Quantifying entanglement in multipartite conditional states of open quantum systems by measurements of their photonic environment

6.4 MB·English

by  Juan Diego Urbina

Quantifying entanglement in multipartite conditional states of open quantum systems
by measurements of their photonic environment
Juan Diego Urbina,1 Walter T. Strunz,2 and Carlos Viviescas3
1Institut fu¨r Theoretische Physik, Universita¨t Regensburg, D-93040 Regensburg, Germany
2Institut fu¨r Theoretische Physik, Technische Universita¨t Dresden, D-01062 Dresden, Germany
3Departamento de F´ısica, Universidad Nacional de Colombia, Carrera 30 No.45-03, Bogota D.C., Colombia
A key lesson of the decoherence program is that information flowing out from an open system
is stored in the quantum state of the surroundings. Simultaneously, quantum measurement theory
showsthattheevolutionofanyopensystemwhenitsenvironmentismeasuredisnonlinearandleads
topurestatesconditionedonthemeasurementrecord. Herewereportthediscoveryofafundamental
relation between measurement and entanglement which is characteristic of this scenario. It takes
the form of a scaling law between the amount of entanglement in the conditional state of the
systemandtheprobabilitiesoftheexperimentaloutcomesobtainedfrommeasuringthestateofthe
2
environment. Using the scaling, we construct the distribution of entanglement over the ensemble
1
of experimental outcomes for standard models with one open channel and provide rigorous results
0
on finite-time disentanglement in systems coupled to non-Markovian baths. The scaling allows the
2
direct experimental detection and quantification of entanglement in conditional states of a large
n class of open systems by quantum tomography of the bath.
a
J
PACSnumbers: 03.65.Ta,03.65.Ud,03.65.Yz,03.67.Bg,03,67,Pp
6
2
Arguable, a defining feature of a real quantum system rule, to the amount of multipartite entanglement of the
]
h is the nature of its unavoidable interactions with its sur- pureconditionalstateofanopenquantumsystem. Aswe
p roundings. It is well known that they are determinant of demonstrate, our findings are universal: They are inde-
- the manners the system can evolve, being even respon- pendentofthenumberanddimensionsoftheconstituent
t
n sible, in occasions, of the total loss of the system quan- subsystems and hold for any of the so-called G-invariant
a
tum character [1]. Less discussed, however, is the fact measuresofentanglement[11,12],e.g.,Wootter’sconcur-
u
thattheyalsodictatethewaysasystemcanbeobserved rence for two-qubit systems [13], characterized by their
q
[ and information can be extracted from it. Thus, for in- invariance under local invertible transformations. No-

stance, a system can be indirectly probed by perform- tably, these results allow for a characterization of multi-
1
ingaprojectivemeasurementonitsenvironment. Ignore partite entanglement in the presence of both Markovian
v
5 the outcome of this measurement, and the information and non-Markovian environments, thus making contact
2 lost implies that a pure state of the system prior ob- with state-of-the-art experimental techniques [14]. Fur-
6 servation is transformed into a statistical mixture [2, 3]. thermore,weshowthatourscalinglawsuggestsaproto-
5
If on the other hand, the result of the measurement is col for a direct measurement of entanglement in a large
.
1 physically stored with full fidelity the state of the cen- class of systems, including all Markovian and all non-
0 tralsystemimmediatelyaftermeasurementisagainpure, demolition baths: As it turn out, the inverse numerical
2
yet conditioned on the particular, stochastic outcome: value of the Husimi function of the bath is a measure of
1
the state, representing our knowledge about the system, the amount of entanglement in the conditional state.
:
v “collapses” [4, 5]. Conditional states not only have an
i Our work is partially motivated by recent results in
X intrinsicimportanceastheoutcomeofphysicalmeasure-
the study of non-classical correlations in unconditional
ments. They provide also an exact representation of the
r states, where a conservative approach based on explicit
a unconditional dynamics, with implications for the field
calculationoftheentanglementmeasuresintheuncondi-
of quantum information [6–8] where characterizing the
tionalmixedstatefacestremendousdifficultiesduetoan
amount and kind of non-classical correlations in uncon-
intrinsic high dimensional optimization problem [15, 16].
ditionalstatesofopensystemsremainsamaintheoretical
The severity of this hurdle has motivated intense re-
challenge [9, 10]. In this order of ideas, a proper under-
search in the general (algebraic) aspects of the interplay
standing of the interplay between measurements on the
between unconditional time evolution and entanglement
bath and entanglement properties of conditional states,
measures. An important step in this direction was the
to our knowledge a problem not yet explored, is clearly
discovery of a scaling property of the G-invariant mea-
necessary.
sures [11, 17], and its experimental verification for two-
qubit systems with one open channel [18].
