Entangled three-particle states in magnetic field: periodic correlations and density matrices

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1 Indian J Phys DOI 0.007/s x ORIGINAL PAPER Entangled three-article states in magnetic field: eriodic correlations and density matrices A Chakrabarti and A Chakraborti * Centre de Physique Théorique, École Polytechnique, 98 Palaiseau Cedex, France Laboratoire de Mathématiques Aliquées aux Systèmes, École Centrale Paris, 990 Châtenay-Malabry, France Received: 7 October 0 / Acceted: January 0 Abstract: We resent a novel study of the time evolutions of entangled states of three sin-/ articles in the resence of a constant external magnetic field, which causes the individual sins to recess and leads to remarkable eriodicities in the correlations and density matrices. The emerging atterns of eriodicity are studied exlicitly for different entangled states and in detail for a articular initial configuration of the velocities. Contributions to recession of anomalous magnetic moments are analysed and general results are also obtained. We then introduce an electric field orthogonal to the magnetic field, linking to the receding case via a suitable Lorentz transformation, and obtain the corresonding Wigner rotations of the sin states. Finally, we oint out for the first time that the entangled states corresonding to well-known ones in the study of -article entanglements, may be classified systematically using a articular couling of three angular momenta. Keywords: PACS Nos.: Quantum entanglement Wigner rotation Lorentz transformation Density matrix 0.65.Ud 0.65.Ca a 0.67.Mn. Introduction Schrödinger had ointed out long time ago [] that quantum entanglement is a crucial element of quantum mechanics. In fact quantum entanglement is one of the most eculiar feature that distinguishes quantum hysics from classical hysics and lies at the heart of what is now quantum information theory []. Quantum hysics allows correlations between satially searated systems that are fundamentally different from classical correlations, and this difference becomes evident when entangled states violate Bell-tye inequalities that lace an uer bound on the correlations comatible with local hidden variable (or local realistic) theories [, 4]. In the last two decades research has been very focused on quantum entanglement because the field of quantum information theory (cf. [5, 6]) has develoed rather quickly to be an imortant one. It is very imortant to get a good understanding of entanglement roerties of the quantum states, under *Corresonding author, anirban.chakraborti@ec.fr effects of accelerations (Lorentz transformations) and magnetic fields (constant and homogeneous), e.g., in studying quantum entangled states of the articles that are roduced and detected in Stern Gerlach exeriments [7]. The issues of quantum entanglement in (constant) external magnetic field were addressed in a revious aer [8], through the aroach of Wigner rotations of canonical sin. We exlored quantum entanglement in the resence of external fields in which the entangled sin states evolved instead of moving away (after being generated in entangled states) as free articles. Considering frames of observers related through Lorentz transformations, it was shown that the entanglement is frame indeendent but the violation of Bell s inequality is frame deendent. Similar features and correlations (or degrees of entanglement) were noted for sins undergoing recession in a magnetic field in that study. However, the study of quantum entanglement in magnetic fields was limited to -article states of total sin zero. The total sin being zero the effects of sin recession in magnetic field cancel (derived exlicitly). Here we remove this restriction and consider entangled three-article states of different total sins. Thus we study the consequences of sin recessions induced by external magnetic fields on three sinning articles, and entangled Ó 0 IACS

