Polynomial method to study the entanglement of pure N-qubit states

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by  H. Mäkelä

Polynomial method to study the entanglement of pure N-qubit states
H. Ma¨kel¨a and A. Messina
Dipartimento di Scienze Fisiche ed Astronomiche,
Universita` di Palermo, via Archirafi 36, I-90123 Palermo, Italy
Wepresenta mappingwhich associates pureN-qubitstates with apolynomial. Theroots of the
polynomial characterize the state completely. Using the properties of the polynomial we construct
a way to determine theseparability and thenumberof unentangled qubitsof pureN-qubit states.
PACSnumbers: 03.67.Mn,03.65.Ud
0 I. INTRODUCTION mechanical system with a polynomial is not new. Al-
1 readyin1932E.Majoranapresentedapolynomial,nowa-
0 days known as the Majorana polynomial, which he used
2 Considerable effort is spent in developing methods for
to show that the states of a spin-S particle can be ex-
the detection and classification of entangled states. One
n pressed as a superposition of symmetrized states of 2S
important aim is to find ways to detect the separabil-
a spin-1 systems [19, 20]. This decomposition, the Majo-
J ity of mixed states consisting of an arbitrary number of 2
rana representation, has been relatively unknown for a
2 subsystems. While ageneral,easilycomputable,method
long time. However, it has recently found applications
1 todetecttheseparabilityofarbitrarymixedmultipartite
in may different fields, such as in studying the symme-
states is still lacking, some partial results exist. Maybe
tries of spinor Bose-Einstein condensates [21–24], in the
] the most famous separability condition for mixed states
h context of reference frame alignment [25], in helping to
isthepositivepartialtransposition,alsoknownasPeres-
p defineanticoherentspinstates[26],andincalculatingthe
- Horodeckicriterion[1,2]. Thismethodissimpleandeasy
nt toapply,butitcanbeusedtodetectonlybipartitesepa- spectrumof the Lipkin-Meshkov-Glickmodel [27,28]. It
has also been used to give a graphical representation for
a rability. Therefore various separability conditions which
the states of an n-level system [29].
u workin an N-partysetting have been developed. Exam-
q plesofthesearepermutationcriteria,wheretheindicesof
[
thedensitymatrixarepermuted[3],theuseofquadratic The states of an N-qubit quantum register can be

2 Bell-typeinequalities[4],algorithmicapproaches[5],and
viewed as the spin states of a particle with spin S =
v the use of positive maps [6]. For a more comprehensive (2N −1)/2. Therefore, expressing the pure states of an
7 list, see [7, 8]. In the case of pure states the situation
N-qubitsystemutilizingtheMajoranarepresentationre-
2
7 is simpler. A pure N-partite state is separable if and quiresthe use of2N−1spin-1 systems. In the approach
only if all the reduced density matrices of the elemen- 2
1 we present in this article only N two-level systems are
tary subsystems describe pure states. Alternatively, in a
. needed to characterize the states of this system. The
8 bipartite case, separability can be determined by calcu-
0 Majoranarepresentationisusefulinstudyingthe behav-
lating the Schmidt decomposition of the state. Unfortu-
9 ior of spin states under spin rotations as a spin rotation
nately,theconceptoftheSchmidtdecompositioncannot
0 ofa spin-S particle is equivalent with rotating the states
: bestraightforwardlygeneralizedtothecaseofN separate of the constituent spin-1 particles [20]. However, when
v subsystems [9, 10]. In addition to these two well-known 2
i discussingthestatesofanN-qubitquantumregister,this
X methods,variousotherapproachesto the purestatesep-
property is not very helpful and therefore the benefits of
arability have been discussed. A separability condition
r theMajoranarepresentationcannotfullybetakenadvan-
a based on comparing the amplitudes and phases of the
tage of. In this case the simplified description presented
components of the state has been discussed in [11, 12].
in this article becomes useful.