Undertheonlyassumptionofastrictlylocaldynamics,
in the present article we derive a scaling law relating the However, the scaling property of unconditional states
probability distribution of experimental outcomes from critically depends on the linearity of the evolution [11]
measurements on the environment, as given by the Born through its Kraus representation [9]. Therefore, an ex-
2
tension of such algebraic properties to the conditional Ifattimet>0ageneralizedmeasurementofthebath
casewithitsintrinsicnonlineardependenceontheinitial is performed, the probability P(a,t) that we obtain |a(cid:105)
state requires both different mathematical methods and is [3, 25]
physical concepts. Our results fill this gap at a moment
P(a,t)
when there is an increasing interest on schemes with a
=(cid:104)ψ(a,t)|ψ(a,t)(cid:105), (3)
less restrictive role for the environment, where the flow P(a,0)
of information from the system to the bath takes an ac-
tive part [6–8, 19–23]. with P(a,0) = Z−1e−|a|2, and the state of the central
system collapses to the conditional state
Open systems in photonic environments |φ(a,t)(cid:105)=(cid:104)ψ(a,t)|ψ(a,t)(cid:105)−1/2|ψ(a,t)(cid:105). (4)
To proceed further, we notice that the unnormalized
Weconsideracentral-system-plus-bathapproach[4,5]
conditional state |ψ(a,t)(cid:105) can be related to the physical
whereamultipartitecentralsystem(CS)madeofN sub-
initial state of the CS by means of [24]
systems with dimensions d(1),...,d(N) is coupled to a
collection of independent photonic baths (channels) act- |ψ(a,t)(cid:105)=Uˆ(a,t)|φ(0)(cid:105) (5)
ing locally on each subsystem. If initially pure, after
transformation to the interaction picture, the state of with
the combined system is given at time t by
Uˆ(a,t)=Te(cid:82)0t[−(cid:126)iHˆCS+(cid:80)k(zk∗(s)Jˆk−Jˆk†Oˆk(s))]ds,
|Ψ(t)(cid:105)=Te−(cid:126)i (cid:82)0t(HˆCS(s)+HˆI(s))ds|φ(0)(cid:105)⊗|0(cid:105). (1)
anon-unitarypropagatordependingonthemeasurement
Intheaboveequation|φ(0)(cid:105)istheinitialstateoftheCS outcomes {a} through zk(s) = (cid:80)λgk,λak,λe−iωk,λs and
and |0(cid:105) the ground state of the bath, T stands for the Oˆ (s) = (cid:82)sα (s−s(cid:48))Oˆ (a ,s,s(cid:48))ds(cid:48). Although in gen-
time ordering operator, HˆCS(s) is the CS Hamiltonian erkal the op0eraktors Oˆ (ak,sk,s(cid:48)) are defined only pertur-
k k
acting locally on the subsystems, and bativelyforeachparticularchoiceofI (ω)andJˆ , exact
k k
HˆI(s)=i(cid:88)gk,λ(cid:16)Jˆkaˆ†k,λeiωk,λs−aˆk,λJˆk†e−iωk,λs(cid:17), cicloasleidnteexrepsrtes[s2i4o–n2s6a].re available for several cases of phys-
k,λ Having a linear relation between |ψ(a,t)(cid:105) and |φ(0)(cid:105),
accounts for the interaction between CS and bath we can turn to the main issue of this work: the amount
through local traceless operators Jˆ , with g the cou- of entanglement G(|φ(a,t)(cid:105)) in the conditional state
k k,λ
pling strength of the system and the λ-th mode to the |φ(a,t)(cid:105) as quantified by a G-invariant measure. As we
k-thchannel,wherephotonsoffrequencyω arecreated now show, the answer to this question follows smoothly
k,λ
orannihilatedbyaˆ† andaˆ ,respectively. Memoryef- from two defining properties of G [11]: (i) It is invari-
k,λ k,λ antunderlocallineartransformationswithunitdetermi-
fectsinthisapproacharecharacterizedbythecorrelation
nant. (ii)Itishomogeneousofdegreetwo,i.e.,G(u|φ(cid:105))=
functions
|u|2G(|φ(cid:105)) for all complex u. Indeed, from property (ii)
(cid:90) ∞
αk(s)=(cid:88)|gk,λ|2e−iωk,λs = Ik(ω)e−iωsdω, and Eqs. (4),(5), the entanglement of |φ(a,t)(cid:105) can be
written as
λ 0
where I (ω) is the spectral density of the bath’s k-th G(|φ(a,t)(cid:105))=(cid:104)ψ(a,t)|ψ(a,t)(cid:105)−1G(Uˆ(a,t)|φ(0)(cid:105)).
k
channel (see Supplementary Information).
We now use property (i), in conjunction with equa-
tion (3), to obtain the central result of this paper,
Universal scaling of multipartite entanglement
G(|φ(a,t)(cid:105)) P(a,0)
x(a,t):= =f(a,t) , (6)
G(|φ(0)(cid:105)) P(a,t)
Following[24–26],thetotalstate|Ψ(t)(cid:105)inequation(1)
can be always written as
where the scaling function
(cid:90)
|Ψ(t)(cid:105)=Z−1 e−|a2|2|ψ(a,t)(cid:105)⊗|a(cid:105)da, (2) f(a,t)=e−(cid:80)k d2((cid:60)k) (cid:82)0t(cid:82)0sαk(s−s(cid:48))Tr[Jˆk†Oˆk(ak,s,s(cid:48))]dsds(cid:48)
where |ψ(a,t)(cid:105) is the (unnormalized) state of the system is independent of the functional form of G(|φ(cid:105)); a con-
relative to the bath (normalized) coherent state |a(cid:105), Z sequence of the use of the identity detTe(cid:82)0tAˆ(s)ds =
is a normalization constant, which will drop out from all e(cid:82)0tTrAˆ(s)ds andthefactthattheHamiltoniansandbaths
(cid:81)
the final results, and we define da:= k,λd(cid:60)ak,λd(cid:61)ak,λ act locally in the subsystems (see Methods section).
