An elementary formula for entanglement entropies of fermionic systems

0.18 MB·English

by  Péter Lévay

An elementary formula for entanglement entropies of fermionic systems
P´eter L´evay, Szilvia Nagy and J´anos Pipek
Department of Theoretical Physics, Institute of Physics,
Budapest University of Technology and Economics, H-1111 Budapest Budafoki u.8.
(Dated: February 1, 2008)
AnelementaryformulaforthevonNeumannandR´enyientropiesdescribingquantumcorrelations
in two-fermionic systems having four single particle states is presented. An interesting geometric
structure of fermionic entanglement is revealed. A connection with the generalized Pauli principle
is established.
PACSnumbers: 03.67.-a,03.65.Ud,03.65.Ta,02.40k
5
I. INTRODUCTION Some problems arise when we calculate the reduced
0
(single particle) density matrix. Regarding the von
0
2 During the pastdecade the characterizationofinsepa- Neumann entropy S as a good correlation measure for
n rablequantumcorrelations,orentanglementhasbecome fermions [14], raises the following puzzling issue. S at-
tains its minimum value S = 1 corresponding to
a one of the most active research fields. The reason for min
J this flurry of activity is two-fold. First the attitude of Slater rank one i.e., nonentangled states. This situation
is to be contrasted with the case familiar for two distin-
5 regarding entanglement like energy as a resource paved
2 the way to the appearance of quantum information sci- guishableparticleswherefornonentangledSchmidtrank
ence including such exciting applications like, teleporta- one states one has Smin = 0. However, as was shown in
1 [15] this contradiction is arising from the fact that the
tion[1],quantumcryptography[2]andmoreimportantly
v quantum computing [3]. Second, entanglement as ”the correlations of the Slater rank one state with Smin = 1
5
are related merely to the exchange properties of the in-
4 characteristic trait of quantum mechanics” [4] is of fun-
distinguishablefermions. Sincethesecorrelationscannot
1 damental importance for a deeper understanding of the
1 conceptual foundations of quantum theory. beusedtoimplementateleportationprocessortoviolate
0 The main problem is how to quantify entanglement. Bell’s inequality they cannot be regarded as manifesta-
5 tions of entanglement.
In this respect as far as entanglement of distinguishable
0
particles is concerned a large number of useful results The aim of the present paper is to study these issues
/
h exists. Entanglementmeasuresfor bipartite [5]andmul- by explicitly working out the example of two correlated
p tipartite [6] pure states have been defined and used in a fermionshavingfoursingle particlestates. Motivatedby
t- wide variety of interesting physical applications. How- geometric considerations after employing a special rep-
n ever, the challenging problem of quantifying also mixed resentation for the complex amplitudes of our fermionic
a state entanglement is still at its infancy. Although the wavefunctionweshowthatthevonNeumannandR´enyi
u
q developmentinthisfieldisapparent,apartfromsystems entropiescanbeexpressedintermsofthemeasureηviaa
: [7], [8], [9] of two qubits, and a qubit and a qutrit no simple formula. Ourelementaryformulais entirelyanal-
v simple sufficient and necessary conditions are known for ogous to the one known for distinguishable particles us-
i
X deciding whether a state is entangled or not. ingtheconcurrence . Moreover,ourconstructionyields
C
r Quantifyingquantumcorrelationsforsystemsofindis- thecanonicalform(Slaterdecomposition)explicitly. Un-
a tinguishable particles is a relativelynew topic. As a first likehowever,thecanonicalformof[10]wheretheexpan-
stepinthis directionSchliemannetal.[10]characterized sion coefficients are complex in our form they are non-
and classified quantum correlations in two-fermion sys- negative real numbers. Having real expansion coefficents
tems having 2K single-particle states. For pure states this decomposition is closer to the spirit of the Schmidt
they introduced in analogy to the Schmidt decomposi- decomposition for distinguishable particles than the one
tion, a decomposition in terms of Slater determinants. presented in Refs. [10] and [11]. As a next step by cal-
States with Slater rank (i.e., the number of Slater de- culating the von Neumann and R´enyientropies it is also
terminantsoccurringinthe canonicalform)greaterthan shownthat in this picture the residualentropySmin =1
one are calledentangled. A sufficient and necessarycon- reflecting the exchangepropertiesof the fermions canbe
dition for a state being entangled was established for reinterpreted as a manifestation of the generalized Pauli
K = 2 in [10], and for arbitrary K later in [11]. For exclusion principle. Finally it is shown that the residual
K =2 a measure 0 η 1 was introduced, Slater rank entropy can be given a nice geometric interpretation in
one (nonentangled)≤stat≤es correspond to η = 0, Slater terms of a nonseparable quadric surface in the five di-
rank two states with maximal entanglement correspond mensional complex projective space.