2 A. Chakrabarti, A. Chakraborti -article states in constant external magnetic fields. The unitary action is indeed local but it is now time deendent. A remarkable new asect emerges: The correlations of the entangled states exhibits eriodicity which we exlore systematically. Different classes of well-known entangled states, aear, increase, decrease and disaear in a eriodic fashion. They can be resent simultaneously, and all of these haen resecting the overall unitarity constraints. To our knowledge such eriodic correlations are obtained and dislayed here for the first time. The recessions being induced by unitary matrices the initial -tangles are conserved. But our aim is to dislay how such a constraint leaves room for rich and subtle atterns of eriodicity in the correlations involved. We emhasize that such effects are not imlemented via some arbitrary and artificial rescrition but in a tyical hysical situation, namely the resence of a magnetic field. New correlations (not resent initially) aear, evolve and disaear eriodically. Then we also introduce an electric field orthogonal to the magnetic field, and obtain the corresonding Wigner rotations of the sin states. Finally, in this aer, we roose a new scheme of systematic classification of the entangled states corresonding to well-known ones in the study of -article entanglements (viz. Greenberger-Horne- Zeilinger jghzi Werner jwi flied Werner ew and their variant states), using a articular couling of three angular momenta. The lan of the aer is as follows: We introduce the notations and formalism in Sect. In Sects. and 4, the emerging atterns of eriodicity in correlations and density matrices, resectively, are studied exlicitly for different entangled states, and articularly for certain initial configuration of the velocities. In Sect. 5, we study the case with an electric field orthogonal to the magnetic field, linking to the receding case via Lorentz transformation, and obtain the corresonding Wigner rotations of the sin states. In Sect. 6, we demonstrate the roosed scheme of systematic classification of the entangled states corresonding to well-known ones in the study of -article entanglements, using the articular couling of three angular momenta. Finally, we make some concluding remarks and give an outlook of future directions of work in Sect. 7.. Sin-/ articles in constant magnetic field: formalism In this section, we describe the formalism and notations: (i) The unitary transformation matrices acting on the sin states of three articles of sin-/ and ositive rest mass will be constructed for arbitrary initial velocities ðv v vþ of the articles (,, ) resectively and a constant magnetic field B and the basic recession equations [8 0] will be used and generalized. (ii) A articularly simle configuration of the initial velocities is then selected for detailed study (later in Sect. ). This would ermit us to dislay the contents of the generalized equations, for a few selected cases of articular interest without obscuring the basic features due to a rofusion of arameters. This will be achieved by restricting the initial