Entangled bases with fixed Schmidt number

0.15 MB·English

by  Yu Guo

Entangled bases with fixed Schmidt number
Yu Guo,1 Shuanping Du,2 Xiulan Li,1 and Shengjun Wu3
1School of Mathematics and Computer Science, Shanxi Datong University, Datong, Shanxi 037009, China
2School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361000, China
3Kuang Yaming Honors School, Nanjing University, Nanjing, Jiangsu 210093, China
An entangled basis with fixedSchmidtnumberk (EBk) is aset of orthonormal basis states with
thesame Schmidtnumberk in a product Hilbert space Cd⊗Cd′. It isa generalization of both the
product basis and themaximally entangled basis. We show here that, for any k≤min{d,d′}, EBk
exists in Cd ⊗Cd′ for any d and d′. Consequently, general methods of constructing SEBk (EBk
with the same Schmidt coefficients) and EBk (but not SEBk) are proposed. Moreover, we extend
5
theconcept of EBk tomultipartitecase andfindout thatthemultipartiteEBk can beconstructed
1
similarly.
0
2
PACSnumbers: 03.67.Mn,03.67.Hk,03.65.Ud.

b
e
F I. INTRODUCTION thereareatleast2(k 1)setsofUEBkwhenbothdand
−
d are the multiples of k. UEBk can be considered as
5 ′
2 Entanglement is a fundamental feature of quantum a generalization of both UPB and UMEB from different
physics, and it is also proved to be a central resource in directions. In Ref. [12, 13], it is shown that there exists
] MEB (maximally entangled basis) in Cd Cd for any d.
h quantum information and quantum computation [1, 2]. ⊗
We remark that the product basis (PB) always exists in
p Consequently, characterizing entanglement is one of the
- fundamentalprobleminthisfield. Oneindispensableap- Cd Cd′ for arbitrary d and d′ and it is a set of pure
nt proach is to analyze the bases of the state space, such stat⊗es whose Schmidt numbers are 1. Furthermore, the
a as the unextendible product basis (UPB) [3], the un- MEB is setofpure states whose Schmidt numbers ared.
u extendible maximally entangled basis (UMEB) [4], and Then, a related question is arisen: is there an entangled
q the unextendible entangled basis with Schmidt number basiswithanyfixedSchmidtnumberinCd Cd′? Inthis
[ k (UEBk) [5], etc. paper, we show that such bases exist in b⊗oth Cd Cd′
2 Bipartite basis (complete or incomplete) in Cd Cd′ and Rd⊗Rd′ for any d and d′, and we provide me⊗thods
v ⊗ to construct them. We also show that our methods can
is a fundamental problem in both quantum physics and
0 bedirectlyextendedtothemultipartitecase. Suchbases
0 mathematics. The unextendible product basis is a set of
couldbeusefulwhenonestudyprojectivemeasurements
4 orthogonal product vectors whose complementary space
onto basis states with a fixed Schmidt number.
6 does not contain product states, which can be used for
The material in this paper is arranged as follows. In
0 constructingboundentangledstates[6]. AUMEBisaset
1. oforthonormalmaximallyentangledpurestatesinatwo- Sec. II, we introduce the concepts of EBk, SEBk and
MEB.InadditiontherelationbetweenEBkandtherank-
0 qudit system consisting of fewer than d2 members which
k Hilbert-Schmidtbasisoftheassociatedmatrixspaceis
5 havenoothermaximallyentangledvectorsorthogonalto
1 all of them. It is shown that there is no UMEB in the illustrated. Sec. III contains methods of constructing
v: two-qubitsystem,asix-memberUMEBexistsinC3 C3 EBk whenever dd′ is a multiple of k. In Sec. IV, we
i anda 12-memberUMEB exists inC4 C4 [4]. Late⊗r,B. discuss the case when dd′ is not a multiple of k by ana-
X ⊗ lyzingthe Hilbert-Schmidt basiswithfixedrankk inthe
Chen andS.-M.Fei provedinRef. [7]that there exists a
ar set of d2-member UMEB in Cd⊗Cd′ (d2′ < d < d′) and sdpisaccuessoefdc.oTeffiheciemnutsltimpaarttriitceesc.asBeoitshdEisBcukssaenddinSESBeck.aVre.
questionedtheexistenceofUMEBsinthecaseofd 2d.
′ ≥ Finally, we conclude in Sec. VI.
Recently,inRef. [8],theauthorsprovedthattheremight
be two sets of UMEBs in any bipartite system, and an
explicit construction of UMEBs is put forward. Some
II. DEFINITION AND PRELIMINARY
properties of UMEBs are given in Ref. [9].