toη =1. Thisquantityinmanyrespectbehavessimilarly The organizationofthis paperis asfollows. InSec.II,
tothewell-knownconcurrence[12]0 1quantifying using a convenient representation the structure of the
≤C ≤
two qubit entanglement for distinguishable particles. In density matrixis elucidatedandthe canonicalformwith
a special case they can in fact be related [13]. real expansion coefficients is achieved. In Sec. III the
2
von Neumann and R´enyi entropies are calculated and where i is the imaginary unit. The geometric mean-
the limiting cases are discussed. Here the connection ing of the representation (hereafter to be called the ”U-
with the generalized Pauli exclusion principle is estab- representation”)arisingfromusingthisunitarytransfor-
lished. In Sec. IV the geometric background underlying mation will be explained later. Straightforward calcula-
ourconstructionisilluminated. Somecommentsandthe tionshowsthatthe transformedmatrixP′ UPUT has
≡
conclusions are left for Section V. the form
II. THE DENSITY MATRIX P′ =UPUT = 1(ε ε)(I aσ+bσ I), (7)
2 ⊗ ⊗ ⊗
AsastartingpointletusassumethattheHilbertspace where
describing the quantum correlations of two fermionic
sHystems(Cw4ithCfo4u)rwshienrgele praerfteircsletostaantetissyims mofettrhiezaftoiormn. a=E+iB, b≡E−iB, ε≡iσ2 =(cid:18) 01 01(cid:19), (8)
H≡A ⊗ A −
An arbitrary element Ψ of has the form
| i H aσ a1σ1+a2σ2+a3σ3 withσj,j =1,2,3thestandard
≡
Paulimatrices,I isthe2 2unitmatrix,andtheoverbar
3 ×
denotes complex conjugation. Notice also that with this
Ψ P c†c† 0 , (1)
| i≡ µν µ ν| i∈H notationwe have at our disposalthe importantrelations
µX,ν=0
wherec† andc , µ=0,1,2,3arefermioniccreationand
annihilaµtionopµeratorssatisfyingtheusualanticommuta- εσε=σ, ε2 = I, (9)
−
tion relations
and according to the normalization condition (3)
{cµ,c†ν}=δµν, {cµ,cν}=0, {c†µ,c†ν}=0, (2) a 2+ b 2 = 1, with a 2 aa, b 2 bb.
|| || || || 2 || || ≡ || || ≡
and 0 is the fermionic vacuum. Due to anticommu-
(10)
| i
tation the 4 4 matrix P with complex elements is an
Since the fermions are indistinguishable the reduced
antisymmetri×c one i.e., we have PT = P. Using these
one-particledensitymatricesareequalandhavetheform
−
relations it can be shown that the normalization condi-
[14]
tion ΨΨ =1 implies
h | i
2TrPP† =1. (3)
ρ=2PP†. (11)
It will be instructive in the following to stress the
similarity with the structure of invariants characterizing Then a calculation using (9) shows that
fermionic entanglement and the ones arising from elec-
trodynamics hence we parametrize our matrix P as
2UρUT =(I aσ+bσ I) I aσ+bσ I . (12)
0 E1 E2 E3 ⊗ ⊗ ⊗ ⊗
(cid:0) (cid:1)
Pµν ≡−EE12 B03 −0B3 BB21, (4) tUhseinngortmhealrizealattioionnc(ounσdi)t(ivoσn)(1=0),(uwve)Iob+taii(nut×hevr)eσsulatnd
−−E3 −B2 B1 −0 
i.e., P0j =Ej,Pjk = ǫjklBl, j,k,l=1,2,3. Itis impor- 1
tant to emphasize at−this point that unlike in electrody- UρU† = (1+Λ), (13)
4
namics here E and B are merely complex three vectors,
i.e., E,B C3. where
∈
As was demonstrated in [10] local unitary transfor-
mations U U with U U(4) acting on C4 C4 do Λ=2 I xσ+yσ I +bσ aσ+bσ aσ , (14)
⊗ ∈ ⊗ ⊗ ⊗ ⊗ ⊗
notchangethefermioniccorrelationsweareintendingto (cid:0) (cid:1)
and
study. Under such transformations P transforms as
P UPUT. (5) x ia a, y ib b, 1 I I. (15)
7→ ≡− × ≡ × ≡ ⊗
A representation convenient for our purposes can be ob-
Noticethatthevectorsxandyarerealonesi.e.,elements
tained by choosing the unitary matrix U ∈U(4) as of R3.
