHzˆaR2t)io=nctro(nHdˆiS2t)io=na2∆+20ba=nd1uthne-
compositesystemconservesandtheenergyistransferred expression (21) for ∆Ie takes the form
R
fromthe”hotter”to”colder”partuntiltheirenergiesbe-
comes equal each other at the end of the measurement. 1 1 (a b)2
Itisclearthatiftheinitialenergiesofthepartsareequal ∆IRe {ρˆS(0)}= 3(cid:20)a2+b2+2|c|2− 2 − −2 (cid:21)
eachother, there is no transfer ofenergy under the mea- 2
surement. We call such a regime the isoenergetic one. = c2.(23)
3| |
Let us compute the main characteristics of the informa-
tion transfer process in the optimal isoenergetic regime. One can see that the maximum amount of information
Asthe matteroffact, inthis case,theformulationofthe transferred to one of the parts under the measurement
problem and the method of computing of the quantities of Oˆ in the isoenergetic regime is determined by the
C
5
non-diagonalelements ofthe densitymatrixρˆ (0)inthe the interference patternof the recoherredbeam with the
S
representation where Hˆ is diagonal. interference pattern of the test beam (obtained for the
S
same ρˆ and ρˆ , but without the ”measuring” magnetic
Thus, we summarize the results of the paper. On the 1 2
atom) one can check all main conclusions and relations
example of the simple model of the measurement in the
of this paper.
compositesystemwehavedemonstratedthepossibilityof
IwouldliketoacknowledgeI.V.Krive,L.A.Pasturand
the exchange of information between its parts. We com-
A.A.Zvyagin for useful discussions of the results of the
pute the main characteristics of such a process for the
paper.
optimal initial states of the subsystems. One should em-
phasize that, in itself, the possibility of the recoherence
of one part of the composite system under the measure-
APPENDIX A
ment is determined entirely by the type of the measured
quantity and by the initial state of the system, and it
does not depend on the simplified assumptions used in Let us present the proof of the statement of the pa-
our consideration. The generalization of the results to per on the monotonical increase of (linear) entropy of
the case of unitary non-equivalent Hamiltonians of the OQS, the interaction of which with the environment is
partsRandSandthestudyoftheinformationexchange described in the framework of the Lindblad equation
process between the subsystems of different dimensions (1) by the operators Rˆ and Rˆ+ that obey the relation
is postponed for further publications. In conclusion, we [Rˆ,Rˆ+]=0.
discussshortlythepossibilityoftheobservationoftheef- UsingEq. (1)andtheexpressionforthelinearentropy
fectpredictedinthispaper-the recoherenceofthe state S[ρˆ]=1 tr(ρˆ2)onecanwritetheexpressionfortherate
−
ofthesubsystemunderthecontinuousmeasurement. For of change of the entropy
thefirsttime,thesuggestiontousetheprocedureofcon-
dS
tinuous measurementofthe energyof a two-levelsystem = 2tr(ρˆρˆ˙)=2γtr(Rˆ+Rˆρˆ2 ρˆRˆρˆRˆ+). (A1)
dt − −
forthemonitoringofthequantumtransitionwasputfor-
wardin Ref. [7]. The experimentalscheme ofrealization As follows from (A1) the monotonic increase of the en-
ofthis idea was alsodescribed in thatpaper. The object tropy (dS/dt 0) takes place if the following inequality
of the measurement is a polarized atom with the tran- is satisfied: ≥
sition excited by resonant pumping. The electron beam
thatscattersontheatom(interactingwithitsdipolemo- tr(Rˆ+Rˆρˆ2) tr(ρˆRˆρˆRˆ+). (A2)
≥
mentum) is used as a meter. Measuring the scattering
To prove the inequality (A2) we use the Cauchy-
angle one can obtain the energy of the probed atom at
Bunyakovsky-Schwarz(CBS) inequality
any time.