velocities to a lane orthogonal to the magnetic field B and assuming them to be of equal magnitude, namely by imosing B vi ¼ 0 vi ¼ v ði ¼ Þ: ðþ Since B v and v are constants of motion, the conditions will hold for all time t. Eight initial entangled -article states would be selected for detailed study in such a context. They are encoded using the following notations (for our three sin-/ articles) jijki ji i ji j ji k ðþ jii ðþ j i jiþ ð j i Þ ðþ and similarly for ðji j ji k Þ: In such notations the states at time t = 0 are ffiffi jiþ ð4þ ffiffi þ exði/þ þ exði/þ ð5þ and ffiffi þ exði/þ þ exði/þ ð6þ ð ¼Þand / ¼ 0 : Not only do these states have direct corresondence to the jghzi jwi ew states familiar in the study of -article entanglements [7, ] (Ref. [] cites original sources), but they have also been constructed long ago [, ] in the study of couling of three angular momenta involving eigenvalues of the oerator Z ¼ðJ JÞJ ð7þ of three angular momenta ðj J JÞ as is briefly exlained in Sect. 6 (iii) We will follow the time-evolution of the states Eqs. (4 6) with secial attention to eriodic oscillations (in Sect. ), and also study the corresonding

3 Entangled three-article states in magnetic field eriodic density matrices (in Sect. 4), dislaying many interesting features. (iv) We will generalize the background field to include a constant electric field E orthogonal to the magnetic field B such that E B ¼ 0 E \ B : This constraint would ermit us to obtain the results via an aroriate Lorentz transformation (in Sect. 5), and adat the results from Ref. [8] (citing original sources) for the resent -article case. First, following the lines of Ref. [8], for a single article we denote ð B v R Þ to be resectively the constant, homogeneous magnetic field, the velocity and the olarization. With unit vectors ð ^B ^vþ and c =, we have B ¼ B ^B v ¼ v ^v c ¼ðv Þ = : ð8þ The anomalous magnetic moment is denoted by ^a ¼ðgÞ=: We define, for mass m and charge e, eb x ¼ mc ^B ^aeb X ¼ mc ðc ^B ðcþð ^B:^vÞ^vÞ: ð9þ ð0þ The equations for ^v and R are d v ¼x v dt d R ¼ðx þ X Þ ðþ R : dt The constants of motion are ðv ^B ^vþ and the moduli x ¼ eb mc X ¼ ^aeb ðþ = c ðc Þð ^B ^vþ mc such that x ¼ x^x X ¼ X^X: Introducing an intermediate set of rotating axes, one can finally obtain the time-deendent unitary transformation matrix acting on sin states of a article moving with velocity v as M ¼ ex i Xt ð^x r Þ ex i xt ð ^B r Þ ðþ r denote the Pauli matrices and the comonents of X are (assuming ð ^B ^vþ 6¼ Þ X ¼ð ^B X Þ¼ aeb mc ðc ðcþð ^B ^vþ Þ X ¼ ð ^B ^vþx qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 0 ð^b ^vþ X ¼ ð ^B ð^b ^vþþ X qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð^b ^vþ ¼ ^aeb qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mc ðc Þð ^B ^vþ ð^b ^vþ : ð4þ Below we leave aside the articularly simle case arising for ð ^B ^vþ ¼: We follow the same rescrition as in Ref. [8], and denote ^B r ¼ðb r þ b r þ b r Þ ^X ð5þ r ¼ðl r þ l r Þ so that b þ b þ b ¼ l þ l ¼ l ¼ 0: We also introduce the notations ðc sþ cos xt sin xt ðc 0 s 0 Þ cos Xt sin Xt so that we can now write M a ib ib a we define a ¼ðc 0 il s 0 Þðc ib sþl s 0 ðb þ ib Þs b ¼ðc 0 il s 0 Þðb ib Þs þ l s 0 ðc þ ib sþ ð6þ ð7þ ð8þ ð9þ which will be the recession arameters. Note that this recession arameter a should not be confused with the anomalous magnetic moment ^a introduced earlier in Eq. (9). Using the notations defined above, we obtain the unitarity constraint M y M ðaa þ bb Þ 0 0 ¼ 0 0 : ð0þ The action of M on the sin states ðþ j i jiþ ji ¼ 0 0 ðþ is given by M ji ¼ aji ib ibjiþ a : ðþ