1 0 0 1 InordertoobtaintheeigenvaluesofρwecalculateΛ2.
1 0 1 i 0 The calculation is easily performed after noticing that
Uµν ≡ √20 1 −i 0 , (6) due to the relations xa = xa = yb = yb = 0 the first
1 0 0 1 two and the last two terms in Λ anticommute with each
 − 
3
other. Astraightforwardcalculationusingthedefinitions Using the theorem above we see that the canonical
(8), the normalization condition (10) and the relations formofourP inEq.(1)isgivenby(20)withK =2and
x 2 = a 4 a2a2, y 2 = b 4 b2b2 shows that
|| || || || − || || || || −
λ+ λ−
Λ2 =(1 64EB2)(I I)=(1 η2)1. (16) r1 =r 2 , r2 =r 2 . (22)
− | | ⊗ −
With this notation
The quantity
0 η 8P01P23 P02P13+P03P12 =8EB 1 (17) |Ψi= 2λ+C0†C1†|0i+ 2λ−C2†C3†|0i, (23)
≤ ≡ | − | | |≤ p p
where
is the measure for fermionic correlations introduced in
[10]. Since DetP = (EB)2 we see that η is invariant 3
under local unitary transformations of the form (5). c† = C†. (24)
µ Vνµ ν
Now Eq. (16) implies that the eigenvalues of Λ are νX=0
1 η2 each of them doubly degenerate. Using this
± − It is clear that for the calculation of the Slater states
repsult in (13) we obtain for the eigenvalues of ρ
(the analogues of the Schmidt states) appearing in (23)
we have to determine the unitary along the lines as
V
1 presented in [16]. Notice that in our case this as a
λ± = 4(cid:16)1±p1−η2(cid:17), (18) function of the complex numbers E and B canVbe ob-
tained explicitly. In order to give some hints notice that
each of them doubly degenerate. according to (16) the 4 4 matrices
In [10] a fermionic analogue of the usual Schmidt de- ×
compositionfordistinguishableparticleswasintroduced.
Adapted to our situation the theorem of [10] states that 1 1
Π 1 Λ , r 1 η2 (25)
there exists a unitary matrix U(4) (not to be con- ± ≡ 2(cid:18) ± r (cid:19) ≡ −
U ∈ p
fused with our U of expression (6)) such that
are orthogonal projectors of rank two, i.e., they satisfy
Π2 =Π , and Π Π =0. Let us define the vectors
± ± ± ∓
0 z1 0 0
Z = P T, where Z =−z1 0 0 0, (19)
U U 0 0 0 z2 v0 =N0Π+e0, v1 =N1Π+e1. (26)
 0 0 −z2 0
where z1 and z2 are complex numbers. When one of the
complex numbers zi i = 1,2 is zero we have Slater rank v2 =N2Π−e2, v3 =N3Π−e3, (27)
one, for both z being nonzero Slater rank two states.
i
According to [10], a fermionic state is called entangled if where eµ µ = 0,1,2,3 are unit vectors corresponding to
andonlyif its Slaternumber is strictly greaterthanone. the columns of the unitary inEq.(6), Nµ are normaliza-
However, according to a theorem of Zumino[16] even tion constants. Then the vµ are normalized eigenvectors
more can be said. ofρ. Withthehelpoftheseeigenvectorswecanbuildup
Theorem. If P is a complex 2K 2K skew symmetric theunitarydiagonalizingthedensitymatrixwiththede-
matrix, then there exist a unitary×transformation pendence on E and B explicitly displayed, the first step
U(2K) such that V ∈ needed for the determination of [16].
