Canonical Form and Separability of PPT States on Multiple Quantum Spaces

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1 Coicl Form d Seprbility of PPT Sttes o Multiple Qutum Spces Xio-Hog Wg d Sho-Mig Fei, 2 rxiv:qut-ph/050445v 20 Apr 2005 Deprtmet of Mthemtics, Cpitl Norml Uiversity, Beijig, Chi 2 Istitute of Applied Mthemtics, Uiversity of Bo, 535 Bo, Germy Abstrct By usig the subtrctig projectors method i provig the seprbility of PPT sttes o multiple qutum spces, we derive coicl form of PPT sttes i C K C N composite qutum systems with rk N, from which sufficiet seprbility coditio for these sttes is preseted. As the key resource i qutum iformtio processig [], qutum etglemet hs resulted i the explosio of iterest i qutum computig d commuictio i recet yers [2]. The seprbility of qutum pure sttes is well uderstood [3]. However for mixed sttes, lthough there re lredy my results relted to the seprbility criterio, the physicl chrcter d mthemticl structure of the qutum etglemet re still fr from beig stisfied. The PPT positive prtil trspose criteri ply very importt roles i the ivestigtio of seprbility. It is of sigificce to ivestigte the geerl form of PPT sttes to study the seprbility d boud etgled sttes. I [4] the PPT mixed sttes o C 2 C 2 C N with rk N hs bee studied d seprbility criterio hs bee obtied. These results hve bee geerlized to the cse of PPT sttes o C 2 C 3 C N with rk N [5], d the cse of high dimesiol triprtite systems [6] d C 2 C 2 C 2 C N [7]. I this ote we summrize our efforts i strivig t some uderstdig of the properties of qutum etglemet for composite systems. We geerlize the results to the most geerl cse of the coicl form of PPT sttes i C K C Km C N composite qutum systems with rk N, d study the seprbility coditio for these sttes i terms of the coicl form. Let C K be K-dimesiol complex vector spce with orthoorml bsis { i }, i = 0,...,K. A geerl pure stte o C K... C Km is of the form Ψ = K i=0 K 2 j=0 K m... k=0 ij...k i,j,...,k,

2 where i,j,...,k = i j... k, ij...k C, ij...k ij...k = deotig complex cojugtio. Ψ issidtobefullyseprbleif ij...k = i j... k forsome i, j,..., k C. ixed stte o C K... C Km is described by desity mtrix ρ, M ρ = p i Ψ i Ψ i, 2 i= for some M IN, 0 < p i, M i= p i =, Ψ i s re pure sttes of the form d Ψ i is the trspose d cojugtio of Ψ i. We cll stte ρ PPT if ρ T l 0, l, where ρ T l is the trspose of ρ with respect to the l-th subspce. I the followig we deote by Rρ, Kρ, rρ d kρ the rge, kerel, rk, dimesio of the kerel of ρ respectively, where, by defiitio Kρ = { φ : ρ φ = 0}, Rρ = { φ : ψ, such tht φ = ρ ψ }. We cosider ow composite qutum systems i C K A C N + with rρ = N, where A i deotes the i-th subsystem, K i stds for the dimesio of the i-th complex vector spce, m, N IN. We first derive coicl form of PPT sttes i C 2 A C 2 C 2 C N + with rk N, which llows for explicit decompositio of give stte i terms of covex sum of projectors o product vectors. Let 0 A, A ; 0 A2, A2 ; ; 0 Am, Am d 0 Am+,, N Am+ be some locl bses of the sub-systems A,,,, d + respectively. I terms of the method used i [4, 5, 6, 7], we hve Lemm. Every PPT stte ρ i C 2 A C 2 C 2 C N + such tht r A, A2,, Am ρ A, A2,, Am = rρ = N, c be trsformed ito the followig coicl form by usig reversible locl opertio: ρ = FT T F, 3 where T = D m I D m I D I, D i, F d the idetity I re N N mtrices ctig o C N + d stisfy the followig reltios: [D i, D j ] = [D i, D j ] = 0, d F = F stds for the trspose d cojugte, i,j =,2, m. Usig Lemm we c prove the the followig Theorem: Theorem. A PPT-stte ρ i C 2 C 2 C 2 C N with rρ = N is seprble if there exists product bsis e A, e A2,, e Am such tht r e A, e A2,, e Am ρ e A, e A2,, e Am = N 2

