The Nature and Realization of Quantum Entanglement

Transcription

1 Appled Physcs Research; Vol. 8, No. 6; 06 ISSN E-ISSN Publshed by Canadan Center of Scence and Educaton The Nature and Realzaton of Quantu Entangleent Bn Lang College of Scence, Chongqng Unversty of Posts and Telecouncaton, Chongqng , Chna Correspondence: Bn Lang, College of Scence, Chongqng Unversty of Posts and Telecouncaton, Chongqng , Chna. E-al: Receved: October 9, 06 Accepted: October 5, 06 Onlne Publshed: Noveber 30, 06 do:0.5539/apr.v8n6p96 URL: Abstract Ths paper analyses the nature of quantu entangleent, proves the quantu entangleent s not acton at a dstance, proposes a schee to realze quantu entangleent, explans that the quantu entangleent s not acton at a dstance and the non-clonng theore of quantu state ensure the quantu echancs s consstent wth relatvty and ake the superlunal councaton could not happened. Keywords: Quantu Entangleent, teleportaton, Quantu Councaton. Introducton Is the quantu entangleent a spooky acton at a dstance? Ths proble has caused heated debate for a long te (Schrödnger, 935; Ensten, Podolsky, & Rosen, 935; L, 000). It s not a sple physcs probles, also nvolves phlosophy. The quantu entangleent s the core concept of quantu nforaton. No real quantu entangleent, no real quantu councaton. The concept of quantu entangleent was put forward by Schrodnger n 935, and s called the essence of quantu echancs (Schrödnger, 935). In the sae year, Ensten, Podolsky and Rosen (EPR) suggested that quantu echancs ght be ncoplete on the bass of that (Ensten, Podolsky, & Rosen, 935), and so caused the n-depth study of people to quantu entangleent and the essence of quantu echancs. However, snce then, the experental results do not support the EPR's vewpont (L, 000; Zhang, 006). It ust be noted that Hdden paraeter theory represented by Bell nequalty (Bell, 964) s essentally deternstc, that s not accord wth the experents proves only agan that the oveent of cro partcles s not deternstc rather than quantu entangleent s an acton at a dstance. Accordng to the Schdt decoposton theore (L, 000; Nelsen & Chuang, 004; Hughes & Henrchs, 004; Zhang, 006) of quantu echancs, the pure state ψ of a coposte syste coposed of two arbtrary subsystes can be expand as ψ = ρ φ φ, () () () () where φ and () () φ s a par of egenstates of densty operator ˆρ and ˆρ () wth the coon egenvalue ρ and called a par of dual states, whle the nuber of ters of the expanson () s called the Schdt nuber, and there s ρ =. () The Schdt expanson to satsfy the equaton above s called coplete, as shown n equaton (), and we dscuss only the coplete Schdt expanson n ths artcle. Accordng to the lteratures of quantu nforaton, f the Schdt nuber s greater than, the easureent to a subsyste of the coposte syste akes the put state ψ becoe nstantaneously nto soe ter n the expanson wth dfferent probablty ρ, and ths s called the quantu entangleent. In other words, the easureent to a subsyste changes nstantaneously the state of another subsyste. So, the quantu entangleent sees to be superlunal. But, the nteracton between aterals s not superlunal accordng to relatvty. The contradcton s sharp, how to solve? 96

2 apr.ccsenet.org Appled Physcs Research Vol. 8, No. 6; 06 In the followng we frstly analyze the nature of quantu entangleent, prove the quantu entangleent s not acton at a dstance and propose a schee to realze quantu entangleent, and then explan that the quantu entangleent s not acton at a dstance and the non-clonng theore of quantu state ensure together the quantu echancs s consstent wth relatvty and ake the superlunal councaton could not happened.. The Nature of Quantu Entangleent () The quantu entangleent s not acton at a dstance. For the faous Schrodnger cat, the Schdt decoposton of the pure state ψ of the coposte syste coposed of a cat and a poson bottle s ψ = bc cl + bo cd, (3) where bc, bo, cl and cd s the state of the closed poson bottle, open poson bottle, lve and dead cat, respectvely. The reason of cat dead s that t contacted the poson gas fro the bottle when the bottle was open. If there s not the poson gas, the state of cat does not change no atter the bottle was open or not. Naely, the entangleent of equaton (3) s only an artfcal atheatcal cobnaton wthout real sgnfcance f there s not the poson gas. It s a par of entangled photons prepared wth SPDC of BBO crystal n Fgure whch s often quoted n the lteratures of quantu nforaton (Nelsen & Chuang, 004; Hughes & Henrchs, 004). Accordng to these lteratures, the two photons flyng respectvely n the top and botto cone are n the entangled state due to the photon ndscernblty wthn the overlappng area of the two cones, ther polarzaton entangleent state s ψ ± = () () () () H V ± V H, (4) where H, V are the horzontal and vertcal polarzaton state of the photons, respectvely. The equaton above shows that as long as the polarzaton state of one photon has been detected, the polarzaton state of another photon would know edately when two photons are separated far. But there s an plct assupton: the polarzaton state of each photon s changeless n oton, and ths ples there s not nteracton between the two photons, they are not really entangled. In fact, the entangleent as shown n the Fgure s only the synony of ndscernblty of photons, and such photons of changeless state can't transfer any nforaton. In a word, the quantu entangleent s real only when there s real nteracton between the subsystes. Snce all of the four nteractons between aterals are not acton at a dstance, so the quantu entangleent s not acton at a dstance. Fgure. The par of entangled photons prepared wth SPDC of BBO crystal () The entangleent does not change the probablty to fnd each qubt state. Assue the coherent state of qubt A and B s A B = α 0 + β, = δ + γ 3, (5) 97