with |a|2 =(cid:80) |a |2. Equation(6)providesuswithanexplicitscalingbetween
k,λ k,λ
3
0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 1
the amount of entanglement in the conditional state of 1.2 1.2 1.2
thecentralsystemandtheprobabilitydistributionofthe 1 1 1 25
outcomes of the measurement on the bath. It expresses 20
0.8 0.8 0.8
universalityofthemultipartiteentanglementinthesense 15
thatforagivenoutcomeofthemeasurement,allnormal- x0.6 0.6 0.6 10
ized G-invariant measures of multipartite entanglement 0.4 0.4 0.4 5
in the conditional state have exactly the same numerical
0.2 0.2 0.2 0
value. Within the standard system-plus-bath approach 00 a0.2 0.4 0.6 0.8 0 b0.2 0.4 0.6 0.8 1
for local systems, our equation (6) is completely general. 1.40 0.2 0.4 0.6 0.81.401.40.2 0.4 0.6 0.8 1
Toillustratetherelevanceofscalinglaw (6), inthere- 1.2 1.2 1.2 q(cid:72)t(cid:76) 15
mainder of the paper we use it to address some crucial 1 1 1
questions in the actual theory of multipartite entangle- 10
0.8 0.8 0.8
ment. x
0.6 0.6 0.6 5
0.4 0.4 0.4
Distribution of multipartite entanglement 0.2 0.2 0.2 0
c d
0
0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 1
Perhaps the most pertinent question at this point is: p p
whatistheprobabilityofhavingagivenamountofentan-
glement in a system just after its photonic environment FIG. 1: Probability distribution of entanglement.
has been measured? Let us remark that the common Probability distribution of scaled multipartite entanglement
approach of using uniform distributions for |φ(cid:105) [27, 28] in the CS over the ensemble of experimental outcomes {a}
does not hold here. Once the measurement process is of an amplitude damping bath for different initial states: a,
ρ = 1/2, ρ = 0, b, ρ = 3/4, ρ = 0, c, ρ = 1/2,
specified, i.e., a choice for the measurement outcomes 11 10 11 10 11
ρ =1/4,d,ρ =1/2,ρ =1/3. Thefigureisthesamefor
is made, unitary invariance as a guiding principle is re- 10 11 10
anyG-invariantmeasureofasystemofquditsifonlyasingle
placed by the specific quantum-mechanical distribution
qubitwithoutinternaldynamicsiscoupledtoazerotemper-
of outcomes P(a,t), as given by the Born rule. Accord- ature Markovian bath. The black continuous line shows the
ingly, the distribution PG(x,t) of the normalized entan- maximum possible value of entanglement xmax(p) while the
glementx(a,t)inthephysicalensembleofpureandran- red dashed line corresponds to the mean entanglement x¯(p).
domconditionalstatesoftheCSmustbecalculatedfrom
(cid:90)
P (x,t)= P(a,t)δ(x−x(a,t))da, (7)
G p(t) = 1−e−γt (see Supplementary Information). In all
cases we observe that the entanglement distribution be-
which can be evaluated using the scaling rule (6) and is comesbroaderastimeincreasestofinallycollapseasymp-
the same for all G-invariant measures. In particular, its totically when entanglement from CS disappears. The
first moment, i.e., the mean entanglement x¯(t), is given sharp boundary exhibit by P (x,p) (black continuous
G
by line) is given by the condition
(cid:90) ∞ (cid:90) √
x¯(t)= PG(x,t)xdx= P(a,0)f(a,t)da. (8) xmax(p)= 1−p(cid:18)1−ρ11p − |ρ10|2(cid:19)−1.
0 ρ ρ ρ2
11 11 11
To exemplify our argument we consider a specific and
Thus, for maximally entangled states, i.e., states with
experimentally relevant case [18]: a multipartite CS Trρˆ2 (0) = 1/2, the maximum possible value of entan-
ch
where only one qubit is coupled to a single open chan- glement xmax(p) is strictly decreasing with xmax(p)<1,
nel (k =“ch”). Hence, the information about the initial
ascanbeseeninpanel1a. However,forinitialstateswith
state|φ(0)(cid:105)isfullyencodedintheinitialreduceddensity Trρˆ2 (0) > 1/2, there are regions in which xmax(p) can
ch
matrix of the coupled qubit, increaseevenuptoxmax(p)>1,asdisplayedinpanel1b.
(cid:18) (cid:19) This behaviour should be contrasted with the evolution
ρ ρ √
ρˆ (0)= 11 10 . oftheaveragednormalizedentanglementx¯(p)= 1−p,
ch ρ∗ ρ
10 00 which is independent of the initial state and strictly de-
Let us start by considering a Markovian bath at zero creasing (red dashed line).