4 A. Chakrabarti, A. Chakraborti So far we have been considering a single article with velocity v : For three articles with velocities ðv v vþ we denote the corresonding recession arameters as ða b Þ ða b Þ ða b Þ generalizing aroriately (a, b) of Eq. (9) Thus, for examle, an initial state at t = 0, jai ðjiþ Þ ðþ will evolve as ðm ðþ M ðþ M ðþ ÞjAi ¼ ffiffiffi a ji ib a ji ib a ji ib þ ffiffi ib jiþ a ib jiþ a ð4þ ib jiþ a or M ðþ M ðþ M ðþ jai ¼ d jiþ d þ d þ d þ d þ d þ d þ d d ¼ ffiffiffi ða a a þ ib b b Þ d ¼ ffiffiffi ða b b þ ib a a Þ d ¼ ffiffiffi ðb a b þ ia b a Þ d ¼ ffiffiffi ðb b a þ ia a b Þ d ¼ ffiffiffi ða a a þ ib b b Þ d ¼ ffiffiffi ðib a a þ a b b Þ d ¼ ffiffiffi ðia b a þ b a b Þ d ¼ ffiffiffi ðia a b þ b b a Þ: ð5þ ð6þ The coefficients above now involve three distinct eriodic time deendences through ðc sþ i cos x it sin x it ðc 0 s 0 Þ i cos X it sin X ð7þ it (i =,, ), determined by the initial velocities ðv v vþ and their orientations with resect to B : Note that we ignore the mutual interactions of the articles assumed to be weak enough as comared to that with a strong magnetic field B : The robability associated to the state jijkiis defined to be P ijk ¼ dijk d ijk: ð8þ We note that in jijki the sum of the robablities of ji i being either ji or must be. Using the relation ða i a i þ b i b i Þ¼ ði ¼ Þ ð9þ systematically, one may check that, for examle, P þ P ¼ ða a a a þ b b b b Þ P ¼ ða a b b þ b b a a Þ ð0þ ðþ P ¼ P þ P ¼ ða a þ b b Þða a þ b b Þ¼ : ðþ Similarly, P ¼ P þ P ¼ : Hence P þ P ¼ : ðþ ð4þ Analogous results hold for (j, k). For the general configuration the interlay of the eriods (see Eq. (7)] 4 4 ði ¼ Þ ð5þ x i X i imlies a rich structure in the variation with t of the coefficients d ijk in Eq. (5) and the correlations Eq. (8). Such variations will of course deend on the velocities and masses involved.. Periodic correlations for -article states in a magnetic field In this section we will select, to start with, a simle case and try to follow closely the eriodicities associated with the initial entangled -article states given by Eqs. (4 6). The constant magnetic field B is taken to be along the x axis, B ¼ðB 0 0Þ: ð6þ Three equal mass sin-/ articles (created, say, by the disintegration of a single one at rest) are assumed to have their initial velocities in the yz-lane and to be of equal magnitude, so that

5 Entangled three-article states in magnetic field B ƒ vðiþ ¼ 0 ƒ vðiþ ¼ v ði ¼ Þ: ð7þ Since v and B v are constants of motion, the velocities will stay in the yz lane and remain of equal magnitude. The three velocities rotate uniformly in the yz-lane. The recession equations Eq. () now simlify (for each case) to d R ¼ðx þ X Þ R dt eb x ¼ mc ^B x ^B ^aeb X ¼ m ^B ¼ ^acx ^B X ^B: and hence ð8þ d R ¼ðxþXÞ^B R : ð9þ dt Now in Eq. (4) the unitary transformation acting on a state jijki is given by the matrix M M M M a ib ib a and a ¼ cos x þ X t b ¼ sin x þ X t : ð40þ ð4þ The corresonding robabilities given by P ijk = f * ijk f ijk are P ¼P ¼ ð a b Þ ¼ ð45þ 4 sin ððx þ XÞtÞ P ¼ 8 ðsin ððx þ XÞtÞÞ consistent with the constraint of X P ijk ¼ : ijk ð46þ ð47þ Figure shows the variations of the robabilities P ijk defined in Eqs. (45) and (46) with h xþx t : The eriodic variations of the coefficients are easy to follow. Let us emhasize some secial oints: (i) At t = 0, one starts with a =, b = 0 jwi X f ijk jijki ¼ ffiffi jiþ : ð48þ ijk (ii) At t ¼ ðxþxþ ða ¼ b ¼ ffiffi ) ex ði=4þ jwi ffiffi ðjiþ Þð þ þ Þð þ þ Þg: ð49þ Note the negative signs before the two trilets, whose consequences have been discussed in Section 4... Case We start by studying the time-evolution of the state Eq. (4). One obtains at time t M M M ðjiþ Þ ¼ f jiþ f þ f þ f þ f þ f þ f þ f ð4þ f ¼ ffiffiffi ða þ ib Þ¼f and f ¼ f ¼ f ¼ f ¼ f ¼ f ¼ ffiffi abða þ ibþ: ð4þ ð44þ Probability Curve Curve θ Fig. Plot of the variations of the robabilities P ijk defined in the text by Eq. (45) (curve ) and Eq. (46) (curve ) with h xþx t

