Geometrically induced singular behavior of entanglement

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by  D. Cavalcanti

Geometrically induced singular behavior of entanglement
D. Cavalcanti,1 P. L. Saldanha,2 O. Cosme,2 F. G. S. L. Branda˜o,3,4 C.
H. Monken,2 S. Pa´dua,2 M. Franc¸a Santos,2 and M. O. Terra Cunha5
1ICFO-Institut de Ciencies Fotoniques, Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain
2Departamento de F´ısica, Universidade Federal de Minas Gerais,
Caixa Postal 702, 30123-970, Belo Horizonte, MG, Brazil
3QOLS, Blackett Laboratory, Imperial College London, London SW7 2BW, UK
4Institute for Mathematical Sciences, Imperial College London, London SW7 2BW, UK
5Departamento de Matem´atica, Universidade Federal de Minas Gerais,
Caixa Postal 702, 30123-970, Belo Horizonte, MG, Brazil
We show that the geometry of the set of quantum states plays a crucial role in the behavior of
8
entanglementindifferentphysicalsystems. Morespecifically it isshownthatsingular pointsat the
0
border of the set of unentangled states appear as singularities in the dynamics of entanglement of
0
smoothly varyingquantumstates. Weillustrate thisresult byimplementinga photonicparametric
2
down conversion experiment. Moreover, this effect is connected to recently discovered singularities

n in condensed matter models.
a
J

6 Entanglement,a genuine quantum correlation,plays a motivation is clear: σ represents a mixture of ρ with the
1 crucial role in different physical situations ranging from random state π, and R (ρ) quantifies how much of this
R

information processing [1] to quantum many-particle noise must be added to ρ in order to obtain a separable
]
h phenomena [2]. Similarly to thermodynamics, smooth state.
p variations of controllable parameters which characterize The main result of this Letter is to show that R can
R
- aphysicalsystemmayleadtosingularbehaviorofentan- be used to investigate the shape of the boundary of S,
t
n glementquantifiers. Insomecases,insimilaritytoquan- ∂S. The principle is to take an entangled state depend-
a
tumphasestransitions[3],thesesingularitiesareattested ing smoothly on one parameter q and compute R as a
u R
q by abrupt changes in the quantum state describing the functionofq. Theone-parameter-dependentdensityma-
[ system. However, unexpected singularities may appear trices ρ(q) can be seen as a curve in the set of quantum

even when the quantum state varies smoothly [4]. Here states as shown in Fig. 1. Singularities at ∂S will show
2
v we demonstrate how the geometryof the set of unentan- up as singularities in RR(ρ(q)). This statement is gen-
1 gledstatescanberelatedtosingularbehaviorinphysical eral for any finite dimension and will be formalized by
0 phenomena. In particular we show that singularities at the contrapositive: if ∂S is non-singular, then R (ρ(q))
R
3 the boundary of this set can be detected by measuring is also non-singular. More precisely:
0
the amount of entanglement of smoothly varying quan-
9. tum states. Proposition 1. Let D be a closed, convex set. Let S
⊂
0 Dalsobeclosedandconvex,withπ apointintheinterior
Entangledstatesaredefinedasthe statesofcomposed
7 of S. If ∂S is a Cm manifold and thestatesρ(q) describe
quantum systems which cannot be written as a convex
0 a Cm curve in D with no points in the interior of S
: sum of products of density matrices for each composing
v and obeying the condition that the tangent vector ρ′(q)
part[5]. Separablestates,ontheotherhand,admitsuch
Xi arepresentationandformaconvex,closedsetwithposi- is never parallel to π ρ(q), then RR(ρ(q)) is also a Cm
−
function.
