Explicit constructions of all separable two-qubits density matrices and related problems for three-qubits systems

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1 Explct constructons of all separable two-qubts densty matrces and related problems for three-qubts systems Y. en-ryeh and. Mann Physcs Department, Technon-Israel Insttute of Technology, Hafa 2000, Israel e-mals: ; STRCT Explctly separable densty matrces are constructed for all separable two-qubts states based on Hlbert-Schmdt (HS) decompostons. For densty matrces whch nclude only two-qubts correlatons the number of HS parameters s reduced to by usng local rotatons, and for two-qubts states whch nclude sngle qubt measurements, the number of parameters s reduced to 4 by local Lorentz transformatons. For both cases we related the absolute values of the HS parameters to probabltes, and the outer products of varous Paul matrces were transformed to densty matrces products. Smple necessary and suffcent condtons for separablty are derved. We dscuss related problems for three qubts. For n-qubts correlaton systems ( n 2) the suffcent condton for separablty may be mproved by local transformatons, related to tensor analyss. Keywords: Separablty, qubts, Hlbert-Schmdt decompostons, local transformatons.

2 . Introducton For systems wth many subsystems, and Hlbert spaces of large dmensons, the separablty problem becomes qute complcated. In the smple case of two-qubts states, t s possble to gve a measure of the degree of quantum correlatons by usng the partal-transpose (PT) of the densty matrx. ccordng to the Peres-Horodeck (P-H) crteron 2 f the PT of the two qubts densty matrx leads to negatve egenvalues of the PT matrx ρ ( PT ), then the densty matrx s entangled, otherwse t s separable. One should take nto account that the densty operator of a gven mxture of quantum states has many ensemble decompostons. The separablty problem for two-qubts states s defned as follows: bpartte system s separable f and only f the densty matrx of ths system can be wrtten n the form: ( ) ( ) p. () ρ = ρ ρ Here: p > 0 and p =. The densty matrx ρ s defned on the Hlbert space H H where and are the two parts of the bpartte system. ( ) ρ and ( ) ρ are densty matrces for the and systems, respectvely. The nterpretaton of such defnton s that for bpartte separable states the two systems are ndependent of each other. The summaton over could nclude large numbers of densty matrces products, but usually t s preferred to lmt ths number to smaller ones, as far as t s possble. The densty matrx for any bpartte system can be gven as ρ ( ) ( k l ) = ρ. (2) k, l, k,, l Here and are summed over a complete set of states of the system, whle k and l states are summed over the system. ρ k, l are complex numbers. The PT of the densty matrx ρ s gven as: 2

3 ρ. () ( PT ) = ρ, ( ) ( l k ) l k, l,, k Here we exchanged the ket and bra states of the system. (lternatvely such exchange can be made for the system). M. Horodeck, et al. showed that the postvty of the PT state s necessary and suffcent for separablty of 2 2 and 2 systems. However, explct expressons for separable densty matrces of two qubts have not been presented. One am of the present work s to exhbt explctly a separable form for any separable 2-qubts state, and dscuss the possbltes to use the present methods for more qubts. densty matrx: 4,5 For a 2-qubts system denoted by and we use the HS representaton of the ρ σ σ σ σ = 4 ( I ) ( I ) + ( r ) ( I ) + ( I ) ( s ) + tm, n ( m ) ( n ) m, n=. (4) Here, I represents the unt 2 2 matrx, the three Paul matrces are denoted by σ whle r and s represent -dmensonal parameters vectors. The normalzed 2-qubts densty matrx s descrbed by 5 real HS -parameters: 6 for r and s, and 9 for t m, n. The quantum correlatons are ncluded n the HS t m, n parameters whle r and s correspond to sngle qubt measurements. ssumng the condtons r = s = 0 n (4), we get the densty matrx 4ρ = ( I ) ( I ) + t m, n ( σ m ) ( σ n ) m, n=. (5) We show n the present work a smple method by whch the HS decompostons (4) and (5) can be transformed to the form of () for all two-qubts densty matrces whch are separable. The analyss for separablty for the densty matrces (4) and (5) becomes n the general case qute complcated due to the large number of parameters nvolved n such analyss (5 for (4), and (9) for (5)). However, the number of parameters descrbng the 2-qubts densty matrces can be reduced by local transformatons. 6 We consder ρ and ρ M to be of the same equvalence class when