6 A. Chakrabarti, A. Chakraborti Now all the 8 ossible states are resent with equal robability P ¼ a4 b f P ¼ a b 4 f P ijk ¼ 8 ði j kþ ¼ or : ð50þ P ¼ b ð a f exði/þþð a f exði/þþ (iii) At t ¼ ðxþxþ ða ¼ 0 b ¼ ) jwi ¼ ffi ðjiþ Þ: ð5þ Note that aart from an overall sign () one is back at the starting oint. P ¼ a ð b f exði/þþð b f exði/þþ P ¼ a ð b f exði/þþð b f exði/þþ P ¼ b ð a f exði/þþð a f exði/þþ P ¼ b ð a f Þ P ¼ a ð b f Þ : ð56þ.. Case We will now study the time-evolution with the next initial state given by Eq. (5): jwi ðt¼0þ ¼ jwi ð0þ ¼ ffiffiffi þ exði/þ þ exði/þ ð5þ ð/ ¼ 0 Þ: We define f ¼ðþexði/Þþexði/ÞÞ ¼ ð/ ¼ 0Þ ¼ 0 / ¼ : ð5þ At time t, the eriodic evolution gives [using Eq. (40)] jwi ðtþ ¼ M M Mjwi ð0þ ¼ c 0 jiþ c 0 þ c þ c þ c þ c þ c þ c : with a ¼ cos x þ X t b ¼ sin x þ X t ð54þ and imlementing systematically a? b =, we have ffi c0 ¼ia bf c0 ¼ab f ffi c ¼ að b f Þ c ¼ ibð a f Þ ffi c ¼ aðexði/þb f Þ c ¼ ibðexði/þa f Þ ffi c ¼ aðexði/þb f Þ c ¼ ibðexði/þa f Þ: ð55þ The correlations have eriodicities determined by (a, b) above and are After this derivation, one notes that for: (i) (/ = 0,f = ) P ¼ a 4 b P ¼ a b 4 P ¼ b ð a Þ P ¼ a ð b Þ : consistent with the constraint Eq. (47) of P ijk P ijk =. ð57þ Figure shows the variations of the robabilities P ijk defined in Eq. (57) with h xþx t : (ii) (/ ¼ f ¼ 0) P ¼ 0 P ¼ b P ¼ a : consistent with the constraint Eq. (47) of P ijk P ijk =. Probability Curve Curve Curve Curve θ ð58þ Fig. Plot of the variations of the robabilities P ijk defined in the text by Eq. (57) with h xþx t P is curve, P is curve, P are curve and P are curve 4

7 Entangled three-article states in magnetic field Case (i) exhibits a richer attern of eriodicity which we analyse now, inoiting some secial values of t. Att = 0, one starts with a =, b = 0: jwi ð0þ ¼ ffiffi þ þ : ð59þ At t ¼ ðxþxþ ða ¼ 0 b ¼ Þ : jwi ¼ i ffiffiffi þ þ : ð60þ we get the flied over comlementary trilet. At t ¼ ðxþxþ ða ¼ b ¼ ffiffi Þ : jwi ¼ i ffiffiffi jiþ þ þ 6 þ ffiffi ð6þ : 6 Now all the ossible jijki (8 in number) aear, groued as above according as the multilicity of the index ji is odd ( or ) or even (0 or ). Other interesting oints are rovided by the extrema of (P P Þ: For a = /, b = /, P ¼ 4 9 P ¼ 9 P ¼ 9 P ¼ 0: ð6þ Considering the ositive roots for examle, a ¼ ffiffiffiffiffiffiffi ffiffi = b ¼ = jwi ¼ i j iþ þ þ : For a ¼ = ffiffi ffiffiffiffiffiffiffi b ¼ = P ¼ 9 P ¼ 4 9 P ¼ 0 P ¼ 9 ð6þ ð64þ and jwi ¼ i ffi ji þ þ þ : ð65þ Aart from the factor i changing lace, the assage from Eqs. (6) to(65) corresonds to ji ji : ð66þ The coefficients for the cases above will be further discussed in Sect. 4. The time-evolution of the initial state jwi ð0þ ¼ þ exði/þ þ exði/þ ð67þ can be studied in closely analogous fashion, but we will not reeat the stes again. The eriods considered above, tyically generated by (a, b), of Eq. (4) (x X) are given by Eq. () can be long but diminish as B and the velocities increase in magnitude. 