r tivevolume(forfinitedimensionalsystems)[6]. Thisset,
a
henceforth designated by S, is a subset of D, the set of
One must remember that a manifold is called Cm if
all density matrices (S D), which is also convex and
⊂ it can be parameterized by functions with continuous
closed. Therefore, a natural geometric way to quantify
derivatives up to order m [25]. The reader can change
entanglement is to see how far - using some definition of Cm bysmooth,intheusualsenseofC∞,withalmostno
distance on the state space - an entangled state is from
loss (actually, we use smooth throughout this Letter in
the set S. This has been carried over for a variety of
thelessprecisesenseof“asregularasnecessary”). Other
notions of distance, generating different measures of en-
topologicalremarksbeforethe proof: the factthatS has
tanglement [7]. One of these geometric quantifiers is the
interior points implies that S and D have the same di-
random robustness, R , defined for any state ρ as the
R mensionality (since there is an open ball of D contained
minimum s (s 0) such that the state
≥ in S), and the proof will use the notion of (topological)
ρ+sπ cone,whichsimplymeanstheunionofallsegmentsfrom
σ = (1) a given point V to each point of a given set A: this is
1+s
called the cone of A with vertex V.
is separable(π =I/d,whereI is the identitymatrix and Proof: Thegeometricalsituationleadstothecone,given
dthetotaldimensionofthestatespace)[8]. Thephysical by (p,q) p π+(1 p)ρ(q), p [0,1]. The condition
7→ − ∈
2
on the tangent vector (together with the fact that π is physical setup? We proceed to answer positively both
interior to S, while ρ(q) has no point in this interior) is questions by showing physical processes where a singu-
sufficient for this cone to be Cm, except at the vertex π, larity in ∂S is revealed by monitoring the entanglement
at least locally in q. of a given system.
As S is bounded and convex, and π is in its interior, First,letusconsiderageneralsystemoffourqubitsa,
every straight line from π crosses ∂S exactly once. As b, A, and B, subject to the following Hamiltonian [19]:
ρ(q)hasnopointintheinteriorofS,thiscrossingalways
happens for 0 p < 1. Denote this crossing value by H =HaA+HbB, (3)
≤
p (q). The curve q p (q)π+(1 p (q))ρ(q) is Cm,
c c c
implying p is a Cm7→function of q. − where
c
The random robustness is given by R (ρ(q))= pc . ω ω g
R 1−pc Hµν = σµ+ σν + (σµσν +σµσν). (4)
As pc <1, we also obtainthat RR is a Cm function ofq. 2 z 2 z 2 − + + −
(cid:3)
We insist on the interpretation: Proposition 1 means Here σ+ = (σx +iσy)/2 and σ− = (σx iσy)/2, where
−
σ ,σ andσ aretheusualPaulimatrices. Thisscenario
that any singularity in R for a well choosen path ρ(q) x y z
R
can be realized in many systems, like cavity QED [20],
reflects singularities in ∂S.
trapped ions [21], and quantum dots [22]. We set the
initial state to be ψ(t=0) = Φ Ψ , where
| i | +iab⊗| +iAB
qubitsabareintheBellstate Φ =(00 +11 )/√2and
+
| i | i | i
qubitsABareintheorthogonalBellstate Ψ =(01 +
+
| i | i
10 )/√2. Hamiltonian (3) induces a swapping process
| i
which leads (in the interaction picture) to the following
temporal evolution for the subsystem AB, obtained by
tracing out the subsystem ab:
FIG. 1: State space. The dotted line represents the path ρAB(t)=q Ψ+ Ψ+ +(1 q) Φ+ Φ+ , (5)
| ih | − | ih |
ρ(q) followed by ρ when parameter q is changed. It is worth
noting that S can present singular points in its shape and to where q =cos2(gt). For this state the negativity reads
remember that the “true” picture is much subtler, given the
large dimensionality of even thesimplest example[9]. (ρ (t))=max 1 2q,2q 1 = 1 2q . (6)
AB
N { − − } | − |
From this point on, we study the situation for two This function presents a singularity for q = 0.5 (gt =
qubits, which is related to the performed experiment nπ/4, with n odd) signaling then a singularity at ∂S.