It has been shown that the separability of pure three-
qubit states can be detected by studying two-qubit den-
sity operators [13] and expectation values of spin opera- Thisarticleisorganizedasfollows. InSec. IIweintro-
tors [14,15]. Separabilitytests basedonstudying matri- duce a mapping between the pure states of an N-qubit
cesconstructedfromthe componentsofthe state vector, quantum register and polynomials. We argue that the
known as coefficient matrices, have gained attention re- roots of a polynomial determine a unique state and vice
cently [16–18]. versa. In Sec. III we calculate the polynomial of sep-
In this article we present a mapping which associates arable pure states and derive a necessary and sufficient
the pure states of anN-qubit systemwith a polynomial. condition for the separability of an arbitrary pure N-
The roots of the polynomial determine the state com- qubit state. We also briefly discuss the generalization of
pletely and vice versa. We show that this polynomial the polynomialapproachto systemscontainingN copies
establishes a simple way to test the separability of pure of an h-level system. In Sec. IV we show how the poly-
N-qubit states and to study the number of unentangled nomial can be used to study the number of unentangled
particles. The idea to associate a state of a quantum qubits. In Sec. V we present the conclusions.
2
II. CHARACTERISTIC POLYNOMIAL firstcalculatethecharacteristicpolynomialofaseparable
state. Any separable pure state φ can be written as
s
Wedenotethebasisofthequbitjby{|0i ,|1i },sothe
j j
basisvectorsofanN-qubitquantumregistercanbecho-
N−1
sen as |i i ···i i ≡ |i i ⊗|i i ⊗···⊗|i i ,
0 1 N−1 0 0 1 1 N−1 N−1 φ = φ
s j
where every i ∈ {0,1}. Each natural number 0 ≤
j Oj=0
i ≤ 2N − 1 can be written using binary notation as
i = Nj=−01ij2j, where ij ∈ {0,1}. Using this we can =N−1(a |0i +b |1i ) a ,b ∈C. (5)
j j j j j j
assocPiate the basis vector |i0i1···iN−1i with |iid. Here Oj=0
the subscript d shows that decimal notation is used to
label the basis states. Let
Assume that |li is a basis state of an L-qubit system
d
2N−1 andthat|mi isthatofanindependentM-qubitsystem.
φ= C |ii , C ∈C, (1) d
i d i Using the binary expressionsfor l andm it is easyto see
Xi=0 that
be some, possibly unnormalized, state vector of an N-
qubit system. We associate this vector with the polyno- |li |mi =|l+2Lmi (6)
d d d
mial
2N−1 holdsforthetensorproductof|li and|mi . Hereandin
P(φ;x)≡ Cixi, (2) what follows we omit the tensor pdroduct sydmbol. Let ξL
Xi=0 andξM bestatesofL-qubitandM-qubitquantumregis-
which we call the characteristic polynomial of φ. By the ters, respectively. Then we can write ξL = i2=L0−1ξiL|iid
fundamental theorem of algebra, this polynomial can be andξM = 2i′M=−01ξi′|i′id. Ifφ∈(C2)L+M cPanbewritten
written in a unique way as as φ=ξLξPM, then
k−1
P(φ;x)=Ck (x−xj), (3) 2L−12M−1
jY=0 φ= ξiLξiM′ |iid|i′id (7)
Xi=0 iX′=0
where {x |j =0,1,...,k−1} are the roots and k is the
j 2L−12M−1
dTehgereseetooffPve(φct;oxr)s.{Icfφk|=c∈0Cw,ecd6=efi0n}edQete−j=r1m0(ixne−saxju)n=iqu1e. = Xl=0 iX′=0 ξiLξiM′ |i+2Li′id, (8)
setofrootsandeachsetofroots{x ,x ,...,x }deter-