4 M ρ ρ M ρ M, M M M = =, (6) where M and M are nvertble. Such equvalence preserves the postvty and separablty of the densty matrces. For densty matrces of the form (5), M and M are gven by orthogonal rotaton matrces so that normalzaton s preserved. For densty matrces of the form (4), M and M are gven by Lorentz transformatons, 7 so the transformed densty matrx should be renormalzed. These local transformatons can reduce the number of parameters for the densty matrx (5) to parameters by usng rotatons, and the number of parameters for the densty matrx (4) to 4 parameters, by local Lorentz transformatons. 7 Whle such transformatons have been analyzed, 7 the explctly separable densty matrces for 2-qubts densty matrces have not been analyzed and such explctly separable densty matrces are treated n the present work, ncludng smple necessary and suffcent condtons for separablty. We dscuss the use of the present method for larger n-qubts systems (n>2). The present paper s arranged as follows: In Sec. 2 we descrbe explctly separable densty matrces of 2-qubts densty matrces of the form (5), where the number of parameters s reduced by local rotatons. We show also that a necessary and suffcent condton for separablty s that the egenvalues of ρ satsfy ( ) λ / 2 =,2,,4. In Sec. () we relate the analyss of separable densty matrces of 2- qubts states of the form (4), by the HS decompostons, to an analyss made prevously by local Lorentz transformatons 7 and show the explct form n the generc case. In Sec. (4) we dscuss certan problems related to the applcaton of the present methods to larger n-qubts correlaton systems ( n > 2). In Sec. (5) we summarze our results. 4

5 2. n explct constructon of separable densty matrces of 2-qubts states descrbed by the HS decomposton of Eq. (5) One may use local rotatons to brng the form t ( σ ) ( σ ) t ( σ ) ( σ ), as follows. Snce the (, 2,) = we may defne: ( σ m ) ( O m, ) ( σ ), ( σ n ) ( O n, ) ( σ ) where ( ), ( ) m, n m n nto dagonal form: m, n= σ = transform as a vector under rotatons, = =, (7) O O are orthogonal matrces. Hence tm, n ( σ m ) ( σ n ) = ( O m, ) tm, n ( O n, ) ( ) σ ( σ ). (8) m, n= m, n,, y the sngular-value decomposton (SVD) 2, we can choose ( O ) and ( O ) so that m, n (, ), (, ) O t O = δ t, (9) m m n n and therefore tm, n ( σ m ) ( σ n ) = t ( σ ) ( σ ). (0) m, n= = We dscard the bars n (0) and analyze the constructon of separable densty matrces for the densty matrx gven as: 4ρ = ( I ) ( I ) + t ( σ ) ( σ ) = It s easy to verfy that ρ of Eq. () may be wrtten as. () 5

6 4ρ = 2 t = ( I sgn t σ ) ( I σ ) ( I + sgn ( t ) σ ) ( I σ ) ( ) ( I ) ( I ) t =. (2) Here for postve t ( ) sgn t =, and for negatve t sgn ( t ) =. Each outer product n the curly brackets on the rght sde of (2) s a separable densty matrx, based on pure states. t / 2 can be consdered as a probablty for each pure state separable densty matrx ncluded n the curly brackets. The unt matrx ( I ) ( I ) (n the last term of (2)) s trvally a separable densty matrx. Therefore, f ts coeffcent s nonnegatve, then ρ s separable. t = Hence t s a suffcent condton for separablty. We wll prove that t s also a necessary = condton. Thus t s equvalent to the P-H crteron for two qubts. To prove that the above condton s necessary assume that ρ s separable. s ponted out by Peres, 2 f ρ s separable then ρ ( PT ) s also a separable densty matrx. In the standard bass 00, 0, 0,, the densty matrx (, 2) s gven by: 4ρ + t 0 0 t t2 0 t t + t2 0 = 0 t + t2 t 0 t t t. () The PT of the densty matrx () s gven by nvertng the sgn of t 2, 4 so + t 0 0 t + t2 0 t t t2 0 4 ρ ( PT ) = 0 t t2 t 0 t + t t. (4) 6