3 apr.ccsenet.org Appled Physcs Research Vol. 8, No. 6; 06 respectvely, where the four quantu states 0 and, and 3 are orthogonal and noralzaton, α + β =, δ + γ =. The entangled state of the two qubts s AB = A B = αδ 0 + αγ 03 + βδ + βγ 3. (6) The probablty to fnd qubt state 0 n the entangled state s = + = + =, (7) 0 AB αδ αγ 3 α δ γ α and ths s dentcal wth the probablty to fnd qubt state 0 n the coherent state A before the entangled. Slarly, the probablty to fnd other qubt state n the entangled state of equaton (6) s dentcal wth that before the entangled. Therefore, the entangleent does not change the probablty to fnd each qubt state. 3. The Realzaton of Quantu Entangleent As sad prevously, the entangleent s real only when there s real nteracton between the subsystes. The followng s a schee to realze quantu entangleent. It s well known that the partcle radated laser pulse transted fro hgh to low energy level, and the partcle absorbed laser pulse transted fro low to hgh energy level. Thereby the laser pulse could be for the use of akng partcles n entangleent state. In fact, the laser pulse had been consdered for the use of change the qubt state n the realzaton schee of quantu coputer, for exaple, on trap schee, QED schee and quantu dots schee (Hughes & Henrchs, 004; De Caro & Garucco, 994; Peres, 996; Kwat et al., 995). Iage the ground and excted state of two dentcal partcles located n two dfferent places s 0 and =,, respectvely, and the entangled state s the nteracton Haltonan s ( = ) ψ = + Hˆ 0 0, (8) = ω e a a, (9) φ + 0 whereω 0 = E E 0 s the dfference of two energy levels of the partcle, naely the energy of the pulse photon. If the energy levels are that of electronc spn, then there s ω 0 = E E 0 = e H (SI), H the agnetc ntensty; and a, a + s the annhlaton and creaton operator of photon, and φ s the phase of the laser. So there s ˆ ω0 H ψ = e φ 0. (0) Obvously, the Haltonan of equaton (9) descrbes that the partcle releases a photon and transts fro the excted state to the ground state, whle the partcle absorbs that photon and transts fro the ground state to the excted state. Ths s the sngle-drecton entangleent. And the Haltonan of double-drecton entangleent φ + φ + ˆ d ω0 H = e aa + e a a, () where the frst ter s just the equaton (9), and the second ter descrbes that the partcle releases a photon and transts fro the excted state to the ground state, whle the partcle absorbs that photon and transts fro the ground state to the excted state, thus the perforance of double-drecton entangleent to the entangled state ψ gves ˆ ω0 φ φ Hd ψ = ( e 0 + e 0 ). () Ths shows that the result of double-drecton entangleent s that the syste has returned to ts ntal state, only the relatve phase of the two ters changes byφ. Obvously, both the equaton (8) and () s the greatest entangled state, and the probablty to fnd each qubt state does not change due to the entangleent. 98