One of the most crucial quantity associated with bi-
Recall that, the Schmidt number of a pure state ψ
puaserdtitteopcuhraerascttaetreiziesatnhde qSuchamntiidfyt tnhuemdbeegrr,ewehoifchbipcaanrtibtee Cd Cd′,denotebySr(ψ ),isdefinedasthelengtho|ftih∈e
⊗ | i
entanglement for pure states directly [10, 11]. In Ref. Schmidt decomposition[14]: if |ψi= mk=−01λk|eki|e′kiis
[5], we introduce the unextendible entangled basis with its Schmidt decomposition, then Sr(ψP)=m. It is clear
| i
gSecnhemraidltwnauymobfecronkst(r2uc≤tinkg<sumchina{bda,dsi′s})w(itUhEaBrbki)traanrdyda tthhaetreSdru(c|ψedi)st=ateraonfkth(ρe1i) =thrpaanrkt.(ρA2)s,tawtheerψe ρiCddenoCteds′
− | i∈ ⊗
and d is proposed. Consequently, it is shown that there is called a maximally entangled state if S (ψ ) =d and
′ r
| i
are at least k r (here r = d mod k, or r = d mod k) λ = λ = = λ . Hereafter, we always assume that
′ 1 2 d
− ···
sets of UEBk whend or d is not the multiple ofk, while d d for simplicity.
′ ′
≤
2
Definition1. Anorthonormalbasis φ inCd Cd′ is 1 0 ), and 1 (0 1 1 0 ) form a MEB. In gen-
{| ii} ⊗ | i| ′i √2 | i| ′i− | i| ′i
called an entangled basis with Schmidt number k (EBk) eral, for the case of d d, d 2, let
if S (φ ) = k 2 for any i. Particularly, it is called a ⊗ ≥
r i
| i ≥
special entangled basis with Schmidt number k (SEBk) d 1
1 −
eifqiutailstoan1E,BakndwiitthisacllaltlheedSacmhmaxidimtaclolyeffienciteanntgsleodfb|ψaisiiss |Ω0,0i= √dXi=0|ii|i′i
√k
(MEB) if φ is maximally entangled for any i.
PB alw|ayisiexists in Cd Cd′ for any d and d. It and
′
⊗
is clear that SEBk reduces to PB (MEB) when k = 1 Wˇ i =ξm(i n) i n , (2)
m,n −
(k = d). That is, SEBk is a generalization of both PB | i | − i
and MEB. MEB is equal to SEBd while an EBd is not where Wˇm,ns are the Weyl operators, ξ = e2dπi and i
a MEB necessarily. By definition, it is clear that φi n i n+d (here, the roman letter ‘i’ denotes|th−e
is an EBk if and only if Ua Ub φi is an EBk, {w|heir}e imia≡gin|a−ryunit,ii n+dmeansi n+dmodd). Thenthe
U is any unitary opera{tor⊗on C|d/di′}. actions of the We−yl operators pr−oduce a MEB [12, 13].
a/b
CdWeCads′s.uLmeet that {|ψii}id=d′0−1 is set of pure states in |Ωm,ni=(Wˇm,n⊗1)|Ω0,0i, (3)
⊗
ψ = a(i) k l , where0 m,n d 1. Infact,Eq.(2)canbesimplified
| ii Xk,l kl| i| ′i as ≤ ≤ −
wofhtehreefi{|rksti}Cadnadn|dl′ithaerestehcoensdtaCndd′a,rrdecspomecptiuvtealyti.oWnaelbwarsietes Wˆm,n|ii=ξmi|i−ni. (4)
Then
A =[a(i)],
i kl
Ωˆ =(Wˆ )Ω (5)
m,n m,n 0,0
then A is a d by d matrix, S (ψ ) = rank(A ) and | i ⊗1 | i
i ′ r i i
| i
tdhhψ′eic|oψimnjnipel=rexpTmrro(aAdtur†iciActejsd)..efiTLnehetednMbMyd×dd×A′db′Beista=hHeTislprb(aeAcret†Bsopf)aafcolelrwdainbthyy wminidetthurci0cea≤allyMm, bE,noBt≤has(d1w−⊗e1llW.aˇlsmo,nfo)|rΩm0,a0iMaEndB(in1C⊗dW⊗ˆmC,dn.)|SΩy0m,0i-
h | i
A, B d d′. It turns out that Ai :rank(Ai)= k is
aHilb∈erMt-Sc×hmidtbasisofthespac{eof d d′ ifando}nly
if {|ψii}id=d′0−1 is an EBk in Cd⊗Cd′. MFor×simplicity, we B. k=d<d′
call {Ai : rank(Ai)= k, Tr(A†iAj)= δij} a rank-k basis
in the following. That is, there is a one-to-one relation We beginwith the simple case of2 3. Itis clearthat
⊗
between the EBk ψ and the rank-k basis A :
i i
{| i} { } 1
φ = (0 0 + 1 1 ),
ψi Ai, ψi Ai . (1) | 1i √2 | i| ′i | i| ′i
| i↔ {| i}↔{ }
1
Therefore, the EBk problem is equivalent to the rank- φ = (0 0 1 1 ),
2 ′ ′
k basis of the associated matrix space. The important | i √2 | i| i−| i| i
role of Eq. (1) will be seen in Sec. IV for the case when 1
φ = (0 1 + 1 2 ),
dd′ is not a multiple of k, although the case when dd′ | 3i √2 | i| ′i | i| ′i
is a multiple of k in the next section can be worked out
1
without using Eq. (1). φ4 = (0 1′ 1 2′ ),
| i √2 | i| i−| i| i
1
φ = (0 2 + 1 0 ),
III. dd′ IS A MULTIPLE OF k | 5i √2 | i| ′i | i| ′i
1
This section is divided into three cases. We firstly |φ6i = √2(|0i|2′i−|1i|0′i)
discuss the case k = d = d, then consider the case
′
k = d < d′, and study the case k < d ≤ d′ at last. is a MEB in C2 ⊗C3. In general, in a d⊗d′ (d < d′)
The methods of constructing EBk can be induced from system, let
the structure of the MEB.
