Network implementation of covariant two-qubit quantum operations

0.15 MB·English

by  J. Novotny

Network implementation of covariant two-qubit quantum operations
J. Novotny´(1), G. Alber(2), I. Jex(1)
(1) Department of Physics, FJFI CˇVUT, Bˇrehov´a 7, 115 19 Praha 1 - Star´e Mˇesto, Czech Republic
(2) Institut fu¨r Angewandte Physik, Technische Universit¨at Darmstadt, D-64289 Darmstadt, Germany
(Dated: February 1, 2008)
Asix-qubitquantumnetworkconsistingofconditionalunitarygatesispresentedwhichiscapable
of implementing a large class of covariant two-qubit quantum operations. Optimal covariant NOT
operationsforoneandtwo-qubitsystemsarespecialcasescontainedinthisclass. Thedesignofthis
quantumnetworkexploitsbasicalgebraic propertieswhichalso shednewlightontothesecovariant
quantumprocesses.
PACSnumbers: 03.67.Mn,03.65.Ud
7
0 I. INTRODUCTION setofcovarianttwo-qubitquantumoperationscanbeim-
0
plemented in quantum networks with the help of simple
2
elementary quantum gates.
It is well known that certain tasks of information pro-
n
cessing cannot be performed perfectly on the quantum
a
level despite the fact that they can be performed per-
J
fectly ona classicallevel[1, 2]. Typically,impossibilities In general, a systematic approach to the problem of
9
of this kind on the quantum level hint on the existence designing elementaryquantum gate sequences whichim-
1 of corresponding no-go theorems. They raise interest- plement a given family of covariant quantum operations
v ingquestionsconcerningtheoptimalityofthesequantum is not known. In the following it is shown that for the
8 processes with respect to particular quality measures. A abovementionedconvexsetofcovarianttwo-qubitquan-
4 prominentexample inthis respectis the copyingofarbi- tum operations this problem can be solved completely.
0
traryquantumstateswhichcannotbeachievedperfectly This is due to the fact that this convex set of quantum
1
[3]. The associatedproblemofdetermining quantumop- operations has special algebraic properties which can be
0
7 erationswhichcanachievethis tasksinthe bestpossible exploited in a convenient way. In addition, these alge-
0 wayhasstimulatednumeroustheoreticalandexperimen- braicpropertiesshednewlightonthepropertiesofthese
/ tal investigations starting with the early work of Buˇzek covariant two-qubit quantum operations. With the help
h
and Hillery [4]. of additional auxiliary qubits it is possible to design a
p
- Another process of this kind is the quantum NOT quantum network which involves a particular sequence
nt transformationwhich is to change an arbitraryquantum of conditional unitary qubit gates. Depending on the
a state into an orthogonal one and which cannot be per- preparation of the auxiliary qubits any covariant quan-
u formed perfectly for arbitrary input states [5, 6]. Re- tumoperationwithinthisconvexsetcanbeimplemented
q cently, the problem of optimizing quantum NOT pro- by this quantum network. One of the advantages of this
v: cesses has been addressed not only for arbitrary pure particular network implementation is that the sequence
i one-qubit input states [5] but also for pure two-qubit ofconditionalunitaryqubitgatesinvolvedisindependent
X
input states of a given degree of entanglement [7]. In ofthe covariantquantumoperationunder consideration.
r thislattercontextthepossibleinputstatesarerestricted
a
to the set of pure two-qubit states of a given degree of
entanglement which does not constitute a linear vector This paper is organized as follows. In Sec. II basic
space. Therefore, the previously mentioned impossibil- definitionsandpropertiesoftherecentlyintroducedcon-
ity arguments concerningquantum NOT operations act- vex set of covariant two-qubit quantum processes [7] are
ing on arbitrary input states do not apply. All optimal summarized. The essential algebraic properties of these
quantum operations could be determined which perform quantum operations which are useful for the subsequent
suchaquantumNOToperationforallpossiblepuretwo- construction of the quantum network are discussed in a
qubitinputstatesofagivendegreeofentanglementwith subsection. Sec. IIIaddressesthemainproblemhowthis
the same quality. It was demonstrated that these op- convex set of quantum operations can be implemented
timal two-qubit quantum NOT operations are members unitarily by a suitable choice of auxiliary quantum sys-
of a convex set of covariant (completely positive) two- temsandbyanappropriatesequenceofelementaryquan-
qubit quantum operations. This convex set is generated tum gates. As a main result it is shown that any covari-
by four elementarytwo-qubitquantum operationswhich ant quantum operation of the convex set discussed in
form the vertices of a three-dimensional polytope. Fur- Sec. II can be implemented by a unitary master trans-
thermore,itcouldbeshownthatonlyinthecaseofmax- formation which is independent of the particular quan-
imallyentangledpuretwo-qubitinputstatesitispossible tum operationunder consideration. A quantum network
toperformsuchacovariantquantumNOToperationper- implementation of this main result involving controlled
fectly. However,sofaritisstillunknownhowthisconvex unitary Pauli operations is discussed in a subsection.