Using the ideas of Ref. [7], we describe the simplest, A 2 B 2 (AB)2, (A3)
fromourpointofview,schemeoftheexperimentinwhich || || ·|| || ≥
the effect of recoherence under the measurement in a thatisvalidinanylinearspacewherethescalarproduct
composite system can emerge. For this purpose, we con- of two vectors A and B ( A √AA) is defined. We
|| || ≡
sider two crossed beams of neutral particles (neutrons) remind that for any pair of linear operators Aˆ and Bˆ
withthespinss =s =1/2thatpropagateclosetoeach actinginavectorspaceoffinitedimensiononecandefine
1 2
other in the area that contains a massivemagnetic atom the operation AˆBˆ tr(AˆBˆ+) that satisfies all axioms of
≡
withthespinS. Weimplytheneutronsofthebeam1are the scalar product [9]. We choose the operator ρˆRˆ as Aˆ
inthemixedstatedescribedbythedensitymatrixρˆ1and and the operator Rˆρˆas Bˆ and write the CBS inequality
theneutronsofthebeam2-inthestatedescribedbythe
(A3) for these operators:
densitymatrixρˆ . Itisassumedthattwobeamsaresyn-
2
chronizedinsuchawaythatintimewhentheneutronof tr(RˆRˆ+ρ2)tr(Rˆ+Rˆρˆ2) [tr(ρˆRˆρˆRˆ+)]2. (A4)
one beammovescloseto the magneticatomthere is also ≥
the neutron of beam 2 near this atom. Since there is the Here we use the possibility to do cyclic permutations of
exchangeinteractionbetweenneutronsofdifferentbeams operators under the trace.
and between neutrons and the magnetic atom, one can Usingthe commutativityofRˆ andRˆ+ andtakinginto
consider that in such an experiment the massive atom account that tr(Rˆ+Rˆρˆ2) = tr(ρˆRˆRˆ+ρˆ) 0 we obtain
≥
provides the continuous measurement of the observable from(A4) the requiredinequality (A2). Thus, the state-
Oˆ =ˆs ˆs of the two-particle system. Let us remind now mentonthe monotonicincreaseoflinearentropyinsuch
1 2
that the operator ˆs ˆs is connected with the operator of OQS is proven.
1 2
spin permutation Tˆ by the relation Tˆ = (1+ˆs ˆs /4)/2
1 2
(see[8]). Takingintoaccounttheforegoingstatementsof
APPENDIX B
the paper we arrive to the conclusion that under proper
choice of the states of the beams ρˆ and ρˆ such an ex-
1 2
perimentshoulddemonstratethe effectofrecoherenceof In this appendix we consider two particular realiza-
the state of neutrons of one of the beams. Comparing tions(cases)ofthealternativeA1(thedecoherenceofthe
6
states ofthe parts under the continuous measurementin the quantities Aˆ and Bˆ in eachof the subsystems and
R S
the composite system). that is why it results in the decoherence of its states.
Case 1. The measurement of the additive variable.
Case2. Themeasurementofthemultiplicativeobserv-
Under such measurement the evolution of the state
able.
ρˆ (t) is described by the master equation of the CQM
C As above, the evolution of ρˆ (t) is determined by Eq.
theory. Thisequation,writteninthedimensionlessform, C
(B1), but the measured quantity Oˆ has the form
reads as C
dρˆ (t) 1
C = Oˆ , Oˆ ,ρˆ , (B1) Oˆ =Aˆ Bˆ . (B7)
C C C C R S
dt −2h h ii ⊗
where One canuse the generalform(B3) of the solution of Eq.
(B1) and write ρˆ (t) in the basis i α , where i
OˆC =AˆR ˆ1S +ˆ1R BˆS. (B2) arethe eigenvectoCrsofAˆ ,and α| iRar⊗e|thieSeigenvec|toirRs
⊗ ⊗ R S
| i
of Bˆ . After simple calculations we find that
One can check directly that the general solution of Eq. S
(B1) has the form
iαρˆ (t)jβ = iαρˆ (0)jβ e−(AiBα−AjBβ)2t, (B8)
∞ C C
ρˆ (t)= 1 dse−s2/2te−iOˆCsρˆ (0)eiOˆCs. (B3) h | | i h | | i
C C
√2πtZ−∞ whereA andB aretheeigenvaluesoftheoperatorsAˆ
i α R
Thefactorexp( iOˆCs)undertheintegralin(B3)canbe and BˆS in the states |iiR and |αiS, respectively.