7 The egenvalues of ρ ( PT ) are gven by: 4 λ ( PT ) = + t t t, 4 λ ( PT ) = t + t t, λ ( PT ) = t t + t, 4 λ ( PT ) = + t + t + t (5) The egenvalues of ρ are gven by 4 λ ( ρ) = t t t, 4 λ ( ρ) = + t + t t, λ ( ρ) = + t t + t, 4 λ ( ρ) = t + t + t (6) We note that n (5, 6) nvertng the sgns of two t merely exchanges the names of the egenvalues. We note also that separablty of ρ ensures that all these eght egenvalues are nonnegatve. One should notce also that by nvertng the sgn of any one of the t parameters the egenvalues of ρ ( PT ) are transformed nto the egenvalues of ρ, and vce versa. The mportant pont s that one has to look at both the egenvalues of ρ and ρ ( PT ). To prove the necessty of the condton t, (7) = We dstngush between two possble cases: Case : The sgn of the trple product t t 2 t s + ( sgn( tt 2t ) = + ). If all t are postve, then by lookng at the egenvalues of ρ we see that λ s proportonal to t so ths expresson s postve (or zero), and snce = t are postve t s the same as t. If one of the = t s postve, and two are negatve, then ether by a π rotaton around the axs of the postve one, or by smply lookng at the egenvalues of ρ we fnd that the smallest egenvalue of ρ s gven agan by t. Indeed, suppose that = t and 7

8 t 2 are negatve and t s postve then λ2 0 s proportonal to: t. So we proved that n = ths case, a necessary condton for separablty s gven by t. = Case : The sgn of the trple product t t 2 t s -( sgn( tt 2t ) = ). If all three t are negatve, then λ ( PT ) s proportonal to: 4 t, hence = t s = nonnegatve. If two t are postve and one negatve, suppose e.g., that t and t 2 are postve t = and t negatve, then s proportonal to λ ( PT ), hence t s nonnegatve. lternatvely we can nvert the sgn of the two postve ones by π rotaton around the negatve one. To sum up, Eq. (7) s a necessary and suffcent condton for separablty of ρ. Further nterestng relatons and even a smpler necessary and suffcent condton for separablty may be obtaned as follows. We can nvert the relatons gven by (6) so that the HS parameters t ( =, 2, ) wll be gven as functons of the egenvalues λ ( ρ) ( =,2,,4), λ = 4 =. Then we get: t = 2 λ ( ρ) 2 λ ( ρ) ; t = 2 λ ( ρ) 2 λ ( ρ) ; t = 2 λ ( ρ) + 2 λ ( ρ). (8) Substtutng these values n (5) we get: ( ) ( ) PT ( ) ( ) ( ) ( ) PT ( ) ( ) λ ( PT ) = / 2 λ ρ ; λ ( ) = / 2 λ ρ ; 2 4 λ ( PT ) = / 2 λ ρ ; λ ( ) = / 2 λ ρ 2. (9) We fnd that f all egenvalues of ρ are less than or equal /2, then ρ s separable. ut f one of the egenvalues of ρ s larger than /2, then ρ s entangled. (Only one egenvalue of λ( PT ) may be negatve because otherwse two egenvalues of ρ wll be larger than /2). We 8

9 fnd that ths crteron s equvalent to (7), but we can fnd the condton for entanglement drectly from the egenvalues of ρ. The use of the relatons gven by (9) may have a practcal advantage over the use of (7), when t wll be easer to calculate drectly the egenvalues of ρ than calculatng the t values, especally when SVD are needed. In fact, gven λ one may obtan the t drectly from (8). The possblty to fnd the condtons for separablty and entanglement drectly from the densty matrx ρ should be of nterest also conceptually. pplcaton of the present methods to a Werner state We analyze by the present methods the smple Werner state 4,5 treated n the artcle by arbery et.al. 5, where they used a densty matrx gven as: C 0 ρ = ; = ( / 4) ( p), = (/ 4) ( + p), C = ( p / 2) 0 C (20) Comparng ρ of Eq. (20) wth the general expresson gven by Eq. () we get: t = t2 = t = p. (2) Hence accordng to Eq. (7) the necessary and suffcent condton for separablty s p. (22) Ths s the same condton as gven by arbery et al.. 5 We get the same result by calculatng the egenvalues of ρ. Then we get: 2 4 ( )( p) ( ) ( ) λ ( ρ ) = λ ( ρ) = λ ( ρ) = / 4, λ ( ρ ) = / 4 + / 4 p. (2) 9

