Time Evolution of Quantum Entanglement of an EPR Pair in a Localized Environment

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by  Jia Wang

Time Evolution of Quantum Entanglement of an EPR Pair in a Localized Environment
Jia Wang, Xia-Ji Liu, and Hui Hu
Centre for Quantum and Optical Science, Swinburne University of Technology, Melbourne 3122, Australia
(Dated: January 25, 2017)
TheEinstein–Podolsky–Rosen(EPR)pairofqubitsplaysacriticalroleinmanyquantumprotocol
applications such as quantum communication and quantum teleportation. Due to interaction with
theenvironment,anEPRpairmightloseitsentanglementandcannolongerserveasusefulquantum
resources. On the other hand, it has been suggested that introducing disorder into environment
might help to prevent thermalization and improve the preservation of entanglement. Here, we
theoretically investigate the time evolution of quantum entanglement of an EPR pair in a random-
fieldXXZspinchainmodelintheAndersonlocalized(AL)andmany-bodylocalized(MBL)phase.
We find that the entanglement between the qubits decreases and approaches to a plateau in the
AL phase, but shows a power-law decrease after some critical time determined by the interaction
7
strengthintheMBLphase. Ourfindings,ononehand,shedlightsonapplyingAL/MBLtoimprove
1
quantuminformationstorage;ontheotherhand,canbeusedasapracticalindicatortodistinguish
0
the AL and MBL phase.
2

n
a An Einstein-Podolsky-Rosen (EPR) pair is a pair of lattices [9–11] and ion traps [12]. These experimentally
J
qubits which are in maximally entangled state. Due to available systems constitute promising platforms for ex-

4 their perfect quantum correlations, EPR pairs lie at the ploring the AL and MBL localization phases and stim-
2
heartofmanyimportantproposalsforquantumcommu- ulated a series of theoretical studies. Many remarkable

nication and computation, such as quantum teleporta- properties of these localized phases have ever since been
]
h tion [1, 2]. In reality, however, due to the unavoidable theoretically predicted: Poisson distributions of energy
p
decoherence induced by the couplings to the surround- gap [13, 14], absences of transportation of charge, spin,
-
t ing environment, an EPR pair might become a state ρ mass or energy even at high temperature [15–17], pro-
n
that lose entanglement after a certain time, making this tecting quantum order and discrete symmetry that nor-
a
u qubitpairnolongerusefulasaquantumresource. Oneof mally only exists in the ground state [18–21] and exis-
q the main tool to overcome the decoherence is a protocol tence of mobility edge [22–24]. In particular, quantum
[ named entanglement distillation [3, 4]. This method can entanglementhasbeendiscoveredtoplayveryimportant

2 be used to transform N copies of less entangled states rolesinidentifyingdifferentphases: energyeigenstatesin
v ρ back into a smaller number m of approximately pure localized phases have area-law bipartition entanglement
5 EPR pairs by using only local operations and classical entropy in contrast to the volume-law entropy of a ther-
6
communication (LOCC), where the ratio m/N depends malizedstate[25,26]. Inaddition,afterasudden(global
5
on the amount of entanglement left in ρ. Therefore, it is or local) quench, the entanglement shows a fast power-
3
0 of great interest to design quantum information storage law spreading in a thermal phase, but only a slow log-
. devicesthatcankeepastrongquantumentanglementfor arithmic spreading in an MBL phase and no spreading
1
a long time to improve the distillation efficiency. In this at all in an AL phase [27–29]. The slow entanglement
0
7 Letter, we study the possibility of preserving quantum spreading in the localized phases is restricted by a vari-
1 entanglement in a localized environment by introducing ant of the Lieb-Robinson bound on the information light
v: strong disorder. cone, which can in principle be observed via out-of-time-
i order correlations (OTOC) [30–32]. This slow spreading
X
The idea that disorder can help protecting initial cor- of entanglement also suggests that the local correlations
r relations and information is first raised by Anderson in might be maintained for a long time in a localized envi-
a
1958. He focused on the behaviors of non-interacting ronment, which has a potential application in quantum
particles experiencing random potentials, which is now informationstorage. Indeed,ithasbeenshownthatdeep
namedasAndersonlocalization(AL)[5]. InAL,thedif- in the localized phase, the quantum coherence of local
fusion of particle’s wave-packet in a disordered environ- degrees of freedom, e.g. a single qubit, has been demon-
ment is absent, implying the initial information of par- strated to be maintained for a very long time [33–35].