2
II. COVARIANT TWO-QUBIT QUANTUM AnoptimalquantumNOToperationtransformsanar-
OPERATIONS bitrarypuretwo-qubitinput statewithagivendegreeof
entanglement into a not necessarily pure output state of
In this section basic aspects of all completely positive its orthogonal complement in an optimal way. Thereby,
quantum process are summarized that transform pure the sets Ωα of pure two-qubit states with a given degree
two-qubit input states of a givendegree of entanglement of entanglement α [0,1/√2] are defined by [11, 12]
∈
inacovariantway. Therecentlydiscussedoptimalquan-
tum NOT operations [7] are special cases thereof. Ω = U U α0 0 +
α n (cid:0) 1⊗ 2(cid:1)(cid:0) | i⊗| i
1 α2 1 1 U ,U SU(2) . (6)
A. Basic definitions and general properties p − | i⊗| i(cid:1)(cid:12)(cid:12) 1 2 ∈ o
(cid:12)
Inthespecialcaseα=0thetwo-qubitstatesaresepara-
Letusconsiderageneralcompletelypositivequantum ble whereas in the opposite extreme case α=1/√2 they
operationΠ which transformsan arbitrarytwo-qubitin- are maximally entangled.
put state ρ in a covariant way according to Let us now summarize some basic properties of such
optimal quantum NOT operations [7]:
Π U U ρU† U† = U U Π(ρ)U† U†. (1)
(cid:16) 1⊗ 2 1 ⊗ 2(cid:17) 1⊗ 2 1 ⊗ 2
There is a characteristic threshold value of entan-
Thereby, the requirement of complete positivity ensures •
glement at α = 1 √1 4K /2 0.1836
that this transformation can be implemented in a uni- 0 q(cid:0) − − (cid:1) ≈
tary way possibly with the help of additional auxiliary with K = 8 3√6 /20. For α α the opti-
0
(cid:0) − (cid:1) ≤
quantum systems which are uncorrelated with the two- mal quantum NOT operation, i.e. USEP, is in-
qubit system initially. If the covariance condition (1) is dependent of the degree of entanglement α and
satisfied for arbitrary unitary one-qubit transformations is characterized by the characteristic parameters
U ,U SU(2) [8], it is guaranteed that the quality of (v = 1/3,x = 1/3,y = 1/9) (compare with
1 2
∈ − −
performance of a quantum NOT operation is the same (3)). This particular quantum operation is iden-
for all possible pure entangled two-qubit input states of tical to two optimal covariant one-qubit NOT op-
a given degree of entanglement [7, 9, 10]. erations u1 [5] applied to each of the input qubits
Recently, it was shown [7] that all possible com- separately, i.e. U =u1 u1 with
SEP
⊗
pletely positive covariant two-qubit quantum operations
1
Π(v,x,y) fulfilling Eq.(1) form a three-parametric set, u1(ρ) = (2I ρ). (7)
i.e. 3 −
3 These one-qubit NOT operations u1 transform an
ρ =Π(v,x,y)(ρ)= K (v,x,y)ρK†(v,x,y),
out X ij ij arbitrarypureone-qubitinputstateintoanorthog-
i,j=0 onal state in an optimal way [5].
