Table Of ContentCasimir Invariants for Systems Undergoing Collective Motion
C. Allen Bishop1, Mark S. Byrd1,2, and Lian-Ao Wu3,4
1Physics Department, Southern Illinois University, Carbondale, Illinois 62901-4401
2Computer Science Department, Southern Illinois University, Carbondale, Illinois 62901
3Department of Theoretical Physics and History of Science,
The Basque Country University (EHU/UPV), P.O. Box 644, 48080 Bilbao, Spain and
4IKERBASQUE–Basque Foundation for Science, 48011, Bilbao, Spain
(Dated: January 20, 2010)
Dicke states are states of a collection of particles which have been under active investigation for
several reasons. One reason is that the decay rates of these states can be quite different from a
set of independentlyevolvingparticles. Anotherreason isthat aparticular class of thesestates are
0 decoherence-freeornoiselesswithrespecttoasetoferrors. Thesenoiselessstates,ormoregenerally
1 subsystems, can avoid certain types of errors in quantum information processing devices. Here we
0 provide a method for calculating invariants of systems of particles undergoing collective motions.
2 These invariants can be used to determine a complete set of commuting observables for a class of
Dickestatesaswellasidentifypossiblelogicaloperationsfordecoherence-free/noiselesssubsystems.
n
Ourmethodisquitegeneralandprovidesresultsforcaseswheretheconstituentparticleshavemore
a
J than two internal states.
0
PACSnumbers: 03.67.Pp,03.65.Yz,11.30.-j,34.80.Pa
2
] I. INTRODUCTION structure. In the physical systems considered by Dicke,
h
p one could imagine evolutions of the states which do not
- changethe essentialfeaturesofthe state(energyortotal
t Decoherence-free/noiseless subsystems (DFS) are now
n angular momentum quantum numbers), but are indeed
part of an arsenal of weapons used to prevent errors in
a nontrivialevolutions. Inthecaseofquantuminformation
u quantum information processing and storage [1–6]. (For processing, these enable quantum computing in a DFS.
q reviews see [7, 8].) DFS are subsystems which are im-
[ mune to certain types of errors. The most common type In both the early analysis of Dicke states and also
found in the literature is a DFS which is immune to quantum computing applications, primarily only two in-
1
collective errors. These types of quantum systems were ternal states of the constituents were considered. How-
v
8 studied earlier by Dicke in a different context [9]. ever, three or more internal states of an atom could
7 There are several types of states which are now called certainly become important in various experiments and
5 could also arise in particle physics where more than two
Dicke states. One such set corresponds to a set of parti-
3 degrees of freedom are associated with both flavor and
cles which undergo a collective motion, are distinguish-
. color symmetries. Recent experiments [11–13] and pro-
1 able, and do not interact with each other. These states
0 posed experiments [14–16] have provided explicit con-
are unchanged by particle interchange, or more gener-
0 structions for these so-calledDicke states using a variety
ally, the interchange of particular constituents [10]. One
1 of physical systems.
particularly clear example is a gas interacting with an
:
v externalfieldwhichhasawavelengthsignificantlylonger Here we carry the consideration of Dicke states to the
Xi thanthecontainerconfiningtheparticles. Thesearealso extreme. We consider collections of particles undergo-
conditionsforcollectivemotion,i.e.,theexternalfieldin- ing some collective motions, for example collective er-
r
a teracts in the same way with each particle. In this case, rors,andaskthefollowingquestion. WhatHamiltonians
if the size of the container R and the wavelength of give rise to evolutions which are compatible with these
∼
the field is λ, then the “Dicke limit” λ R is said to be motions? Ourresultsarenotrestrictedtoanyparticular
≫
satisfied. In this limit, when the external influence gives numberofinternalstatesforeachoftheconstituents,nor
rise to errors in a quantum computing device, the errors are they restricted to any number of particles. We then
are called collective, whether they describe an evolution answerthequestionbyusingaconstructionofinvariants
of each particle which is unitary or not. analogous to Casimir’s construction of invariants for Lie
algebras and Lie groups.
Sinceerrorsarethegreatestobstacletobuildingafully
functionalquantumcomputingdevice,anymethodwhich InSectionIIwereviewthestandardCasimirconstruc-
aids in the preventionof errorsis quite important. How- tion for single-particle invariants. In Section III we ex-
ever, for the practical use of a DFS/NS for quantum in- tend the construction to sets of N particles each with
formation processing one requires the ability to perform d internal states. Section IV discusses the physical im-
universal computing on these states. This requires find- plications of our results. In particular, we discuss the
ing evolutions which do not take the states out of the use of these invariants for Dicke state identification as
protected subspace during gating operations [5]. We re- well as the manipulation of decoherence-free or noiseless
fer to suchoperationsas being compatible with the DFS subsystem. Section V concludes.
