Quantum tomography and nonlocality

0.47 MB·English

by  Evgeny V. Shchukin

Quantumtomographyandnonlocality
Evgeny V. Shchukin∗
InstituteofPhysics,Johannes-GutenbergUniversityofMainz,Staudingerweg7,55128Mainz,Germany
Stefano Mancini†
School of Science and Technology, University of Camerino, 62032 Camerino, Italy
& INFN Sezione di Perugia, I-06123 Perugia, Italy
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Wepresentatomographicapproachtothestudyofquantumnonlocalityinmultipartitesystems.
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2 Bellinequalitiesfortomogramsbelongingtoagenerictomographicschemearederivedbyexploiting

n
toolsfromconvexgeometry. Then,possibleviolationsoftheseinequalitiesarediscussedinspecific
a
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tomographicrealizationsprovidingsomeexplicitexamples.

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1

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h
p
I. INTRODUCTION
-
t
n
a
The Bell inequalities [1] demonstrate paradigmatic difference of quantum and classical worlds. They
u
q
wereoriginallywrittenfordichotomic(spin−1)variables[2]. Spin−1 operatorsrealizetheLiealgebraof
[ 2 2

1 the SU(2) group. For several spin particles their spin operators form Lie algebra of the tensor product of
v
2 theLiealgebras. DuetoalgebraicequivalenceoftheoperatorssatisfyingcommutationrelationsoftheLie
8
5 algebraconstructedfromparticlespinoperatorsandconstructedfromcreationandannihilationoperatorsof
2
0 a field, one can obtain Bell inequalities also for the case of continuous variables besides discrete ones [3].
.
1
Beyond the specific operators involved in the Bell inequalities, their possible violations obviously depend
0
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onthestateunderconsideration.
1
:
v Fora(multipartite)classicalsystemwithfluctuations, thesystemstateisdescribedbymeansofajoint
i
X
probability distribution function of random variables corresponding to the subsystems. In contrast, for a
r
a
(multipartite) quantum system the state is described by the density matrix. In view of this difference the
calculations of the system’s statistical properties (including correlations) are accomplished differently in
classicalandquantumdomains.
Recently, a probability representation of quantum mechanics has been suggested [4]. This representa-
tion, equivalent to all other well known formulations of quantum mechanics (see, e.g. [5]), goes back to
quantumtomography,atechniqueusedforquantumstatereconstruction[6]. Theapproachmakesuseofa
∗Electronicaddress:[email protected]
†Electronicaddress:[email protected]
2
set of fair probabilities, tomograms, to “replace” the notion of quantum state. It has also been understood
[7] that for classical statistical mechanics the states with fluctuations can be described as well by tomo-