d 1
1 −
Ω = i i
0,0 ′
| i √d | i| i
A. k=d=d′ Xi=0
and
For the two-qubits case, the Bell basis states,
√12(|0i|0′i + |1i|1′i), √12(|0i|0′i − |1i|1′i), √12(|0i|1′i + W˜m,n|i′i=ξmi|(i−n)′i, (6)
3
mwheaernesξi=ne2+dπidanmdo|di−dn).iT≡h|ein−n+d′i (here i−n+d′ |φ7i = √13(|2i|3′i+ξ|3i|4′i+ξ2|0i|2′i),
′ ′
−
1
|Ω˜m,ni=(1⊗W˜m,n)|Ω0,0i (7) |φ8i = √3(|2i|3′i+ξ2|3i|4′i+ξ4|0i|2′i),
with 0 m d 1 and0 n d 1 induce a MEB in 1
Cd C≤d′. ≤ − ≤ ≤ ′− |φ9i = √3(|1i|3′i+|2i|4′i+|3i|5′i),
T⊗hat is, SEBd always exists in Cd Cd′ with d d′. 1
We can also show that there is EBd⊗but not SEB≤d in φ10 = (1 3′ +ξ 2 4′ +ξ2 3 5′ ),
Cd Cd′. Let | i √3 | i| i | i| i | i| i
⊗ 1
φ = (1 3 +ξ2 2 4 +ξ4 3 5 ),
2−d 2 2 ··· 2 | 11i √3 | i| ′i | i| ′i | i| ′i
2 2 d 2 2
Od = d1 22... −22... 2−2... d ···.···.···. 2−2... d, ||φφ1132ii == √√1133((||00ii||33′′ii++|ξ1|1i|i4|4′i′i++|2ξi2||52′ii|)5,′i),
then Od is a d by d orthogonal matrix and Ud = iOd is φ = 1 (0 3 +ξ2 1 4 +ξ4 2 5 ),
14 ′ ′ ′
a unitary matrix. Replacing the coefficients in Eq. (5,7) | i √3 | i| i | i| i | i| i
by the entries in the columns (or rows) of the Ud or Od 1
φ = (3 0 + 0 4 + 1 5 ),
respectively,onecangettheexistenceofEBdinanyd⊗d′ | 15i √3 | i| ′i | i| ′i | i| ′i
system.
1
Infact, anyd by disometricmatrix X =[xkl]without |φ16i = √3(|3i|0′i+ξ|0i|4′i+ξ2|1i|5′i),
zero entries can induce an EBd since we can replace the
1
coefficients in Eq. (5,7) by the entries in the columns of φ = (3 0 +ξ2 0 4 +ξ4 1 5 ),
17 ′ ′ ′
any d by d isometric matrix without zero entries. That | i √3 | i| i | i| i | i| i
is, we replace the coefficient ξmi of ξmi i (n i) by 1
xn+1,m+1. Here, an m by n matrix A i|si|an −isom′eitric |φ18i = √3(|2i|0′i+|3i|1′i+|0i|5′i),
matrix if A A = I , I is the n by n identity matrix.
† n n 1
Sincethereareinfinitelymanyisometricmatrices,wecan φ19 = (2 0′ +ξ 3 1′ +ξ2 0 5′ ),
| i √3 | i| i | i| i | i| i
constructinfinitelymanyEBds. Alsonotethat,thereare
1
many ways of constructing MEB since we can replacing φ20 = (2 0′ +ξ2 3 1′ +ξ4 0 5′ ),
the coefficient ξmi by any other ones that guarantee the | i √3 | i| i | i| i | i| i
orthogonality. 1
φ = (1 0 + 2 1 + 3 2 ),
21 ′ ′ ′
| i √3 | i| i | i| i | i| i
1
C. k<d≤d′ φ22 = (1 0′ +ξ 2 1′ +ξ2 3 2′ ),
| i √3 | i| i | i| i | i| i
1
We now consider the EBk in a d d system with dd φ = (1 0 +ξ2 2 1 +ξ4 3 2 ).