(2)
with the Kraus operators
Forα>α theoptimalNOToperationsdependon
0
•
1 1 thedegreeofentanglementαandarecharacterized
K (v,x,y) = (1+3x+3v+9y)2 I I,
00 4 ⊗ by the parameters (compare with (3))
1 1
Ki0(v,x,y) = (1+3x v 3y)2 σi I, 1 2 31α2β2+20α4β4
4 − − ⊗ y = − ,
1 1 −3 2 35α2β2+100α4β4
K (v,x,y) = (1 x+3v 3y)2 I σ , (3) − −
0i 4 − − ⊗ i 2 4 29α2β2 20α4β4
x+v = − − ,
1 1 3 2 35α2β2+100α4β4
K (v,x,y) = (1 x v+y)2 σ σ , i,j 1,2,3 ,
ij i j − −
4 − − ⊗ ∈{ } 1
x,v (8)
theunitoperatorI andthePaulispinoperatorsσ1 =X, ≥ −3
σ = Y, and σ = Z. The possible values of the three
2 3
parameters x,v and y are restricted by the requirement with β =√1 α2.
−
of non negativity of the prefactors entering (3), i.e.
It can be shown that perfect NOT operations
1+3x+3v+9y 0, 1+3x v 3y 0, • can be constructed for maximally entangled input
≥ − − ≥
1 x+3v 3y 0, 1 x v+y 0. (4) states only. These perfect covariant NOT oper-
− − ≥ − − ≥
ations form a one-parameter family specified by
Inaddition,tracepreservationofthe quantumoperation
characteristic parameters fulfilling the conditions
Π(v,x,y) implies
y = 1,x+v = 2 with x,v 1.
−3 3 ≥−3
3
K†(v,x,y)K (v,x,y)=I. (5) All completely positive covariant two-qubit pro-
X ij ij •
cesses (1) form a three-dimensional convex set [7].
i,j=0
3
Any of these processes Π(a) can be represented in covariant quantum operation a a a a
00 11 01 10
the form U2 1/9 4/9 2/9 2/9
SEP
Π(a) = a00I+a11USEP +a01UM(1)E +a10UM(2)E (9) UUM((12))E22 11//33 00 20/3 2/03
ME
wtuimthoapmenra≥tio0nasnd Pm,n∈{0,1}amn =1. The quan- USEPUM(1)E =UM(1)EUSEP 0 2/3 0 1/3
U U(2) =U(2) U 0 2/3 1/3 0
SEP ME ME SEP
(1) (2) (2) (1)
U(1) = Π(v =1,x= 1/3,y= 1/3), UMEUME =UMEUME 0 1 0 0
ME − −
U(2) = Π(v = 1/3,x=1,y = 1/3) (10) TABLE I: Convex decompositions of products of elementary
ME − −
covariantquantumoperationswhichconstitutetheverticesof
are members of the one-parameter family of per- thepolytope (9).
fectNOToperationsformaximallyentangledinput
states. They are characterized by the additional
property that they leave the reduced density op- theresultingquantumoperationisagainoftheform(9).
erators of the first (U(1) ) or second (U(2) ) qubit Thus, these quantum operations form a half group. The
ME ME
unchanged. The convex set of quantum processes coefficients of the convex decompositions of some prod-
(9) forms a three dimensional polytope whose ver- ucts of the elementaryquantum operationsUSEP, UM(1)E,
ticesaregivenbythequantumoperationsI,USEP, and U(2) are summarized in Table I. According to this
ME
U(1) ,andU(2) . Thispolytope containsalsoother table we have the relation
ME ME
interestingquantumoperations,suchastheuniver-
saltwo-qubitNOTprocess NOT studiedinRef.[6]. UM(1)EUM(2)E =USEP. (14)
G
This latter process is the optimal NOT operation
forallpossiblepuretwo-qubitinputstatesirrespec- Furthermore,itisapparentthatthequantumoperations
tiveoftheir degreeofentanglement. Its convexde- USEP,UM(1)E,andUM(2)E commute. Finallyletuspointout
composition is given by thatthe consideredcovariantprocesses(9)havenontriv-
ial limit expressions for Π(a)n for n .
=0.6 U +0.2 U(1) +0.2 U(2) . (11) →∞
GNOT SEP ME ME
III. QUANTUM NETWORK
B. Algebraic properties IMPLEMENTATION
Let us now explore further algebraic properties of the In this section it is shown how an arbitrary covariant
covariant two-qubit processes of Eqs.(1) and (9). two-qubit quantum operation (9) can be implemented
The vertices U , U(1) , U(2) of the polytope (9) in a six-qubit quantum network by an appropriate se-
SEP ME ME quence of controlledunitary gates. For this purpose it is
are orthogonaland the operatorsrepresentingthese pro-
demonstrated first that any covariant two-qubit process
cesses are traceless, i.e.
