Coarse graining the phase space of $N$ qubits

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by  Olivia Di Matteo

CoarsegrainingthephasespaceofN qubits
Olivia Di Matteo,1,2 Luis L. Sa´nchez-Soto,3,4 Gerd Leuchs,4,5 and Markus Grassl4,5
1DepartmentofPhysicsandAstronomy, UniversityofWaterloo, OntarioN2L3G1, Canada
2Institute for Quantum Computing, University of Waterloo, Ontario N2L 3G1, Canada
3DepartamentodeO´ptica, FacultaddeF´ısica, UniversidadComplutense, 28040Madrid, Spain
4Max-Planck-Institut fu¨r die Physik des Lichts, Staudtstraße 2, 91058 Erlangen, Germany
5Institutfu¨rOptik,InformationundPhotonik,Universita¨tErlangen-Nu¨rnberg,Staudtstraße7/B2,91058Erlangen,Germany
WedevelopasystematiccoarsegrainingprocedureforsystemsofN qubits. Weexploittheunderlyingge-
ometrical structures of the associated discrete phase space to produce a coarse-grained version with reduced
effectivesize. Ourcoarse-grainedspacesinheritkeypropertiesoftheoriginalones. Inparticular,ourproce-
durenaturallyyieldsasubsetoftheoriginalmeasurementoperators,whichcanbeusedtoconstructacoarse
discreteWignerfunction. Theseoperatorsalsoconstituteasystematicchoiceofincompletemeasurementsfor
thetomographerwishingtoprobeanintractablylargesystem.
7
1
0
2 I. INTRODUCTION phasespaceaspecificrank-oneprojectionoperatorrepresent-
ingapurequantumstate. Foreachpointofthegrid,asuitable
n
quasi-probabilityastheWignerfunctioncanbedirectlycom-
a Recently, the understanding of many-body quantum sys-
J tems has dramatically progressed. Nowadays we are achiev- putedfromthemeasurementofthestatesassociatedwiththe
linespassingthroughthatpoint. Wecoarsegrainbycombin-
0 ing an amazing degree of control over larger and larger sys-
3 tems [1, 2]. Therefore, verification during each stage of ex- inggroupsoftheselinesintothicklines,whichwewillshow
tobelinesinthephasespaceofaneffectivelysmallersystem.
perimental procedures is of utmost importance; quantum to-
] Ourcoarse-grainedphasespacesareendowedwithmanynice
h mographyistheappropriatetoolforthatpurpose.
properties.
p Thegoalofquantumtomographyistoreconstructthestate
- ofasystembyperformingmultiplemeasurementsonidenti- Most notably, our procedure systematically and naturally
nt cally prepared copies of the system. Once the experimental revealsasubsetofmeasurementswhichonecouldusetoper-
a form incomplete tomography. In addition, using the coarse-
data are extracted, a numerical procedure determines which
u grained points and lines, we show that one can define a dis-
density matrix fits best the measurements. This estimation
q crete Wigner function in largely the same way as it is de-
[ can be performed using different approaches, such as maxi-
mum likelihood estimation [3], or Bayesian methods [4–7]. finedintheoriginalspace.Whenplotted,thecoarsefunctions
1 resemble smoothed out versions of the originals, preserving
However,tomographybecomesharderasweexploremorein-
v manyoftheirprominentvisualfeatures.
tricatesystems. Ifwelookatthesimple,yetillustrativecase
0
of N qubits, which will serve as the consistent thread in this
3
6 paper,onehastomakeatleast2N+1measurementsindiffer-
8 entbasesbeforeonecanclaimtoknoweverythingaboutana II. PHASESPACEOFNQUBITS
0 prioriunknownsystem. Withsuchanexponentialscalingin
1. the number of qubits, it is clear that current methods rapidly Aqubitisatwo-dimensionalquantumsystem,withHilbert
0 becomeintractableforpresentstate-of-the-artexperiments. space isomorphic to C2. It is customary to choose two nor-
7 As a result, more sophisticated tomographical techniques
malized orthogonal states, say {|0(cid:105),|1(cid:105)}, as a computational
1
are called for. New protocols try to simplify the process
: basis. Theunitarymatrices
v by making an educated guess about the nature of the state.
i Among other assumptions, this includes rank deficiency [8–
X σ =|0(cid:105)(cid:104)0|−|1(cid:105)(cid:104)1|, σ =|0(cid:105)(cid:104)1|+|1(cid:105)(cid:104)0|, (2.1)
z x
12],extrasymmetries[13–15],orGaussianity[16]. Whileall
r
these approaches are extremely efficient, their pitfall is that
a generate the Pauli group P , which consists of all the
whenthestartingguessisinaccurate,theyproducesignificant 1
Pauli matrices plus the identity, with multiplicative factors
systematicerrors.
±1,±i[19].