V
R= P T, where R=diag[R1,R2,...RK], (20) III. ENTROPY
V V
with Having the eigenvalues and the canonial form at our
disposal we can now write down the explicit form of
0 r
R = i , (21) entropies used in quantum information theory. These
i (cid:18)−ri 0(cid:19) are the von Neumann and the quantum counterpart of
R´enyi’s α (α=2,3,...) entropies [17] defined as
where r i = 1,2,...K are nonnegative real numbers.
i
Notice that unlikethe decompositionof[10]the one pre-
sented in (20) is closerto the spirit of the usual Schmidt 1
decomposition where the expansion coefficents are non- S1 ≡−Trρ log2ρ, Sα ≡ 1 αlog2Trρα α>1.
negative real numbers. − (28)
4
Noticethatforconveniencewehavechosen2forthebase also holds. Applying (33)
of the logarithm, and the von Neumann entropy can be
regUasridnegd(a1s8)thweeαobtetanidnstthoe1exdpelcirceitasfionrgmlyullaimit of Sα. −S2 =log2M−1λ2µ ≤log2M−1λµN1 =−log2N (36)
µX=0 µX=0
and using (35) finally leads to
S1(η)=1 xlog2x (1 x)log2(1 x), (29)
− − − −
log2N S1 log2M, log2N S2 log2M, (37)
≤ ≤ ≤ ≤
which is clearly a generalization of (32) for an arbitrary
1
S (η)=1+ log (xα+(1 x)α), α>1 (30) particle number N with α=1,2.
α 2
1 α −
− Notice that Sα = 1 iff η = 0. These are the states
havingSlaterrankone,i.e., Ψ inthiscasecanbetrans-
where
| i
formedvialocalunitariesU U withU U(4)toasingle
1 Slaterdeterminant. Mathem⊗aticallythi∈smeansthatP
x (1+ 1 η2), (31) µν
≡ 2 − of(1)is aseparable bivector,i.e.,there existfour-vectors
p
u and v µ,ν =0,1,2,3 such that P =u v u v .
µ ν µν µ ν ν µ
with η given by (17). All of our entropies satisfy the −
In order to study in our formalism the S =2 (η =1)
α
inequalities
casecorrespondingtoSlaterranktwostatessatisfyingan
additional requirement we introduce some terminology.
Let us define the matrix
1 S (η) 2, α 1. (32)
α
≤ ≤ ≥
Note, that the left hand side inequality is a consequence 1 0 0 0
of the antisymmetry property of the two-particle state 0 1 0 0
g  − . (38)
Ψ as it was shown in [15]. This statement, however, ≡ 0 0 1 0
|is ai special case of a more general result obtained from 0 0 −0 1
 − 
the so-calledPauliprinciple for density matrices,related
to the N-representability problem of ρ. For N-particle Then a short calculation using (6) shows that
fermionic systems the following question is of physical
relevance. Given a one-particle density matrix ρ, does
UgUT =ε ε. (39)
there exist an N-particle density matrix ρ with the
N ⊗
usual properties and which is antisymmetric with re-
We use g to raise and lower indices in the usual way
spect to the exchangeof particles, satisfying the relation
hence for example we have Pµν gµκgν̺P , in short
ρ = Tr2,...,NρN? Operation Tr2,...,N is a partial trace a quantity like gPg corresponds t≡o P with bκo̺th indices
on the particle indices 2,...,N. Clearly, a physical re-
raised.
duced density matrix should satisfy this requirement, in
Now let us define the dual ∗P of P as
this case it is called N-representable. If, furthermore,
ρ = Ψ Ψ, ρ is called pure state N-representable.