(9) can be implemented with the help of four auxiliary
(1) (2) qubits by a master unitary operation. This master uni-
Tr(U ) = 0, Tr U =0, Tr U =0,
SEP (cid:16) ME(cid:17) (cid:16) ME(cid:17) tary operation is independent of the particular covari-
Tr U(1) U = 0, Tr U(2) U =0, ant two-qubit quantum operation under consideration.
(cid:16) ME SEP(cid:17) (cid:16) ME SEP(cid:17) A particular covariant two-qubit process is selected by
Tr U(1) U(2) = 0. (12) preparing the auxiliary four-qubit quantum system in a
(cid:16) ME ME(cid:17) suitably chosen quantum state. In a second step a se-
quenceofconditional(unitary)Pauligatesisconstructed
Therefore, according to Eq.(9) the coefficients a
≡ which implements this unitary master transformation in
(a ,a ,a ,a ) of an arbitrary covariant two-qubit
00 01 10 11 this six-qubit quantum network.
quantum operation Π(a) are given by
1 Tr(Π(a)U )
a = Tr(Π(a)), a = SEP , A. Unitary representation with auxiliary qubits
00 4 11 Tr(U2 )
SEP
Tr Π(a)U(1) Tr Π(a)U(2) For the purpose of implementing the covariant quan-
(cid:16) ME(cid:17) (cid:16) ME(cid:17)
a01 = Tr(cid:16)UM(1)E2(cid:17) , a10 = Tr(cid:16)UM(2)E2(cid:17) . (13) tquumbitospleertautisonfisrs(t9o)fuanliltainrtilryodwuictehstohmeehuelspefuolf naoutxaitliiaorny.
InadditiontothefourdimensionalHilbertspace ofthe
H
Another interesting feature of the covariant two-qubit two-qubitinputstatesweintroducefourauxiliaryqubits
quantum operations (9) concerns repeated applications. whoseHilbert space is sixteendimensional. The
ancilla
H
If two such quantum operations are applied successively quantum states ijkl with i,j,k,l 0,1 are assumed to
| i ∈
4
form an orthonormal basis in this latter Hilbert space. B. Network implementation with conditional Pauli
We start from the observation that apart from normal- gates
ization factors the Kraus operators of (3) are unitary.
Therefore, it is convenient to introduce the correspond- Let us now implement the unitary master transforma-
ing sixteen renormalized unitary two-qubit operators tion (16) by a quantum circuit in the six-qubit quan-
tum network which involves four auxiliary qubits. Ac-
F2i+j 2k+l = σ2i+j σ2k+l (15) cording to Eq.(16) the quantum circuits have to be de-
⊗
signed in such a way that, whenever the four auxiliary
with σ = I and i,j,k,l 0,1 . From these latter qubits are preparedin a particular quantum state of the
0
∈ { }
unitarytwo-qubitoperatorswecanconstructtheunitary computationalbasis ijkl (i,j,k,l 0,1 ),the unitary
| i ∈{ }
master transformation transformation F2i+j 2k+l is acting onto the two target
qubitsofthemainsystemwithHilbertspace . Inorder
H
= F ijkl ijkl (16) to achieve this goal let us introduce elementary condi-
U X 2i+j 2k+l⊗| ih | tional unitary five-qubit quantum gates C(U) which in-
i,j,k,l∈{0,1}
volvefourcontrolqubits andonetargetqubitandwhose
action on an arbitrary quantum state ψ of the target
whichoperatesonallsix-qubitsofthe Hilbertspace
| i
. Let us assume that initially the four auxilHiar⊗y qubit anda quantum state of the computationalbasis of
ancilla
H the four control qubits ijkl is given by
qubits are prepared in the mixed quantum state
| i
Σ(a) = asgn(i+j) sgn(k+l) ijkl ijkl (17) C(U)|ψitarget⊗|ijklicontrol=Ui·j·k·l|ψitarget⊗|ijklic(o2n0t)rol
X 3sgn(i+j)+sgn(k+l)| ih |
i,j,k,l∈{0,1} (compare with Fig. (1)). In other words, the unitary
with the normalization a + a + a + a = 1
00 01 10 11 u
and with sgn(x) = x/x denoting the signum-function
| |
(sgn(0) = 0). Depending on the values of the coeffi- u
cientsa (a ,a ,a ,a )anycovariantquantumpro-
00 01 10 11
cess Π(a≡) can be implemented unitarily with the help of u
the unitary master transformation(16) by preparingthe
u
auxiliary four qubits in the quantum state (17) initially
and by disregarding these four auxiliary qubits after the
U
unitary transformation, i.e.