′ ′ 23 ′ ′ ′
⊗ | i √3 | i| i | i| i | i| i
a multiple of k, and k < d d. For clarity, we give an
′
example of EB3 in C4 C6≤. Let ξ = e23πi, it is obvious With the same spirit in mind, in general, we let
⊗
that the following states constitute an EB3 in a 4 6
⊗
system. i i , 0 i<d,
1 |γii=(cid:26)||rii||(′ti⊕r)′i, i=≤td+r,0≤r<d,
φ0 = (0 0′ + 1 1′ + 2 2′ ),
| i √3 | i| i | i| i | i| i where t r means t+r mod d. If dd = sk, 2 k <d,
′ ′
φ1 = 1 (0 0′ +ξ 1 1′ +ξ2 2 2′ ), then ⊕ ≤
| i √3 | i| i | i| i | i| i
k 1
|φ2i = √13(|0i|0′i+ξ2|1i|1′i+ξ4|2i|2′i), |Ω˘m,ni= √1k Xl=−0ξml|γnk+li, (8)
1
|φ3i = √3(|3i|3′i+|0i|1′i+|1i|2′i), constitute an EBk, where ξ = e2kπi, 0 m k 1,
≤ ≤ −
0 n s 1.
1
φ4 = (3 3′ +ξ 0 1′ +ξ2 1 2′ ), ≤The≤EBk−aboveisalsoaSEBk. Inordertoconstructan
| i √3 | i| i | i| i | i| i
EBk that is not a SEBk, one can replace the coefficients
1
φ = (3 3 +ξ2 0 1 +ξ4 1 2 ), in Eq. (8) by the entries of any k by k isometric matrix
5 ′ ′ ′
| i √3 | i| i | i| i | i| i X = [x ] without zero entries, namely, we can replace
ij
1 the coefficient ξml in Eq. (8) by x . From the
φ = (2 3 + 3 4 + 0 2 ), l+1,m+1
| 6i √3 | i| ′i | i| ′i | i| ′i discussion in Subsec. A and Subsec. B we know that
4
we can constructinfinitely many suchEBks that arenot haverank-3bases. Ingeneral,foranyk,ifd=sk+rand
SEBks, and we also know that there are other methods d′ =s′k+r′,thentheHilbertspace d d′ isadirectsum
of constructing SEBks. of three subspaces which are equivMalen×t to d (s′ 1)k,
(s 1)k (k+r′) and (k+r) (k+r′) respectiveMly.×We−only
nMeed−toc×onsiderrankM-kbase×sinthefollowingspaces(as-
IV. dd′ IS NOT A MULTIPLE OF k sume that there exists EBl (l <k))
x 0 0 0
In this section, we assume that dd is not a multiple 1
′ ···
0 x 0 0
of k. We discuss an EBk that is not a SEBk firstly and  2 ··· 
theWnedbeaegliwnitwhitShEtBhkesc.ase k =2. If dd′ is not a multiple Lk+1 = 0... 0... ... 0... x... ,
 k 
toahfnrd2e,etshuebnsptrahecesepsHewcitlhbiviecerhlty,asrwpeaitcehequpMi,vqadl×end′t1.tisoFaoMrde2ipxr×eac3mt,pMsluem,d×th2oeqf  0 0 ······ 0 xk+1 
3 3
spaceMof× is a direct sum of ≥ and : x 0 0 0
M3×5 M3×2 M3×3 01 x ··· 0 0
 2 
 ∗∗ ∗∗ ∗∗ ∗∗ ∗∗=∗∗ ∗∗ 000 000 000 ⊕ 000 000 ∗∗ ∗∗ ∗∗; Lk+2 = 0... 0... ·.·.·. 0... x...k ,
 ∗ ∗ ∗ ∗ ∗ ∗ ∗   ∗ ∗ ∗  0 0 ··· 0 x 
 k+1 
the3sp3:ace of M5×5 is a direct sum of M5×2, M2×3 and  0 0 ······ 0 xk+2 
M ×
0 0 0
∗ ∗ ∗ ∗ ∗ ∗ ∗ ···
0 0 0
   
∗ ∗ ∗ ∗ ∗ ∗ ∗
= 0 0 0 x 0 0 0
 ∗∗∗ ∗∗∗⊕∗∗∗∗∗∗0000∗∗∗0000000 000000∗∗∗∗∗∗⊕00000000000000∗∗∗0 ∗∗∗0 ∗∗∗0. L2k−1 = 0000...1 x000...2 ·····.·····.·····. 0000... xxxkk0...++k12 .
Swiencoenlryannkee-2dbtoa0sdisi0scau∗∗lwssa∗∗yMs∗∗e3×xi3s.tsOibn0seMr0v2e0pt×h0sata0nd Mt×2q, In fact, for the space0o...f L0...k+s·.,·.·.1≤0... sx2≤k...−k1−1, we take
0 0 0 1
∗∗ ∗∗ ∗∗ =∗ ∗0 ∗∗ ⊕∗0 ∗ 0, xp(i) = √k+sξ(p−1)(i−1), ξ =ek2+πis, (9)
0 0 0
∗ ∗ ∗  ∗ ∗   ∗ 
1 p,i k+s. That is, x(i)s are the coefficients of the
we thus only need to check ≤ ≤ p
i-th element of MEB in (k +s) (k +s) system as in
⊗
0 Eq. (5), i=1, 2, ..., k+s. Let
∗
2+1 = 0 
L  0 ∗  x(i) 0 0 0
 ∗  01 x(i) ··· 0 0 
haverank-2basessince0 ∗0 ∗haverank-2bases(it  ... 2... ·.·.·. ... ... 