√
temperature where Jˆ = γσˆ = Oˆ(a,s,s(cid:48)) [25], with A quite different picture arises when the channel cor-
−
σˆ the two-level deexcitation operator, γ the decay rate, responds to a non-demolition bath. Thus Hˆ(ch) ∝ Jˆ=
− CS
and I(ω) = π−1. In order to focus on decoherence ef- ∆σˆ = Oˆ(a,s,s(cid:48)) [25], σˆ is the third of the Pauli’s ma-
2 z z
fects, we assume frozen internal dynamics. In Fig. 1 trices, and ∆2 the “phase-flip” rate. In this so-called de-
we show P (x,p) in terms of ρˆ (0) and the scaled time phasingchannel,thedistributionP (x,t)isindependent
G ch G
4
0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 1
a b
3.5 p(cid:61)0.15 (cid:230) 0.0 Ω(cid:61)1
1 1 1 10 3.0 pp(cid:61)(cid:61)00..3405 (cid:45)0.1 ΩΩ(cid:61)(cid:61)23
0.8 0.8 0.8 7.5 2.5 p(cid:61)0.60 Ω(cid:61)4
x 0.6 0.6 0.6 5 (cid:72)(cid:76)x,pG12..50 ppp(cid:61)(cid:61)(cid:61)000...799508 (cid:230) (cid:72)(cid:76)ogx (cid:45)(cid:45)00..32
0.4 0.4 0.4 2.5 P 1.0 (cid:230) l (cid:45)0.4
0.2 0.2 0.2 0 0.5 (cid:230) (cid:230) (cid:45)0.5
(cid:230)
a b (cid:230)
(cid:45)0.6
0 0.0 0.2 0.4 0.6 0.8 1.0 0 1 2 3 4
0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8 1
p p x (cid:68)2t
FIG. 3: Evolution of entanglement distribution and
FIG. 2: Probability distribution of entanglement.
mean entanglement. a, Probability distribution of multi-
Probability distribution of scaled multipartite entanglement
partite entanglement at different values of the scaled time p.
intheCSovertheensembleofexperimentaloutcomes{a}ofa
The black dots mark the mean entanglement x¯(p). b, Loga-
non-demolition bath for different initial states: a, ρ =1/2,
11 rithm of the average entanglement x¯ as a function of ∆2t for
b,ρ =1/4. ThefigureisthesameforanyG-invariantmea-
11
ohmic(continuouslines)andsuperohmic(dashedlines)envi-
sure of a system of qudits if only a single qubit is coupled to
ronmentsanddifferentvaluesoftheeffectiveDebyefrequency
a non-demolition bath. The black continuous line shows the
maximum possible value of entanglement xmax(p) while the (ω¯ =ωd/∆2). Both behaviors demonstrate a transition from
exponentialtomuchsloweralgebraicdecay,alsocharacteristic
red dashed line corresponds to the mean entanglement x¯(p).
of the approach p(t→∞)→1.
ofρ andhasasharp,timeindependentboundarygiven
by x10max = (4ρ ρ )−1/2. For maximally mixed ρˆ (0) The parameterization p(t) = 1−e−∆2q(t) for the time
11 00 ch is of most convenience in this case, leading to the mean
the upper limit is xmax = 1, while xmax has square-root √
entanglement x¯(p) = 1−p. The function p(t) rules
divergences for pure initial states ρ = 1,0. In Fig. 2
11 the actual speed at which the distribution and its first
weplotP (x,p)fortheMarkoviancasewithI(ω)=π−1
G moment attain their asymptotic form. It is given for
and rescaled time p(t)=1−e−∆2t.
exponential cut-off at Debye frequency ω by
d
For this class of environments the maximum of
(cid:40)
PG(x,t), which signals the most likely values of the mul- 1−(1+ω2t2)1/πω¯e −∆2t (ohmic),
tipartite entanglement, do not lie near the average value p(t(cid:29)ω−1)= d
d 1−1/(1+ω2t2)1/πω¯ (superohmic),
x¯(t). Instead, the distributions peaks aroundthe bound- d
ary x=xmax, where it has an integrable divergence (See whereweintroducedω¯ =ω /∆2. Theaboverelationim-
d
Fig. 2). This is clearly seen in the Markovian case with
pliesthatthedecayoftheaveragedentanglementisonly
ρ =1/2, where
11 algebraic for superohmic environments (see Fig. 3). Re-
(cid:115) markably, however, the approach to the limit ω t → ∞
d
2(1−p) 1
P (x,p)= √ issuchthatfordephasingchannelstheaverageentangle-
G π|log(1−p)|x 1−x2 ment is never zero for finite times.
 (cid:16) √ (cid:17)
log2 1+ 1−x2
x
×exp−  ,
2|log(1−p)| Finite disentanglement time
and that we show in Fig. 3. When time increases a sec- As discussed above, the most likely values of mutipar-
ond maximum appears near (but not exactly at) x = 0, titeentanglementinconditionalstatesareingeneralnot
showing that the average x¯(p) (red dashed line in Fig. 2 related to their average value. A prime exception takes
and points in Fig. 3) misses the most likely values of place when there exists a finite time t = τ such that
multipartite entanglement. x¯(τ)=0, sinceit entails that at τ all entanglement mea-
Inthiscase,thedependenceofourresultsonthemem- suresofthemultipartiteconditionalstatesvanishidenti-
ory scale of the environment is easily checked. For this cally over the whole ensemble of experimental outcomes.
weintroducenon-Markovianeffectsusingaspectralden- The conditions for this event to occur as well as the de-
sity I(w) (cid:54)= π−1 [4] (see Supplementary Information), pendenceofthedisentanglementtimeτ withtheparame-
and find that this amounts only to a redefinition of time tersofthesystemareissuesofobviouspracticalinterest.