Here,wepursueadifferentapproach,inspiredbyanotion
ForN qubits,thecorrespondingHilbertspaceisthetensor
from statistical mechanics: coarse graining [17]. This oper-
productC2⊗···⊗C2=C2N.Acompactwayoflabelingboth
ation transforms a probability density in phase space into a
statesandelementsofthecorrespondingPauligroupP isby
“coarse-grained”densitythatisapiecewiseconstantfunction, N
usingthefinitefieldF . InAppendixAwebrieflysumma-
a result of density averaging in cells. This is the chief idea 2N
rizethebasicnotionsoffinitefieldsneededtoproceed.
behind the renormalization group [18], which allows a sys-
tematic investigation of the changes of a physical system as Let |ν(cid:105), ν ∈F2N, be an orthonormal basis in the Hilbert
viewedatdifferentscales. space C2N (henceforth, field elements will be denoted by
Inourcase,weconsiderasystemofqubitsandlookatthe Greek letters). The elements of the basis can be labeled by
associated phase space, which turns out to be a discrete grid powersofaprimitiveelementσ (i.e.,arootofanirreducible
of2N×2N points. Weassigntoeachsuitablydefinedlinein primitivepolynomial): {|0(cid:105),|σ(cid:105),...,|σ2N−1=1(cid:105)}. Nowthe
2
equivalentversionof(2.1)is[20–22] where f(x)=∑ x2i+2j,whichensuresthattheoper-
0≤j<i≤m−1
atorsdefinedinEq.(3.3)belowarerank-oneprojections.
Z =∑χ(αν)|ν(cid:105)(cid:104)ν|, X =∑|ν+β(cid:105)(cid:104)ν|, (2.2)
α β Onthephasespacegridonecanintroduceavarietyofge-
ν ν
ometrical structures with much the same properties as in the
sothat continuous case [27–29]. The simplest are the straight lines
passingthroughtheorigin(alsocalledrays),withequations
Z X =χ(αβ)X Z , (2.3)
α β β α
whichisthediscretecounterpartoftheWeyl-Heisenbergalge- α =0, or β =λα. (2.9)
braforcontinuousvariables[23]. Here,theadditivecharacter
χ is defined as χ(α)=exp[iπtr(α)] and the trace of a field The rays have a very remarkable property: the monomials
element(wedistinguishitfromthetraceofanoperatorbythe D(α,β)belongingtothesameraycommute,andthus,havea
lowercase“tr”)isdefinedinAppendixA.Moreover,Z and commonsystemofeigenvectors{|ψ (cid:105)},
α ν,λ
X arerelatedthroughthefiniteFouriertransform[24]
β
D(α,λα)|ψ (cid:105)=exp(iξ )|ψ (cid:105), (2.10)
ν,λ ν,λ ν,λ
1
F = √ ∑χ(νν(cid:48))|ν(cid:105)(cid:104)ν(cid:48)|, (2.4)
2N ν,ν(cid:48) where λ is fixed and exp(iξν,λ) is the corresponding eigen-
value,so|ψ (cid:105)=|ν(cid:105)areeigenstatesofZ (displacementop-
ν,0 α
sothatXα =FZαF†. eratorslabeledbytherayβ=0,whichwetakeasthehorizon-
The operators (2.2) generate the Pauli group PN of N talaxis). Theprojectionoperatorsassociatedwiththelinesof
qubitsand,withasuitablechoiceofbasis,theycanbefactor- equal slope are the projections onto these eigenvactors. In-
izedintoatensorproductofsingle-qubitPaulioperators. To deed,wehavethat
thisend,itisconvenienttoconsiderF asanN-dimensional
2N
linear space over Z . It is spanned by an abstract basis 1
{θ1,...,θN},sothat2givenafieldelementα theexpansion |(cid:104)ψν,λ|ψν(cid:48),λ(cid:48)(cid:105)|2=δλ,λ(cid:48)δν,ν(cid:48)+2N(1−δλ,λ(cid:48)), (2.11)
N
and, in consequence, they are mutually unbiased bases
α =∑a θ , a ∈Z , (2.5)
i i i 2 (MUBs)[30].
i=1
Now suppose for each ray we disregard the origin (0,0),
allowsustheidentificationα ⇔(a1,...,aN). Thebasis{θi} whosemonomialistheidentityoperator. Thisleavesuswith
canbechosentobeorthonormalwithrespecttothetraceop- 2N−1 commuting operators. If we then consider the whole
eration; i.e., tr(θiθj)=δij. This is a self-dual basis, which bundle of 2N +1 rays (which are obtained by varying the
alwaysexistforthecaseofqubits. Inthisway, weassociate “slope”λ overallofF ),wecanconstructacompletesetof
2N
each qubit with a particular element of the self-dual basis: MUBoperatorsarrangedina(2N−1)×(2N+1)table[31].