0 1 k−1
minesthevectorφuptonormalizationandphase. There-
fore we have a bijective map between the pure states of where we have used Eq. (6). Consequently, the charac-
an N-qubit quantum register and the roots of complex teristic polynomial of φ becomes
polynomials of degree k ≤ 2N −1 [31]. Explicitly, the
components of φ are determined by the roots through
the formula P(φ;x)=2L−12M−1ξiLξiM′ xi+2Li′
C =(−1)k−i x x x ···x , (4) Xi=0 iX′=0
i j0 j1 j2 jk−1−i
j0<j1<j2X<···<jk−1−i = 2L−1ξiLxi 2M−1ξiM′ (x2L)i′
where i = 0,1,2,...,k−1 and we have chosen Ck = 1. (cid:16) Xi=0 (cid:17) iX′=0
The roots contain the same amount of information on =P(ξL;x)P(ξM;x2L). (9)
the system as the state vector φ. In particular, all the
entanglement properties of φ are encoded in the set of
roots corresponding to φ. With the help of the roots Therefore, if the state of the quantum register is the
the state φ can be given a geometrical representation as product of an L-qubit state and an M-qubit state, the
2N −1 points on the Bloch sphere, see Ref. [30]. characteristicpolynomialfactorizesastheproductofthe
polynomials of the two states. In the polynomial of the
M-qubit state the variable x is replaced by x2L. Using
III. SEPARABLE PURE N-QUBIT STATES Eq. (9) it is easy to calculate the characteristic poly-
nomial P(φ ;x) of a separable state φ ≡ φ φ ···φ
s s 0 1 N−1
Inthissectionweshowhowtheseparabilityofφcanbe given by Eq. (5). By defining φ ≡ φ φ ···φ ,
j;N j j+1 N−1
detected with the help of P(φ;x). In order to do so, we so that φ = φ φ , and using Eq. (9) repeatedly
j;N j j+1;N
3
we get Eq. (12) it is easy to see that now a /b = C /C .
j j k−2j k
Ontheotherhand,ifk =0,then(k+2j) =k +δ and
j l l jl
P(φs;x)=P(φ0,x)P(φ1;N,x2) Eq. (12) gives bj/aj =Ck+2j/Ck =0. Summarizing,
=P(φ ,x)P(φ ,x2)P(φ ,x4)
0 1 2;N a C
=P(φ0,x)P(φ1,x2)P(φ2,x4)P(φ3;N,x8) bj = Ck−2j if kj =1, (13)
j k
=··· b = 0 if k =0.
j j
N−1
= P(φ ,x2j) Using Eq. (11) we immediately see that the k roots of
j P(φ ;x) are
jY=0 s
N−1 1/2j
= (aj +bjx2j). (10) xjm = −Ck−2j ei22πjm, m=0,1,...,2j−1,
(cid:18) C (cid:19)
jY=0 k
(14)
We see that the characteristic polynomial of a separable wherejtakesthosevaluesforwhichkj =1. Ontheother
state can always be written in the form of (10). On the hand, if the roots and their multiplicities are known, the
other hand, there always exists a separable state whose polynomial can be determined up to a multiplying con-
characteristic polynomial is given by Eq. (10), namely stant. Inparticular,ifx=0is a root,thenits multiplic-
thestateφ . FromthedefinitionofP(φ;x)itfollowsthat ityhastobeequaltothelowestpowerofthepolynomial.
s
if P(φ;x) = P(φ˜;x), then necessarily φ = φ˜. Therefore Inconclusion,anarbitrarypurestateφisseparableifand
φ istheuniquevectorwhichgivesrisetothepolynomial only if
s
of Eq. (10). In conclusion, a pure N-qubit state φ is
(Ia) All the numbers x given by Eq. (14) are
separable if and only if P(φ;x) can be written as in Eq. jm
roots of P(φ;x).
(10). The roots of this equation are
(Ib) The number of x equaling zero is equal to
jm
xjm = −aj 1/2jei22πjm, m=0,1,...,2j −1, (11) the lowest power of P(φ;x).