∗∗ 0 ∗∗ A(i) = 0 0 ··· 0 xk(i) , (10)
is indeed a 2 3 case). Similarly, for k = 3, the Hilbert k+s  0 0 0 x(i) 
rsaeprseapceeeqctuMiivvedal×lyed,n′wt⊗ihsteoareMdpird,e×qc3tp,s1uM,m13qo×f(3st+h,trt)eeans2ud.bWMspea(3ct+ehssu)s×w(oh3n+iclthy)  0... 0... ··.··.··. 0... xk(ki...++)21 
need to show ≥ ≤ ≤  0 0 0 x(i) 
 ··· k+s 
0 0
L3+1 =0∗0 0∗0 0∗0 and L3+2 =000∗ ∗00 0∗  tthheaTnthEe{ABcokk(i+e)effisx}citiisesnitnasariannnyEkb-qki.pa(b9ra)tsiitcseaonsfybsLteekmr+es.p.laWceedtbhyusthperoevne-
0 0   ∗ 
 ∗ 0 0  triesinthecolumnsofanyk+sbyk+sisometricmatrix
   ∗ 
5
without zero entries respectively. That is, there are in- systems or systems that can be divided into 3 p, 4 q,
⊗ ⊗
finitely manyEBksinanyd d system. Ofcoursethere r 4 and s 3 systems with q,r 4 and p,s 3. We
′
⊗ ⊗ ⊗ ≥ ≥
alsoexistothertypesofEBk(butnotSEBk). Forexam- now can conclude the following.
ple, Theorem 2. If dd is a multiple of k, then there exist
′
infinitely many SEBks in Cd Cd′.
1 √3 ⊗
|ψ1i = 2|0i|0′i+ 2 |1i|1′i, anTdhdeo=remsk3+. Irf,dtdh′einstnhoetreaemxiustltsipSlEeBokf kin, dCd= sCkd+′ irf
′ ′ ′
√3 1 thereexistsa(k+rˇ) (k+rˇ)isometricmatrixX⊗=[x ],
|ψ2i = 2 |0i|0′i− 2|1i|1′i, rˇ=min r,r′ , such×that each column ij
{ }
1 1
ψ3 = 0 1′ + 1 0′ , (x ,x , ,x )t
| i √2| i| i √2| i| i 1j 2j ··· k+s,j
1 1
ψ = 0 1 + 1 0 , (here, At denotes the transpose of A) satisfies:
4 ′ ′
| i −√2| i| i √2| i| i 1)eitherthereareonlyknonzeroentriesandthemod-
1 1 ulus of them are 1 or
ψ5 = 1 2′ + 2 0′ , √k
| i √2| i| i √2| i| i 2) x = 1 when1 i k 1and x 2 = 1.
| ij| √k ≤ ≤ − i k| ij| k
1 1 Thatis,the SEBk problemis reducedPto≥the construc-
ψ6 = 1 2′ 2 0′ ,
| i √2| i| i− √2| i| i tion of the special isometric matrices. However,whether
1 1 ornotthereexistsk+sbyk+sisometricmatrixX =[x ]
ij
|ψ7i = √2|0i|2′i+ √2|2i|1′i, that satisfying the condition 1) or 2) is unknown when
k 3. With the increase of dimensions d and d and k,
1 1 1 ≥ ′
ψ8 = 0 2′ 2 1′ + 2 2′ , the verificationof the existence of SEBk becomes harder
| i √3| i| i− √3| i| i √3| i| i
and harder.
1 1 √2
ψ9 = 0 2′ 2 1′ 2 2′ (11)
| i √6| i| i− √6| i| i− √3| i| i
V. MULTIPARTITE CASE
constitute an EB2 in C3 C3, but it is not a SEB2. We
⊗
thus obtain the following result.
In this section, we consider the multipartite case. We
Theorem 1. For any d, d and 1 k d, there exist
infinitely many EBks in Cd′ Cd′. ≤ ≤ firstlyextendtheconceptofEBktomultipartitesystems.
⊗ In a product Hilbert space Cd1 Cd2 CdN with
Now we discuss SEBk. We begin with k = 2 and d = ⊗ ⊗···⊗
N 3,onlyspecificpurestatesadmitaSchmidtdecom-
d′ = 3. It is straightforwardto show that, for any rank- pos≥ition form [15, 16]. Now we discuss whether a basis
2 basis of , if it corresponds to three elements of a
L2+1 can be constructed fromsuch specific states for the mul-
SEB2, then it must admit the following form
tipartite case. Hereafter, we always assume with no loss
eix1 0 eiy1 0 eiz1 0 of generality that d1 d2 dN.