2
II. IDENTIFYING INVARIANTS whichcommute with the setofcollective motions. A ba-
sis for these motions is given by the set of operators of
the form
A Casimir Operator is a member of the center of the
universal enveloping algebra meaning such an operator S = λ(α), (II.6)
j j
will commute with every element of the universal en-
α
velopingalgebra. Formatrixrepresentationsofquantum X
where the sum is taken over the particles in the system.
evolutions,whichwewillconsiderhere,the universalen-
These types of operatorsalso form a basis for the collec-
velopingalgebraisthealgebraofallproductsofLiealge-
tive errors acting on a DFS/NS and linear combinations
brabasiselements. Itismostimportantforourpurposes
give the stabilizer elements. (See Sec. IVB for the defi-
that the Casimir operators commute with every genera-
nition and discussion.) An element of the algebra (with
tor of the Lie algebra and the collective errors form a
realcoefficients)whichcommuteswiththeseprovidesthe
representationofthe Lie algebra(which is the algebraof
Hamiltonians which are compatible with a DFS/NS.
Hermitian matrices). Once we find such invariants, we
will have the set of Hamiltonians which commute with
collectiveerrorsandarethereforecompatibletransforma-
III. EXPLICIT FORMS FOR THE INVARIANTS
tions. WebeginbyreviewingtheconstructionofCasimir
invariants.
In this section we will find a set of independent oper-
Let a basis for the Lie algebra of SU(d) (hereafter de-
ations for which each element of the set commutes with
noted (SU(d))) begivenbyaset λ withthe normal-
i
L { } allmembersofthealgebraformedbytheS . Denotethe
ization and properties described in the Appendix. The j
algebra of the S by .
Casimir operators of SU(d) are known. The most famil- j A
Note that the Casimir operators formed from the ele-
iar, the quadratic Casimir, is proportional to the sum of
ments S form a representation of (SU(d)) if the λ do
the squares of the elements, j L i
[19]. Therefore these are invariants of the algebra , i.e.
A
they commute with elements of this algebra. However,
C λ λ . (II.1)
2 ∝ i i this is not an irreducible algebra. Thus the construction
i
X must rely on the identification of the irreducible compo-
nents.
This along with all other Casimir operators can be ob-
To proceed, we first calculate the Casimir invariants
tained using the formula [17, 18]
of (SU(d)). Then, noting that linear combinations of
L
I =Tr(ad ad ad )λ λ ...λ . (II.2) these invariants are also invariants, we extract reducible
n λa1 ◦ λa2 ◦···◦ λan a1 a2 an components of the invariants. From a physical perspec-
For example, tive,thismeansidentifyingn-bodyinteractionswhichare
contained within the m-body interactions where n m.
≤
The quadratic Casimir operator for the algebra is
C2 = fa1,b1,b2fa2,b2,b1λa1λa2, (II.3) A
a1,aX2,b1,b2 J2 = fijkfkliSjSl SjSj. (III.1)
∝
i,j,k,l j
which reduces to Eq. (II.1) using Eq. (A.7). It turns X X
outthattheformulagiveninEq.(II.2)doesnotproduce Expanding this in terms of the basis elements λi gives
{ }
independentinvariantsforthecollectiveerrors. However,
2
theindependentinvariantscanbe obtained[17]andmay (α)
J λ
be written in terms of the totally symmetric d-tensor. 2 ∝ i !
i α
For example, the cubic Casimir invariant is X X
= (λ(α))2+2 λ(α)λ(β) . (III.2)
C3 = dijkλiλjλk. (II.4) i i i
i α α<β
ijk X X X
X
Note that the firstterm of the lastexpressionis the sum
HigherorderCasimiroperatorscanbeconstructedusing ofsingle-particleCasimirinvariants. Thisallowsustoin-
the general formulation ferthatthesecondterminEq.(III.2)isalsoaninvariant
quantity. Furthermore, the only nontrivial contributions
Cn = di1,i2,i3di3,i4,i5,...din−4,in−3,in−2 appearing in the commutator iλ(iα)λi(β),Sl have the
i1,iX2,...,in form h i
d λ λ λ ...λ λ .(II.5) P
× in−2,in−1,in i1 i2 i4 in−1 in (α) (α) (β) (α) (β) (β)
[λ ,λ ]λ +λ [λ ,λ ], (III.3)
i j i i i j
The sum is over all elements of the algebra.
