Entanglement dynamics and relaxation in a few qubit system interacting with random collisions

0.27 MB·English

by  Giuseppe Gennaro

epl draft
Entanglement dynamics and relaxation in a few qubit system in-
teracting with random collisions
Giuseppe Gennaro1, Giuliano Benenti2 and G.Massimo Palma1
8
01 NEST - CNR (INFM) & Dipartimento di Scienze Fisiche ed Astronomiche,
0Universita` degli Studi di Palermo, via Archirafi 36, I-90123 Palermo, Italy
2
2 CNISM, CNR (INFM) & Center for Nonlinear and Complex systems,

nUniversita` degli Studi dell’Insubria, via Valleggio 11, I-22100 Como, Italy
a
& Istituto Nazionale di Fisica Nucleare, Sezione di Milano, via Celoria 16, I-20133 Milano, Italy
J

9

h] PACS 03.65.Yz–Decoherence; open systems; quantumstatistical methods
p PACS 03.67.-a–QuantumInformation
- PACS 03.67.Mn–Entanglement production, characterization, and manipulation
t
n
Abstract. - The dynamics of a single qubit interacting by a sequence of pairwise collisions with
a
u an environment consisting of just two more qubits is analyzed. Each collision is modeled in
q terms of a random unitary operator with a uniform probability distribution described by the
[ uniform Haar measure. We show that the purity of the system qubit as well as the bipartite