qubit ⇔θ. Usingthisbasis,wehavethefactorization
i i To round up the scenario, we need to represent states in
Z =σa1⊗···⊗σaN, X =σb1⊗···⊗σbN, (2.6) phasespace. ThediscreteWignerfunction[32]istheappro-
α z z β x x
priatetool. Itcanbeconsideredasaninvertiblemapping
wherea =tr(αθ)andb =tr(βθ)arethecorrespondingex-
i i i i
pansioncoefficientsforα andβ intheself-dualbasis. 1
W (α,β)= Tr[ρ∆(α,β)], (2.12)
Wenextrecall[25,26]thatthegriddefiningthephasespace ρ 2N
forNqubitscanbeappropriatelylabeledbythediscretepoints
(α,β),whicharepreciselytheindicesoftheoperatorsZ and sothat
α
X :α isthe“horizontal”axisandβ the“vertical”one. Inthis
β
gridwecanintroducethesetofdisplacements ρ = ∑∆(α,β)Wρ(α,β). (2.13)
α,β
D(α,β)=Φ(α,β)Z X , (2.7)
α β
Theoperationalkernelisdefinedas
where Φ(α,β) is a phase required to avoid plugging extra
factors when acting with D. A sensible choice for the case
1
ofqubitsisΦ2(α,β)=χ(αβ),whichensurestheHermitic- ∆(α,β)= ∑ χ(αα(cid:48)−ββ(cid:48))D(α(cid:48),β(cid:48)), (2.14)
2N
ity of the displacement operators. In addition, we impose α(cid:48),β(cid:48)
Φ(α,0)=1andΦ(0,β)=1,whichmeansthatthedisplace-
mentsalongthe“position”axisαandthe“momentum”axisβ which,inviewofequation(2.4),canbeinterpretedasadou-
arenotassociatedwithanyphase. Thesedisplacementopera- ble Fourier transform of D(α,β). One can check that this
torsshiftphasespacepoints,sotheactionofD(α(cid:48),β(cid:48))maps kernel has all the good properties [33]: it is Hermitian, nor-
(α,β)(cid:55)→(α+α(cid:48),β+β(cid:48)), justifying their designation. Note malizedandcovariantunderthePauligroup. Asaresult,for
that we still have to fix the sign of the phase Φ(α,β). We eachpointonthegrid,thecorrespondingvalueoftheWigner
choosethephaseas functioncanbecomputedfromtheprobabilitiesofmeasuring
the pure states associated with the lines passing through that
Φ(α,β)=itr(αβ)(−1)f(αβ), (2.8) point.
3
15 14
14 12
13 11
12 3
11 13
10 9
9 7
8 6
7 8
6 4
5 2
4 1
3 15
2 10
1 5
0 0
0 1 2 3 4 5 6 7 8 9101112131415 0 510151 2 4 8 6 7 9133111214
FIG.1. Graphicalsketchofcoarsegraining. Hereweconsiderdimension16,anditsdiagonalray,β =α. Thefirstpanelplotsallthelines
oftheformβ =α+γ,parametrizedbytheshiftγ. Pointsonthesamelinehavethesamecolour. Axislabelscorrespondtopowersofthe
primitiveelementofF ,withtheconventionthatσ0 isdenotedby0andσ15=1. Themiddlepanelshowstheoriginalgridwiththeaxis
16
labelspermutedsuchthatcosetelementsaregroupedtogether. Wecanseethatthisleadstodistinct4×4blockscontainingpointsofexactly
fourdifferentcolours.Theseareshownexpandedoutinthesmall,lowerfourgrids.Onenoticesthatthese“coarse”blocksformthediagonal
rayandallitstranslatesindimension4,whichweshowsuperimposedinthelastpanel.
III. COARSEGRAINING intercept. A large, coarse-grained line is denoted as |L(λ)(cid:105),
Cτ
wherenowtheinterceptisawholecoset.
AsheraldedintheIntroduction,ourgoalistotailorapro- Toeachlineinthefine-grainedphasespacewecanassign
cedure that allows us to coarse grain the phase space of a aprojector|(cid:96)(λ)(cid:105)(cid:104)(cid:96)(λ)|,constructedasalinearcombinationof
γ γ
multiqubit system; i.e., to break it down into simpler sub- thedisplacementoperators. Wechooseasourconventionfor
components. therays(γ =0)theall-positivesum
Tothisend,weconsiderthenumberNofqubitstobecom-
posite,i.e.N=mn.Let{µ0,...,µn−1}beabasisofF2mn with |(cid:96)(λ)(cid:105)(cid:104)(cid:96)(λ)|= 1 ∑D(α,λα). (3.3)
respecttoF2m. Wedefine 0 0 2mn α
(cid:40) (cid:41)
n−1 These lines are eigenstates with eigenvalue +1 for all dis-
C0= ∑τjµj | τj∈F2m , (3.1) placement operators in the sum. Projectors with nonzero in-
j=1 tercepts are obtained by conjugating that of the ray with an
i.e., the subspace made of linear combinations of basis ele- appropriatedisplacementoperator.
mentsµ1,...,µn−1withcoefficientsinthebasefieldF2m. We The coarse lines are produced by grouping together lines
canusethissetC ,whichwehenceforthrefertoastheinitial withinterceptsinthesamecoset:
0
coset,todecomposethefieldF2mn intocosets: |L(λ)(cid:105)(cid:104)L(λ)|= ∑ |(cid:96)(λ)(cid:105)(cid:104)(cid:96)(λ)|. (3.4)
Cτ =τµ0+C0, τ ∈F2m. (3.2) Cτ Cτ γ∈Cτ γ γ
The coarse-grained space will be labeled according to these Thepossiblechoicesofslopefortheselineswillbelimitedto
cosets. elementsofthesubfieldF2m,asthesehavenaturalanalogues
Wecanimaginetheprocessofcoarsegrainingaspartition- betweenthetwofields.
ing the grid F2mn×F2mn in such a way that we superimpose AsdiscussedinmoredetailinAppendixB,thecoarserays
a grid of size 2m×2m on top, with each superimposed point ofEq.(3.4)canbesimplifiedandrewrittenasthesumofdis-
indexed by cosets rather than field elements in the original placementoperators
grid. Each point in the coarse grid then contains a sub-grid
(cid:34) (cid:35)
the same size as Fn−1×Fn−1. To provide some intuition for 1
2m 2m |L(λ)(cid:105)(cid:104)L(λ)|= ∑ ∑ χ(γα) D(α,λα). (3.5)
this, we show a visual example of this process in action in C0 C0 2mn
Fig.1. λ γ∈C0
Ourprocedureforcoarse-grainingthegridarisesnaturally Onecancheckherethattheinnersumovertheelementsof
from consideration of the line structure of phase space. We C will cause some of the displacement operators to vanish.