10 λ ( ) ρ s larger than / 2 when p > / and under ths condton the densty matrx s entangled. We get the same result by usng ether the relaton (7) or (9) where both relatons should be of nterest.. Explctly separable densty matrces of 2-qubts states descrbed by the general case of HS decomposton gven by Eq. (4) Let us assume a two-qubts densty matrx n whch the matrx t mn has been dagonalzed but t ncludes also sngle qubts elements. Ths densty matrx s gven as: ρ σ σ σ σ = 4 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) I I + a I + I b + t =, (24) where a and b are certan parameters- vectors satsfyng the relatons a densty matrx can be wrtten as: and b. Ths t I + sgn( t ) σ I + σ I sgn( t ) σ I σ 4ρ = 2 + = a σ I I,(25) b σ a I b I I I a b t + / 2 / 2 ( ) ( ) a b = where a = a, b = b. suffcent condton for separablty s gven by a b t 0. (26) = However, t s not necessary unless a = b = 0. To obtan a necessary and suffcent condton and explctly separable form for the densty matrx we note that the general densty matrx (4) can be wrtten n the HS representaton as 6 0

11 4ρ = R α, β α, β = 0 σ α σ β. (27) ( ) ( ) Here the summaton extends from 0 to, σ 0 s the 2 2 dentty matrx, and σ, σ 2, σ are the Paul spn matrces. Verstraete et al., 7, studed how the densty matrx s changed under local quantum operatons and classcal communcatons (LQCC) of the type ( ) ρ ( ) ρ '. (28) They 7, have shown that the 4 4 matrx R αβ can be gven as R = L Σ L. (29) T 2 Here L and L 2 are proper local Lorentz transformatons, and Σ s ether the dagonal form ( 0,, 2, ) Σ = dag s s s s (the generc case wth s0 s s2 s where s s postve or negatve) or of the form a 0 0 b 0 d 0 0 Σ = 0 0 d 0 c 0 0 a + c b. (0) Here a, b, c, d are real parameters satsfyng one of the four relatons: a. For b = c = a / 2, ρ s always entangled. b. For the other cases, ρ s separable and has a smple form gven explctly by them. For the generc case, dagonal Σ, the stuaton s very smlar to that whch we analyzed n Secton 2. (Note that n Secton 2 we could have chosen t t2 t ). Therefore n ths case

12 ρ ( I + σ ) ( I + σ ) ( I σ ) ( I σ ) 2 s = + = ( I + σ ) ( I + sgn( s ) σ ) ( I σ ) ( I sgn( s ) σ ) s ( I ) ( I ) + s ( s + s + s ) The necessary and suffcent condton for separablty s. () s + s2 + s s0. (2) lso ths condton may agan be wrtten n terms of the egenvalues of ρ as λ / 2 (,2,,4) s0 =. () 4. Suffcent condtons for separablty and entanglement for correlated systems of -qubts three-partte system s separable f and only f the densty matrx of ths system can be wrtten n the form: ( ) ( ) ( ) p C. (4) ρ = ρ ρ ρ Here: p > 0 and p =. We are nterested n genune qubts correlatons densty matrces lke that of the twoqubts correlatons gven by the densty matrx (5). We assume a qubts correlaton densty matrx gven by: ( ) ( ) ( ) ( ) ( ) 8 ρ = ( I ) I I + G σ σ σ. (5) C a, b, c a b c C a, b, c= We fnd that the GHZ densty matrces 6 and brad densty matrces 7 are specal forms of (5). These examples are, however, based on pure states whle the present analyss for the 2