t→q(t) in the purely ohmic result by means of Here we answer these questions for a large class of sys-
tems where at least one channel k =l satisfies
t2 (cid:90) ∞ (cid:18)sinωt/2(cid:19)2
q(t)= 2 0 I(ω) ωt/2 dω. Oˆl(al,s,s(cid:48))=Oˆlul(s,s(cid:48)) (9)
5
for some function ul(s,s(cid:48)), and operator Oˆl. This class a b 7
ofsystemsincludesallMarkovianandallnon-demolition 6.5
6
baths,theJaynes-Cummingsmodelandthedampedhar-
5
6.0
monic oscillator among others [25]. Τ
We notice two immediate consequences of the form of ΓΤ 5.5 (cid:144)12Ω(cid:76)d34
equation (8) for the mean entanglement: First, if (9) Γ
(cid:72) 2
holdsforallchannels, themeanentanglementreducesto 5.0
1
x¯(t)=e−(cid:80)k d2((cid:60)k)Tr[Jˆk†Oˆk](cid:82)0t(cid:82)0sαk(s−s(cid:48))uk(s,s(cid:48))dsds(cid:48). (10) 2 3 4 5 6 7 0 2 3 4 5 6 7
Μ Μ
Second, in the Markovian case, u(s,s(cid:48)) = 1, Oˆ = Jˆ,
x¯(t) is independent of the internal dynamics and strictly FIG.4: Disentanglementtimes. a,Dependenceofthedis-
exponential;nofiniteτ existsandthemeanentanglement entanglement times τ with the effective coupling µ = 2γ/ωd
for an arbitrary multipartite system where a qubit without
disappears only asymptotically in time.
internaldynamicsiscoupledtoazero-temperaturebathwith
More generally, the product form of both P(a,0) and
a characteristic memory scale ω−1. b, Same as in a showing
f(a,t)impliesthatthevanishingofthefactorassociated d
that for large coupling the natural scale for the disentangle-
with a single channel will automatically disentangle the ment time τ is given by (γω )−1/2.
d
whole system. Let us focus on this particular channel
and assume that it couples a qubit with frozen inter-
naldynamicstoanon-Markovianzero-temperaturebath
with memory kernel α(s) = (ωd/2)e−ωd|s|. The exact of entanglement G(ρˆunc(t)) is defined by a convex roof
dynamics of the total system for arbitrary α(s) is ob- construction, we have
√
tained with the choice Oˆ = γσˆ as long as the func-
−
tion u(s,s(cid:48)) satisfies u(s,s(cid:48))=c(s)/c(s(cid:48)) and dc(s)/ds= G(ρˆunc(t))
≤x¯(t).
−γ(cid:82)sα(s−s(cid:48))c(s(cid:48))ds(cid:48), which can be solved for the case G(ρˆunc(0))
0
weareconsidering(seeSupplementaryInformation). Af-
Therefore all measures of multipartite entanglement of
ter substituting our result into (10) we obtain
an open system interacting with an arbitrary photonic
(cid:26) (cid:90) t ωd sinhΩs (cid:27) environment must be zero at a finite time if at least one
x¯(t)∝exp −γ ωd sin2hΩΩs+coshΩsds , qubit is strongly coupled with a non-Markovian bath at
0 2Ω zerotemperature. Theimportantextensionofthisresult
√
tofinite-temperaturebathsispresentlyunderstudy[35].
where Ω = (ω /2) 1−µ and µ = 2γ/w . We notice
d d
that for weak and strong coupling (µ ≤ 1) the average
entanglement is not zero for all finite times. In contrast,
intheultrastrongcouplingregime(µ>1)wegetx¯(τ)= Tomography of entanglement
0 for
µ (cid:16) (cid:112) (cid:17) To conclude, in this last section we show how scaling
γτ = √ π−arctan µ−1 , law (6) provides us with a geometrical and operational
µ−1
interpretation of the amount of entanglement in condi-
rigorouslyshowingtheexistenceoffinitedisentanglement tional states of multipartite systems. This follows from
times in all conditional states of arbitrary multipartite theobservationthat,associatedwiththepuretotalstate
systems where at least one single qubit is coupled with |Ψ(t)(cid:105), the unconditional density matrix describing the
a non-Markovian bath. Remarkably, the disentangle- environment ρˆE(t) = TrCS|Ψ(t)(cid:105)(cid:104)Ψ(t)| [36] has the fol-
ment time increases with µ for strong enough coupling, lowing representation in coherent states
as shown in Fig. 4.
Ourlastconsiderationsaredirectlyextendedtotheun- Q(a,t)=Z−1(cid:104)a|[TrCS|Ψ(t)(cid:105)(cid:104)Ψ(t)|]|a(cid:105),
conditional dynamics of the open system (entanglement
whereQ(a,t)istheso-calledHusimifunctionofthebath
sudden death [29–34]). The mixed state describing the
[37]. Using equation (2) then leads to
unconditional evolution is given in complete generality
by an average of the pure conditional states [24–26]
Q(a,t)=P(a,t),
(cid:90)
ρˆunc(t)= P(a,t)|φ(a,t)(cid:105)(cid:104)φ(a,t)|da.
and therefore equation (6) relates the possible values of
the entanglement of the conditional state with the mor-
Therefore,theset{|φ(a,t)(cid:105),P(a,t)da}constitutesavalid phology of the Husimi function of the bath; a quantity
convex decomposition of ρˆunc(t), and since the amount accessible to direct measurement.