(1) (2) FIG. 1: Quantum circuit representation of the elementary
a I+a U +a 1U +a U (ρ)
(cid:16) 00 01 ME 10 ME 11 SEP(cid:17) ≡ controlledunitaryoperationC(U)whichinvolvesfourcontrol
Π(a)(ρ)=Tr ρ Σ(a) † . (18) andonetargetqubit. Thereby,U denotesaunitaryoperation
ancilla
(cid:8)U ⊗ U (cid:9) acting on the single target qubit which is performed if and
only if thecontrol qubitsare in state |1111icontrol.
This unitary implementation of the covariant quantum
operations (9) is a main result of our paper. It can be
operation U acts on the target state ψ if and
provedinastraightforwardwaybyinsertingEqs.(16)and | itarget
only if the four control qubits are prepared in the state
(17) into Eq.(18).
1111 . Universalquantumgateswhicharecapable
control
Before addressing the general problem of imple- | i
of implementing such controlledunitary operations were
menting an arbitrary quantum operation of the form
studied extensively in Ref. [13], for example.
of Eq.(18) by elementary quantum gates in this six-
With the help of the controlled unitary operations
qubit quantum network let us consider the unitary
C(U) also other controlled operations can be realized in
implementation of the covariant quantum operation
a straightforwardway. Suppose one wants to implement
Π(a00 =0=a10 =a11,a01 =1) = UM(1)E as an ex- a five-qubit quantum gate in which the target qubit is
ample. For this purpose the auxiliary four-qubit transformedby aunitarytransformationU ifandonlyif
quantum system has to be prepared in the mixed the first, second, and third (control) qubits are in state
quantum state Σ(a00 = 0 = a10 = a11,a01 = 0 andthefourthcontrolqubitisinstate 1 ofthecom-
1) = (1/3) 0001 0001 + 0010 0010 + 0011 0011 . |puitational basis. As apparent from Fig. 2|tihis quantum
{| ih | | ih | | ih |}
Thus, Eq. (18) yields gatemayberealizedbyactingwithaPaulispinoperator
X ontothecontrolqubitsone,two,andthreebeforeand
F F† after the application of the controlled unitary quantum
Π(a =0=a =a ,a =1) = 01ρ 01 +
00 10 11 01 √3 √3 gate C(U).
Also multi-target conditional unitary quantum gates
F F† F F†
02ρ 02 + 03ρ 03 = U(1) . (19) can be realized with the help of the elementary quan-
√3 √3 √3 √3 ME tumgateC(U). Suchmulti-targetgatesarenaturalgen-
5
e u FIG.3: Circuitimplementationofatwo-targetquantumgate
X X
whichperformsanoperationU onthefifthqubitandanoper-
e X u X ationV onthesixthqubitconditionalonthefirsttwoqubits
beinginstate|0i andqubitsthreeandfourbeinginstate|1i
e = X u X of the computational basis.
u u
U U
FIG.2: Controlledunitaryoperationwithaunitaryoperation
U acting on the target qubit if and only if the first, second,
andthirdcontrolqubitsareinstate|0iandthefourthcontrol
qubit is in state |1i of the computational basis.
eralizations of the one-qubit controlled quantum gates
justintroduced. Inagenerald-targetconditionalunitary
quantum gate a set of unitary operations, say U d ,
{ i}i=1
are performed on d target qubits simultaneously if and
onlyifthecontrolqubitsarepreparedinprescribedquan-
tum states. In Fig. 3 a two-target conditional quantum
With the help ofsuch two-targetconditionalquantum
gateisdepictedinwhichtheunitaryoperationsU andV
gates a simple sequence of conditional two-target Pauli
are performed on the first and the second target qubit if
gates can be designed in our six-qubit quantum system
andonlyifthefirstandthesecondcontrolqubitsarepre-
which performs the master unitary transformation (16).
paredinstate 0 andthethirdandfourthcontrolqubits
| i The circuit scheme of this network is depicted in Fig. 4.
are prepared in state 1 of the computational basis.
| i The first four qubits constitute the control qubits of the
auxiliary quantum system. According to Eq.(18) these
e e e
auxiliaryqubitshavetobepreparedinthequantumstate
e e e (17)initially. Thetwoinputqubitsofthemainquantum
system are prepared in an arbitrary quantum state ρ.
u u u The dynamics of the composite six-qubit quantum sys-
tem are governed by the master unitary transformation
u = u u
(16) which is implemented by the network displayed in
figure 4. The action of this dynamics on the two qubits
U U
of the main quantum system after having discarded the
four auxiliary qubits is given by the quantum operation
V V
(18).