0
willusethethinlinesinF2mn tocreatethicklinesinthecoarse The sum in brackets in Eq. (3.5) is either zero or a positive
phasespace,bygroupingtogetherlineshavingthesameslope, constant. Hence, the projection associated to the thick lines
and with intercepts in the same coset. We write thin lines in areasumoverasubsetofthedisplacementoperatorsassoci-
thebigfieldF2mn as|(cid:96)(γλ)(cid:105),whereλ istheslope,andγ isthe ated with the thin lines. This leads us to the key idea of our
4
work: rather than measuring all the displacement operators,
we measure only those which are present in the rays of the
coarse-grainedspace.
We note here that the choice of C is not unique, and will
0
ultimatelydeterminetheresultantsetofdisplacementopera-
tors. Forexample,aspecialcaseoccurswhenthedimension
ofthesystemissquare. Inthiscase,wecanconsidertherela-
tionshipbetweenthefieldsasaquadraticfieldextension,i.e.
whenn=2. InthiscasewecanpartitionF22m intoF2m×F2m.
Wecanthenchoosetheinitialcosetasthecopyofthesubfield
FIG.3.Resultantoperatorsfromcoarse-grainingadimension8sys-
F2m ⊂F22m: temdowntodimension2.(Leftpanel)Coarsegrainingusingtheba-
sis{1,σ,σ2}. Theresultantmeasurementsareunitarilyequivalent
C0={σi(2m+1), i=0,...,2m−1}, (3.6) toacasewheretwoofthequbitsremainuntouched. (Rightpanel)
Resultant operators when the coarse-graining uses the initial basis
whereσ isaprimitiveelementofF2mn andweusethenotation {σ,σ4,σ5}. Hereweobtaintheinterestingresultthatallresultant
σ0 for0. Thesubsequentcosetsareobtainedadditivelyfrom operatorscommute.
thissubfieldusingtherepresentativesτ =σ2m(i−1)+i.
i
Finally, the coarse-grained phase space inherits a coarse-
grainedWignerfunction. Acoarsekernelcanbeconstructed Another basis for F4/F2 is {1,σ}. Taking all scalar mul-
by grouping together kernel operators from the same coset, tiples of µ1 =σ from the prime field gives us C0 ={0,σ}.
i.e. WethenobtainC1=1+C0={1,σ2}. Foreachray,wecan
list the operators which survive in the inner sum over C in
0
D(C ,C )= ∑ ∑ ∆(α,β). (3.7) Eq. (3.5). Moreover, we can label the points of the coarse-
τ ξ
α∈Cτβ∈Cξ grained grids by those displacement operators. Disregarding
the identity operator, the resulting set {11X,11Z,11Y} consti-
Desired properties of a Wigner function all follow from the tutes the appropriate measurements to be performed to de-
originalkernel. Aswasthecasewiththedisplacementoper- termine which coarse-grained line they are in. They are es-
ators,differingchoicesofthesubsetC willleadtodiffering sentiallyPauli measurementson one ofthe twoqubits inthe
0
Wignerfunctions. system.
Alternatively,thedimensionisasquare,sowecanchoose
as our initial coset the subfield F : C ={0,1}. This yields
2 0
IV. EXAMPLES the secondcoset C ={σ,σ2}. Weonce againcompute the
σ
surviving operators using Eq. (3.5). The final result now is
{XX,YY,ZZ}. Here, we see that we are making a measure-
We illustrate the previous ideas with some relevant exam-
mentwiththesamePaulioperatoronbothqubits,therebycap-
ples. WehavewrittenaPythonsoftwarepackagecapableof
turing the full correlations between the two qubits. Figure 2
generatingallthefollowingresults,whichwemakeavailable
showsbothpartitioningmethodssidebyside.
online[34]
Thefirstnontrivialinstancewecanhaveisthecaseoftwo Ournextexampleisthecaseofdimension8. Wechooseσ
qubits,sodimension4. Usingtheirreducibleprimitivepoly- arootoftheirreducibleprimitivepolynomialx3+x+1=0,
nomialx2+x+1=0,wehavethatF ={0,1,σ,σ2=σ+1}.
4
Theself-dualbasisis{σ,σ+1},andweuseittoproducethe
displacementoperators.
FIG. 4. Resultant operators from coarse-graining a dimension 16
systemdowntodimension4. Theleftpanelcontainsthesurviving
operatorsfromthegeneralbasismethod,therightpanelfromchoos-
ingthesubfieldasC . ThecosetsarelistedinEqs.(4.4)and(4.5)
0
FIG.2.Resultantoperatorsfromcoarse-grainingadimension4sys- respectively.Inthecaseoftheleftpanel,theseoperatorsareunitarily
temdowntodimension2.Coloursareindicativeofparticularcoarse equivalenttoasetwheretwoqubitsareuntouchedandthe2-qubit
rays. TheleftimagecoarsegrainsbytakingC0={0,σ}, whereas MUBoperatorsareappliedtotherest. Therightpanelhasnosuch
thelowerimageusesthesubfieldC0={0,1}. transformation.