c 0 aeix2 ,c 0 aeiy2 ,c 0 aeiz2 , Definition 2. An o≤rtho≤no·r·m·a≤l basis ψi in Cd1
0 beix2 0 beiy2 0 beiz2 Cd2 CdN is a N-partite EBk (1 {|k i}d1) if ⊗
      ⊗···⊗ ≤ ≤
where a > 0, b > 0, a2+b2 = 1, c = √2, x , y , z (i = k 1
1,2) are real numbers that should sati2sfy tihe ifolloiwing |ψii= − λj(i)|e(j1)i|e(j2)i···|e(jN)i (12)
equalities Xj=0
x1 y1 =x2 y2 π, for any i, 0 i d d d 1, where e(l) is an
 xy11−−zz11 ==yx22−−zz22±±ππ., orthonormal≤set o≤f Cd1l,2λ·j(·i·) >N0−, j(λj(i))2{=| 1j,i1}≤ l ≤
− − ± N, N 3. Particularly, it is a NP-partite SEBk if it is a
However, theabove three equalities contradict each N-part≥iteEBkwithallthe coefficientsλ(i)sequalto 1 .
other. Therefore, there is no SEB2 in 3 3 systems ac- j √k
cordingtoourScenario(maybeSEB2can⊗beconstructed EBWkeofbCegdi1n wCitdh2,tahnedtbreipwarrtiittteencainset.heLSetch{m|ψidiit}debceoman-
byotherapproaches). Thespaceof canbedecom- ⊗
posed as M4×4 position form as |ψii = jk=−01λj(i)|eji|e′ji, of which the
l-th product term e e Pis denoted by ψl for conve-
0 0 0 0 | li| ′li | ii
nience. We define
∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
0 0 0 0
∗ ∗ ∗ ∗ = ∗ ∗ ∗  ∗ .
∗∗ ∗∗ ∗∗ ∗∗   ∗∗ ∗∗ ∗0 ∗0⊕ 00 00 0∗ 0∗ |ψi(,2j+1)i:=Xkl=−01λl(i)|ψili|(j+l)3i, (13)
It is obvious that diag(0,1,1,1), diag(1,0,-1,1),
diag(1,1,0,-1) and diag(1,-1,1,0) form a rank-3 ba- wherej+l means j+l mod d , j is anorthonormal
3 3
sis of {diag(∗,∗,∗,∗)}. Thus, SEB3 exists in any 4⊗d′ basis of Cd3, 0 ≤ j ≤ d3 −1. T{h|eni}{|ψi2,+j1i} with 0 ≤
6
i d d 1 and 0 j d 1 is a tripartite EBk in ψ(2+1) = a(0) ψ0 0 +a(1) ψ1 1 ,
C≤d1 1C2d2− Cd3. ≤ ≤ 3 − | 2,0 i 2 | 2i| i 2 | 2i| i
Ge⊗nerall⊗y, we denote by |ψli = |e(l1)i|e(l2)i···|e(lm)i |ψ2(2,1+1)i = a(20)|ψ20i|1i+a(21)|ψ21i|0i,
for convenience provided that ψ = ψ(2+1) = a(0) ψ0 0 +a(1) ψ1 1 ,
jk=−01λj|e(j1)i|e(j2)i···|e(jm)i with {|e(js)i} |isi an or- |ψ3(2,0+1)i = a(30)|ψ30i|1i+a(31)|ψ31i|0i
tPhonormal set of Cds, 1 s m. If ψi is an | 3,1 i 3 | 3i| i 3 | 3i| i
m-partite EBk of Cd1 Cd≤2 ≤ Cdm, th{e|n i}
⊗ ⊗···⊗ area3-partiteEB2inC2 C2 C2 viarelationEq.(13).
⊗ ⊗
ψ(m+1) :=k−1λ(i) ψl (j+l) (14) For the case C3⊗C3⊗C3,
| i,j i l | ii| m+1i
Xl=0 1 √3
ψ(2+1) = 0 0 0 + 1 1 1 ,
with 0 j dm+1 1 form an m+1-partite EBk of | 0,0 i 2| i| i| i 2 | i| i| i
Cbads1i⊗s Cofd≤2C⊗d·m·≤+·⊗1,Cjd+m+l−1,mwehaenrse{j|j+ml+1mi}odisdamn+or1t,h0on≤ormi ≤al |ψ0(2,1+1)i = 12|0i|0i|1i+ √23|1i|1i|2i,
d d d 1. That is, (m + 1)-partite EBk can be
1 2 m
··· − 1 √3
obtained from m-partite EBk for any m 2. Together ψ(2+1) = 0 0 2 + 1 1 0 ,
with Theorems 1-3, we thus get the follow≥ing. | 0,2 i 2| i| i| i 2 | i| i| i
Proposition 1. For any 1 k d , there exist in- √3 1
finitely many N-partite EBks≤in Cd≤1 1Cd2 CdN. |ψ1(2,0+1)i = 2 |0i|0i|0i− 2|1i|1i|1i,
⊗ ⊗···⊗
Proposition 2. If d1d2 dN is a multiple of k, then √3 1
there exist infinitely man·y··N-partite SEBks in Cd1 ψ(2+1) = 0 0 1 1 1 2 ,
Cd2 CdN. ⊗ | 1,1 i 2 | i| i| i− 2| i| i| i
⊗···⊗ √3 1
Proposition 3. Ifd1d2 dN isnotamultipleofk,then ψ(2+1) = 0 0 2 1 1 0 ,