−
presented as Let us compute the rate of change of the entropy of
the subsystem R
exp( iOˆ s)=exp( i(Aˆ ˆ1 +ˆ1 Bˆ )s)
C R S R S
− − ⊗ ⊗
=(exp(−iAˆRs)⊗ˆ1S)·(ˆ1R⊗exp(−iBˆSs)). (B4) dSR = dtr(ρˆ2(t))= 2 ρR(t)ρ˙R(t)
dt −dt R − ij ji
Accordingto the quantumtheory,ρˆR(t), the state ofthe Xij
part R, is determined by the relation =2 e−(Bα2+Bβ2)(Ai−Aj)2t(Ai−Aj)2Bβ2
αXβij
ρR(t)=tr (ρˆ (t)) iαρˆ (t)jα , (B5) iαρˆ (0)jα jβ ρˆ (0)iβ (B9)
ij S C ≡ h | C | i ×h | C | ih | C | i
Xα
If one assumes that at the starting time of the measure-
where iα i α is the orthogonalbasis in C.
| i≡| iR⊗| iS ment of Oˆ the state of the parts R and S are not cor-
Using the expressions (B3), (B4) and the definition C
related (ρˆ (0) = ρˆ (0) ρˆ (0)), the expression (B9) is
(B5) we find C R S
⊗
reduced to
∞
ρRij(t)= √21πtZ−∞=dse−s22t1Xα hiα∞|ed−siOeˆ−Css2ρˆ/C2t(0i)ee−iOˆiACˆRs|sjαki ddStR =2αXβij(Ai−Aj)2Bβ2e−(Bα2+Bβ2)(Ai−Aj)2t
Xkl √2πtZ−∞ h | | i ×|ρRij(0)|2ρSαα(0)ρSββ(0). (B10)
kγ ρˆ (0)lγ l eiAˆRs j
C
· h | | ih | | i
Xγ One can see directly from (B10) that dSR/dt 0. The
∞ ≥
= 1 dse−s2/2t e−iAˆRsρˆ (0)eiAˆRs (B. 6) relation dSS/dt ≥ 0 can be obtained by the same way.
√2πtZ−∞ (cid:16) R (cid:17)ij Thus, we have proven the statement of the paper on the
decoherenceofthe stateofthe subsystemRandSunder
Thus, one can see that the measurement of the additive the measurementofthe multiplicative observableOˆ for
C
quantity Oˆ is, in fact, reduced to the measurement of the case of uncorrelated initial states of R and S.
C
[1] W. H. Zurek, Phys. Rev D 24, 1516 (1981); Phys. Rev D mechanics, Princeton, 1955.
26, 1862 (1982). [5] L. Brillouin, Science and information theory,N.-Y,1956.
[2] M. B. Mensky, Zh. Exp. Teor. Fiz. 77, 1326 (1979) [Sov. [6] R.P.Poplavskii, Thermodymamics of information pro-
Phys.JETP50,667(1979)];Phys.RevD20,384(1979); cesses. Moskva, Nauka,1981 (in Russian).
Usp. Fiz. Nauk 168, 1017 (1998). [7] J. Andretsch,M.B.Mensky, Phys.Rev A 56, 44 (1997).
[3] G. Lindblad, Commun. Math. Phys. 48, 119, (1976). [8] R.Feynman,R.Leighton,M.Sands.TheFeynmanlectures
[4] J. von Neumann, Mathematical foundations of quantum on physics.V.3, Massachusetts-London, 1963.
7
[9] M. Marcus, H. Minc. A Survey of matrix theory and ma-
trix inequalities, Boston, 1964.