5
0.25 0.25
0.04 0.20 0.20
0.02 0.15 0.15
0.00 00..0150 00..0150
Fourier basis Computational− b0a.s0is2 Fourier basis Computational 0ba.0s0is Fourier basis Computational 0ba.0s0is
00..0024 0.10 00..12
Fourier basis Computational− −b000a..s.000i0s42 Fourier basis Computational 0b0a..0s00i5s Fourier basis Computational −b−a000s...i021s
FIG.5.(Top)Coarse-grainedWignerfunctionforthestate 1(|00(cid:105)+|11(cid:105))⊗(|00(cid:105)+|11(cid:105)).(Left)TheoriginalWignerfunctionindimension16.
2
Thex-axisrepresentsthecomputationalbasis,inthestandardordering|0000(cid:105),|0001(cid:105),|0010(cid:105),etc. TheFourierbasis,asdefinedviaEq.(2.4),
isonthey-axisandissimilarlyordered. (Centre)CoarsegrainingoverF4withthepolynomialbasis{1,σ}. Heretheaxesarenotlabeledby
singlestates,butratherbyasetofstatesassociatedwitheachcoset. (Right)Coarsegrainingwiththesubfieldastheinitialcoset. (Bottom)
Thesamecoarsegrainingprocedureasabove,butappliedtothestate 1(|0001(cid:105)+|0010(cid:105)+|0100(cid:105)+|1000(cid:105)).
2
andobtainaself-dualbasis{σ3,σ5,σ6}. Anobviouschoice x+σ(cid:48) overF wherewedenoteaprimitiveelementofF as
4 4
forabasisofF /F isapolynomialbasis{1,σ,σ2}. Tocon- σ(cid:48). We know from Eq. (3.6) that σ(cid:48) =σ5, where σ is the
8 2
structC ,wemusttakeallpossiblelinearcombinationsofσ primitive element in F . Then F in F can be written as
0 16 4 16
andσ2withcoefficientsinF . Thisproduces {0,σ5,σ10,σ15=1}.
2
Forthegeneralcase,wechoosethebasis{1,σ}.Takingall
C0={0,σ,σ2,σ4}. (4.1) F4-multiples of σ, we obtain C0 ={0,σ,σ6,σ11}. The full
setofcosetsis:
Weobtainthesecondcosetbyaddingtheremainingsubfield
element1toC0: C0={0,σ,σ6,σ11}, Cσ5 ={σ5,σ2,σ9,σ3},
C ={σ10,σ8,σ7,σ14}, C ={1,σ4,σ13,σ12}.
σ10 1
C ={1,σ3,σ5,σ6}. (4.2) (4.4)
1
Proceedinginthestandardway,andtakingintoaccountthat
The traces of all elements in C0 are 0, and the traces for all aself-dualbasisis{σ3,σ7,σ12,σ13},weobtaintheoperators
elementsinC1 are1. Thesurvivingfouroperatorsareshown inFig. 4. Whatis(un)interestingabouttheseoperatorsisthat
inFig.3. we can transform them all into operators which completely
Using a Clifford transformation, we can “trace out” two ignore two of the qubits. In particular, consider the follow-
of the qubits. The sequence of CNOT gates: CNOT12 ingsequenceofoperations: CNOT43 –CNOT32 –CNOT31 –
– CNOT13 – CNOT21 – CNOT31 transforms the set into CNOT14 – CNOT24. Application of this to the operators of
{X1111,Z1111,Y1111}, so we see that this partitioning is, after thefirstpanelofFig.4yieldsanewsetofoperatorswherethe
a global change of basis, equivalent to measuring each Pauli lasttwoqubitscontainonly11,andthefirsttwoqubitscontain
ononlyasinglequbit. thefullsetofMUBoperatorsontwoqubits.
If we choose instead the basis {σ,σ4,σ5} to build our Alternatively, we can choose our initial coset as the sub-
cosets,wegetamoreinterestingresult: field, andthecosetrepresentativesasτ =σ4(i−1)+i. Weob-
i
tainthecosets
C ={0,1,σ4,σ5}, C ={σ,σ2,σ3,σ6}. (4.3)
0 σ
C ={0,1,σ5,σ10}, C ={σ,σ4,σ2,σ8},
0 σ
The operators that survive have the form Z X , α,β ∈ C ={σ6,σ13,σ9,σ7}, C ={σ11,σ12,σ3,σ14}.
α β σ6 σ11
{0,σ4},yieldingtheoperatorsinFig.3,whichallcommute. (4.5)
In this case, we are already ignoring one of the three qubits. Using Eq. (3.5) we get the table shown in the right panel of
However,itisnotpossibletofindaCliffordwhichwilltrace Fig.4. Unlikeinthepreviouscase,thereisnotransformation
outaremainingoneaswasthecasewiththepolynomialbasis whichwillleadtous‘tracingout’twoofthequbits.However,
case. So, in a sense, using this partitioning we are ignoring wecanbringtheseoperatorsintoamorebasicformbyapply-
fewerqubitsthanbefore. ingthesequenceCNOT13–CNOT24. Theresultantoperators
Dimension 16 is perhaps the first really interesting case. havethepropertythatonthefirsttwoqubits,weonlyhaveX,
Firstofall, wecanconsideritintwoways: m=1,n=4, or andonthelasttwoqubitsonlyZ,sothattheyallcommute.
m=2,n=2. Essentially,todothepartitioning,wecanlook Toconclude,wepresentsomeofthecoarse-grainedWigner
atF asaquarticextensionoverF ,oraquadraticextension functions we obtain using our method. Those in dimensions
16 2
over F . We consider the quadratic case, so we can coarse 4 and 8 are somewhat trivial, so we focus on dimension 16.