grams related to standard probability distributions in classical phase-space. A comparison of classical and
quantumtomogramscanbefoundinRef.[7,8].
Thus,intheprobabilityrepresentation,tomogramsturnedouttobeauniquetooltodescribebothclassi-
calandquantumstates. Asaconsequencetheyrepresentanaturalbedwheretoplaceinequalitiesmarking
the boarder-line between quantum and classical worlds. Tomograms can be either continuous or discrete
variablefunctionsdependingonthetomographicscheme(realization). Inbothcasestheymightbedirectly
used to test nonlocality. This possibility was described for symplectic tomography [9] in bipartite system
[10]andspintomography[11]stillinbipartitesystem[12].
Here we shall derive Bell inequalities for multipartite systems in terms of tomograms belonging to a
generic tomographic scheme. Then, we shall discuss the possibility to violate such inequalities depending
onthetomographicrealization.
Thelayoutofthepaperisthefollowing. InSectionIIweformalizequantumtomographyinamultipar-
tite setting. Then, in Section III we derive the Bell inequalities in terms of tomograms. In Section IV we
providesomeevidencesofviolationsofsuchinequalitiesforspin−1 systemsaswellasforfieldmodesand
2
finallydrawtheconclusionsinSectionV.
II. QUANTUMTOMOGRAPHY
Herewebrieflyreviewthegeneralquantumtomographyapproachforasinglesystem,bydetailingthree
relevantcases(optical[13],spin[11]andphoton-numbertomography[14])andthenextendtheformalism
tomultipartitesystems.
ThebasicingredientsofanytomographicschemeareaHilbertspaceHassociatedwithspaceofthesys-
temunderconsiderationandapairofmeasurablesets(X,Λ)withmeasuresµ(x)andν(λ)correspondingly.
Moreprecisely,thesetofsystemstatesisthesetS(H)ofHermitiannon-negativetrace-classoperatorson
H with trace 1. Usually the set X is the spectrum of an observable of the system and the set Λ plays the
roleoftransformations.
We use the notation P(X) for the set of probability distributions on X, i.e. the set of nonnegative
measurablefunctionsp : X → Rnormalizedtooneinthefollowingsense(cid:82) p(x)dµ(x) = 1.
Both sets S(H) and P(X) are closed with respect to the convex combinations: if (cid:37)ˆ,σˆ ∈ S(H) (resp.
p(x),q(x) ∈ P(X))anda ∈ [0,1]then
a(cid:37)ˆ+(1−a)σˆ ∈ S(H) (resp. ap(x)+(1−a)q(x) ∈ P(X)).
3
Definition1 A map T : S(H) → RX×Λ is called tomographic map if the following three conditions are
satisfied:
1. for any (cid:37)ˆ ∈ S(H) the image T((cid:37)ˆ) : X × Λ → R restricted on the set X × {λ} is a probability
densityonX
T ((cid:37)ˆ) ∈ P(X) ∀λ ∈ Λ, where T ((cid:37)ˆ) = T((cid:37)ˆ)| : X → R.
λ λ X×{λ}
2. themapT preservesconvexcombinations
T(a(cid:37)ˆ+(1−a)σˆ) = aT((cid:37)ˆ)+(1−a)T(σˆ), ∀(cid:37)ˆ,σˆ ∈ S(H),a ∈ [0,1].
3. themapT isone-to-one
T((cid:37)ˆ) = T(σˆ) ⇔ (cid:37)ˆ= σˆ.