there exists N-partite SE··B·k in Cd1 Cd2 CdN if | 1,2 i 2 | i| i| i− 2| i| i| i
tiheNcon,dditio=nsofkT+hreo.rem 3 holds w⊗ith rˇ=⊗m··in·⊗{ri :1 ≤ |ψ2(2,0+1)i = √12|0i|1i|0i+ √12|1i|0i|1i,
i i i
≤ }
In other words, the key point of multipartite case is 1 1
ψ(2+1) = 0 1 1 + 1 0 2 ,
in fact the same as the bipartite one. In addition, from | 2,1 i √2| i| i| i √2| i| i| i
our approach, the N-partite EBk can be constructed in
1 1
many ways. For example, if d d is a multiple of k, one ψ(2+1) = 0 1 2 + 1 0 0 ,
can either construct an EBk (2or3SEBk) of Cd2 Cd3 or | 2,2 i √2| i| i| i √2| i| i| i
constructanEBk(orSEBk)ofCdp Cdq ((p,q)=⊗(2,3)), ψ(2+1) = 1 0 1 0 + 1 1 0 1 ,
thenaN-partiteEBk(orSEBk)is⊗obtainedby6Eq.(14). | 3,0 i −√2| i| i| i √2| i| i| i
Thebasisvector ψ ofanyN-partiteEBk ψ isin 1 1
| ii {| ii} ψ(2+1) = 0 1 1 + 1 0 2 ,
fact a GHZ-like state, namely, it is local unitarily equiv- | 3,1 i −√2| i| i| i √2| i| i| i
alent to the form
1 1
ψ(2+1) = 0 1 2 + 1 0 0 ,
k 1 | 3,2 i −√2| i| i| i √2| i| i| i
|GHZk(N)i=Xj=−0λj|j1i|j2i···|jNi, |ψ4(2,0+1)i = √12|1i|2i|0i+ √12|2i|0i|1i,
wheλre2 ={|j1l,i}1 islanNo.rthonormal set of Cdl, λj > 0, |ψ4(2,1+1)i = √12|1i|2i|1i+ √12|2i|0i|2i,
j j ≤ ≤
cPasAetCla2st⊗, wCe2g⊗iveCt2w, oitexisameapslyestfoorchileluckstrtahtaiotnt.hFeoerigthhet |ψ4(2,2+1)i = √12|1i|2i|2i+ √12|2i|0i|0i,
3-qubits GHZ states deduced from the four Bell states
1 1
via relation Eq. (13) form a 3-partite SEB2 (or MEB), ψ(2+1) = 1 2 0 2 0 1 ,
| 5,0 i √2| i| i| i− √2| i| i| i
and a general 3-partite EB2 (that is not a SEB2) can
be obtained through Eq. (13) from any EB2 in C2 ψ(2+1) = 1 1 2 1 1 2 0 2 ,
C2. Namely, if ψ = a(0) ψ0 + a(1) ψ1 : 0 i ⊗ | 5,1 i √2| i| i| i− √2| i| i| i
3, (a(0))2+(a(1)){2|=ii1 is ain|EBii2 of tiwo|-qiuibits s≤ystem≤, ψ(2+1) = 1 1 2 2 1 2 0 0 ,
theni i } | 5,2 i √2| i| i| i− √2| i| i| i
1 1
|ψ0(2,0+1)i = a(00)|ψ00i|0i+a(01)|ψ01i|1i, |ψ6(2,0+1)i = √2|0i|2i|0i+ √2|2i|1i|1i,
|ψψ0((22,1++11))i == aa((000))|ψψ000i|01i++aa((011))|ψψ011i|10i,, |ψ6(2,1+1)i = √12|0i|2i|1i+ √12|2i|1i|2i,
| 1,0 i 1 | 1i| i 1 | 1i| i 1 1
|ψ1(2,1+1)i = a(10)|ψ10i|1i+a(11)|ψ11i|0i, |ψ6(2,2+1)i = √2|0i|2i|2i+ √2|2i|1i|0i,
7
1 1 1
ψ(2+1) = 0 2 0 2 1 1 + 2 2 1 , tite case and we thus have provided a complete char-
| 7,0 i √3| i| i| i− √3| i| i| i √3| i| i| i acterization of EBk in both the bipartite case and the
1 1 1 multipartitecase. Ineitherbipartitecaseormultipartite
ψ(2+1) = 0 2 1 2 1 2 + 2 2 2 ,
| 7,1 i √3| i| i| i− √3| i| i| i √3| i| i| i case,the basisstates ofEBk(or N-partiteEBk)arejust
the pure states that admit the Schmidt decomposition
1 1 1
ψ(2+1) = 0 2 2 2 1 0 + 2 2 0 , withthesamelength. Althoughonlysomespecialmulti-
| 7,2 i √3| i| i| i− √3| i| i| i √3| i| i| i
partite pure states have the Schmidt decomposition, the
ψ(2+1) = 1 0 2 0 1 2 1 1 √2 2 2 1 , basiswithsuchaspecialstructurealwaysexists. Ourre-
| 8,0 i √6| i| i| i− √6| i| i| i− √3| i| i| i sultsprovideamathematicaltoolforstudyingprojective
measurements onto basis states that admit a Schmidt
1 1 √2
ψ(2+1) = 0 2 1 2 1 2 2 2 2 , decomposition with the same length.