4
grain in two ways. We work with F as constructed by the Wignerfunctionsforthestates 1(|00(cid:105)+|11(cid:105))(|00(cid:105)+|11(cid:105))and
16 2
irreducibleprimitivepolynomialx4+x+1overF ,andx2+ 1(|0001(cid:105)+|0010(cid:105)+|0100(cid:105)+|1000(cid:105))arepresentedinFig.5.
2 2
6
RecallinSectionIIthatwecouldassociatetheelementsof A commutative ring is a nonempty set R with two binary
F with a basis in our Hilbert space. Then, in the coarse operations, called addition and multiplication, such that it is
2N
Wigner functions, when we group the field elements into an Abelian group with respect to addition, and the multipli-
cosets,wecanconsiderthisalsoasgroupingtogethertheas- cationisassociative. Themosttypicalexampleistheringof
sociatedbasisstates. Hence,theprobabilitiesintheseWigner integersZ,withthestandardsumandmultiplication. Onthe
functions become distributed over the cosets which contain otherhand,thesimplestexampleofafiniteringisthesetZ
n
theconstituentbasisstatesofourtargetstate. Asaresult,the ofintegersmodulon,whichhasexactlynelements.
coarseWignerfunctionsresemble‘smoother’versionsofthe A field F is a commutative ring with division, i.e., such
originalonetovaryingdegrees. that 0 does not equal 1 and all elements of F except 0 have
a multiplicative inverse (note that 0 and 1 here stand for the
identity elements for the addition and multiplication, respec-
V. CONCLUSIONS tively,whichmaydifferfromthefamiliarrealnumbers0and
1). Elements of a field form Abelian groups with respect to
addition and multiplication (in this latter case, the zero ele-
Compared to the continuous Wigner function, the discrete
mentisexcluded). NotethatthefiniteringZ isafieldifand
Wignerfunctionisanadolescentformulation,slowlydevelop- d
onlyifd isaprimenumber.
ingintoadultmaturity. Discretephasespaceimposesseveral
The characteristic of a finite field is the smallest positive
new challenges, which leads to an intricate mapping of the
integerd suchthat
Wignerfunction.
Ourcoarsegrainingprocedureshowsawaytofacilitateour
1+1+...+1=0 (A1)
understandingwhenthenumberofqubitsishigh. Whileitis (cid:124) (cid:123)(cid:122) (cid:125)
alwayspossibletoignorepartofthesystemandtodetermine dtimes
thefullWignerfunctionoftheresultingreduceddensityma-
and it is always a prime number. Any finite field contains a
trix,ourapproachallowsmorechoicesregardingwhichinfor- prime subfield Z and has dn elements, where n is a natural
d
mationofthewholesystemismeasured. Inanotherextremal
number. Moreover, the finite field containing dn elements is
case,thecoarse-grainedWignerfunctioniscompletelydeter- uniqueuptoisomorphismandiscalledtheGaloisfieldFdn.
minedbyasetofcommutingoperatorsthatcanbemeasured
We denote as Z [x] the ring of polynomials with coeffi-
simultaneously. d
cients in Z . If P(x) is an irreducible polynomial of degree
d
However, several open questions remain. An obvious n(thatis,onethatcannotbefactorizedoverZ ),thequotient
d
next step would be to extend the coarse graining procedure spaceZd[X]/P(x)providesanadequaterepresentationofFdn.
to multi-qudit systems. Furthermore, knowing the coarse-
Its elements can be written as polynomials that are defined
grainedfunction,doesthereexistanothersubsetofmeasure-
modulo the irreducible polynomial P(x). The multiplicative
ments which will allow us to zoom in on specific areas of it group of Fdn is cyclic and its generator is called a primitive
andgainmoreinformation? Alogicalfirstchoicewouldbeto
elementofthefield.
extendthesetofmeasurementssuchthattheyincludeallop-
As a trivial example of a nonprime field, we consider the
eratorsthatcorrespondtoslopesinthesubfield. Forexample, polynomialx2+x+1=0,whichisirreducibleoverZ .Ifσ is
inthedimension16case,wewouldmeasurealloperatorsfor 2
arootofthispolynomial,theelements{0,1,σ,σ2=σ+1=
theraysα=0andβ =λα,λ ∈{0,σ5,σ10,σ15},ratherthan σ−1}formthefinitefieldF andσ isaprimitiveelement.
justthreefromeach. Thisstrategywouldallowustooptimize 22
Abasicmapisthetrace
measurementsinaverysubtleway. Workalongtheselinesis
inprogress.
tr(α)=α+αd+...+αdn−1. (A2)
The image of the trace is always in the prime field Z and
d
VI. ACKNOWLEDGMENTS
satisfies
O.D.M.isfundedbyCanada’sNSERC.Sheisalsograteful tr(α+α(cid:48))=tr(α)+tr(α(cid:48)). (A3)
for hospitality at the MPL. IQC is supported in part by the
GovernmentofCanadaandtheProvinceofOntario. L.L.S.S. Intermsofitwedefineanadditivecharacteras
acknowledges financial support from the Spanish MINECO
(cid:20) (cid:21)
2πi
(GrantNo. FIS2015-67963-P). χ(α)=exp tr(α) , (A4)
d
whichpossessestwoimportantproperties:
AppendixA:Finitefields
χ(α+α(cid:48))=χ(α)χ(α(cid:48)), ∑ χ(αα(cid:48))=dnδ .