with all other terms vanishing. Since this can be rewrit-
To show independence, one may begin with Eq. (II.2)
ten as
andreducetheexpressionsusingthe identitiesintheap-
pendix. Here our objective is to find a set of operators 2if (λ(α)λ(β) λ(α)λ(β))=0, (III.4)
ijk k i − k i
3
we find that functions ofseparateparticlesto requiresymmetrization
ofthe states. Itwasalsoassumedthateachparticlecou-
I2(α,β) = λ(iα)λ(iβ) (III.5) pled to the common radiationfield via an electric dipole
i interaction. Ingeneral,the interactionenergyof the αth
X
particle with the field can be written as
is also an independent invariant for each pair (α,β).
Now consider
HI(α) =−A(rα)·(e1σx(α)+e2σy(α)), (IV.1)
J = f f f S S S
3 ijk klm mni j l n for some constant real vectors e1 and e2.
X In the caseof a gas confinedto a smallregionof space
=Xfijkfklmfmni Xα λ(jα)!Xβ λ(lβ) Xγ λn(γ)!. tdheepevnedcetonrt pfuontcetnitoinalocfatnheeffspecattiivaellcyoboredcinoantseidserrαe.dIannthinis-
(III.6) approximationthe total interaction energy becomes
Expanding the sums over the particle (Greek) indices, H =c σ(α)+c σ(α), (IV.2)
I 1 x 2 y
and reducing the results, three types of terms are ob- α α
X X
tained. First, if all three superscripts are the same, for
example λ(α)λ(α)λ(α), the term reduces to the quadratic wherec1 andc2 denoteconstants. Therearetwodegrees
i j k of freedom associated with the internal energy of any
Casimir invariant for particle α. Since any linear com-
given particle. The energy eigenvalues of the jth parti-
bination of invariants is invariant, the sum of all terms
having this form is also invariant. Second, if two are cle,correspondingto the diagonaloperatorσz(α), takeon
the same, e.g. λ(iα)λ(jα)λ(kβ), then the result reduces to tghiees,vtaolugeesth±er~ωw/it2h. tThheetrsaunmslaotfioanllailnetneernrgayl poafrtthieclgeaesnHer-
I2(α,β), thus terms of this form are also invariant quanti- and the interaction with the field, provides a complete0
ties. Third, if all three are different, we obtain description of a gaseous system consisting of mutually
noninteracting particles.
I(α,β,γ) = f λ(α)λ(β)λ(γ), (III.7)
3 ijk i j k TheHamiltonianforthissystemcanbebrokenupinto
Xijk two parts,
as an independent invariant. Notice this case is different
from the ordinary Casimir construction where no such H =H + c σ(α)+c σ(α)+~ω/2 σ(α) ,
0 1 x 2 y z
independentinvariantarisesforatermofthe formofJ3. α α α !
DefiningandexpandingJ producesonenewinvariant, X X X (IV.3)
4
where the first part describes the translational energy of
(α,β,γ) (α) (β) (γ)
I4 = dijkλi λj λk . (III.8) the system and thus depends solely on the spatial posi-
Xijk tions rα while the second is a quantity independent of
these coordinates. As a result, these two parts commute
Continuing with this will iteratively produce a set of in-
implying the existence of simultaneous eigenfunctions of
dependent invariants for collective motions of particles.
the two contributions. Let us denote these energy eigen-
For three qutrits this set, I ,I ,I is complete [20].
2 3 4 states
ψ =U (r ,r ,...,r )Φ , (IV.4)
pq p 1 2 N q
IV. PHYSICAL IMPLICATIONS
where U depends on the spatial coordinates and Φ
p q
After the motivation in the introduction and the con- is a function of the internal coordinates. The oper-
structionof the invariants,we now consider more explic- ators S = σ(α) (i = x,y,z) not only individu-
i α i
itly the implications of our findings. ally commute with the spatially independent quantity
S2 = S2 +SP2 +S2, but also satisfy the same commu-
x y z
tation relations (up to a multiplicative scaling factor) as
A. Motion of Dicke States
the three components of angular momentum. In other
words, they form a representation of the SO(3) algebra.