13 suffcent condtons for separablty wll be more general, as we nclude n (5) mxed densty matrces. We study the relatons between the real HS parameters G a, b, c, whch can be ether postve or negatve, wth the suffcent condtons for separablty of (5). 8ρ C ( / 4) + = separable-lke form for the densty matrx (5) can be gven as: a, b, c= a, b, c Ga, b, c { ( I ) } ( I ) a, b, c= G {( I ) ( σ a ) } {( I ) ( σ b ) } {( I ) C sgn ( Gabc )( σ ) } c C {( I ) ( σ a ) } {( I ) ( σ b ) } {( I ) C sgn ( Gabc )( σ ) } c C {( I ) ( σ a ) } {( I ) ( σ b ) } {( I ) C sgn ( Gabc )( σ c ) C} {( I ) ( σα ) } ( I ) + ( σ β ) { I sgn G σ } , (6) + + { } ( ) ( )( ) C abc c C { } { ( I ) } c Each expresson n the curly brackets of (6) represents a pure state densty matrx multpled by 2. We get accordng to (6) that a suffcent condton for separablty s gven by Ga, b, c suffcent condton for separablty. (7) a, b, c= The crucal pont here s that the form Ga, b, c s not nvarant under orthogonal a, b, c= transformatons and we can mprove the condton for separablty by usng orthogonal transformatons whch wll mnmze ths separablty form. For the densty matrx (5) we can use for each qubt a rotaton whch wll nclude parameters. For the general mxed -qubts systems we have 27 parameters, but the rotatons of the -qubts system wll ntroduce only 9 parameters. So we are lookng for mnmzng the functon Ga, b, c,by varyng 9 rotaton a, b, c= parameters, whch s a very complcated problem. The stuaton s qute dfferent for the case of two qubts where the correspondng separablty form s gven by tm, n. We get 9 m, n=

14 parameters and a suffcent condton for separablty s gven by t mn m, n=. For two qubts there are 6 rotatons parameters, and by varyng these parameters we reduced t to three dagonal parameters t ( =, 2, ), where we have obtaned Eq. (7) as suffcent and necessary condton for separablty. The P-H crteron, that ρ ( PT ) has a negatve egenvalue, s suffcent for entanglement. Eq. (7) gves a suffcent condton for separablty. Hence f the egenvalues of ρ ( PT ) are postve and therefore the P-H crteron does not work, Eq. (7) may be tred as a suffcent condton for separablty. Of course, usually there wll be a regon where both crtera wll not help. The present approach can be generalzed to larger correlaton n-qubts systems (n>) but the analytcal analyss wll be extremely complcated. One can use, however, numercal analyss for mnmzng the separablty forms. 5. Summary In the present artcle we have exhbted an explctly separable form for all separable two qubts densty matrces (Eqs. (2), ()) and suggested new necessary and suffcent condtons for separablty (Eqs. (7), (9),(2),()). For -qubts systems we have exhbted a possble separable form (Eq. (6)) for all densty matrces satsfyng a suffcent condton for separablty (Eq. (7). We dscussed orthogonal transformatons whch can mprove the condton for separablty. References. R. Horodeck, P. Horodeck, M. Horodeck and K. Horodeck, Rev. Mod. Phys. 8 (2009) Peres, Phys. Rev. Lett. 77 (996) 4.. M. Horodeck, P. Horodeck and R. Horodeck, Phys. Lett. 22 (996). 4

15 4. I. engtsson, K. Zyczkowsk, Geometry of Quantum states: n Introducton to entangled states (Cambrdge Unversty Press, Cambrdge, 2006). (In ths book HS decomposton s defned as Fano Form ). 5. Y. en-ryeh,. Mann and.c. Sanders, Foundatons of Physcs 29 (999) F. Verstraete, K. udenaert,. De Moor, Phys. Rev. 64 (200) F. Verstraete, J. Dehaene,. De Moor, Phys. Rev. 64, (200) J. E. vron, G. sker, O. Kenneth, J. Math. Phys. 48 (2007) J. E. vron, O. Kenneth, nnals of Physcs 24 (2009) M. Teodorescu-Frumosu, G. Jaeger, Phys. Rev. 67 (200) F. Verstraete, J. Dehaene,. De Moor, Phys. Rev. 65 (2002) R.. Horn, C. R. Johnson, Matrx nalyss (Cambrdge Unversty Press, Cambrdge, 99).. W. H. Press,. P. Flannery, S.. Teukolky, W. T. Vetterlng, Numercal Recpes (Cambrdge Unversty Press, Cambrdge, 987). 4. R. F. Werner, Phys. Rev. 40, (989) M. arbery. F. De Martn, G. D. Nep, P. Matalon, Phys. Rev. Lett. 92 (2004) D. ouwmeester, J-W. Pan, D. M. Danell, H. Wenfurther,. Zelnger, Phys. Rev. Lett. 82 (999) Y. en-ryeh, Internatonal Journal of Quantum Informaton (204)