6
In the following we will consider systems where equa- must satisfy
tion (9) holds. In this case we easily obtain
G(Uˆ(a,t)|φ(0)(cid:105))=f(a,t)G(|φ(0)(cid:105))
x(a,t) Q(a(cid:48),t)
=e−(Na−Na(cid:48)) , (11) with
x(a(cid:48),t) Q(a,t)
Uˆ(a,t)=Te(cid:82)0t[−(cid:126)iHˆCS+(cid:80)k(zk∗(s)Jˆk−Jˆk†Oˆk(s))]ds.
where N = |a|2 is the mean number of photons in the
a
state|a(cid:105). Hence,thenumericalvalueoftheentanglement Assuming that the Hamiltonian Hˆ , the Linblad op-
CS
of the conditional state after a single measurement with erators Jˆ , and the convolutionless operators Oˆ (s) act
k k
experimental outcome {a(cid:48)} together with the uncondi- locallyonthesubsystems,theoperatorUˆadmitsthefac-
tionalHusimifunctionofthebathQ(a(cid:48),t)automatically torization
predicts all the entanglement measures of all possible
N
multipartite conditional states. Quantum tomography of Uˆ(a,t)=(cid:89)Uˆ(i)(a(i),t).
the bath is actually measuring entanglement of the con-
i=1
ditional states.
Finally, known upper bounds for the Husimi function Here
m[37en]tcan be used to provide a lower bound to entangle- Uˆ(i)(a(i),t)=Te(cid:82)0t(cid:104)−(cid:126)iHˆC(iS)+(cid:80)k(i)(cid:16)zk∗(i)(s)Jˆk(i)−Jˆk†(i)Oˆk(i)(s)(cid:17)(cid:105)ds
x(a,t)≥e−Nax¯(t), is a local operators acting on the ith subsystem, and
{k(i)}denotestheindexesoftheLindbladoperatorsact-
which is particularly useful in the experimentally acces- ingonthesamesubsystem. Wenowintroduceoperators
sible case where |a(cid:105) lies close to the ground state.
Uˆ(i)(a(i),t)=|det Uˆ(i)(a(i),t)|−1/d(i)Uˆ(i)(a(i),t)
with |det Uˆ(i)| = 1, since d(i) is the dimension of the
Summary
Hilbert space of the ith subsystem, and use the homo-
geneity of the function G to obtain
The derivation of a scaling law relating the amount of
eanfttearnigtlsempheonttoinnictehnevsirtoantemoefntaismmuletaipsaurrteidte, tsoysttheemprjoubst- G(Uˆ(a,t)|φ(0)(cid:105))=(cid:89)N (cid:12)(cid:12)(cid:12)det Uˆ(i)(a(i),t)(cid:12)(cid:12)(cid:12)2/d(i)
abilitydistributionoftheexperimentaloutcomes,consti- i=1
tutes the central result of this paper. It indicates that (cid:32)(cid:34)N (cid:35) (cid:33)
for bosonic environments, multipartite entanglement is ×G (cid:89)Uˆ(i)(a(i),t) |φ(0)(cid:105) .
severely constrained by indirect measurement. Specifi- i=1
cally, we use equation (6) to demonstrate that all pos- Since by definition G(|φ(cid:105)) is invariant under local trans-
sible normalized measures of multipartite entanglement formations with determinant one, we can identify
in the conditional state have the same distribution over
the ensemble of measurement outcomes of the bath. We f(a,t)=(cid:89)N (cid:12)(cid:12)det Uˆ(i)(a(i),t)(cid:12)(cid:12)2/d(i).
constructthefulldistributioninstandardcaseswithone (cid:12) (cid:12)
open channel and show that for non-demolition environ- i=1
ments non-Markovian effects appear only through a re- Toproceedwefocusontheindividualtermsintheprod-
definition of the time variable. We rigorously prove dis- uct. Under mild conditions, the determinant of Uˆ(i) ad-
entanglementinbothpureconditionalandmixeduncon- mits the representation
ditionalstatesofmultipartitesystemswhereonequbitis
coupled with a zero temperature non-Markovian bath, det Uˆ(i)(a(i),s)=eTr log Uˆ(i)(a(i),s),
with a counter intuitive dependence of the disentangle-
and therefore
menttimesonthecouplingstrength. Finally, weshowed
thatduetothescaling,foralargeclassofsystems,multi- (cid:12) (cid:12)2/d(i) 2(cid:60)
log (cid:12)det Uˆ(i)(a(i),s)(cid:12) = Tr log Uˆ(i)(a(i),s).
partite entanglement of the conditional states is directly (cid:12) (cid:12) d(i)
measured by quantum tomography of the bath through
To evaluate this last expression we start from the equa-
the Husimi function.
tion of motion for Uˆ(i),
∂ i
Methods Uˆ(i)(a(i),s)=− Hˆ(i)Uˆ(i)(a(i),s)
∂s (cid:126) CS
(cid:34) (cid:35)
Scaling function. Here we calculate explicitly the + (cid:88)(cid:16)z∗ (s)Jˆ −Jˆ† Oˆ (s)(cid:17) Uˆ(i)(a(i),s),
k(i) k(i) k(i) k(i)
scaling function in equation (6). By definition, f(a,t)
k(i)
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∂ log (cid:12)(cid:12)det Uˆ(i)(a(i),s)(cid:12)(cid:12)d(2i) =−2(cid:60)Tr (cid:88)Jˆ† Oˆ (s), Dthaeviddyonvaicmhi,csL.ofaenndtaSnoguletmo-eRnitb.eSircoi,enPc.eH3.24D,e1te4r1m4-i1n4in17g
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f(a,t)=e−(cid:80)k d2((cid:60)k) (cid:82)0tTr[Jˆk†Oˆk(s)]ds, c(2a0l1c1o)n.tinuous measurements. Phys. Rev A. 83, 022311
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ingdiscussionswithChristopherEltschkaandJensSiew-
ert. C.V. is thankful for the hospitality extended to him
by Institut fu¨r Theoretische Physik at Universit¨at Re-
gensburg. J. D. U. acknowledges financial support from
the Deutsche Forshungsgemeinschaft through Forscher-
gruppe 760: Scattering Systems with Complex Dynam-
ics. C. V. and J. D.U acknowledge Klaus Richter for his
support to this project.