IV. CONCLUSION quantum operations. Analogous approaches exploiting
similaralgebraicpropertiesmayalsoturnouttobeuseful
A six-qubit quantum network implementation of all fornetworkimplementationsofothercovariantquantum
possible two-qubit quantum operations was presented processes.
whichtransformallpuretwo-qubitinputstatesofagiven
degree of entanglement in a covariant way. An advan-
tage of this particular implementation is that it is based
Acknowledgments
on a sequence of conditional Pauli gates which does not
depend on the quantum operation under consideration.
A particular covariant quantum operation is selected by Financial support by GACˇR 202/04/2101, by the
preparing the four auxiliary qubits in an appropriate DAAD (GACˇR 06-01) and by projects LC 06002 (J.N.)
quantum state. The implementation presented rests on andMSM6840770031(I.J.)oftheCzechMinistryofEd-
special algebraic properties of these covariant two-qubit ucation is acknowledged.
6
- eu ee eu ee ee ee ue ue ue ee ee ee ue ue ue
(cid:10)(cid:10)(cid:30)
- ee eu eu ee ee ee ee ee ee ue ue ue ue ue ue
(cid:10)
(cid:17)3
Σ(a) (cid:10)(cid:17)XXz - ee ee ee eu ee eu ue ee ue ue ee ue ue ee ue
@
@@R- ee ee ee ee eu eu ee ue ue ee ue ue ee ue ue
-
(cid:8)* X Y Z X Y Z X Y Z X Y Z QQs
ρ (cid:8) Π(a)(ρ)
HHj - (cid:0)(cid:0)(cid:18)
X Y Z X X X Y Y Y Z Z Z
FIG. 4: Network consisting of 15 conditional unitary one- and two-target Pauli gates which performs the unitary master
transformation (16) on a six-qubit quantum system. Initially the four auxiliary qubits are prepared in state Σ(a) of (17) and
the two qubits of the main quantum system are prepared in an arbitrary quantum state ρ. After the application of these
quantumgates thetwo qubitsof themain quantumsystem are in thequantumstate Π(a)(ρ) of (18).
[1] M. A. Nielsen, I. l. Chuang, Quantum Computation sachusetts, 1981)
andQuantum Information(CambridgeUniversityPress, [9] R. F. Werner, Phys. Rev. A 58, 1827 (1998); M. Keyl
Cambridge, 2000). and R.F. Werner, J. Math. Phys. 40, 3283-3299.
[2] A.K. Pati, Phys. Rev.A 66, 062319 (2002). [10] L.-P. Lamoureux, P. Navez, J. Fiur´aˇsek, and N. J. Cerf,
[3] W.K.WoottersandW.H.Zurek,Nature(London)299, Phys. Rev.A 69, 040301-1 (2004).
802 (1982). [11] J. Novotny´, G. Alber, and I. Jex, Phys. Rev. A 71,
[4] V.Buˇzek and M. Hillery, Phys.Rev.A 54, 1844 (1996). 042332 (2005).
[5] V. Buˇzek, M. Hillery, and R. F. Werner, Phys. Rev. A [12] R. Demkowicz-Dobrzanski, M. Lewenstein, Aditi Sen,
60, R2626 (1999). Ujiwal Sen, and D. Bruß, Phys. Rev. A 73, 032313
[6] P.Rungta,V.Buˇzek,C.M.Caves,M.Hillery,andG.J. (2006).
Milburn, Phys.Rev.A 64, 042315 (2001). [13] A.Barenco,Ch.H.Bennett,R.Cleve,D.P.DiVincenzo,
[7] J. Novotny´, G. Alber, I. Jex, Phys. Rev A 73, 062311 N. Margolus, P. Shor, T. Sleator, J. Smolin, H. Wein-
(2006). furter, Phys.Rev.A 52, 3457-3467 (1995).
[8] L. C. Biedenharn and J. D. Louck, Angular momentum
in Quantum Physics (Addison-Wesley, Reading, Mas-