N
| ih |
It is a result of Coleman [18] that a necessary and 1
∗P ǫ Pκ̺. (40)
sufficient condition for N-representability can be formu- µν ≡ 2 µνκ̺
lated using the eigenvalues λ0,...,λM−1 of ρ (here M
Here ǫ is the fourthordertotally antisymmetricten-
stands for the dimension of the basis set of the one- µνκ̺
particle Hilbert space). The reduced density operator sor defined by the condition ǫ0123 =1. Then we see that
∗E= B and ∗B =E. Moreover for the U(4) invariant
ρ is N-representable iff −
η occurring in our formulae for the entropy we have
0 λ 1/N for any µ=0,...,M 1. (33)
µ
≤ ≤ −
η =2∗P Pµν 2Tr(∗PgPg). (41)
The above condition is known as the generalized Pauli | µν |≡ | |
principle in the literature and is obviously satisfied by
Comparing this with (3) we see that η =1 iff
the eigenvalues (18) with N =2, M =2K =4.
Consideringnowtheentropyexpressions(28)Jensen’s
inequality results in the standard relations P =eiθg(∗P)g, (42)
0 S1 log2M, 0 S2 log2M, (34) where eiθ is an arbitrarycomplex phase factor. In terms
≤ ≤ ≤ ≤
of E and B this means that η 1 iff
≡
moreover,it can be shown [19] that
S2 S1 (35) E=eiθB. (43)
≤
5
Transformingthisequationwiththeunitary(6)weget
P′ =eiθ(ε ε)∗P′(ε ε). (44) σA2B′ =−σ2AB′ = √12(cid:18)0i −0i(cid:19), (49)
⊗ ⊗
Withanabuseofnotationwecanomittheprimeandwe
caanndsoanylythifat in the U-representation of Eq. (7) η = 1 if σA3B′ =σ3AB′ = √12(cid:18)10 01(cid:19). (50)
−
Here A,B′ = 0,1 are the matrix (spinor) indices of the
∗P =eiθP˜, (45) Pauli matrices. The quantities σAB′ and σµ µ =
µ AB′
0,1,2,3can be usedto convertvector andspinor indices
wherewehaveintroducedthespinflipoperationofWoot-
back and forth. For example for a four vector a we can
ters [9] playing a crucial role in the definition of the en- µ
tanglement of formation for two-qubit systems, (recall formthefourcomponentspinorialobjectaAB′ =σAµB′aµ
wheresummationoverµis understood. Writing outthis
that iσ2 =ε)
relation explicitly we have
P˜ ≡σ2⊗σ2Pσ2⊗σ2. (46) a00′ 1 0 0 1 a0
Hence in the U-representation for states with maximal a01′= 1 0 1 −i 0 a1. (51)
fermionic entanglement their duals are equal to their a10′ √2 0 1 i 0 a2
spin-flipped transforms (up to a phase). This result has a11′ 1 0 0 −1a3
to be compared with the similar one obtained in [11].
Comparing this with Eq. (6) we see that our use of the
Here we also uncoveredthe instructive connectionofdu-
U-representation amounts to reverting to the spinorial
alization and its connection with the spin flip operation
analogue of our tensorial quantities. In particular the
(i.e., time reversal) of quantum information theory. No-
transformationofEq.(5)inthisformalismtakestheform
tice also that in the original (42) representation taking
the spin-flip transform amounts to complex conjugation
ffroollmowsepdecbiyalrareisliantgivbitoyt.hTinhdeicroesotwsiothf tthhiesmcoartrreisxpgonkdneonwcne Pµν 7→PAA′BB′ =σAµA′σBνB′Pµν. (52)
will be revealed in the next section. Moreover,Eq.(39)becomes oneofthe basicidentities of
the spinorial formalism
IV. THE GEOMETRY OF FERMIONIC
ENTANGLEMENT εABεA′B′ =σAµA′σBνB′gµν. (53)
Inthissectionweclarifythegeometricmeaningofour Ourdecomposition(7)inthis picture correspondsto the
U-representation,andthe residualentropyS =1. To well-known one in the spinor formalism of Penrose and
min
begin with, notice that our U-representation is a vari- Rindler [21]
antof the method of expressingquantities insteadof the
computational base in the so called ”magic base” of Hill
and Wootters [12]. The use of this base has its roots PAA′BB′ =εABψA′B′ +ϕABεA′B′. (54)
in the group theoretical correspondences (SL(2,C)
× Notice that the symmetric spinors ψ and ϕ correspond
SL(2,C))/Z2 ≃ SO(4,C), (SU(2) × SU(2))/Z2 ≃ to 1ε(aσ)and 1ε(bσ),respectively. Itisstraightforward
SO(4). These correspondences have been used succes- 2 2
to check that ∗a = ia and ∗b = ib. Tensors satisfy-
fully forestablishingexactresultsforthe behaviorofthe ing ∗P = iP are called [21] self d−ual and anti self-dual,
entanglement of formation [9], [20]. Here we would like ±
respectively. Hence our decomposition (7) is in terms of
to provide a different insight on the effectiveness of this
theself-dualandanti-selfdualpartsofourmatrixP. As
base provided by geometry.