These conditions have simple meaning: (i) means that the tomogram T((cid:37)ˆ) of any state (cid:37)ˆis a probability
distributiononX parameterizedbythepointsofΛ,(ii)isthelinearitycondition,and(iii)requiresthatthe
tomogramofeachstatebeunique,or,inotherwords,thatanystatecanbeunambiguousreconstructedfrom
itstomogram.
Inthepresentworkwedealwithtomographicmapsofthefollowingform
(cid:16) (cid:17)
T((cid:37)ˆ)(x,λ) ≡ p (x,λ) = Tr (cid:37)ˆUˆ(x,λ) , (1)
(cid:37)ˆ
where Uˆ(x,λ) is a family of operators on H parameterized by points (x,λ) of the set X × Λ. In the
examples considered below the state (cid:37)ˆcan be reconstructed from its tomogram p (x,λ) according to the
(cid:37)ˆ
formula
(cid:90) (cid:90)
(cid:37)ˆ= p(x,λ)Dˆ(x,λ)dµ(x)dν(λ), (2)
X×Λ
fortheappropriate(x,λ)-parameterizedfamilyofoperatorsDˆ(x,λ)onH.
The set X is the spectrum of an observable Oˆ and the set Λ is a group equipped with a representation
(ingeneralprojective)π : Λ → HinH. TheoperatorsUˆ(x,λ)havethefollowingform
Uˆ(x,λ) = π(λ)|x(cid:105)(cid:104)x|π†(λ), (3)
where |x(cid:105) is an eigenstate of the observable Oˆ. For a group theoretical approach to quantum tomography
see[15]. Seealso[16]forarelationtogrupoids.
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A. Spintomography
Letusconsiderasystemwithspinj. Inthiscasewehave: H = C2j+1,X = {−j,−j+1,...,j−1,j}
and Λ = SO(3,R). We denote the elements of the sets X and Λ as s and Ω respectively. The measure
on X is equal to one on each element, so the corresponding integral is simply the finite sum over 2j +1
terms. The measure on SO(3,R) is Haar’s one. For the group SO(3,R), parameterized with Euler angles
Ω ≡ (ϕ,ψ,θ)themeasureν(Ω)readsν(Ω) ≡ ν(ϕ,ψ,θ) = sinψdϕdψdθandtheoperatorUˆ of(3)takes
theform
Uˆ(s,Ω) = Kˆ(Ω)|j,s(cid:105)(cid:104)j,s|Kˆ†(Ω). (4)
Here the vectors |j,s(cid:105), s = −j,−j +1,...,j −1,j are the basis of the space C2j+1 (eigenvectors of the
spin projection sˆ ) and the operators Kˆ(Ω) are the operators of the irreducible representation of SO(3,R)
z
inC2j+1. Theirmatrixelementsaregivenby
(cid:115)
(j +s(cid:48))!(j −s(cid:48))!
(cid:104)j,s|Kˆ(Ω)|j,s(cid:48)(cid:105) = ei(sθ+s(cid:48)ϕ)
(j +s)!(j −s)!
× coss+s(cid:48)(ψ/2)sins(cid:48)−s(ψ/2)P(s(cid:48)−s,s(cid:48)+s)(cosψ), (5)
j−s(cid:48)
(α,β)
withP (x)theJacobipolynomials.
n
Thenthetomogramp(s,Ω) ≡ p(s,ϕ,ψ,θ)of(1)is
p(s,Ω) = (cid:104)j,s|Kˆ(Ω)(cid:37)ˆKˆ†(Ω)|j,s(cid:105). (6)
Due to the property (cid:104)j,s|Kˆ(Ω)|j,s(cid:48)(cid:105) = (−1)s(cid:48)−s(cid:104)j,−s|Kˆ(Ω)|j,−s(cid:48)(cid:105), the tomogram does not depend on
theangleθ,i.e. p(s,ϕ,ψ,θ) ≡ p(s,ϕ,ψ).
Finally,theoperatorDˆ of(2)results
j
(cid:88)
Dˆ(s,Ω) = (cid:104)j,n|Dˆ(s,Ω)|j,m(cid:105)|j,n(cid:105)(cid:104)j,m|,
n,m=−j
wherethematrixelements(cid:104)j,n|Dˆ(s,Ω)|j,m(cid:105)aregivenbythefollowingexpression
(−1)s+m (cid:88)2j
(cid:104)j,n|Dˆ(s,Ω)|j,m(cid:105) = (2j +1)2
8π2 3
j3=0
  