4 apr.ccsenet.org Appled Physcs Research Vol. 8, No. 6; Quantu Entangleent and Quantu Councaton We know that the key of realzaton of quantu councaton s the realzaton of quantu teleportaton, and the key of realzaton of quantu teleportaton s the realzaton of quantu entangleent. The explanaton s as follows (L, 000; Nelsen & Chuang, 004; Hughes & Henrchs, 004; Zhang, 006). Assue each of Alce and Bob has one qubt of the greatest entangled state ( 00 ) φ + = +, and Alce has the frst qubt. The unknown qubt state of Alce wants to transt to Bob s α = a 0 + b, (3) where both abs, the unknown coeffcent. Thus, the ntal state of the syste conssted of the three qubts s ( ) Ψ 0 = α φ+ = a + b + a + b. (4) Snce there exsts quantu entangleent, Alce perfors the CNOT operaton to the frst two qubts of the above equaton, naely α and her own qubts, and t gves ( ) Ψ = a + b + a + b. (5) And then the H door operaton to the frst qubt of the equaton above gves Ψ = ( 0 ) 00 ( 0 ) 0 ( 0 ) ( 0 ) 0 a + + b + a + + b a a b b a a b b = ( a b ) ( a b ) ( a b ) ( a b ) = , (6) where the followng forula s used: H 0 = 0 +, H = ( 0 ) Now Alce easures each of the frst two qubts before one of the four superposton states of the equaton (6) once, and Ψ becoes one of the four states wth equal probablty due to the exstence of quantu entangleent. But the four states also could be obtaned by the operatons to the state α as follows: I α = a 0 + b, (7) Z α = a 0 b, X α = a+ b 0,. (8) Y α = a b 0. Therefore, the nforaton of the frst two qubt states of each ter of equaton (5) s correspondent wth the above each operaton one to one. Naely the above local operaton of Alce has projected the Bob's qubt to one of the above four states due to the exstence of quantu entangleent. And then Alce sends the obtaned nforaton about the frst two qubt states to Bob through classcal councaton, and Bob has known that he should do the nverse transforaton of whch equaton of the equaton (8) accordng to the nforaton fro Alce and obtaned the state α. In the transfer process above, the state α, naely the coeffcent a and b are unknown always for Alce, so the transfer process s called the quantu teleportaton. Obvously, the realzaton of the seres of operatons above s possble wthout the real quantu entangleent, therefore, the key of realzaton of quantu teleportaton s the realzaton of quantu entangleent. 99

5 apr.ccsenet.org Appled Physcs Research Vol. 8, No. 6; Concluson: Superlunal Councaton s Ipossble The above analyses the nature of quantu entangleent, proves the quantu entangleent s not acton at a dstance, proposes a schee to realze quantu entangleent, and shows the key of realzaton of quantu teleportaton s the realzaton of quantu entangleent. Also t ust be noted that the quantu entangleent s not acton at a dstance and the non-clonng theore (Wooters & Zurek, 98; Nelson & Chuang, 000; Pan, Chen, Żukowsk, Wenfurter, & Zelnger, 008; Xong, Yu, Zhan, & Zhang, 06) of quantu state ensure together the quantu echancs s consstent wth relatvty, and ake the superlunal councaton could not happened. The explanaton s as follows. If the quantu entangleent s a spooky acton at a dstance, when Alce wants to send the one bt of nforaton to Bob, she could ake nstantaneously the qubts of Bob n the correspondng quantu state as long as she perfors the proper operaton to her own qubts. If Bob could clone relably these unknown quantu states fro Alce, he could know the nforaton fro Alce by copy and easureent for any tes, and thus the superlunal councaton s realzed. But, snce the quantu entangleent s not acton at a dstance and the unknown quantu state could be not relable cloned, the quantu echancs s consstent wth relatvty, the superlunal councaton s possble. References Bell, J. S. (964). Physcs Long Island Cty. NY, (95),. De Caro, L., & Garucco, A. (994). Relablty of Bell-nequalty easureents usng polarzaton correlatons n paraetrc-down-converson photon sources. Physcal Revew A, 50(4), R803. Ensten, A., Podolsky, B., & Rosen, N. (935). Can quantu-echancal descrpton of physcal realty be consdered coplete?. Physcal revew, 47(0), 777. Hughes, R., & Henrchs, T. (004). A quantu nforaton scence and technology roadap. Kwat, P. G., Mattle, K., Wenfurter, H., Zelnger, A., Sergenko, A. V., & Shh, Y. (995). New hgh-ntensty source of polarzaton-entangled photon pars. Physcal Revew Letters, 75(4), L, C. Z. (000). Quantu councaton and quantu coputaton. Natonal unversty of defense technology press. Nelsen, M. A., & Chuang, I. L. (004). Quantu Coputaton and Quantu Inforaton (Cabrdge Seres on Inforaton and the Natural Scences). Nelson, M. A., & Chuang, I. L. (000). Quantu coputaton and quantu nforaton. Cabrdge Unversty Press. Pan, J. W., Chen, Z. B., Żukowsk, M., Wenfurter, H., & Zelnger, A. (008). Mult-photon entangleent and nterferoetry. arxv preprnt arxv: Peres, A. (996). Separablty crteron for densty atrces. Physcal Revew Letters, 77(8), 43. Schrödnger, E. (935). De gegenwärtge Stuaton n der Quantenechank. Naturwssenschaften, 3(49), Wooters, W. K., & Zurek, W. K. (98). Quantu no-clonng theore. Nature, 99, 80. Xong, P. Y., Yu, X. T., Zhan, H. T., & Zhang, Z. C. (06). Multple teleportaton va partally entangled GHZ state. Fronters of Physcs, (4), -8. Zhang, Y. D. (006). The physcal prncple of quantu nforaton. Scence press, Chna. Copyrghts Copyrght for ths artcle s retaned by the author(s), wth frst publcaton rghts granted to the journal. Ths s an open-access artcle dstrbuted under the ters and condtons of the Creatve Coons Attrbuton lcense ( 00