(cid:18) b (cid:19) An alternative formulation is that φ is separable if and
j
only if the quantity
whereb hastobenonzero. Ifb iszerothedegreeofthe
j j
polynomial is decreased by 2j from the maximal degree 2j−1
2N −1. S(φ)≡ |P(φ;x )| (15)
jm
The separability of a state φ can be determined by j,Xkj=1mX=0
calculating the roots of P(φ,x) and checking if they are
of the form given by Eq. (11). These calculations can equals zero and Condition (Ib) holds. Note that if k =0
in practice turn out to be very complicated. It may be the state is separable.
computationally demanding to achieve accurate enough Ifastateisfoundtobeseparable,thentheone-particle
results in order to reliably see how the roots are dis- states it consists of can be explicitly constructed with
tributed in the complex plane. This is partly related the help of the ratios aj/bj given by Eq. (13). We now
to the fact the degree of the polynomial P(φ;x) can be presentsome examplesof the detectionof separabilityof
2N−1,whichgrowsrapidlywithN,renderingthecalcu- states with several freely varying components.
lationofrootstime-consuming for largeN. However,we
willshownextthattherootsofP(φ;x) canbe expressed
in a simple way in terms of the components {C } of the A. Example 1
i
state vector if φ is separable. Let φ be the separable
s
state given by Eq. (5). When this vector is written in As the first example we consider a state defined as
theformφ = 2N−1 C |ii ,thecomponentsC areeas-
s i=0 i d i ξN =C |0i +C |1i +···+C |k−2i +C |ki , (16)
ily obtained byPnoting that ij = 0 (ij = 1) corresponds 0 d 1 d k−2 d k d
to aj (bj): where C0,Ck 6= 0 and k is odd. Since k is odd k0 = 1
and Eq. (14) shows that x = C /C = 0. Because
00 k−1 k
N−1
P(ξN;x ) = C 6= 0, ξ cannot be a separable state. In
C = [(1−i )a +i b ]. (12) 00 0
i j j j j a three-qubit case we see that, for example,
jY=0
ξ3 =C |000i+C |100i+C |010i+C |110i
Here we have used the binary form of i, that is, we have 0 1 2 3
written i= N−1i 2j. We assume that C 6=0,C = +C4|001i+C5|101i+C7|111i (17)
j=0 j k k+1
··· = C2N−P1 = 0, so that the degree of P(φs;x) is k. where C C 6=0 cannot be separable.
By writing k = Nj=−01kj2j we see that if kj = 1, then In ord0er7to compare our approach with other separa-
(k−2j) = k −Pδ , l = 0,1,...,N −1. Using this and bility tests, we now check the separability of ξ3 using an
l l jl
4
alternative method. There exist various (partial) multi- Inthecaseofathree-qubitsystemthisresultmeansthat
partite separability criteria for mixed states (see, for ex-
ample, [3–6]). While these are useful when mixed states ξ3 = C0|000i+C1|100i+C3|110i+C4|001i
are studied, in the case of pure states the most conve- +C |101i+C |011i, C 6=0, (20)
5 6 6
nient separability check is usually the standard method
of calculating the reduced single-qubit density matrices cannot be a product state if C0,C1, or C3 is nonzero.
of the N-qubit state. This view is supported by the fact If C4 = 0 we have x10 = x11 = 0, which means that
that alternative pure state separability tests require ex- all xjm are equal to zero. Then ξ3 cannot be separable
aminingthepropertiesofmatricesthatarehigherdimen- unlessallCiexceptC6arezero. Thereducedsingle-qubit
sional than the two-by-two dimensional reduced single- density matrices ρ0,ρ1, and ρ2 can be straightforwardly
qubit density matrices [13, 18] or require the calculation calculated and are not presented here. The determinant
oftheexpectationvaluesofoperatorsexpressedastensor of ρ2 is