× (cid:88)j3 (cid:104)j,k|Kˆ(Ω)|j,0(cid:105) j j j3 j j j3 , (7)
n −m k s −s k
k=−j3
intermsofWigner3j-symbols.
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B. Opticaltomography
Here we have : H = L (R), X = R and Λ = {eiθ|θ ∈ [0,2π]}. The measures on X and Λ are
2
Lebegue’sones. TheoperatorcorrespondingtoEq.(3)reads
Uˆ(X,θ) = Rˆ(θ)|X(cid:105)(cid:104)X|Rˆ†(θ), (8)
whereRˆ(θ)istherotationoperator
(cid:18) (cid:19)
θ
Rˆ(θ) = exp i (xˆ2+pˆ2) ,
2
actingonandthecanonicalpositionxˆandmomentumpˆoperatorsas
    
xˆ cosθ −sinθ xˆ
Rˆ(θ) Rˆ†(θ) =   .
pˆ sinθ cosθ pˆ
In other words, Uˆ(X,θ) of (8) is the projector on the rotated eigenvector |X(cid:105) of the position operator xˆ.
Thetomogramp(X,θ)of(1)isthediagonalmatrixelement
p(X,θ) = (cid:104)X|Rˆ(θ)(cid:37)ˆRˆ†(θ)|X(cid:105). (9)
Furthermore,theoperatorDˆ of(2)results
1 (cid:90) (cid:16) (cid:17)
Dˆ(X,θ) = |r|exp −ir(X −cosθxˆ−sinθpˆ) dr.
4π
C. Photon-Numbertomography
Here we have: H = L (R), X = Z = {0,1,...} and Λ = C. We denote the elements of the sets X
2 +
andCasnandαrespectively. ThemeasureonX isequaltooneoneachelementandthemeasureonCis
(1/π)d2α, where d2α = dReαdImα is the Lebegue’s measure on the real plane. Here, the operator Uˆ is
theprojectorontothedisplacedFockstate
Uˆ(n,α) = Dˆ(α)|n(cid:105)(cid:104)n|Dˆ†(α), (10)
with
(cid:20)α−α∗ α+α∗ (cid:21)
D(α) ≡ exp √ xˆ−i √ pˆ .
2 2
From(1)thetomogramp(n,α)reads
p(n,α) = (cid:104)n|Dˆ(α)(cid:37)ˆDˆ†(α)|n(cid:105). (11)
Furthermore,theoperatorDˆ of(2)becomesinthiscase
+∞
(cid:88)
Dˆ(n,α) = 4(−1)n (−1)mDˆ(α)|m(cid:105)(cid:104)m|Dˆ†(α).
m=0
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D. Tomographyformulti-partitesystems
Thegeneralizationformulti-partitesystemsisstraightforward.
Definition2 Consider a n-partite system with the state space H⊗n and n tomographic schemes, one for
each part with sets (X ,Λ ) and operators Uˆ (x ,λ ) and Dˆ (x ,λ ), k = 1,...,n. The tomographic
k k k k k k k k
schemeforthewholesystemisthenconstructedasthedirectproductoftheseschemes,byusing
n n
(cid:89) (cid:89)
X ≡ X , Λ ≡ Λ ,
k k
k=1 k=1
n n
(cid:79) (cid:79)
Uˆ(x,λ) ≡ Uˆ (x ,λ ), Dˆ(x,λ) ≡ Dˆ (x ,λ ), (12)
k k k k k k
k=1 k=1
where x ≡ (x ,...,x ), λ ≡ (λ ,...,λ ) and the measures µ(x), ν(λ) on X, Λ are direct products of
1 n 1 n
µ (x ),...,µ (x )andν (λ ),...,ν (λ )respectively. Thetomogramp(x,λ)ofastate(cid:37)ˆ(generalizing
1 1 n n 1 1 n n
(1))is
(cid:16) (cid:17)
p(x,λ) = Tr (cid:37)ˆUˆ(x,λ) . (13)
(cid:82)
Foranyλ ∈ ΛitisaprobabilitydistributiononX,thus p(x,λ)dµ(x) = 1.
X
Remark. Fromthedefinition(12)oftheoperatorUˆ(x,λ)itimmediatelyfollowsthatthetomogramp(x,λ)
ofafactorizedstate
(cid:37)ˆ= (cid:37)ˆ ⊗...⊗(cid:37)ˆ (14)
1 n
isalsofactorized,i.e.
p(x,λ) = p (x ,λ )...p (x ,λ ), (15)
1 1 1 n n n
wherep (x ,λ )isthetomogramofthestate(cid:37)ˆ . Moregenerally,thetomogramofaseparablestate
k k k k
+∞ +∞
(cid:37)ˆ= (cid:88)a (cid:37)ˆ(i)⊗...(cid:37)ˆ(i), a (cid:62) 0, (cid:88)a = 1 (16)
i 1 n i i
i=0 i=0
isalsoseparableinthefollowingsense
+∞
p(x,λ) = (cid:88)a p(i)(x ,λ )...p(i)(x ,λ ), (17)
i 1 1 1 n n n
i=0
(i) (i)
wherep (x ,λ )isthetomogramofthestate(cid:37)ˆ .
k k k k
7
III. BELLINEQUALITIESFORTOMOGRAMS
Let us consider a n-partite system in tomographic representation, whith each subsystem supplied by a
tomographic map T , k = 1,...,n. The tomogram p(x,λ) of a state (cid:37)ˆis a function of 2n arguments and
k
with respect to one half of them it is a probability distribution. We will show that in general it cannot be
consideredasaclassicaljointprobability.
Definition3 Foranyk = 1,...,nletY andZ betwomeasurablesetssuchthat
k k
(cid:91) (cid:92)
X = Y Z , Y Z = ∅,
k k k k k
(cid:81)
andforanyλ ∈ Λ letA (λ )beadichotomicrandomvariableonX = X suchthat
k k k k k k
(cid:16) (cid:17)
P(A (λ ) = 1) = (cid:82) Tr (cid:37)ˆUˆ (x ,λ ) dµ (x ),
k k Y k k k k k
k
(cid:16) (cid:17)
P(A (λ ) = −1) = (cid:82) Tr (cid:37)ˆUˆ (x ,λ ) dµ (x ). (18)
k k Z k k k k k
k
SymbolicallythevariablesA (λ )canbewrittenas
k k