ticle’s position is “remembered”. Extending this concept However, to the best of our knowledge, whether disorder
toaninteractingsystem,namelymany-bodylocalization can also help to protect quantum entanglement between
(MBL), has attracted many people’s interest including qubits has never been directly studied. In this Letter,
Andersonhimself. Recently,thisfieldattractsanintense we focus on studying the time evolution of the quantum
attraction[6–8],partiallyduetothelatelyrapidprogress entanglement between an EPR pair shared by two ob-
in ultracold atomic experiments that has made quantum servers namely Alice and Bob and coupled to a localized
isolatedmany-bodysystemswithtunableinteractionand environment.
disorder available, including ultracold atoms in optical
2
As a concrete example, we conduct our analysis in a
prototypeHamiltonianthathasbeenstudiedextensively
in the MBL literature: a one-dimensional (1D) s = 1/2
spin chain XXZ Hamiltonian with nearest neighbor in-
teractions
H =(cid:88)J(cid:0)sxsx +sysy (cid:1)+∆szsz +h sz, (1)
i i+1 i i+1 i i+1 i i
i
where J and ∆ are both constant, and h are random
i
fields uniformly distributed over [−h,h]. The total mag-
netization S ≡ (cid:80)sz is a good quantum number, and
z i
i
hencewewillrestrictourcalculationforS =0hereafter.
z Figure1. Asketchofthetwoscenariosthatstudiedhere. (a)
We want to emphasize here that the spin-spin interact- The first scenario, where both Alice and Bob’s spins are im-
ing Hamiltonian is chosen not only because its localized mersed in the same disordered environment. (b) The second
phase has been well studied, but also because in some scenario, where Bob is isolated from the environment that
reality cases, the main source of decoherence for qubits Alice experiences.
are from their interaction with unwanted environment
spins. We also remark here that, the Hamiltonian in Eq.
(1) can be mapped into a Fermi-Hubbard model using a from the environment, which resembles a quantum com-
Jordan-Wigner transformation, where J is equivalent to munication or quantum teleportation situation; in the
the hopping coefficient and ∆ is equivalent to the inter- second scenario illustrated in Fig. 1(b), both Alice and
actionstrength. Thuswithstrongenoughdisorderh,the Bobareincontactwiththesameenvironmentmimicking
spin chain is expected to be in the AL (MBL) phase for a quantum calculation realization. Our numerical result
∆=0 (∆(cid:54)=0) respectively. shows that the entanglement evolutions in both scenar-
Here, we prepare an initial state in the form of ios have similar qualitative behavior. Therefore, we fo-
|Ψ(0)(cid:105) = |EPR(cid:105) ⊗|NEEL(cid:105) , where the subscript A cus on discussing the logarithmic negativity for scenario
AB E
stands for Alice’s spin, B stands for Bob’s, and E stands one here. These discussion and conclusions are however
for all the other spins serving as an environment. The applicable for other entanglement measurements such as
environment is prepared in a Néel’s state mimicking a concurrence and entanglement of formation in both sce-
high-temperatureenvironmentandisinitiallynot entan- narios [39].
gled with the EPR pair of the spin A and B. We then Figure 2 shows our main result, the logarithmic nega-
study the time evolution of this state under the Hamil- tivitybetweenAliceandBobasafunctionoftimeinthe
tonian in Eq. (1) using exact diagonalization, obtaining scenarioone,withthedisorderstrengthh=3J fordiffer-
the reduced density matrix ρ of the spin pair A and B entnumbersofspinsL(includingAliceandBob’sspins)
by tracing out all the environment spins and calculat- and ∆ = 0 (10−2) for the AL (MBL) phases. Initially,
ing quantum entanglement measurements that are usu- theentanglementispreparedatmaximumSN(0)=1. At
ally averaged over many realizations of disorder (typ- around t≈1/J, the entanglement in both AL and MBL
ically 1000 times). The quantum entanglement mea- phases shows a power-law decay following some oscilla-
surement we focus here is the logarithmic negativity is tions,whichhasbeenrecognizedasthediffusionofinitial
given by S = log (N +1), where N, namely negativ- state to a size of the localization length. After about a
N 2
ity,isameasurerelatedtothePeres-Horodeckicriterion: critical time tc ≈ 1/∆, the entanglement of MBL and
N = 2(cid:80) max(0,−µ ) [36, 37]. Here µ ’s are the eigen- ALshowsdramaticallydifferentbehavior. Theentangle-
i i i
values of ρΛ who is the partial transpose of ρ. We would mentinALphasesconvergestoaplateauindependentof
liketoremarkherethat,thelogarithmicnegativity,even the spin chain size L, where all curves for different L are
though lacks convexity, is a full entanglement monotone visually overlapping. In contrast, the entanglement in
that does not increase on average under a general posi- MBL phases shows a power law decay ∼t−v with v >0,
tivepartialtranspose(PPT)preservingoperationaswell which is emphasized by the linear behavior on a log-log
aslocaloperationsandclassicalcommunication(LOCC) scale in Fig. 2.