products of the Paulispin matrices [14, 15]. This results
det(ρ )=(|C |2+|C |2+|C |2)(|C |2+|C |2+|C |2)
in a complex calculation if a state that contains many 2 0 1 3 4 5 6
freely varying components, such as ξ3, is studied. For −|C C∗+C C∗|2 (21)
0 4 1 5
these reasons we now examine the separability of ξ3 by
≥|C |2(|C |2+|C |2+|C |2)
usingthemethodofpartialtraces. Hereandinwhatfol- 6 0 1 3
lows we denote the reduced single-qubit density matrix +|C3|2(|C4|2+|C5|2)+(|C0C5|−|C1C4|)2,
pertaining to qubit j by ρ . Now the indexing of qubits (22)
j
runs from 0 to N −1. The vector ξ3 is separable if and
where we have obtained a lower bound for the determi-
only if any two of the three density matrices ρ ,ρ , and
0 1
nantinthe samefashionasinthe previousexample. We
ρ describe pure states. The state ρ is pure if and only
2 j reproduce the earlier result that ξ3 is necessarily entan-
if det(ρ ) = 0, so if det(ρ ) 6= 0 for at least one j, then
j j
ξ3 is entangled. As an example we determine det(ρ ). gledifC1,C2 orC3 isnonzero. Inordertodeterminethe
0
separabilityconditionsinthe caseC =0onehasto cal-
Asimple calculationshowsthatthe single-qubitreduced 4
culate det(ρ ) and repeat the above calculation for this
density matrix of the first qubit is 0
quantity. The result agrees with the one obtained using
ρ0=(cid:18) C|C0∗C0|12++C|C2∗2C|23++|CC4∗4C|25 |C1C|20+C1∗|C+3|C22+C|3∗C+5|4C+4C|C5∗7|2 (cid:19) tCh4e=po0l,ynthoemniaolnalyppCro6accahn, tbheatnoisn,zeifroξ.3Wisesespeeartahbalet aalnsdo
(18)
in this case the polynomial approach provides an easier
Using the inequality Re(C) ≤ |C|, where C is an arbi-
wayto check the separability than the method of partial
trarycomplexnumber,itcanbeshownthatthefollowing
traces.
inequality holds for the determinant of ρ
0
det(ρ )≥|C |2(|C |2+|C |2+|C |2)+(|C C |−|C C |)2
0 7 0 2 4 0 3 1 2 C. Example 3
+(|C C |−|C C |)2+(|C C |−|C C |)2.
0 5 1 4 2 5 3 4
(19) As the final example we study a state given by
This is bounded below by |C C |2 > 0, confirming the 2N−1 2N−2−1
aforementioned result concern0ing7 the separability of ξ3. ξN = |iid+eiθ |4iid
Thereforeanecessaryconditionforthe separabilityofξ3 Xi=1 Xi=0
i6=0,4,8,...,2N−4
can be straightforwardly obtained using partial traces.
However,the polynomialmethodprovidesasimplersep- 2N−1 2N−2−1
= |ii + eiθ−1 |4ii . (23)
arability test in the present example. Even more so if d d
instead of ξ3 the separability of the N-qubit state ξN is Xi=0 (cid:0) (cid:1) Xi=0
studied.
Now C /C =1 for all j, so Eq. (11) gives
(2N−1)−2j 2N−1
xjm =ei(2m2+j1)π, m=0,1,...,2j −1, (24)
B. Example 2
wherej =0,1,2,...,N−1. Usingthesumformulaofge-
In the second example we choose ξN such that the ometric series we find that the characteristic polynomial
degree of P(ξN;x) is k = 2N −2. Then k = 1−δ . can be written as
j 0j
We assume that C2N−1−2(= Ck−2N−1) = 0, from which x2N −1 x2N −1
it follows that x = 0 for m = 0,1,...,2N−1 − P(ξN;x)= +(eiθ−1) . (25)
(N−1)m x−1 x4−1
1. According to Condition (Ib) the lowest order of the
polynomial has to be at least 2N−1 for the state to be a It is easy to see that for j =2,3,...,N −1
productstate. Thus,ifC 6=0foratleastoneisuchthat
i
0≤i<2N−1, i6=2N−1−2, thenξN mustbe entangled. P(ξN;x )=0, m=0,1,...,2j −1, (26)
jm
5
while andan(N−1)-qubitstate. Wewriteφ=φ φN−1,where
j
φ =a |0i +b |1i is the state of the qubit j and φN−1
j j j j j
P(ξN;x00)=P(ξN;x10)=P(ξN;x11)=2N−2(eiθ−1). gives the state of the rest of the qubits. As before, the
(27) degree of the polynomial is denoted by k. Using Eq.