In Ref. [9] Dicke examined the spontaneous radiation Stationary states of this system can therefore be identi-
of photons emitted from a gas consisting of two-level fiedwiththoseeigenstatesthatconservethesquareofthe
particles. Gasses of both small and large extent were total angular momentum operator, i.e., Φ Φ , with
q jm
treatedseparately,the scalebeingdeterminedrelativeto S2Φ =j(j+1)Φ and m j N/2. C≡onsequently,
jm jm
| |≤ ≤
the wavelength λ of an externally applied field. Taking the stationary states of a gaseous system confined to a
Rto be the spatialextentofthe container,the twocases small region of space can be expressed as
correspond to λ R or λ R. In both cases it was
assumed that the≫re was insu≪fficient overlap of the wave ψ =U (r ,r ,...,r )Φ . (IV.5)
pjm p 1 2 N jm
4
Sincetheindividualparticleswhichformthegasallex- or, equivalently
perience a common interaction with the radiation field,
the system as a whole evolves in a collective manner. v US U† = v′S . (IV.9)
j j j j
However, while this collective motion is occurring on
j j
X X
thesestates,theymaystillundergoothernon-trivialevo-
lutions. Such operations conserve energy and angular Taking the derivative of both sides of this equation with
momentum. Hamiltonians corresponding to these non- respect to time yields a sufficient condition for a Hamil-
trivial evolutions commute with the collective operators tonian to generate a compatible transformation
and thus can be constructedfrom the previouslyderived
invariants. Furthermore,the number ofinternalstates is [H,Sj]=0, Sj. (IV.10)
∀
not restricted to two, but can be arbitrary. Many inter-
nalstatesmaybeundergoingsimultaneoustransitionsto SincetheCasimiroperatorsforthealgebra satisfythis
A
other internal states, collectively, while still undergoing condition, they can be used to generate nondissapative
this evolution. transformations of a DFS/NS encoding. We will discuss
In the next section we consider a particular type of the implications of these results for the case of a three
Dicke state which is actually invariant under these col- qudit encoding next, with a particular emphasis on the
lectivemotions. Althoughtheargumentfollowstheusual abilityoftheseoperationstogenerateuniversalquantum
treatmentregardingthecompatibilityoftransformations computation.
of a collective DFS/NS, it applies to the present case as
well since DFS/NS states suitable for quantum informa-
tion processing correspond to degenerate Dicke states. C. Three Qudits
As mentioned earlier, a basis for the collective errors
B. DFS-Compatible Hamiltonians
is given by the set
Let us suppose that the Dicke states corresponding to S = λ(α), (IV.11)
i i
a collective DFS/NS are spanned by the set λ µ ,
withλ=1,...,dandµ=1,...,n. Here the{λ| ’is⊗di|stiin}- Xα
| i
guish a particular basis state of an encoded d-state sys- where the subscript indicates the type of error and the
tem and the µ ’s label the n orthogonalelements which superscriptlabelstheparticleonwhichtheoperatoracts.
| i
spaneachquditdimension. Whenacteduponbythecol- The invariants I , I , and I not only commute with
2 3 4
lective errors S these DFS/NS states have the property every element of this set, but can also be used to form
j
that a representation of the Lie algebra of SU(2) [20]. It has
beenshownthat the encoded, orlogicalanaloguesofthe
n
Sj λ µ = Mµµ′,j λ µ′ . (IV.6) Pauli matrices acting on an encoded qubit can be given
| i⊗| i | i⊗| i in terms of these invariants by the relations
µ′=1
X
In other words, these encoded qudit states remain unaf- X¯ = 1 I(2,3) I(1,3) , Y¯ = I3 , (IV.12)
fectedbythepresenceofsuchnoisesincetheymapevery 2√3 2 − 2 2√3
λ to itself. One can parameterize the collective errors h i
| i
using a set of time-independent complex numbers v , and
j
{ }
Z¯ = I(2,3)+I(1,3) 2I(1,2) /6. (IV.13)
2 2 − 2
D(v ,v ,...)=exp v S . (IV.7)
1 2 j j h i
j Allthreeofthesegeneratorscanbeexpressedintermsof
X
two body interactions since I can be decomposed into
3
The DFS/NS states are not the only accessible states
products of I . In fact, the invariant I alone suffices
2 2
inherenttoasystem. Therearesomeorthogonaltothese
to perform universal computation using encoded qubits
which cannot protect against collective noise. When
thatarecomprisedofthreephysicalquditssincetheyare
information is leaked into these regions of the systems
abletogenerateanysinglequbitrotation,andcanalsobe
Hilbert space it may be permanently lost. Gates which
combined in such a way as to implement an entangling
are used to manipulate the state of an encoded qudit
CNOT gate as well. This is due to the fact that the
should therefore operate in a manner such that they
stateswhichwereusedinRef.[21]fortheCNOTarealso
map DFS/NS states to other DFS/NS states. It can
present in the expansions of the logical states encoded
be shownthat asufficient conditionfora transformation
into qudits having d 3.