4. Periodic density matrices We have seen how, starting with a restricted initial state, sin recessions in a magnetic field lead to eriodic aearances and disaearances of all the ossible eight states jijki of three sin-/ articles. Since such eriodicities are induced through local unitary transformations, acting searately on each state ji the sum of the varying correlations remains unity. Moreover, the basic invariant measures of entanglement (-tangle, -tangles) must also be conserved. The corresonding constrained eriodic variations of the elements of the density matrices is briefly studied below for secial cases. For jwi ¼ f 0 jiþ f þ f þ f j i þ g 0 þ g þ g þ g ji ð68þ tracing out the index, the density matrix for the Eq. () subsystem is a 00 a 0 a 0 a 0 a q ¼ 0 a a a a 0 a a a ð69þ a 0 a a a a 00 ¼ f 0 f 0 þ g g a 0 ¼ f 0 f þ g g a 0 ¼ f 0 f þ g g a 0 ¼ f 0 f þ g g 0 a ¼ f f þ g g a ¼ f f þ g g a ¼ f f þ g g 0 a ¼ f f þ g g a ¼ f f þ g g 0 a ¼ f f þ g 0g 0 : ð70þ Similarly, one can obtain q, q. The -tangle (invariant under ermutations of articles,, ) is obtained as follows [4]: We define fq ¼ 0 i i 0 0 i i 0 q 0 i i 0 0 i i 0 : ð7þ

8 A. Chakrabarti, A. Chakraborti For the subsystems considered (q fq ) has at most two non-zero eigenvalues. Let (k, k ) be the square roots (ositive) of these two. Then the -tangle is given by s ¼ 4k k : ð7þ This can also be directly exressed in terms of the coefficients of Eq. (68) above: s ¼ 4jd d þ 4d j ð7þ [in our notations of Eq. (68)] d ¼ðf 0 g 0 Þ þðf g Þ þðf g Þ þðf g Þ d ¼ f 0 g 0 ðf g þ f g þ f g Þ þðf g f g þ f g f g þ f g f g Þ d ¼ f 0 f g g þ f f g 0 g : ð74þ The conversion of Eq. (7) to Eq. (7) ermits us to relate directly certain central features of the eriodicities studied in Sect. to constraints imosed by the invariance of Eq. (7). In articular, let us now evaluate the crucial role of the negative signs signalled below Eq. (49). For the initial state Eq. (48), at t = 0 f 0 ¼ g 0 ¼ ffiffi f ¼ f ¼ f ¼ g ¼ g ¼ g ¼ 0 and thus d ¼ 4 d ¼ d ¼ 0: ð75þ ð76þ Hence from Eq. (7) s ¼ ð77þ a well-known result for GHZ states Eq. (48). At t ¼ ðxþxþ from Eqs. (49) to(74) ex i=4 ðd d d Þ¼ ð ffiffi ð4 6 Þ: Þ 4 ð78þ Hence again (as exected), s ¼ 4 j4 :6 4:j ¼ : ð79þ 64 If all the terms have the same sign, as in jwi 0 ¼ ffiffiffi ðjiþ þ þ Þ þ ffiffi ð80þ ð þ þ þ Þ then d changes sign and s ¼ 4 j4 :6 þ 4:j ¼ 0: ð8þ 64 Thus the two negative signs in Eq. (49) alters s from a minimum (0) to maximum (). For the initial state Eq. (5), using Eqs. (5 55) the f s in Eq. (68) have cubic eriodic terms given by (ab,a b), a ¼ cos x þ X t b ¼ sin x þ X t : ð8þ Hence P in Eqs. (69) and (70) has sixth order eriodic terms (a b 4, a 4 b, a b ). Such eriodicities indeed turn out to be comatible with [5]: s ¼ 0 s ¼ s ¼ s ¼ 4 9 : ð8þ One realizes now more fully how elaborate and subtle atterns of eriodicities are comatible with constraints of local unitary transformations involved in sin-recessions. 5. Constant orthogonal electric and magnetic fields We briefly indicate below how the eriodicities studied so far, for a magnetic field alone are affected by the resence of an electric field E satisfying E : B ¼ 0 E \ B : ð84þ Consider a Lorentz transformation corresonding to the 4-velocity = u 00 ¼ E B E B ^E ^B ð85þ denoting E ¼ E ^E B ¼ B ^B ð ^E ¼ ¼ ^B Þ: In the transformed frame the tensor ð E B Þ reduces to ð 0 0 E B Þ 0 0 E = E ¼ 0 B ¼ B ð86þ such that B B ¼ B E ¼ B E ð87þ 0 0 B E ¼ 0 ¼ B E : So in this frame, one finds back the situation studied in the Sects. 4, with B 0 ¼ðB E Þ = : ð88þ The velocities and the sins of the articles are transformed according to standard rules [8, 9]. We now recaitulate some essential oints. A 4-velocity u is transformed by a Lorentz transformation corresonding to u 00 to u 0 such that