described here. In this case, Ref. [10] shows that the Anotherphysicalprocesswhichalsoproducesthefam-
Random Robustness is proportional to the Negativity ily of states (5) is the following simple quantum commu-
( (ρ))[11], given by the absolute value of the sum of nicationtask: Alice preparesaBellstate Φ+ andsends
N | i
the negative eigenvalues of the partial transposed state. one qubit to Bob through a quantum channel; if this
The negativityisa monotoneunder localoperationsand channel has a probability q of introducing a bit flip, and
classical communication [12] and has the operational in- 1 q of no error at all, the state (5) is the output of the
−
terpretation of a cost function under a certain class of process [27].
operations [13, 14]. To illustrate the dynamics given by Eq. (5), we have
Atthesametime,entanglementcanbemeasuredwith performed an optical experiment, shown in Fig. 2. In
the help of entanglement witnesses [15]. These are Her- ourexperiment,twinphotonsmaximallyentangledinpo-
mitian operators with positive mean value for all sep- larization are generated in a non-linear crystal [23] and
arable states, but with a negative mean value for some sent to an unbalanced Michelson interferometer, which
entangledstates[16]. Infact,manygeometricalentangle- is used to simulate the channel described above. The
mentquantifiersaredirectlyrelatedtowitnessoperators experiment works as follows: we produce a two-photon
[17]. In the particular case of two qubits [26], we have Ψ+ state, send one of the photons directly to the de-
| i
that for every entangled state ρ [10], tectionstage,andtheothertothe(unbalanced)interfer-
ometer. One of the arms of this interferometer does not
2 (ρ)=RR(ρ)= 2 min Tr(Wρ), (2) change the polarization of the photon, and if the pho-
N − W∈W
ton went through this path the two photons would be
where is the set of entanglement witnesses W with detected in Ψ . However if the photon went through
+
W | i
TrW =2. theotherpathitspolarizationwouldberotatedinsucha
At this point we might ask some natural questions. way that the final two-photon state would become Φ .
+
| i
Is there in fact any singularity in the shape of S? In Wehavemadeatomographiccharacterizationofthepho-
the affirmative case, does this singularity appear in any tonic states corresponding to these two extremal points.
3
4 A. 4 B.
3 3
2 2
1 1
0.4 0.4
0.2 0.2
0.0 0.0
1 1
2 2
3 3
4 4
4 C. 4 D.
3 3
2 2
1 1
0.4 0.4
0.2 0.2
FIG. 2: Experimental setup: The state source is com-
0.0 0.0
posed by a 2mm-thick BBO (β-BaB2O4) nonlinear crystal
1 1
(C1)pumpedbyacwkryptonlaseroperatingat413nm,gen- 2 2
3 3
erating photon pairs at 826nm by type II spontaneous para- 4 4
metric down-conversion. Crystal C1 is cut and oriented to
generate either one of the polarization entangled Bell states FIG. 3: (Color online) The reconstructed density matrices
|Ψ−i or |Ψ+i. Walk-off and phase compensation is provided corresponding (ideally) to the states |Φ+i (A. real and B.
bythehalf-waveplateH0followedbya1mm-thickBBOcrys- imaginary parts) and |Ψ+i (C. real and D.imaginary parts).
tal(C2) [23],together with two1mm-thickcrystalline quartz Theattainedfidelityforthesestatesare,respectively,FΦ+ ≡
plates(Z)insertedinoneofthedown-convertedphotonpaths. hΦ+|ρ|Φ+i≈(92±3)%andFΨ+ ≡hΨ+|ρ|Ψ+i≈(96±3)%.