[38]. Inaddition,thelogarithmicnegativityservesasthe Due to the finite size of our system, the entanglement
upper bound of distillable entanglement that limits the inMBLphaseswilleventuallyalsosaturatetosomecon-
amount of nearly maximally entangled qubit pairs that stantsafteraverylongtime. Nevertheless, asillustrated
can be asymptotically distilled from N copies of ρ via intheinsetofFig. 2,thefinalsaturatedvaluesareshown
quantum distillation [3, 4]. to decrease exponentially as a function of spin chain size
In our current set-up, two scenarios can be studied: ∼exp(−βL),whereβ isaconstant. Fromtheseobserva-
in the first scenario shown in Fig. 1(a), Bob is isolated tions, onecanexpectthattheentanglementinALphase
3
ment of AL phase and MBL phase only become different
abruptlyafterthecriticaltimet ,therefore,if∆issmall,
c
1 the entanglement can still be preserved in the AL level
0.9 AL
0.8 before tc.
0.7 Therefore, we can conclude that the AL phase is ideal
0.6
for creating quantum storage devices to preserve quan-
0.5 tc 1/ MBL, L=8
tum entanglement between a qubit pair. On the other
(t) 0.4 0.5 MBL, L=10 hand, if a weak interaction strength ∆ is unavoidable
N
S 0.4
0.3 MBL, L=12 in the system, the MBL phase can still be applied to
)
S(0N.3 MBL, L=14 preserve entanglement but with an expiration time tc.
0.2 However, one must carry an entanglement distillation to
0.2
MBL, L=16 use the qubit pair before conducting quantum protocols.
-v We also wish to emphasize here that the preservation
8 10 L12 14 16 t of entanglement between two qubits in a localized envi-
0.1
-5 -3 -1 1 3 5 7 9 11 ronment is, of course, not better than in a completely
10 10 10 10 10 10 10 10 10
decoupled environment. However, in a realistic situation
t(1/J)
where coupling between the qubit pair and the environ-
Figure 2. Logarithmic negativity in AL (∆ = 0J) and MBL ment cannot be eliminated, our study provides a generic
(∆=10−2J)phasesasafunctionoftimefordisorderstrength way of preserving entanglement without a specific fine
h=3anddifferentspinchainlengthL. Thedashedlinehelps tuningoftheHamiltonianbutsimplyintroducingstrong
tohighlightthepower-lawdecaybehaviorintheMBLphases. enough disorder into the environment.
The inset shows the final saturation values of entanglement
Our results can also be directly applied to identify
S (∞) in MBL phases as a function of L, where the error
N the localized phase being AL or MBL. Most of previous
bar shows the variance from averaging over different disor-
such studies have been focused on studying the bipartite
der realizations. The dashed line shows a exponential fit of
S (∞)∼exp(−L). entanglement (, i.e. dividing the system into two sub-
N
systems,) of an initial product state after global quench
oranenergyeigenstateafteralocalquench[27–29]. Nev-
ertheless, the experimental observation of bipartite en-
tanglementinprinciplecanbeverychallengingforalarge
11
0.10 system and may even be impossible in the thermody-
10 namiclimit. Ontheotherhand,measuringentanglement
betweenlocaldegreesoffreedominanopticallattice[40]
9 0.08
and trap ions [41, 42] has been reported lately. There-
8
J) fore,studiesofentanglementbetweentwositeshavebeen
g(t10 7 0.06 investigated recently, motivated by the fact that such
lo 6 entanglement between local degrees of freedom is much
0.05
more experimentally accessible [43–45]. These studies
5
usuallyfocusonthecasewheretheinitialstateisaprod-
4 0.03 uct state, and a temporary entanglement generated due
3 to initial diffusion. The AL and MBL features are ana-
0.01 lyzed by the following decay of this temporary entangle-
2
8 10 12 14 ment that decreases exponentially as a function of dis-
tances between sites in the deep localized phase. There-
L
fore, these studies are usually limited to entanglement
Figure 3. Color plot of ∆S(L) defined in the main text. The between nearest few sites. Our methods, using an ini-
N
redcolorindicatesalargevalueof∆S(L),whichisconstrained tial prepared EPR pair, can in principle overcomes these
N
by a logarithmic light cone. limitations and be experimentally accessible.
