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 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. Tóth, 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).

Entangled bases with fixed Schmidt number

Similar Entangled bases with fixed Schmidt number