The unconverted laser beam transmitted by crystal C1 is
discarded by means of a dichroic mirror (U). The detection
stages are composed by photon counting diode modules D1 sponding optimal witness given by
and D2, preceded by 8nm FWHM interference filters F1 and
I 2 Φ Φ , for 0 q 1/2,
Fan2dceAn2teorefd3a.0tm82m5n∅m. ,SainngdlebyancidrccuolainrcaidpeenrtcuerecsouAn1tsofw1i.t6hm5mn∅s Wopt =(cid:26)I−−2||Ψ++iihhΨ++|| , for 1/≤2≤≤q ≤1. (7)
resolving time are registered by a computer controlled elec-
For the family of generated states these two observables
tronic module (CC). Polarization analyzers are composed by
quarter-wave plates Q1 and Q2, half-wave plates H1 and H2, are the only candidates of optimal entanglement wit-
followed by polarizing cubes P1 and P2. The State Source nesses, so they are the only ones to be measured. In
produces state |Ψ−i. For each pair, the photon emerging in a more general situation, if less is known about the pre-
theupperpathgoesstraighttothepolarization analyzerand pared state, much more candidate witnesses should be
to the detection stage 1. The lower path photon is directed
measured. The results are displayed in Fig. 4. The blue
bymirror M3 throughthecircularapertureA3 intothestate
curve in the figure shows the witnessed negativity mea-
mixer (an unbalanced Michelson interferometer), composed
surementanditsedgeindicatestheexistenceofsingular-
by the beam splitter BS, mirrors M4 and M5, quarter-wave
platesQ4 andQ5,variablecircularaperturesA4 andA5,and ities at ∂S. This experimental result shows the abrupt
by the half-wave plate H3, whose purpose is to compensate change in the optimal witness at the value q = 21, which
for an unwanted slight polarization rotation caused by the heralds the singularity in ∂S. As a proof of principles,
beam splitter. The quarter-wave plate Q4 is switched off each operator W is measured for the whole range of q,
which means that if the lower photon follows path labeled
which yields the points bellow zero in Fig. 4. Note that
4, there is no change to its polarization and the half-wave
the singularity occurs exactly for R =0 (q =1/2). Ac-
plate H3 changes the state to |Ψ+i. On the other hand, if R
cordingtoourgeometricalinterpretation,thismeansthe
the lower photon follows path labeled 5, Q5 is oriented with
thefast axisat45◦ in ordertoflipitspolarization. Thepath pathfollowedbytheparameterizedstateρ(q)touchesthe
length difference, 130mm, is much larger than the coherence borderofS. Thisresultmustnotbeasurprise,sinceitis
length of the down-converted fields, ensuring an incoherent wellknownthatinthetetrahedrongeneratedbytheBell
recombination at BS. The pair detected by CC is in state states(which weaccessin ourexperiment)the separable
q|Ψ+ihΨ+|+(1−q)|Φ+ihΦ+| where q is defined by the rela- states form a inscribed octahedron [24].
tivesizes of apertures A4 and A5.
The geometrical properties of entanglement discussed
here give new insight into singularities found recently in
the entanglement of condensed matter systems. Striking
examples, dealing with entanglement properties of cer-
tain spin-1 models subjected to a transverse magnetic
2
The reconstructed density matrices are displayedin Fig. field h, are described in Refs. [4]. In these works, the
3. These two possibilities are then incoherently recom- two-qubit reduced state shows a singularity in entangle-
bined, thus allowing the preparation of state (5). Each ment for a particular field value h far from the critical
f
preparation yields a different value for q with the corre- field of the respective model. As correlation functions,
4
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ham and funding from CNPq, Fapemig, PRPq-UFMG, [26] An optimal entanglement witness Wopt satisfying (2) is
proportional to the partial transposition of the projec-
andBrazilianMilleniumInstitute forQuantumInforma-
torovertheeigenspaceofthenegativeeigenvalueofρT2,
tion.
where ρT2 denotesthe partial transposition of ρ [18].
[27] The simplest way of drawing the complete line repre-
sentedbyEq.(5)istoconsiderthreedifferentinitialcon-
ditions: from |Φ+i one obtains q ∈ [0,1/2), from |Ψ+i,
q∈(1/2,1],andq=1/2isafixedpointofthisdynamical
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