 1 if x ∈ Y ,
k k
A (λ ) = (19)
k k
−1 if x ∈ Z
k k
in the coordinate system deformed by the operator Uˆ (x ,λ ). The joint probability distribution of the
k k k
randomvariablesA (λ ),...,A (λ ),namely
1 1 n n
p (λ ,...,λ ) = P(A (λ ) = ε ,...,A (λ ) = ε ),
ε1,...,εn 1 n 1 1 1 n n n
whereε = ±1,isgivenby
k
(cid:90) (cid:90)
p (λ ,...,λ ) = ... p(x,λ)dx, (20)
ε1,...,εn 1 n
W1 Wn
with

 Y if ε = 1,
k k
W =
k
Z if ε = −1.
k k
ThecorrelationfunctionofA (λ ),...,A (λ )results
1 1 n n
(cid:10) (cid:11)
E(λ ,...,λ ) ≡ A (λ )...A (λ )
1 n 1 1 n n
(cid:88)
= p (λ ,...,λ )ε ...ε . (21)
ε1,...,εn 1 n 1 n
ε1,...,εn=±1
8
(1) (2)
Definition4 Letusfixtwoparametersλ andλ fork = 1,...,nanddenote
k k
E(j ,...,j ) ≡ E(λ(j1),...,λ(jn)), j = 1,2. (22)
1 n 1 n k
Since each index j can take 2 values independently on all the other indices, there are 2n correlation
k
functions(22). Thenwedefineby
(cid:16) (cid:17)
e ≡ E(j ,...,j ) ∈ R2n. (23)
1 n
thevectorofthesecorrelationfunctionswithsomeorderofmulti-indices(j ,...,j ).
1 n
It is convenient to enumerate the functions E(j ,...,j ). For this purpose we use the binary base with
1 n
“digits”1and2insteadof0and1. Thismeansthatweusethefollowingone-to-onecorrespondence
{1,...,2n} (cid:51) j ↔ (j ,...,j ), j = 1,2,
1 n k
wherej and(j ,...,j )arerelatedtoeachotheraccordingto
1 n
j = (j −1)2n−1+...+(j −1)2+j . (24)
1 n n
Byvirtueofsuchanordering,thevectore(23)canbewrittenas
(cid:16) (cid:17) (cid:16) (cid:17)
e = E(1),...,E(2n) = E(1,...,1),...,E(2,...,2) ∈ R2n. (25)
What region Ω ⊂ R2n fills the vector e of (25)? Due to the fact that each observable has only two
n
outcomes±1itfollowsthateachcorrelationfunction(22)isboundedbyonebyabsolutevalueand,so,the
setΩ isasubsetof2n-dimensionalcube[−1,1]2n.
n
Supposethatitispossibletomodeltheresultofthemeasurementbyarandomvariable,A (j ),which
k k
cantaketwovalues±1. Weassumethattheserandomvariablescanbearbitrarycorrelated.
Definition5 Letusdefineby
(cid:16) (cid:17)
p(i (1), ...,i (2)) ≡ P A (1) = i (1),...,A (2) = i (2) , (26)
1 n 1 1 n n
the joint probability distribution for random variables A (j ), with i (j ) = ±1. Since each index i (j )
k k k k k k
cantakeindependently2valueswehave22n numbers(26)whichcompletelydescribestatisticalcharacter-
isticsoftherandomvariablesunderconsideration. Weenumeratethemwithasinglenumberi = 1,...,22n
usingthesameruleasforthecorrelationfunctionsE(j),namely
{1,...,22n} (cid:51) i ↔ (i (1),...,i (2)),
1 n
9
whereiand(i (1),...,i (2))arerelatedtoeachotheraccordingto
1 n
i = (i (1)−1)22n−1+(i (1)−1)2+i (2). (27)
1 n n
Enumeratedinsuchawaytheprobabilities(26)forma22n-dimensionalvector
p ≡ (p ,...,p ) ∈ R22n. (28)
1 22n
Thepoint(28)liesinthestandardsimplex
 