it is well-known spinorial methods proved to be of basic
Letusconsiderthequantities(Infeld–vanderWaerden
importance for a Petrov type of classification of curva-
symbols)
ture tensors in general relativity [21]. It is interesting to
notethatthesemethodsprovedtobeofrelevanceforthe
σA0B′ =σ0AB′ = √12(cid:18)10 01(cid:19), (47) cgllaesmsiefincta,ttioonoo[2f2t]h.reTe-hqeubbiats(icaniddepaobsseihbilnydn-tqhuisbiatp)pernotaacnh-
to n-qubit entanglement is to convert spinorial indices
reflecting transformation properties under the group of
σA1B′ =σ1AB′ = √12(cid:18)01 10(cid:19), (48) tni-cfollodcatlenospoerraptrioodnuscatsndofcSlaLs(s2ic,aCl)corempmreusnenictaintigonstoocfhtahse-
6
entangled parties to vectorial ones (or vice versa) and Twounnormalizedqubitsarecharacterizedbyfour com-
then use the techniques as developed in twistor theory. plexnumbers,hencetherelevantspaceofraysinthiscase
Finally,letusdiscussthegeometricmeaningofS = is the three dimensional complex projective space CP3.
min
1characterizingnonentangledfermionicstates. FromEq. Nonentangled states are the ones for which the concur-
(1) it is clear that an unnormalized fermionic state Ψ rence is zero. It can be shown [12] (again by using the
| i C
can be characterized by six complex numbers, i.e., ele- magicbase)thatfourcomplexcoordinatesw0,w1,w2,w3
ments of C6. However, the space of states of a quantum can be introduced such that for vanishing they satisfy
C
systemis the space ofrays a spaceobtainedby identi- the relation
P
fying states Ψ and Φ ifthey arerelatedby Ψ =cΦ
| i | i | i | i
where 0 = c C. In our case is the five dimensional
complex6proj∈ective space i.e., P CP5. Alternatively, w02+w12+w22+w32 =0. (58)
P ≃
one can consider the space of normalized states which
This equation defines the four real dimensional quadric
icsasteheC1P15dcimanenaslisoonablesrpehgearredeSd11as⊂thCe6s≃pacRe1o2.f eIqnutivhais- Q2(C) in CP3. Now as in the fermionic case nonentan-
lenceclassesofnormalizedstatesdefineduptoacomplex egnletdansgtaletdesoanreespbaerlaomngettroizietdsbcoymthpelepmoeinnttsinofCQP2(3C. ),and
phase. (S11 has elevenrealdimensions, a complex phase
It was shown in [24] that for distinguishable qubits a
of unit magnitude is the circle S1 which has one real
dimension, and CP5 has 11−1=10 real dimensions.) mspeaacseuorferoafysenCtaPn3gcleamnebnetecqaunipbpeeddewfiinthedthaesFfuolbloinwi-sS.tTudhye
Let us now consider the constraint η = 0 which gives
metric [25], which is induced by the standard Hermitian
riseaccordingtoEqs.(29)–(31)toSmin =1forallofour scalarproductonC4. Letusfixanentangledstateoffthe
entropies. This condition is
(58) quadric. Thenthe measure of entanglementfor this
state is related to the length of the shortest arc of the
P01P23 P02P13+P03P12 =0, (55) geodesic with respect to the Fubini-Study metric, con-
− nectingthestateinquestionwiththe(58)quadric. More
whichisthePlu¨ckerrelationamongthesixcomplexcoor- precisely we have
dinates characterizing separable bivectors. As it is well-
known, a bivector (an antisymmetric 4 4 matrix) is
separable if and only if condition (55) h×olds. This re- cos2s = 1(1+ 1 2) (59)
2 2 −C
sult dates back to the work of Plu¨cker and Klein in the p
middle of the 19th century and was rediscovered in the where0 1istheconcurrence,andsisthegeodesic
≤C ≤
contextoffermionicentanglementin[10]. Theseparabil- distance. The separable states corresponding to the two
ity conditions for an arbitrary dimensional bivector can pointsofintersectionofthisgeodesicwithQ2(C)arejust
be found in Penrose and Rindler [21], or in connection the ones occurring in the Schmidt decomposition [26].