Ourstudyalsogivesaninterestinginsightintothena-
ture of entanglement spread in MBL phases. The power
will never reduce to zero, but a constant depend only law decay and the saturated values of entanglement in
on disorder strength even in the thermodynamic limit MBL phases suggest that we can define a saturation
L → ∞. On the other hand, no matter how small the time scale t , where the entanglement in MBL phases
s
interaction strength ∆ is, the entanglement will be com- is about to be saturated, as vlog(t ) ∼ L, which re-
s
pletely dissipated after infinite long time in the thermo- sembles the logarithmic light-cone found in previous bi-
dynamic limit. However, this dissipation is very slow if partite entanglement studies [27–29]. This logarithmic
the disorder is strong enough. In addition, the entangle- light-cone can be understood from the modified Lieb-
4
1 1
1 0.8 (b)
0.9 =10-2J,10-3J,10-4J,10-5J,10-6J,0J 0.8 (a) ¥)0.6
0.8 (N0.4
0.6 S
0.7 0.7 1-
0.2
0.6
(t)N 0.5 0.6 S(t)N 0.4 2 3 4 5 6
S S0N.5 1 h(J)
0.4
0.4 0.2
v| (c)
0.3 0.3 | 0.1
0.2 0 2 4 6 8
0.2 10 10 1t0 10 10 10-5 10-3 10-1 101 103 105 107 109 1011 0.01 2 3 4 5 6
10-5 10-3 10-1 101 103 105 107 109 1011 t(1/J) h(J)
t(1/J) Figure 5. (a) Entanglement decay for different disorder
strength h with L = 12 and ∆ = 10−2J. The red thin
Figure 4. Entanglement decay for different interaction
(blue thick) curves represents AL (MBL) phases for h =
strength ∆. Here, the size of spin chain L=12 and h=3J.
2J,3J,4J,5J,6J frombottomtotop. (b)Thesaturatedvalue
The inset shows the long-time behavior of entanglement as a
1−S (∞) as a function of h, the red circles (blue squares)
universal function of t∆. N
representAL(MBL)phase,wherthesolidlinesarepower-law
fit 1−S (∞)∼hc . (c) The symbols are the decay index v
N
as a function of h, and the solid line is a power-law fit.
Robinson bound of information spreading (in this case
entanglement spreading) [46]. One convenient and com-
monwaytodescribetheLieb-Robinsonboundistocom- pairs. Our numerical results also show that the decrease
parethetime-evolutionofalocalobservableAunderthe of entanglement at infinite long time 1−S (∞) and the
N
full Hamiltonian with its time-evolution under a trun- decay index v are both has a power-law dependence on
cated Hamiltonian that only includes interactions con- h, as shown in Fig. 5 (b) and (c). This analysis can be
tained in a region of distance no more than L. Even interpretedasastrongerdisorderandweakerinteraction
thoughentanglementistechnicallynotanobservable,we isbeneficialforpreservingquantumentanglement,which
study the quantity ∆S(L) = S(L=16) −S(L) under the is consistent with our expectation.
N N N
same spirit. In scenario two, this quantity can be inter- In summary, we studied the time evolution of quan-
preted as the differences of entanglement between a full tumentanglementofanEPRpaircouplingtoalocaliza-
spin chain of L = 16 and a truncated spin chain L near tion environment. This study allows us to explored the
Alice’s spin. The result is shown in Fig. 3, where one possibility and limitation of applying localization phase
can directly see that the significant differences are con- to preserve quantum entanglement between qubit pairs.
strained within a logarithmic light cone. This is a direct Our results can also be regarded as an experimentally
evidence that entanglement is spreading logarithmically accessible protocol to discriminate AL and MBL phases,
in an MBL phase suggested by previous OTOC studies and understand the nature of entanglement propagation
[30–32]. in these systems.
We further take a qualitative analysis of the effects of This research was supported under Australian Re-
interaction strength ∆ and disorder strength h on the search Council’s Future Fellowships funding scheme
decayofentanglement. Figure4showstheentanglement (project number FT140100003) and Discovery Projects
decay for different ∆, confirming that entanglement in funding scheme (project number DP170104008). The
ALandMBLphasesonlybecomedifferentabruptlyafter numericalcalculationswerepartlyperformedusingSwin-
the critical time t ≈ 1/∆. Furthermore, the saturated burnenewhigh-performancecomputingresources(Green
c
value in the MBL phases does not variate appreciably II).