(cid:12) 22n
S22n−1 = (p1,...,p22n) (cid:12)(cid:12) (cid:88)pi = 1,pi (cid:62) 0 ⊂ R22n. (29)
(cid:12)
 
i=1
What region Ω ⊂ R2n fills the vector e (25) when the point p (28) runs over the simplex S
n 22n−1
(29)? Toanswerthisquestionwearegoingtoexplicitlyrelateeandpassumingtheformerexpressedlike
classicaljointprobabilities. Then,thecorrelationfunctionE(j)isintendedasasimplelinearcombination
ofp withpropercoefficients. Lookingat(21)weconsidersuchcoefficients,E(j,i),givenbytheproduct
i
E(j,i) = i (j )...i (j ), (30)
1 1 n n
where j and i (j ) are “digits” of the numbers j and i in the binary representations (24) and (27). The
k k k
numbersE(j,i),j = 1,...,2n,i = 1,...,22n forma2n×22n matrixE andtherelationbetweeneandp
n
canthenbewrittenas
e = E p. (31)
n
WeseethattheregionΩ istheimageofthestandardsimplexS
n 22n−1
Ω = E (S ), (32)
n n 22n−1
where we do not distinguish the linear map E : R22n → R2n and its matrix E in the standard bases of
n n
R22n andR2n.
Thus, we have reduced the problem of finding Bell inequalities to find the set Ω . It means that the
n
problem of finding Bell inequalities boils down to a standard problem of convex geometry, referred to
as convex hull problem: given points c find their convex hull, or facets of maximal dimension of the
i
correspondingpolytope(fornotionsofconvexgeometrysee,e.g. [17]).
NowwewillgettheBellinequalitiesexplicitly. NotethatpermutationsofthecolumnsofthematrixE
n
do not change their convex hull and that they correspond to permutations of the components of the vector
pordifferentorderingsoftheprobabilities(28),soonecansafelypermutecolumnsofE withoutaltering
n
(31).
10
Theorem1 ThesetΩ isspecifiedbythe(Bell)inequalitiesforthevectorofthecorrelationfunctions
n
(e,H c) (cid:54) 2n, ∀c = (±1,...,±1). (33)
2n
ThematrixH istheHadamardmatrixrecurrentlydefinedas
2n
 
1 1
H2n = H2⊗...⊗H2, H2 =  .
(cid:124) (cid:123)(cid:122) (cid:125) 1 −1
n
Proof. ThekeyfactinderivingtheBellinequality(33)isthatthematrixE canbewritteninthefollowing
n
blockform
(cid:16) (cid:17)
En = H2n −H2n ... H2n −H2n (34)
(cid:124) (cid:123)(cid:122) (cid:125)
2n
after appropriate arrangement of its columns. One can rewrite the r.h.s. of (34) as the product of two
matrices
(cid:16) (cid:17)
En = H2n E2n −E2n ... E2n −E2n = H2nAn,
whichmeansthatthelinearmapE : R22n → R2n canbedecomposedintotwomaps
n
E = H ◦A , A : R22n → R2n, H : R2n → R2n.
n 2n n n 2n
Accordingtothisdecomposition(31)reads
e = H q, (35)
2n
wherethevectorq = A p ∈ R2n isexplicitlygivenbythefollowingexpression
n
 
p −p +...−p
1 2n+1 (2n−1)2n+1
 . 
q =  .. . (36)
 
p −p +...−p
2n 2·2n 22n
DefinethefollowingconvexpolytopeO ⊂ RN
N
O = {x ∈ RN|(x,c) (cid:54) 1, ∀c = (±1,...,±1)}. (37)
N
As one can easily see the image of the standard simplex S is exactly the polytope O , that is
22n−1 2n
A (S ) = O . Fromthisfactwehave
n 22n−1 2n
Ω = H (O ). (38)
n 2n 2n