3 . Proof. Accordig to Lemm the PPT stte ρ c be writte s the form of 3. Sice ll the D i d D j commute, they hve commo eigevectors f. Let, 2,, m be the correspodig eigevlues of D, D 2,, D m respectively. We hve [ ] f ρ f = m m m m where = e A, e A2,, e Am e A, e A2,, e Am. We c thus write ρ s ψ = N ρ = ψ ψ φ φ ω ω f f, = m, φ = m,, ω = Becuse the locl trsformtios re reversible, we c ow pply the iverse trsformtios d obti decompositio of the iitil stte ρ i sum of projectors oto product vectors. This proves the seprbility of ρ. By usig Lemm, Theorem d the method i [6], we c geerlize the results to multiprtite qutum systems i C K A C N + with rk N. Let 0 Ai, Ai,, K i Ai, i =,2,,m; d 0 Am+,, N Am+ be some locl bses of the sub-systems A i, i =,2,,m, + respectively. Lemm 2. Every PPT stte ρ i C K A C Km C N + such tht r K A,,K m Am ρ K A,,K m Am = rρ = N,cbetrsformed ito the followig coicl form by usig reversible locl opertio: ρ = FT T F, 4 where T = D K D I D 2 K 2 D2 I D m K m D m I, D t i, F d the idetity I re N N mtrices ctig o C N + d stisfy the followig reltios: [D t i t, D q j q ] = 0, d F = F, i t,j t =,2, K t, t,s,p,q =,2, m. Proof. We prove the lemm by iductio o m. It is lredy proved for the cses m = [8] d m = 2 [6]. Now we cosider the cse of geerl m. Suppose tht for the cse m the result is correct. I the cosidered bsis desity mtrix ρ c be lwys writte s: S S-prtitioed mtrix, where S = K K 2 K m, with the i-row j-colum etry E ij, re S S = N. 3.

4 Theprojectio K A ρ K A givesrisetostte ρ = K A ρ K A which is stte i C K 2 C N + with r ρ = rρ = N. The fct tht ρ is PPT implies tht ρ is lso PPT, ρ 0. By iductio hypothesis, we hve ρ = FT T F, 5 wheret = D 2 K 2 D 2 I D m K m D m I,with[D t i t, D q j q ] = 0, i t,j t =,2,, K t, t,s,p,q = 2,, m. Similrly, ifwecosidertheprojectio K 2 A2 ρ K 2 A2,, K m Am ρ K m Am, by iductio hypothesis, we hve m reltios like 5. I fct, the projectio K j Aj ρ K j Aj, j = 2,,m, is ρ j = FT j T j F, 6 where T j = D K D I D j K j D j I D j+ K j+ D j+ I D m K m D m I, with [D t i t, D q j q ] = 0, i t,j t =,2,,K t, t,s,p,q =,,j,j+, m. Tkig ito ccout ll these projectios d usig the kerel vectors of ρ we c determie prt of the etries E ij. The by usig the PPT property of the prtil trspose of ρ relted to the subsystems: the kerel vectors of ρ re lso the kerel vectors of the prtil trsposeofρ; ife jj iskow, therest etriesofρ, E ij, i < j, relsodetermied. Forthe digol elemets like E, we defie = E S S, where S = D K D2 K 2 Dm K m, to determie E by provig tht = 0. It is strightforwrd to prove tht there exist mtrices D i s i, s i =,2,,K i, i =,2,,m stisfyig the reltio 4. From Lemm 2 we hve the followig coclusio: Theorem 2. A PPT-stte ρ i C K A C Km C N + with rρ = N is seprble if there exists product bsis e A, e A2,, e Am such tht. r e A, e A2,, e Am ρ e A, e A2,, e Am = N WehvederivedcoiclformofPPTsttesi C K C N composite qutum systems with rk N, together with sufficiet seprbility criterio from this coicl form. Geerlly the seprbility criterio we c deduce here is weker th i the cses of C 2 C 2 C N d C 2 C 3 C N, s the PPT criterio is oly sufficiet d ecessry for the seprbility of biprtite sttes o C 2 C 2 d C 2 C 3. Besides the 4

5 discussios of seprbility criterio, the coicl represettio of PPT sttes c shde light o studyig the structure of boud etgled sttes. Oe c check if these PPT sttes re boud etgled by checkig whether they re etgled or ot. Refereces [] C. H. Beet, Phys. Scr. T 76, [2] M. Nielse d I. Chug, Qutum Computtio d Qutum Iformtio Cmbridge Uiversity Press, Cmbridge, Egld, [3] A. Peres, Qutum Theory: Cocepts d Methods, Kluwer Acdemic Publishers 995. [4] S. Krs d M. Lewestei, Phys. Rev. A 64, [5] S.M. Fei, X.H. Go, X.H. Wg, Z.X. Wg d K. Wu, Phys. Rev. A [6] S.M. Fei, X.H. Wg, Z.X. Wg d K. Wu, O PPT Sttes i C K C M C N Composite Qutum Systems, to be pper i Commu.Theor. Phys [7] S.M. Fei, X.H. Go, X.H. Wg, Z.X. Wg d K. Wu, Commu. Theor. Phys [8] S.M. Fei, X.H. Go, X.H. Wg, Z.X. Wg d K. Wu, It. J. Qut. Iform.,