9
SUPPLEMENTARY INFORMATION CS Hamiltonian
Open system Hamiltonian Hˆ →Hˆ −(cid:88) gλ2 (cid:16)α∗Jˆ+αJˆ†(cid:17) (13)
CS CS w
λ
λ
In this section we derive within the usual system-plus-
allow us to work with traceless Lindblad operators in
bath approach (see Refs. 4, 24, 25 and 26) the Hamil-
full generality. Since this change, up to a constant, is
tonian of an open system appearing in equation (1) in
the only consequence of the transformation, we stick to
the main text. For this, we consider the system dynam-
theinitialnotationandwritethetotalHamiltonianasin
ics in a convenient interaction picture and show how to
equation (12) but with Jˆinstead of Lˆ, and keep in mind
redefine the Lindblad operators to make them traceless.
renormalization (13) for the CS Hamiltonian.
Forsimplicityweconsiderasinglechannelforthederiva-
tion; the generalization to arbitrary number of channels Theinteractionpictureforthetotaltime-evolutionop-
isstraightforwardandwillbepresentedattheendofthe erator Uˆ(t) = e−(cid:126)iHˆt is defined via the unitary transfor-
section. mation
OurstartingpointisthetotalHamiltonianHˆ describ-
ingacentralsystem(CS)linearlycoupledwithabosonic Uˆint(t)=e(cid:126)iHˆBtUˆ(t)e−(cid:126)iHˆBt,
bath (B),
and absorbs the explicit dependence on the Hamilto-
Hˆ =Hˆ +Hˆ +Hˆ (12) nian of the bath into a redefinition of the bath operators
CS CSB B
aˆiλnt(t) = aˆλe−iwλt, rendering the transformed Hamilto-
with environment and coupling terms given by nian explicitly time-dependent. Formal solution of the
Schr¨odinger equation in the interaction picture yields
Hˆ =(cid:88)w aˆ†aˆ ,
B λ λ λ λ Uˆint(t)=Te−(cid:126)i (cid:82)0t(HˆCS(s)+HˆI(s))ds,
Hˆ =i(cid:88)g (cid:16)Lˆaˆ† −Lˆ†aˆ (cid:17).
CSB λ λ λ where Hˆ (s) is the renormalized CS Hamiltonian, T is
λ CS
the time-ordering operator, and
We consider a bosonic bath with frequency ω in its λth
λ
mode,forwhichaλ anda†λ correspondtoanihilationand HˆI(s)=i(cid:88)gk,λ(cid:16)Jˆkaˆ†k,λeiωk,λs−aˆk,λJˆk†e−iωk,λs(cid:17)
creation operators, respectively. The operator Lˆ acts on
k,λ
the CS and is usually dubbed Lindblad operator. To-
getherwiththerealcouplingconstantsg itdescribesthe is the interaction Hamiltonian. In this last step we al-
λ
interactionofthecentralsystemwithallthemodesofthe ready introduced the result for the general case with an
bath. The total Hilbert space H=H ⊗H of the sys- arbitrary number of channels indexed by k.
CS B
tem is spanned by product states |e (cid:105)⊗|a ,...,a ,...(cid:105),
i 1 λ
where {|e (cid:105)} is any basis of the CS Hilbert space, and
i
|a(cid:105)=|a ,...,a ,...(cid:105)isanormalizedcoherentstateofthe Entanglement distributions
1 λ
environment, i.e., a common eigenstate of the annihila-
tion operators of the bath, aˆλ|a(cid:105)=aλ|a(cid:105) with (cid:104)a|a(cid:105)=1. Here we evaluate the probability distributions of mul-
Introduce now the transformations tipartite entanglement for conditional states in the CS
g over the ensemble of experimental outcomes of measure-
aˆ =ˆb −i λα,
λ λ ω mentsontheenvironmentsstudiedinthemaintext. Us-
λ
ing the scale law stated in equation (6) of the main text,
Lˆ =Jˆ+α,
the probability distribution of entanglement, as given by
equation (7) of the main text, takes the manifestly uni-
with complex constant α. Observe that the coherent
states |a(cid:105) are also eigenstates of ˆb. Then, up to a physi- versal form
callyirrelevantconstantterm,Hamiltonian(12)becomes
1 (cid:90) f(a,t)e−|a|2
P (x,t)=
Hˆ =Hˆ −(cid:88) gλ2 (cid:16)α∗Jˆ+αJˆ†(cid:17)+(cid:88)w ˆb†ˆb . G Z (cid:32) x (cid:33)
CS w λ λ λ f(a,t)
λ λ λ ×δ x− da,
Tr ρˆ (0)Uˆ(a,t)†Uˆ(a,t)
CS
InordertoobtainatracelesscouplingoperatorJˆ,wetake
advantage of the freedom introduced by α and choose where we used equations (3) and (5) from the main text
α = d−1Tr Lˆ, where d is the dimension of the Hilbert for simplifications. The particular dependence of the
space of the CS. Thus the hermitian redefinition of the scaling function f(a,t) and Uˆ(a,t)†Uˆ(a,t) on the bath
10
measurement outcomes {a} must be found in each par- the bath defining the open channel. The above equation
ticularcaseofinterestfromtheirdefinitions(seeMethods can be solved by introducing the interaction picture in
section in the main text): nˆ and exploiting the commutator [nˆ ,σˆ ] = ±σˆ , the
− − ± ±
result is
Ufˆ((aa,,tt))==Te−e(cid:80)(cid:82)0tk[−d2((cid:126)i(cid:60)kH)ˆC(cid:82)0St+(cid:82)0(cid:80)sαkk((zsk∗−(ss)(cid:48)J)ˆTkr−[JJˆˆk†k†OOˆˆkk((ask)),s]d,ss(cid:48).)]dsds(cid:48), Uˆ(a,t)=e−γ2nˆ−te√γ(cid:82)0tz∗(a,s)e−γ2sdsσˆ−.