and the tripartite entanglement reach time averaged equilibrium values characterized by large
1
instantaneous fluctuations. These equilibrium values are independent of the order of collision
v
amongthequbits. Therelaxationtoequilibriumisanalyzedalsointermsofanensembleaverage
4
1 ofrandomcollision histories. Suchaverageallowsforaquantitativeevaluationandinterpretation
4 ofthedecayconstants. Furthermoreadependenceofthetransientdynamicsontheinitialdegreeof
1 entanglementbetweentheenvironmentqubitsisshown toexist. Finally thestatistical properties
. of bipartite and tripartite entanglement are analyzed.
1
0
8
0
:
v
i Introduction. – The repeated collision model has unitary operators [7–12].
X
rbeenrecently used inliterature to analyze the irreversible Random unitary operators have received considerable
adynamicsofaqubitinteractingwithareservoirconsisting attention in quantum information theory, mainly be-
of a large number of environmental qubits. In particular cause they find applications in various quantum proto-
processeslikethermalization[1]andhomogenization[2–5], cols [13–16]. Unfortunately the implementation of a ran-
have been analytically investigated. The same model has dom unitary operatoracting on the n-qubitHilbert space
ben used recently also to analyze the dynamics of a qubit requires a number of elementary quantum gates that is
interacting with a very small environment consisting of exponential in the number of qubits. On the other hand
just two qubits [6]. The interest for such system is due to sequences of random two-qubit gates (collisions) generate
thefactthat,atvariancewithwhathappensinthecaseof pseudo-random unitary operators which approximate, to
anenvironmentwithalargenumberofdegreesoffreedom, the desired accuracy, the entanglement properties of true
the systemdynamicscannotbe describedbyaMarkovian n-qubit random states [17–23]. This approach has given
master equation. Indeed, due to the fact that the sys- verygoodresults,showingthatpseudo-randomstatescan
tem qubit collides repeatedly with the same environment be generated efficiently, that is polynomially in n [17–23].
qubits,thedynamicsischaracterizedbylargefluctuations
Ourchoicetodescribethepairwisecollisionsintermsof
and only when the sequence of collision is random a time
random two qubit unitary operators is motivated by the
averaged equilibrium is reached. While in all the above
fact that often a precise modelization of the interactionis
mentionedpapers the - elastic - collisionshave been mod-
very hard and that, on the other hand, a good descrip-
eled by a partial swap unitary operator, in the present
tion of the approach to equilibrium can be obtained by
paper we will analyze the system dynamics in the case in
suitable averages of the quantities of interest, as we will
which the two-qubit collisions are described by random
describebelow. Furthermoresuchcollisionmodelexhibits
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G.Gennaro, G.Benenti and G.M.Palma
interesting features ranging from memory effects to the unpolarizedmixedstate. Asalreadymentioned,duetothe
efficient entanglementgenerationbetween system and en- small number of environment qubits, the instantaneous
vironment. system purity undergoes large fluctuations regardless of
the initial state of both the system and the environment
The model. – In order to illustrate the approach of
and regardless of the sequence of collisions as shown in
ourwork,wefirstreviewtherepeatedcollisionmodel. Let
Fig.1 (firstrow). The purity,however,approachesa time
us consider a set of N +1 qubits, the first of which is the
averaged equilibrium state. To see this we calculate the
system qubit and the remaining N are the reservoir. The
time averagedpurity (t) as
TA
system-environment interaction is due to pairwise colli- P
sions between the system and a singe reservoir qubit. Af- 1 t
tertcollisionstheoverallstateofthesystemplusreservoir PTA(t)= t+1 XTr(cid:2)ρ2S(t′)(cid:3). (2)
is t′=0
̺(StE) =Uit···Ui2Ui1̺(S0E)Ui†1Ui†2···Ui†t, (1) AsshowninFig.1(secondrow),PTA(t)reachesthesame
equilibrium value regardless of the initial entanglement
where ̺ is the total density operator and the sequence
SE of the environment qubits, a natural consequence of the
i i specify the order with which the environment
1··· t random nature of the collisions.
qubits collide with the system one. As in [6] we have
concentrated our attention to the case in which the en-
1.0 1.0
vironment consists of just two qubits. In the following
0 will label the system qubit while 1,2 will label the en-
vironment qubits. At variance with the previous work,
P P
however, we have considered here the case in which each
collision is described by a random unitary operator U
i
picked up from the uniform Haar measure on the group 0.1500 102 104 0.1500 102 104
U(4). Inparticularinourcalculationswehavefoundcon- t t
venientto parametrizeeachrandomunitarymatrixU ,in 0.9 0.9
i
terms of the Hurwitz representation of the unitary group
U(4) [12,20,26]. Such different choice of the collision uni-
TA TA
tary operators has several consequences. First of all the P P
collisions are of course no longer elastic and, regardless
of the number of environment qubits, homogenization is
0.55 0.55
no longer achieved. However, even for a few qubits en- 100 102 104 100 102 104
t t