The state ξN is separable if and only if θ = 2πn for (6)wesee thatthe characteristicpolynomialofthe basis
some integer n. If ξN is separable Eq. (13) shows that states reads
ξN =⊗N−1(|0i +|1i ).
j=0 j j P(|ii ;x)=P(|i i ···i i;x)
Now the N reduced single-qubit density matrices of d 0 1 N−1
ξN can be straightforwardlydetermined. Lengthy calcu- =xi020xi121xi222···xiN−12N−1. (30)
lation shows that det(ρ ) = det(ρ ) = 22N−3(1−cosθ)
0 1
and det(ρ ) = 0 when j = 2,3,...,N − 1, confirming We write the (N −1)-qubit state as
j
the earlierresult. Inthe presentexamplethe polynomial
methoddoesnotseemtoprovideasobviouscalculational φN−1 = Ci0···ij−1;ij+1···iN−1|i0···ij−1ij+1···iN−1i,
simplification as in the previous two examples. il∈{X0,1},l6=j
(31)
so using Eq. (30) we find that
D. Generalization to h-level systems
P(φ;x)=b (x2j −x2j ) C
j jm i0···ij−1;ij+1···iN−1
X
We now briefly discuss a generalization of the separa- il∈{0,1},l6=j
bilitytesttoasystemconsistingofN copiesofanh-level ×xi020+···+ij−12j−1+ij+12j+1+···+iN−12N−1 (32)
system. We write the basis of a single h-level system as
{|0ih,|1ih,...,|h−1ih} and choose the basis vectors for where we have assumed that bj 6= 0, which is equivalent
the N-partite system as |ii = |i i ···i i , where tok =1. Wehavealsowritten(a +b x2j)=b (−x2j +
d 0 1 N−1 h j j j j jm
i= Nj=−01ijhj andij ∈{0,1,2,...,h−1}. Anarbitrary x2j). Notethatx2jmj isindependentofm. Ifbj =0,weget
purePstate can be expressed as anexpressionwhichisobtainedbymultiplyingthesumof
Eq. (32)bya . Equation(32)showsthatthepolynomial
j
hN−1 P(φ;x)/(x2j−x2j )containsonlythosepowersofxwhich
φ= C |ii . (28) jm
i d do not have 2j in their binary representation and that
Xi=0 x is a root of P(φ;x) for each m = 0,1,...,2j −1.
jm
Let φh =φhφh···φh be a separablestate where φh = Therefore,if kj =1, necessary conditions for the qubit j
s 0 1 N−1 j
a |0i +b |1i +c |2i +···+q |h−1i . Astraightforward to be unentangled with respect to the rest of the qubits
j h j h j h j h
calculation shows that are
(IIa) P(φ;x )=0 for every m=0,1,...,2j −1.
N−1 jm
P(φh;x)= a +b xhj +···+q x(h−1)hj (.29) (IIb) 2j does not appear in the binary representations
s jY=0 (cid:16) j j j (cid:17) of the exponents of x in P(φ;x)/(x2j −x2jmj ).
If k =0 there is only one condition, namely,
Inordertoestablishaseparabilitytest,onehastoexpress j
the roots of this polynomial in terms of the coefficients
(IIc) 2j does not appear in the binary representations
C ,C ,...,C . This is possible but complicated if
0 1 hN−1 of the exponents of x in P(φ;x).
2 < h < 6. If h ≥ 6, the roots cannot be calculated an-
alytically and therefore cannot be written using the co- It is easy to see that these are also sufficient conditions.
efficients Ci. Thus the separability test can be extended The number of unentangled qubits can be obtained by
to systems containing less than six levels, but it is more checking Conditions (IIa) and (IIb) for every qubit j for
complicated to apply than in the two-level case. An ex- which k = 1 and Condition (IIc) for the rest of the
j
tension is not feasible if the number of levels is equal to qubits. It is possible to extract information about the
or larger than six. number of unentangled qubits without using Conditions
(IIb)and(IIc),namely,anupperboundforthisquantity
can be obtained by adding to the number of qubits for
IV. NUMBER OF UNENTANGLED QUBITS which (IIa) holds the number of indices j for whichk =
j
0. Thiscorrespondstoassumingthateither(IIb)or(IIc)
Entangledstatescanbeclassifiedbasedonthenumber holds for every qubit.