Explicit calculation using the algebraic properties of the
Further simplifications in the expression for the en-
Pauli matrices then gives
tanglement distribution are obtained if we assume that
only a single subsystem is coupled to a channel and Uˆ(a,t)†Uˆ(a,t)=Iˆ+(cid:0)e−γt−1+|(cid:126)aτ(cid:126)g(t)|2(cid:1)nˆ
−
use the locality of Hˆ . In this case, the part of
CS +[(cid:126)aτ(cid:126)g(t)σˆ +((cid:126)aτ(cid:126)g(t))∗σˆ ],
Tr ρˆ (0)Uˆ(a,t)†Uˆ(a,t)whichtracesoverthesubsystems + −
CS
not coupled with the open channel can be calculated,
where
leaving only the reduced density matrix describing the
initialstateofthecoupledsubsystem(denotedby“ch”). √ (cid:90) t
(cid:126)aτ(cid:126)g(t)= γ z∗(a,s)e−γ2sds
If in addition we suppose that
0
Oˆ(ch)(a ,s,s(cid:48))=Oˆ(ch)u(ch)(s,s(cid:48)), √ (cid:88) e−i(ωλ−iγ2)t−1
(ch) =i γ a g .
λ λ ω −iγ
λ λ 2
for some operator Oˆ and function u(s,s(cid:48)), then f(a,t)=
f(t) is independent of {a}, and all variables a in Consequently, F(a,t) depends on the integration vari-
(k(cid:54)=ch)
theprobabilitydistributioncanbeintegratedoutbyper- ables a only through the linear combination y(a,t) =
λ
formingtheGaussianintegralleftforthem. Alltogether, (cid:126)aτ(cid:126)g(t). The distribution function for y = y(a,t) is read-
the key technical difficulties we must face to obtain the ily calculated,
entanglementdistributionaretheevaluationofthefunc-
tion (cid:90) e−|a|2
P (y,t)= δ(y−(cid:126)aτ(cid:126)g(t))da
y Z
F(a(ch),t)=Trch ρˆch(0)Uˆ(ch)(a(ch),t)†Uˆ(ch)(a(ch),t), 1 (cid:18) |y|2 (cid:19)
= exp − ,
π|(cid:126)g(t)|2 |(cid:126)g(t)|2
and the calculation of the integral
f(t)(cid:90) e−|a(ch)|2 (cid:18) f(t) (cid:19) yielding a Gaussian function with width given by
P (x,t)= δ x− da .
G x Z(ch) F(a(ch),t) (c(h1)4) |(cid:126)g(t)|2 =γ(cid:88)ω2 +g(λ2γ/2)2 (cid:16)e−γt−2e−γ2tcosωλt+1(cid:17),
We now follow this program for the two physically rele- λ λ
vant situations with one open channel considered in the
which can be express in terms of the spectral density of
main text.
the bath. For the Ohmic case we get
Markovian channel at zero temperature p(t):=|(cid:126)g(t)|2 = γ (cid:90) ∞ e−γt−2e−γ2tcosωt+1dω
2π ω2+(γ/2)2
−∞
=1−e−γt.
For a Markovian channel at zero temperature coupled
√
toaqubit, Jˆ= γσˆ withσ thedeexcitationoperator
− − After collecting our results, the probability distribu-
andγ thedecayrate,Oˆ(a,s,s(cid:48))=Jˆ(seeRefs.24,25and
tion of entanglement (14) can be recast into
26), and f(a,t)=e−γt/2.
Hˆ(Tcha)ki=ng0,fotrheseimqupalitciiotny oaf mfrootzioenn foinrtetrhneanlodny-unnaimtaircys P (x,p)= e−γ2t (cid:90) e−|py(|t2) δ(cid:18)x− e−γ2t(cid:19)dy
prCoSpagator is G x πp(t) F˜(y)
∂ (cid:104)√ γ (cid:105) with
Uˆ(a,s)= γz∗(a,s)σˆ − nˆ Uˆ(a,s)
∂s − 2 − F˜(y)=(cid:60)Tr ρˆ (0)(cid:104)Iˆ+(cid:0)e−γt−1+|y|2(cid:1)nˆ +yσˆ (cid:105).
ch − +
with initial condition Uˆ(a,0) = Iˆ. Here nˆ = σˆ σˆ ,
− + −
and z(s) = (cid:80)λgλaλe−iωλs is a sum over the modes of This last integral can be evaluated to finally obtain