vironment, a time averaged equilibrium state with large
0.8
fluctuations is reached regardless of the order with which
2/3
the qubits collide (in the numericaldata shownbelow, we
consider random sequences i , i ). Furthermore such
1 t EA EA
··· P P
equilibrium state is independent of the initial state of the
qubits. In the following we will characterize some aspects
2/3
of such approach to a time averagedequilibrium state.
0.65 0.6
0 10 20 30 0 10 20 30
Random qubit-environment interactions have been re- t t
cently considered [24,25], for high dimensional environ-
ments. However in our work we deal with small environ-
ments,suchthattheinformationacquiredbytheenviron- Fig. 1: From top to bottom: instantaneous, time averaged
ment on the system can flow back to the system and a andensembleaveraged purity,foranenvironmentinaninitial
Markoviandescriptionofourmodelis surelynotpossible. product (left) or maximally entangled (right) state. The time
evolutionoftheensembleaveragedpurityisinbothcaseswell
Moreover, there is no weak coupling parameter and the
state of the environment is significantly affected by the fitted by the exponential curve PEA(t)−PL ∝ exp(−0.36t),
interactionateachcollision,sothatalsotheBornapprox- with PL = 23 (asymptotic horizontal line).
imation does not apply.
Further insight into the approach to equilibrium of the
Approach to equilibrium.– Wehavecharacterized system qubit is obtained if rather than the time average
thedecoherenceofthesystemqubitintermsofthepurity purity we evaluate the ensemble averaged purity (t)
EA
P
. We remind the reader that the purity is defined as over the uniform Haar measure. As we shall see, an anal-
P
=Tr ̺2 , where ̺ =Tr [̺ ] is the reduced density ysisofthetimeevolutionofsuchaveragedquantityallows
P (cid:2) S(cid:3) S E SE
operator of the system qubit. The purity is a decreasing ustodeterminethetimescalefortherelaxationtoequilib-
function of the degree of statistical mixture of the qubit rium. Toevaluate (t)wehavegeneratedalargenum-
EA
P
and takes values in the range 1 1 where = 1 berofsequencesofrandomcollisions,eachonedrawnfrom
2 ≤ P ≤ P
corresponds to pure states and = 1 to the completely theuniformmeasure,andwehaveaveragedthepurityover
P 2
p-2
Entanglement dynamics and relaxation in a few qubit system interacting with random collisions
the different histories after t steps. Also such ensemble of two qubits, the tangle τ is defined as
i|j
average shows an irreversible behaviour of the qubit dy-
namics. Regardless of the initial state of the environment τ (ρ)=[max 0,α α α α ]2, (4)
i|j 1 2 3 4
{ − − − }
qubits, (t)decaystothevalue 2,whichcoincideswith
PEA 3
the average purity PL predicted by Lubkin [28] for true where {αk} (k =1,..,4) arethe squareroots of the eigen-
overall (system-environment) random states. Note that values (in non-increasing order) of the non-Hermitian op-
IPfTµA(atn)daνndarPeErAes(pt)ectteinvedlytothtehedismaemnesioansysmofpttohteicHvilablueret. eorpaetroartoρ¯rijan=dρρi∗ijj(σisy t⊗heσyc)oρm∗ijp(σleyx⊗coσnyj)u,gaσtyeiosftρhiej,yi-nPatuhlei
space of the system and of the environment one has eigenbasis of the σz σz operator. The concurrence C is
⊗
defined simply as Cij = √τi|j. The tangle τi|j, or equiv-
µ+ν alently the concurrence Cij can be used to quantify the
= . (3)
PL µν+1 entanglement between the pair of qubits i,j for an arbi-
trary reduced density operator ρ . Furthermore, when
ij
In our case µ = 2 and ν = 4, and therefore = 2. the overall state of the system is pure, the amount of
PL 3 entanglement between qubit i and all the remaining can
However,asshowninFig.1(thirdrow)thetimeevolution
of EA(t) exhibits a clear dependence on the degree of be quantified by the tangle τi|rest = 4detρi. The tan-
entPanglementoftheinitialstateoftheenvironmentqubits. gle τ0|rest between the system qubit and the environment
conveys the same information as the purity . Indeed
Notablytheexponentialapproachtoequilibriumisforthe P
firstcollisionsadecreasing(increasing)functionoftforan it is easy to show that τ0|rest = 2 −2P. We have nu-
merically computed the tangles τ , τ , and τ of the
environment in an initial product (maximally entangled) 0|1 0|2 1|2
two-qubit reduced density matrices and the three-tangle
state. In both cases a numerical fit shows that (t)
EA
P − τ = τ τ τ , where i,j,k can be any permu-
L exp( λt) with λ 0.36. i|j|k i|jk − i|j − i|k
P ∝ − ≈ tation of 0,1,2 and where the tangle τ measures the
The asymptotic relaxation to equilibrium can be com- i|jk
entanglement between the ith qubit and the rest of the
puted analytically following the approach developed in
system, i.e., qubits j,k. The three-tangle τ is a mea-
[21–23]. Each pure three-qubit state ρ can be 0|1|2
SE sureofthe purelytripartiteentanglementandisinvariant
expanded over products of Pauli matrices: ρ =
SE under permutations of the three qubits [31].
c σα0 σα1 σα2, where σαi denotes a
PPaαu0l,iαm1,αa2triαx0aαc1αti2ng0on⊗the1ith⊗qu2bit, withα i 0,x,y,z The instantaneous dynamics of the tangles τi|j and of
and σ0 = I. The purity then reads (t) =i ∈{ c2 (t)}. the three-tangle τ0|1|2 is similar to the dynamics shown
P Pα0 α000 in the Fig. 1 (top) for the purity, that is, these quanti-
Thepuritydecaycanthereforebeobtainedfromtheevolu-
ties are characterizedby large instantaneous fluctuations.
tionintimeofthecoefficientsc2 . Fortherandomcolli-