9 Entangled three-article states in magnetic field u 0 0 ¼ðu 0u 00 0 þ u u 00 Þ: ð89þ Define a ¼ð þ u 0 Þð þ u 00 0 Þð þ u0 0 Þ b ¼ð þ u 0 þ u 00 0 þ u0 0 Þ ð90þ and cos d ¼ b ffiffiffiffiffi : ð9þ a The sin states are conserved if u u 00 ¼ 0: Otherwise they undergo a Wigner rotation d about the axis ^k ¼ u u 00 ð9þ u u 00 such that jþi jþi 0 ¼ cos d þ i ^k sin d j jþi þ ið ^k i ^k Þ sin d ji i ji 0 ¼ ið ^k þ i ^k Þ sin d jþi þ cos d i ^k sin d ji: ð9þ The inverse rotation ðd dþ exresses ðþ j i jiþ in terms of ðþ j i 0 ji 0 Þ: The crucial oint to note is that in Eq. (5) aart from the ðc 0 ^v 0 Þ corresonding to the transformed velocities [deending on the initial velocity v and u 00 given by Eq. (85)], B is relaced by B 0 = (B - E ) /. The magnitudes ðx 0 X 0 Þ thus obtained determine the modified eriodicities. This is the consequence of the resence of E in the initial frame. In the transformed frame our receeding results (for E ¼ 0) can be imlemented systematically along with the velocities transformed corresonding to Eq. (85). Then the inverting of Eq. (9) gives the results for the initial frame ( E 6¼ 0). 6. Classification scheme of -article entangled states In this section, we roose a new classification scheme of the -article entangled states, by using the eigenstates of Z ¼ðJ JÞJ of three angular momenta. One can systematically construct eigenstates [, ] of three couled angular momenta (J J J) by diagonalizing the oerators ðj þ J þ JÞ ðj ð0þ þ J ð0þ þ J ð0þ Þ ð94þ and Z ¼ðJ JÞJ J (0) i,(i =,, ) are the rojections on the z axis. The states are denoted by the resective eigenvalues of the above oerators (jðj þ Þ j ð0þ f) asjjmfi: Not only one obtains a comlete mutually orthogonal set of eigenstates for each j but along with reduction with resect to the rotation grou one obtains simultaneously a reduction with resect to S, the ermutation grou of three articles. This is in shar contrast with the usual -ste reduction via -j coefficients such ermutations lead to 6-j coefficients. For our uroses, we need here only the results for j ¼ j ¼ j ¼ : In the Table, the states on the left corresond to eigenvalues (j m f) resectively of the oerators Eq. (94) and those on the right to the values of of (m, m, m ) of J (0) i, (i =,, ) denoted by ( ji Þ: This rovides the comlete set of 8 states sanning the sace of ossible values of (m, m, m ). Thus ffiffi 0 0 ¼ ðji Þ ð95þ giving the jghzi states as a doublet. The others corresond directly to the Werner ( jwi) and the flied Werner ( ew ) and their variants with relative hases ex i which we had mentioned in the remarks below Eq. (6) in Sect.. Of the three oerators in Eq. (94) the first two are invariant under all ermutations of the articles (,, ). The remaining one (Z) is invariant under circular ermutations of (,, ) and just changes sign under Eqs. (), () and (). This is in shar contrast to the standard - ste coulings one has to ass from one -j couling scheme to another under ermutations. This is at the root of Table The states on the left corresond to eigenvalues (j m f) resectively of the oerators Eq. (94) and those on the right to the values of of (m, m, m ) of J (0) i, (i =,, ) denoted by ( ji ) jjmfi jm m m i 0 ji 0 0 ð þ þ Þ 0 ð þ þ Þ ffiffi E ffiffi 4 ðexði Þ þ exði Þ þ Þ ffiffi E ffiffi ðexði Þ þ exði Þ þ Þ 4