with fermionic entanglement in [11]. The proof of this theorem in [24] can be trivially gen-
Using the coordinates eralized for an arbitrary quadric Qn−1(C) in CPn. In
particular for n=5 we get the result
(z0,z1,z2,z3,z4,z5) (a1,a2,a3,ib1,ib2,ib3) (56)
≡ s 1
cos2 = (1+ 1 η2). (60)
where the components of a and b are related to the 2 2 −
p
Plu¨ckercoordinatesP byEqs.(4)and(8), the Plu¨cker
µν
Hence the measure of nontrivialfermionic correlationsis
relations can be written as
related to the geodesic distance between the fermionic
state in question and the Klein quadric of nonentangled
z2+z2+z2+z2+z2+z2 =0. (57) states by Eq. (60). Notice also that the Slater decompo-
0 1 2 3 4 5
sition of Eq. (23) can be reexpressed as
This equation is homogeneous of degree two and defines
a quadric surface Q4(C) (the so called Klein quadric) in s s
CP5. As far as we know the (57) quadric has made |Ψi=cos2C0†C1†|0i+sin2C2†C3†|0i, (61)
its debut to physics as early as 1936 in the seminal
workofDirac[23]ofconformalgeometryandwaveequa- whichforvariablesdescribesafamilyofentangledstates
tions. Here the Klein quadric is the eight real (four lying on a horizontal [26] geodesic. The normalized sep-
complex)dimensionalmanifoldcharacterizingnonentan- arable Slater states C†C† 0 and C†C† 0 are on the
0 1 2 3
gled fermionic states. Q4(C) is a submanifold of CP5. quadric Q4(C). They can|ailso be calcu|laited by a La-
StatesofCP5 notlyinginQ4(C)areexhibitingnontriv- grange multiplier technique as in [24] giving a geometric
ial fermionic correlations hence they are entangled. meaningto the Slaterdecompositionofafermionicstate
Inordertogainmoreinsightonthenatureoffermionic having four single particle states.
entanglement let us compare these results with the cor- Now the question arises: is there any basic differ-
responding ones known for two distinguishable qubits. ence between the quadrics Q2(C) and Q4(C) that can
7
account for the different physical situations as reflected theseeigenvaluescanbeexpressedintermsoftheinvari-
by the different minimum values of their respective en- ant η of [10] via an elementary formula analogous to the
tropies? The answer to this question is surprisingly yes. one well-known for distinguishable qubits. In this way
Itis a theoremindifferentialgeometrythatthe quadrics wemanagedto representourentangledstatein acanon-
Qn−1(C) in CPn parametrized by homogeneous coordi- ical form (the so called Slater decomposition) with real
nates Z0,Z1,...Zn satisfying the additional constraint nonnegative expansion coefficents. This decomposition
n Z2 = 0 are symmetric spaces that can be repre- is closer to the spirit of the standard Schmidt decom-
j=0 j
Psented in the form [25] position thran the one presented in [10], and [11] with
complex expansion coefficents.