for different ∆. In fact, the logarithmic entanglement
for t (cid:29) t is a universal function of t∆ as evidenced by
c
the inset of Fig. 4. Finally, Fig. 5(a) shows the entan-
glement decay for different h, where the decay rate be-
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6
√ √ √
SUPPLEMENTAL MATERIAL concurrence is given by C = max( λ − λ − λ −
√ 1 2 3
λ ), where λ ’s are the eigenvalues of ρρ˜ in decend-
4 i
It is well known that von Neumann entropy is not a ing order and ρ˜ ≡ σy ⊗ σyρ∗σy ⊗ σy. Concurence is
valid entanglement measurement if the collective states monotonically related to the entangelment√of formation
(cid:2)(cid:0) (cid:1) (cid:3)
are in mix states. However, several entanglement mea- by the Wootters formula SF = h 1+ 1−C2 /2 ,
surement have been found for a pair of qubits, including where h(x) = −xlog2x−(1−x)log2(1−x). Negativ-
concurrence,negativityandtheircloserelatives: entropy ity is a measure related to the Peres-Horodecki criterion:
(cid:80)
of formation and logarithmic negativity. These entan- N =2 imax(0,−µi), where µi’s are the eigenvalues of
glement measurements are “good” in the following sense: ρΛ who is the partial transpose of ρ. The logarithmic
(i) for a maximally entangled state, i.e. an EPR pair, negativity is then given by SN =log2(N +1).
these measurements reach their maximum values (equal In the main text, we show that the time evolution of
oneinourdefinitions);(ii)forcollectiveseperablestates, logarithmic negativity between a pair of qubits that ini-
these measurements vanish; (iii) is a continuous function tially prepared to be as an EPR pair in scenario one.
of density matrices of the two-qubit states. Notice that Here, we present results of other entanglement measures
these measurements, however, does not nessesarily give in scenario two and show that these measurements are
same ordering for different entangled states. qualitativelysimilar,andservethesameroleinouranal-
Let us first give the defination of the entanglement ysis. Therefore, the discussions and conclusions of loga-
measurements mentioned above. Denoting the collective rithmic negativity in scenario one are also applicable for
state for two selected qubits by a density matrix ρ, the all measurements in scenario two.
7
1
0.8 AL
0.6
MBL, L=8
C
MBL, L=10
0.4
MBL, L=12
MBL, L=14
MBL, L=16
0.2
-5 -3 -1 1 3 5 7 9 11
10 10 10 10 10 10 10 10 10
t(1/J)
Figure 6. Concurrence as a function of time for disorder
strength h = 5 and interaction strength ∆ = 0J(10−3J) for
AL (MBL) phases. The results for AL phases with different
spinchainlengthLarealmostoverlappingeachother,andthe
results for different MBL phases are indicated in the figure.
1
0.9
0.8
AL
0.7
0.6
0.5
(t)F 0.4 MBL, L=8
S MBL, L=10
0.3
MBL, L=12
0.2
MBL, L=14
MBL, L=16
0.1
-5 -3 -1 1 3 5 7 9 11
10 10 10 10 10 10 10 10 10
t(1/J)
Figure7. Entropyofformationasafunctionoftimefordisor-
der strength h = 5 and interaction strength ∆ = 0J(10−3J)
for AL (MBL) phases. The results for AL phases with dif-
ferentspinchainlengthLarealmostoverlappingeachother,
and the results for different MBL phases are indicated in the
figure.
8
1
0.9
0.8 AL
0.7
0.6
N(t) 0.5 MBL, L=8
0.4 MBL, L=10
MBL, L=12
0.3
MBL, L=14
MBL, L=16
0.2
-5 -3 -1 1 3 5 7 9 11
10 10 10 10 10 10 10 10 10
t(1/J)
Figure8. Negativitysafunctionoftimefordisorderstrength
h=5andinteractionstrength∆=0J(10−3J)forAL(MBL)
phases. The results for AL phases with different spin chain
length L are almost overlapping each other, and the results
for different MBL phases are indicated in the figure.
1
0.9
0.8 AL
0.7
0.6 MBL, L=8
(t)N 0.5
S MBL, L=10
0.4 MBL, L=12
0.3
MBL, L=14
MBL, L=16
0.2
-5 -3 -1 1 3 5 7 9 11
10 10 10 10 10 10 10 10 10
t(1/J)
Figure9. Logarithmicnegativitysafunctionoftimefordis-
orderstrengthh=5andinteractionstrength∆=0J(10−3J)
forAL(MBL)phases. TheresultsforALphaseswithdiffer-
ent spin chain length L are almost overlapping each other,
and the results for different MBL phases are indicated in the
figure.