0,α
Inthisappendixwebrieflyrecallsomebackgroundneeded α(cid:48)∈Fdn
for this paper. The reader interested in more mathematical (A5)
details is referred, e.g., to the excellent monograph by Lidl Any finite field Fdn can be also considered as an n-
andNiederreiter[35]. dimensionallinearvectorspaceoveritsprimefieldF . Given
d
7
abasis{θ },(j=1,...,n)inthisvectorspace,anyfieldele- projectorsfortherays,
j
mentcanberepresentedas
1 1
|(cid:96)(λ)(cid:105)(cid:104)(cid:96)(λ)|= ∑D(α,λα)= ∑Φ(α,λα)Z X .
n 0 0 2mn 2mn α λα
α α
α = ∑ajθj, (A6) (B1)
j=1 AsmentionedinSec.II,theprojectorsfortheshiftedlinescan
beobtainedbyapplyinganappropriatedisplacementoperator
withaj ∈Zd. Inthisway, wemapeachelementofFdn onto toinduceatransformation. Letusignorefornowtheraywith
anorderedsetofnaturalnumbersα ⇔(a1,...,an). infiniteslope,α=0.Thenfortherestoftherays,wecanshift
Twobases{θ1,...,θn}and{θ1(cid:48),...,θn(cid:48)}aredualwhen themverticallybyapplyingthedisplacementoperatorsofthe
formD(0,γ):
tr(θ θ(cid:48))=δ . (A7)
k l k,l 1
|(cid:96)(λ)(cid:105)(cid:104)(cid:96)(λ)|= ∑D(0,γ)D(α,λα)D†(0,γ)
γ γ 2mn
A basis that is dual to itself is called self-dual. A self-dual α
basisexistsifandonlyifeitherd isevenorbothnandd are = 1 ∑Φ(α,λα)X Z X X , (B2)
odd. 2mn γ α λα γ
α
There are several natural bases in Fdn. One is the polyno- wherewerecalltheconventionthatallthephasesΦ(0,γ)=1.
mialbasis,definedas
Here,wecanmakefurtheruseofthecommutationrelation
inEq.(2.3). Weobtain
{1,σ,σ2,...,σn−1}, (A8)
1
|(cid:96)(λ)(cid:105)(cid:104)(cid:96)(λ)|= ∑Φ(α,λα)χ(γα)Z X
where σ is a primitive element. An alternative is a normal γ γ 2mn α λα
α
basis,constitutedof 1
= ∑χ(γα)D(α,λα). (B3)
2mn
{σ,σd,...,σdn−1}. (A9) Itisthenstraightforwardtoαseethatthethickrays,whichare
obtainedbysummingoverallinterceptsγ incosetC ,canbe
0
Theappropriatechoiceofbasisdependsonthespecificprob- writtenas
lem at hand. For example, in F22 the elements {σ,σ2} are (cid:34) (cid:35)
bothrootsoftheirreduciblepolynomial. Thepolynomialba- |L(λ)(cid:105)(cid:104)L(λ)|= 1 ∑ ∑ χ(γα) D(α,λα). (B4)
sis is {1,σ} and its dual is {σ2,1}, while the normal basis C0 C0 2mn
{σ,σ2}isself-dual. λ γ∈C0
Finally,wementionthatfortheinfiniteslopetheanalysispro-
ceedsinexactlythesameway,butthatthelinesaretranslated
by displacement operators of the form D(γ,0) and Eq. (2.3)
AppendixB:Derivationofequationforlineoperators
givesusχ(γβ)instead.
Only those operators which have a non-zero term in the
Herewepresentthederivationofourequationforthesur- sum will contribute, thus we consider them as the effective
viving displacement operators. We begin by considering the displacementoperatorsinthecoarsephasespace.
[1] I. Bloch, J. Dalibard, and W. Zwerger, “Many-body physics [8] D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, and J. Eis-
withultracoldgases,”Rev.Mod.Phys.80,885–964(2008). ert,“Quantumstatetomographyviacompressedsensing,”Phys.
[2] R. Blatt and C. F. Roos, “Quantum simulations with trapped Rev.Lett.105,150401(2010).
ions,”Nat.Phys.8,277–284(2012). [9] M.Cramer,M.B.Plenio,S.T.Flammia,R.Somma,D.Gross,
[3] M.G.A.ParisandJ.Rˇeha´cˇek,eds.,QuantumStateEstimation, S.D.Bartlett,O.Landon-Cardinal,D.Poulin, andY.K.Liu,
Lect.Not.Phys.,Vol.649(Springer,Berlin,2004). “Efficientquantumstatetomography,”Nat.Commun.1,149EP
[4] V.Buzˇek,R.Derka,G.Adam, andP.L.Knight,“Reconstruc- (2010).
tionofquantumstatesofspinsystems:FromquantumBayesian [10] S. T. Flammia, D. Gross, Y.-K. Liu, and J. Eisert, “Quan-
inference to quantum tomography,” Ann. Phys. 266, 454–496 tumtomographyviacompressedsensing:errorbounds,sample
(1998). complexityandefficientestimators,”NewJ.Phys.14,095022
[5] R. Schack, T. A. Brun, and C. M. Caves, “Quantum Bayes (2012).
rule,”Phys.Rev.A64,014305(2001). [11] O.Landon-CardinalandD.Poulin,“Practicallearningmethod
[6] F.Husza´randN.M.T.Houlsby,“AdaptiveBayesianquantum for multi-scale entangled states,” New J. Phys. 14, 085004
tomography,”Phys.Rev.A85,052120–(2012). (2012).