| 8,1 i √6| i| i| i− √6| i| i| i− √3| i| i| i
It is worth mentioning here that real space is different
1 1 √2
ψ(2+1) = 0 2 2 2 1 0 2 2 0 fromthecomplexspace. FromtheargumentsinSec. III-
| 8,2 i √6| i| i| i− √6| i| i| i− √3| i| i| i IV,weconjecturethatthereisnoSEB2inthe realspace
Rd Rd′ when dd is not a multiple of 2 (due to the fact
′
constitute an EB2 in C3 C3 C3, ψ(2+1) s are in fact tha⊗ttherealnumberswithmodulus1areonly1and-1).
⊗ ⊗ | i,j i
three-qubits GHZ-like states. It is clear that ψ(2+1) s Moreover, we conjecture that: i) MEB exists only when
| i,j i d = 2s with s 0; ii) SEBk exists only when k = 2s,
above are derived from Eq. (11) via relation Eq. (13). ≥
s 0, and dd is a multiple of k. In addition, from the
Thereis noSEB2in3 3system,sothere isno SEB2in ≥ ′
3⊗3⊗3systemeither.⊗AnEB3orSEB3inC3⊗C3⊗C3 norotthSogEoBnkalemxiasttsrixinORnd, weRcda′nfocronacnlyuddeatnhdatdEB(okftchoautrsies
can be obtained via relation Eq. (13) from any EB3 or ⊗ ′
SEB3 in C3 C3 respectively. there are other types of EBks). Similar results for the
⊗ multipartite case can be obtained from the method in
Sec. V.
VI. CONCLUSIONS
We have introduced the concept of EBk and showed
that EBk exists in Cd Cd′ for any d and d. In our dis- Acknowledgments
′
⊗
cussion we have also proposed methods of constructing
EBks by analyzing the structure of the matrix space of We thank Valerio Scarani for useful remarks. Guo is
the coefficients matrices. We showed that the existence supported by the National Natural Science Foundation
of EBks is equivalent to the existence of the special iso- of China under Grants No. 11301312 and 11171249,the
metric matrices. Our result is a complement of the UB Natural Science Foundation of Shanxi under Grant No.
problem(here, UB contains UPB,UMEB andUEBk) to 2013021001-1and2012011001-2,andtheResearchstart-
someextent. Bynow,thebasisproblems(namely,MEB, up fund for Doctors of Shanxi Datong University under
EBk,PB,UPB,UMEB andUEBk)aresomehowsettled Grant No. 2011-B-01. Du is supported by the National
down: any kind of basis (complete or incomplete) exists Natural Science Foundation of China under Grant No.
inanybipartitespaceCd Cd′. Itisalittlepitythatthe 11001230 and the Natural Science Foundation of Fujian
⊗
SEBk problem is left open when dd is not a multiple of under Grants No. 2013J01022 and 2014J01024. Wu is
′
k. supported by the National Natural Science Foundation
Furthermore,our approachescan be used in multipar- of China under Grants No. 11275181and No. 11475084.
[1] R. Horodecki, P. Horodecki, M. Horodecki, and K. [9] Y.-L. Wang, M.-S. Li, and S.-M Fei, Phys. Rev. A 90,
Horodecki, Rev.Mod. Phys. 81, 865 (2009). 034301 (2014).
[2] O.Gu¨hne and G. T´oth, Phys. Rep.474, 1 (2009). [10] M. J. Donald, M. Horodecki, and O. Rudolph, J. Math.
[3] C.H.Bennett,D.P.DiVincenzo,T.Mor,P.W.Shor,J. Phys. 43, 4252–4272 (2002).
A. Smolin, and B. M. Terhal, Phys. Rev. Lett. 82, 5385 [11] J. Sperling and W. Vogel, Phys. Scr. 83, 045002 (2011).
(1999). [12] B. Baumgartner, B. Hiesmayr, and H. Narnhofer, Phys.
[4] S. Bravyi and J. A. Smolin, Phys. Rev. A 84, 042306 Rev. A 74, 032327 (2006).
(2011). [13] B.Baumgartner,B.HiesmayrandH.Narnhofer,J.Phys.
[5] Y.Guo and S. Wu,Phys. Rev.A 90, 054303 (2014). A: Math. Theor. 40, 7919–7938 (2007)
[6] A.O. Pittenger, Lin.Alg. Appl.359, 235–248 (2003) [14] E. Schmidt,Math. Ann.63, 433 (1907).
[7] B.ChenandS.M.Fei,Phys.Rev.A88, 034301 (2013). [15] A. Peres, Phys.Lett. A 202, 16–17 (1995).
[8] M.-S.Li,Y.-L.Wang,andZ.-J.Zheng,Phys.Rev.A89, [16] A. V.Thapliyal, Phys. Rev.A 59, 3336 (1999).
062313 (2014).