10 A. Chakrabarti, A. Chakraborti the simultaneous reduction under S via the imlementation of Z. This also hels to exlain the direct relations of the Z-eigenstates with -tangles invariant under ermutations of (,, ). 7. Conclusions An external (constant) magnetic field induces a recession of the sin of each article of the entangled states considered, through local unitary transformations. Note that for two article states of total sin zero, studied in earlier aer, the eriodic recession of individual sins was not resent. Here we analysed three article states such eriodicities ( and 4 eriods resectively for the articular cases illustrated) in correlations and density matrices were remarkably dislayed and intertwined. The atterns of eriodicity thus emerging were shown to be remarkably rich and subtle. Such a study was resented for the first time. The individual recessions being imlemented by local unitary matrices, the initial -tangle was conserved. But simly the verification of this fact was not the aim of the analyses of Sect. and 4. We went beyond that and dislayed in details how the conservation left scoe for comonent eriodic correlations to aear, increase and decrease. We inointed the crucial roles of the signs of the coefficients of different comonents, again for the first time. The generalization to include orthogonal electric field was resented in Sect. 5, using the Wigner rotation. For E = B, the Lorentz transformation via Eq. (85) is not welldefined and for E [ B it becomes comlex. The limiting case E = B (e.g. for a lane wave field) will be studied else using exact solutions [6] of the Dirac-Pauli equations in such fields. The remarkable and systematic corresondence of famous entangled states to a secific couling scheme for -angular momenta was resented in Sect. 6, and a new classification scheme was roosed. We intend to study this asect in details else. Finally one may note that unitary transformations may be non-local when induced via unitary braid matrices [7, 8]. Acting on ure roduct states, they can then generate entanglement. In the resent work we started with states already entangled and then followed their eriodic ramifications as they evolved in the magnetic fields, which is comletely different. The unitary action here is indeed local but it is now time deendent. Different classes of well-known entangled states, aear, increase, decrease and disaear in a eriodic fashion. They can be resent simultaneously, and all of these haen resecting the overall unitarity constraints. References [] E Schrödinger Die Nat. 807, 8, 844 (95) [] R Horodecki, P Horodecki, M Horodecki, and K Horodecki Rev. Mod. Phys (009) [] J S Bell Rev. Mod. Phys (966) [4] L E Ballentine Am. J. Phys (986) [5] C Macchiavello, G M Palma and A Zeilinger Quantum Comutation and Quantum Information Theory (Singaore: World Scientific Publishing Co. Pte. Ltd.) (000) [6] M A Nielsen and I L Chuang Quantum Comutation and Quantum Information (Cambridge: Cambridge University Press) (00) [7] D M Greenberger, M A Horne, A Shimony and A Zeilinger Am. J. Phys. 58 (990) [8] A Chakrabarti J. Phys. A (009) [9] A Chakrabarti Fortschr. Phys (988) [0] G B Malykin Usekhi (006) [] E Jung, M R Hwang and D Park Phys. Rev. A (009) [] A Chakrabarti Annales de l Institut Henri Poincaré 0 (964) [] J M Levy Leblond and M Nahas J. Math. Phys. 6 7 (965) [4] V Coffman, J Kundu and W K Wootters Phys. Rev. A (000) [5] H A Carteret and A Sudbery J. Phys. A 498 (000) [6] A Chakrabarti Il Nuovo Cimento A (968) [7] B Abdesselam and A Chakrabarti J. Phys. Math. P00804 (00) [8] A Chakrabarti, A Chakraborti and A Jedidi J. Phys. A (00)