[7] ChristopherGranade,JoshuaCombes, andDGCory,“Practi- [12] T.Baumgratz,D.Gross,M.Cramer, andM.B.Plenio,“Scal-
calBayesiantomography,”NewJ.Phys.18,033024(2016). ablereconstructionofdensitymatrices,”Phys.Rev.Lett.111,
8
020401(2013). [24] A.B.Klimov, L.L.Sa´nchez-Soto, andH.deGuise,“Multi-
[13] G.To´th,W.Wieczorek,D.Gross,R.Krischek,C.Schwemmer, complementaryoperatorsviafiniteFouriertransform,”J.Phys.
andH.Weinfurter,“Permutationallyinvariantquantumtomog- A38,2747–2760(2005).
raphy,”Phys.Rev.Lett.105,250403(2010). [25] W.K.Wootters,“Picturingqubitsinphasespace,”IBMJ.Res.
[14] T.Moroder,P.Hyllus,G.To´th,C.Schwemmer,A.Niggebaum, Dev.48,99–110(2004).
S.Gaile,O.Gu¨hne, andH.Weinfurter,“Permutationallyinvari- [26] K.S.Gibbons,M.J.Hoffman, andW.K.Wootters,“Discrete
antstatereconstruction,”NewJ.Phys.14,105001(2012). phase space based on finite fields,” Phys. Rev. A 70, 062101
[15] A. B. Klimov, G. Bjo¨rk, and L. L. Sa´nchez-Soto, “Opti- (2004).
malquantumtomographyofpermutationallyinvariantqubits,” [27] A.B.Klimov,J.L.Romero,G.Bjo¨rk, andL.L.Sa´nchez-Soto,
Phys.Rev.A87,012109(2013). “Geometricalapproachtomutuallyunbiasedbases,”J.Phys.A
[16] J. Rˇeha´cˇek, S. Olivares, D. Mogilevtsev, Z. Hradil, M. G. A. 40,3987–3998(2007).
Paris,S.Fornaro,V.D’Auria,A.Porzio, andS.Solimeno,“Ef- [28] A.B.Klimov,J.L.Romero,G.Bjo¨rk, andL.L.Sa´nchez-Soto,
fectivemethodtoestimatemultidimensionalGaussianstates,” “Discrete phase-space structure of n-qubit mutually unbiased
Phys.Rev.A79,032111(2009). bases,”Ann.Phys.324,53–72(2009).
[17] P.Castiglione,M.Falcioni,A.Lesne, andA.Vulpiani,Chaos [29] C.Mun˜oz,A.B.Klimov, andLLSa´nchez-Soto,“Symmetric
andCoarseGraininginStatisticalMechanics(CambridgeUni- discrete coherent states for n-qubits,” J. Phys. A 45, 244014
versityPress,Cambridge,2008). (2012).
[18] S. R. White, “Density matrix formulation for quantum renor- [30] W.K.WoottersandB.D.Fields,“Optimalstate-determination
malizationgroups,”Phys.Rev.Lett.69,2863–2866(1992). bymutuallyunbiasedmeasurements,”Ann.Phys.191,363–381
[19] I.ChuangandM.Nielsen,QuantumComputationandQuan- (1989).
tum Information (Cambridge University Press, Cambridge, [31] S. Bandyopadhyay, P. O. Boykin, V. Roychowdhury, and
2000). F.Vatan,“Anewprooffortheexistenceofmutuallyunbiased
[20] M.Grassl,M.Ro¨tteler, andT.Beth,“Efficientquantumcircuits bases,”Algorithmica34,512–528(2002).
for non-qubit quantum error-correction codes,” Int. J. Found. [32] G.Bjo¨rk,A.B.Klimov, andL.L.Sa´nchez-Soto,“Thediscrete
Comput.Sci.14,757–775(2003). Wignerfunction,”Prog.Opt.51,469–516(2008).
[21] A.Vourdas,“QuantumsystemswithfiniteHilbertspace,”Rep. [33] R.L.Stratonovich,“Ondistributionsinrepresentationspace,”
Prog.Phys.67,267–320(2004). Sov.Phys.JETP31,1012–1020(1956).
[22] A.Vourdas,“QuantumsystemswithfiniteHilbertspace: Ga- [34] https://github.com/glassnotes/balthasar.
loisfieldsinquantummechanics,”J.Phys.A40,R285–R331 [35] R. Lidl and H. Niederreiter, Introduction to Finite Fields and
(2007). their Applications (Cambridge University Press, Cambridge,
[23] E. Binz and S. Pods, The Geometry of Heisenberg Groups 1986).
(AmericanMathematicalSociety,Providence,2008).