Using these results we have computed the von Neu-
Qn−1(C) SO(n+1)/SO(2) SO(n 1). (62) mannandR´enyientropiesthatcanalsobe usedto char-
≃ × −
acterize fermionic correlations. These entropies satisfy
Fortheveryspecialvaluen=3,SO(4) SU(2) SU(2) the bound 1 S (η), α = 1,2,.... This inequality is
i.e., this group exhibits a product s∼tructure.× Since to be contrast≤ed wαith the corresponding one 0 S ( )
SO(2) U(1) and SU(2)/U(1) S2 one can show that known for distinguishable qubits, where is the≤conαcuCr-
Q2(C)≃ S2 S2,i.e., the direct≃productoftwo-spheres. rence. We have shown that the differencCe in the bounds
≃ ×
These spheres are just the Bloch spheres corresponding can be traced back to fact that the so-called Pauli prin-
to the distinguishable qubits in a separable state. The ciple for density matrices has to hold. We related the
embedding of Q2(C) in the form S2 S2 ֒ CP3 is the special values for the entropies satisfying the lower or
× →
special case of the so called Segr´e embedding having al- upper bounds to the algebraic properties of the matrix
readybeenusedingeometricdescriptionsofseparability P.
for distinguishable particles [27].
We have also clarified the geometric meaning of the
For n 4 the corresponding symmetric spaces are
U-representation. An interesting and useful connection
≥
irreducible [25], hence they cannot be represented in a
with the spinor formalism of Penrose and Rindler hith-
product form of two manifolds. Hence we can conclude
ertousedmerelywithintheratherexoticrealmoftwistor
that the manifold of nonentangled states representing
theory was pointed out. Next we initiated the study of
quantum systems of distinguishable or indistinguishable quadrics Qn−1(C) embedded in the space of rays CPn
constituents exhibits different topological structure. For
forrevealingthegeometricaspectsofentanglement. The
nonentangled distinguishable particles (S = 0) we
min cases n = 3 and n = 5 correspond to the simplest cases
havea productstructureofthe state spaceQ2(C)which ofentanglementfor systems with distinguishable andin-
conforms with our expectations coming from classical
distinguishable constituents. We noticed that previous
mechanics. However, for nonentangled indistinguishable
results connecting the measureof entanglementwith the
fermions (S = 1) no product structure of the state
min geodesic distance between states and the quadric can be
space Q4(C) can be observeddue to correlationsreflect- generalized for the fermionic case as well. Finally we
ingtheexchangepropertiesofthefermions. Thesecorre-
have proved that the different physical situations show-
lations are of intrinsically quantum in nature. However,
ing up as the occurrenceof different minimum values for
they are not to be confused with the correlations that
the entanglement entropies, are also reflected in the dif-
can be regarded as true manifestations of entanglement.
ferent topological properties of the quadrics Q2(C) and
Representative states of this kind belonging to the com- Q4(C). Q2(C) exhibits a product structure S2 S2 of
plement of Q4(C) can be used to implement quantum twoBloch-sphereswhichconformswithourexpec×tations
information processing tasks.
based in classical physics. However, for Q4(C) i.e., the
Kleinquadric,as the manifoldof nonentangledstates no
product representation is available. This fact can be re-
V. CONCLUSIONS
garded as a geometric manifestation of the Pauli princi-
ple,showingthe existenceofcorrelationsrelatedentirely
In this paper we studied the nature of quantum corre- to the exchange properties of fermions. The main con-
lations for fermionic systems having four single particle cern of quantum information science can be the use of
states. Though these are the simplest systems among states off the Klein quadric. These are the ones that can
the fermionic ones exhibiting such correlations,but they be used to implement quantum information processing
clearlyshowsomeofthebasicdifferencesbetweenentan- tasks.
glementpropertiesofquantumsystemswithdistinguish-
able and indistinguishable constituents.
As a starting point, we employed a comfortable, the
VI. ACKNOWLEDGEMENTS
so-called U-representation for the 4 4 antisymmetric
×
matrix P containing the six complex amplitudes repre-
senting our fermionic system. This representation en- FinancialsupportfromtheOrsz´agosTudom´anyosKu-
abledanexplicitconstructionofthereduceddensityma- tata´siAlap(OTKA),(grantnumbersT032453,T038191,
trix,itseigenvaluesandeigenstates. Wehaveshownthat and T046868)is gratefully acknowledged.
8
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