U =exp( iHt)tosatisfythis compatibilityrequirement ≥
− In addition, the invariant I(α,β) can also be used to
is that 2
performthegeneralizedexchangeinteractionbetweenthe
UD(v ,v ,...)U† =D(v′,v′,...), (IV.8) states p (α) q (β) associatedwithparticlesαandβ since
1 2 1 2 | i | i
5
it has been shown in Ref. [20] that two internal states.
Acknowledgments
exp i(π/4) λ λ αβ = iexp(πi/2d) βα ,
j j This material is based upon work supported by the
− ⊗ | i − | i
Xj National Science Foundation under Grant No. 0545798.
(IV.14)
for α,β =1,2,...,d.
Clearly these are linear combinations of the two-body Appendix A: The algebra of SU(d)
(α,β)
interactionswhicharecomprisedofthe invariantsI .
2
Three-body and higher order interactions are less often
We have chosen the following convention for the nor-
experimentallycontrollable,butarealso,inprinciple,vi-
malization of the algebra of Hermitian matrices which
able candidates forquantumgates. Forexample the log-
are generators of SU(d).
ical Y interaction for qudits is proportionalto I .
3
Tr(λ λ )=2δ . (A.1)
i j ij
V. CONCLUSIONS The commutation and anticommutation relations of
the matrices representing the basis for the Lie algebra
For quantum systems containing many particles, each can be summarized using the following equation:
having a number of internal states, the system could be
2
inavastarrayofpossiblestatescorrespondingtoalarge
λ λ = δ +if λ +d λ , (A.2)
i j ij ijk k ijk k
Hilbert space dimension. The evolution of such states d
can be fairly simple however, as in the case of a system
where here, and throughout this appendix, a sum over
undergoing collective motion. Such motions occur, for
repeated indices is understood. The sums are written
example, when λ R so that each particle feels the
≫ explicitly for clarity only in a few cases.
same field. If states, or subsystems, of a collection of
As with any Lie algebra we have the Jacobi identity:
particles are invariantunder collective motions, they are
decoherence-free, or noiseless with respect to any collec-
f f +f f +f f =0. (A.3)
ilm jkl jlm kil klm ijl
tive operation, unitary or not. This leads to the promis-
ingmethodforerrorprevention–encodinginoneofthese
There is also a Jacobi-like identity,
subspaces to avoid collective errors. To take advantage
of such an encoding for the purposes of quantum infor- f d +f d +f d =0, (A.4)
ilm jkl jlm kil klm ijl
mation processing, one requires a complete set of logical
operationstobeperformedonthesesubsystemswhichis which was given by Macfarlane, et al. [23].
compatible with the encoding. We have provided a way The following identities, also providedin [23], are use-
inwhichtofindthesetofHamiltoniansforthis purpose. ful
However, we also note that since collective motions
commutewiththe invariantoperatorswe havepresented d = 0, (A.5)
iik
here, the invariants may be measured while the system
d f = 0, (A.6)
ijk ljk
undergoes these collective errors. This allows one to de-
f f = dδ , (A.7)
scribethesystembythevaluesoftheseoperators. Indeed ijk ljk il
one of the originalmotivations for studying these invari- d2 4
d d = − δ , (A.8)
ants was to find a complete set of commuting observ- ijk ljk d il
ables to completely specify a quantum system. (See for
and
example Ref. [22] and references therein.) Not all of the
invariants presented here will commute with each other, 2
but they each commute with the collective motions. A fijmfklm = (δikδjl δilδjk)+(dikmdjlm djkmdilm)
d − −
subset of these invariant operators which also mutually (A.9)
commute will help provide a complete set of commuting and finally
operators along with the energy and total angular mo-
mentum. d2 4
d d f = − f , (A.10)
Ourworkisquitegeneralandcanbeappliedtoanyset piq qjr rkp 2d ijk
of d-state systems undergoing collective motions. There- d2 12
fore,wehaveextendedtheDicke-statedescriptionexplic- dpiqdqjrdrkp = − dijk. (A.11)
2d
itly to the general case leading the way to the descrip-
tionofsetsofparticlesundergoingcollectivemotionsand The proofs of these are fairly straight-forward, but we
their manipulation when the particles have more than omit them here.
6
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