of unentangled one-qubit states. The state φ is said to
contain n unentangled qubits if it can be written as a
productofnsingle-qubitstatesφ andan(N−n)-qubit A. Example 1
l
stateφN−n. Inordertostudythenumberofunentangled
particles, we determine the characteristic polynomial of As an example of the use of this method we consider
a state whichseparatesas a productof a one-qubitstate thestategivenbyEq. (23). Nowk =2N−1andtherefore
6
k =1foreveryj. Equations(26)and(27)togetherwith mial determine the state completely and vice versa. The
j
Condition (IIa) show that the number of unentangled structure of the polynomial is inspired by the one used
qubits is at most N −2 (N) if θ 6= 2πn (θ = 2πn). In intheMajoranarepresentation[19,20]. Theseparability
order to simplify the polynomial P(ξ;x) we note that of a state can be studied by examining the properties of
therootsofthecorrespondingpolynomial. Inparticular,
x2N −1=(x2−1)(x2+1)(x4+1)···(x2N−1+1). (33) we have presented a method which establishes a neces-
sary and sufficient condition for a given pure N-qubit
With the help of this and Eq. (25) we get state φ to be separable. This method provides a new
point of view to the pure state separability and gives an
P(ξN;x)=(x+1)(x2+1)(x4+1)···(x2N−1 +1) alternative to the conventional separability test of cal-
+(eiθ −1)(x4+1)(x8+1)···(x2N−1 +1) (34) culating the reduced single-qubit density matrices of the
state. The separabilityofφ canbe determinedbycheck-
ing whether the numbers x , defined in equation (14),
Now x2j −(xjm)2j = x2j +1 when j ≥ 2 and using the arerootsofthepolynomialPjm(φ;x)ofequation(2). Both
aboveequationonecanseethatCondition(IIb)holdsfor thenumbersx andthepolynomialP(φ;x)canbeeas-
jm
j =2,3,...,N −1 regardless of the value of θ. Further- ily obtained asa function ofthe components ofthe state
more, (IIb) holds for every j if θ = 2πn. In conclusion, φ. We have illustrated through examples that in some
the qubits j = 2,3,...,N − 1 are always unentangled casesthe polynomialseparabilitytestiseasierandfaster
with respect to the rest of the qubits and if θ = 2πn to apply than the method of reduced single-qubit den-
the state is separable. The same result can be obtained sity matrices. We have also shown how the number of
using the reduced single-qubit density matrices ρj. The unentangled qubits can be obtained with the help of the
number of unentangled qubits is equal to the number of polynomialP(φ;x). Itseems,however,thatfor this task
ρj for which det(ρj) = 0. The values of these determi- themethodofsingle-qubitdensitymatricesispreferable.
nants have been presented in Example 3 and reproduce
the aforementioned result.
Anecessarystepinthecalculationofthenumberofun-
entangledqubits is toapply Condition(IIa) toallqubits
j for which k = 1. This is equivalent to checking the
j
separability of the state. In addition to this, Conditions
(IIb)and(IIc)havetobecontrolled. Ontheotherhand,
in the case of single-qubit reduced density matrices ρ ,
j
the determination of the number of unentangled qubits
doesnotrequireanyadditionaloperationsincomparison
Acknowledgments
withtestingtheseparability. Inbothcasesdet(ρ )hasto
j
be calculated. This suggests that the method of reduced
single-qubit density matrices is preferable if the number
of unentangled qubits is studied. The authors are grateful to V.I. and M.A. Man’ko for
helpful discussions. A.M. acknowledges partial support
by MIUR Project II04C0E3F3 Collaborazioni Interuni-
V. CONCLUSIONS versitarie ed Internazionali Tipologia C. H.M. wants to
thank E. Kyoseva, B.W. Shore, and N.V. Vitanov for
We have defined a mapping which associates pure N- comments on an earlier version of the manuscript and
qubitstateswithapolynomial. Therootsofthispolyno- EC Projects CAMEL and EMALI for financial support.
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