α000 However the time averaged tangles (τ ) (t), defined in
sionmodelthecolumnvectorc2 ofthecoefficientsc2 i|j TA
evolvesaccordingtoaMarkovchaindynamicsofthαe0fαo1rαm2 analogywithEq.(2),approachthesamelimitingvalueτP.
Again we have numerically evaluated the ensemble
c2(t+1) = Mc2(t) [23]. The matrix M is obtained af-
average (with respect to the Haar measure) tangles
ter averaging over the two possible couplings 01 and 02:
(τ ) (t). As shown in Fig. 2 (bottom) the pairwise
M = 1(M(2)+M(2)), with M(2) acting non trivially (dif- i|j EA
2 01 02 ij tangles(τ0|1)EA(t)and(τ0|2)EA(t)approachexponentially
ferently from identity) only on the subspace spanned by
the same equilibrium value τ 0.367. The numeri-
P
qubits i and j. Averaging over the uniform Haar mea- ≈
cal data in Fig. 2 (bottom) are well fitted by the curves
sure on U(4) one can see that Mi(j2) preserves identity (τi|j)EA(t) τP exp( λijt). When the environment is
(pσrio0d⊗ucσtj0s σ→αiσi0σ⊗αjσ[j02)3]a.nTdhuenmifoartmrixlyMmihxaessatnheeigoethnevral1u5e iwnhitiilaelflyorsaepn−airnaibtila∝ellwyemoab−xtaiminaλll0y1e≈ntaλn02gl≈ed0s.t7a6t,eλs1o2f≈th0e.e4n4-,
i ⊗ j
equalto 1 (with multiplicity 2)andallthe other eigenval- vironment we have λ λ 0.44, λ 0.36. There-
01 02 12
≈ ≈ ≈
uessmallerthan1. Thereforetheasymptoticpuritydecay fore, initially entangled environment qubits limit the rate
is determined by the gap ∆ in the Markov chain, namely ofgenerationofbipartite entanglementbetweenthe qubit
by the second largest eigenvalue 1 ∆ of the matrix M. system and a single environment qubit, even though the
−
We have EA(t) L (1 ∆)t = exp [ln(1 ∆)]t . amountofpairwiseentanglementobtainedasymptotically
P −P ≍ − { − }
In our model 1 ∆=0.7 (eigenvalue with multiplicity 2) isalwaysthesame. Wehavefurthermoreverifiedalsothat
−
andtherefore ln(1 ∆) 0.357,inverygoodagreement the average tangle (τ ) approaches the same limiting
− − ≈ 1|2 EA
with the value λ 0.36 obtained from our fit. value τ even though the environment qubits do not col-
P
≈
lide directly.
Entanglementdynamics.– Thedynamicsofbipar- As one would expect from the above discussion, also
tite and multipartite entanglement between the qubits of the mutipartite entanglement approaches an equilibrium
our model shows interesting features. Such dynamics has value. This can be seen by the time and ensemble aver-
been conveniently characterized in terms of the concur- ages three-tangle τ shown in Fig. 3. The approach to
0|1|2
rence and of the tangles [30,31]. We remind the reader the equilibrium valueτ is exponentialalsofor this quan-
T
that, given the density operator ρ of a bipartite system tity and we can extract the convergence rate from the fit
ij
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G.Gennaro, G.Benenti and G.M.Palma
0.4 1
0.4 0.4
A A
A A T T
τ()i|jT τ()i|jT0.4 τ)0|1|2 τ)0|1|2
( (
0 0 0 0
100 102 104 100 102 104 100 102 104 100 102 104
t t t t
0.5 0.5
0.367 0.367
τ()0|1EA τ()0|1EA τ()0|1|2EA0.33 τ()0|1|2EA0.33
0.25 0.25 0
0 10 20 30 0 10 20 30 0 10 20 30 0 10 20 30
t t t t
0.5
0.367
Fig. 3: Time (top) and ensemble (bottom) averages for the
A A
)2E )2E three-tangleτ0|1|2foranenvironmentinaninitialproduct(left)
τ1| τ1| or maximally entangled (right) state. For the ensemble aver-
( ( aged quantities also thefit (τ0|1|2)EA(t)−τT ∝exp(−0.36t) is
0.367
shown.
0 0.35
0 10 20 30 0 10 20 30
t t
500 500
Fig.2: Time(firstrow) andensemble(secondand thirdrows) 400 400
averages for the pairwise tangles τ0|1 (solid line in the first 300 300
row),τ0|2 (dashedlineinthefirstrow)andτ1|2 (dottedlinein 200 200
thefirstrow),foranenvironmentinaninitialproduct(left)or
maximally entangled (right) state. For the ensemble averaged 100 100
quantities also the fit (τi|j)EA(t)−τP ∝exp(−λijt) is shown. 0 0
Inthesecondrowonlythe(τ0|1)EA(t)isshownduetothesame 0 0.5 1 0 0.5 1
figureof (τ0|2)EA(t)
Fig. 4: Statistical distributions of the three-tangle τ0|1|2, for
separable (left) and entangled (right) initial state of the envi-
(τ ) (t) τ exp( λ t), with λ = 0.36 re-
ga0r|d1|l2esEsAonw−hetTher∝the ini−tia0l1s2tate ofthe0e1n2vironmentis ronment. In both cases τ0|1|2 =0.33±0.19.
entangled or separable.
Finally, we note that not only the time and ensemble arethesameforallpossiblequbitpairs. Incontrastto[6],
averagesof the various tangles convergeto the same limit residual entanglement (three-tangle) is generated regard-
valuesregardlessoftheinitialconditions,butalsotoawell less of initial entanglement of the environment qubits.
defined limit distribution. In particular the concurrences
are distributed in accordancewith the distribution shown
∗∗∗
in in Fig.3 of [27] for random pure 3-qubit states. Fi-
nallythenumericallycalculateddistributionsofthethree- G.B.acknowledgessupportfromthePRIN2005”Quan-
tangle in our collision model are shown in Fig. 4, for sep- tum computation with trapped particle arrays, neutral
arable and for entangled initial environment state. and charged”. G.M.P. and G.G. acknowledge support
from the PRIN 2006 ”Quantum noise in mesoscopic sys-
Conclusions. – In summary, we have shown that re- tems”. G.B. acknowledges useful discussions with Marko
laxation(intimeaverage)tostatisticalequilibriumispos- Zˇnidariˇc.
sible for a system of just three qubits undergoing purely
random unitary evolution, regardless of the order of the
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