Entanglement distillation with local common reservoirs

(1)PHYSICAL REVIEW A 79, 022306 共2009兲. Entanglement distillation with local common reservoirs D. F. Mundarain1 and M. Orszag2. 1. Departamento de Física, Universidad Simón Bolívar, Apartado 89000, Caracas 1080A, Venezuela 2 Facultad de Física, Pontificia Universidad Católica de Chile, Casilla 306, Santiago, Chile 共Received 13 November 2008; revised manuscript received 8 January 2009; published 5 February 2009兲 In this work, we study a system composed of four two-level particles, two source entangled qubits AB, and two ancilla entangled qubits A⬘B⬘, in the presence of two local common reservoirs, i.e., one reservoir for subsystem AA⬘ and another for BB⬘. For initial maximal entangled states, we study the entanglement of the source if after evolution the ancilla is measured at time t. DOI: 10.1103/PhysRevA.79.022306. PACS number共s兲: 03.67.Bg, 03.65.Ud, 03.65.Yz. I. INTRODUCTION. Entanglement is an essential tool for many interesting features of quantum theory of information. Quantum cryptography and quantum teleportation require maximal entanglement for optimal operation 关1,2兴. It is known that after evolution in the presence of reservoirs, the entanglement is lost. Some interesting proposals for recovering it or for diminishing the entanglement loss rate include filtering operation on one copy of the system or distillation protocols on many copies. Entanglement purification is an important concept in quantum-information theory. It was originally suggested by Bennett et al. in 1996 关3兴. In their proposal, they showed that it is possible to convert two copies of a less entangled state into one copy of a more entangled state, using only local operations and classical communication. In particular, they increase the entanglement in one pair of entangled source particles coupling it to an ancilla entangled pair via some local bilateral operations and local measurements on the ancilla. It was shown by Horodecki et al. 关4兴 that in a 2 ⫻ 2 system, the entanglement of any entangled system can be enhanced by a combination of filtering operations and Bennett’s distillation protocol. Also, Deutsch et al. 关5兴 introduce the notion of quantum privacy amplification that was developed to account for the noise in the quantum communication channel over which information must be sent when making use of a cryptographic protocol. This scheme uses also an entanglement purification procedure requiring only a few controlled-NOT 共CNOT兲 and single-qubit operations. When the initial fidelity is close to 0.5, this procedure becomes more efficient than the Bennett protocol. So far, experimental realizations of entanglement purification schemes have been limited to linear optics 关6兴 and trapped ions 关7兴. Particularly, this last reference uses the protocol of Bennett et al. for trapped ions, and they report an efficient entanglement purification with atomic quantum bits, where two noisy entangled pairs were created and distilled into one higher-fidelity pair for further use, with a success probability above 35%. In this work, we are going to compare the evolution of the source entanglement using the original Bennett protocol with one in which all the operations that couple the source and ancilla locally are substituted by the simpler interaction via local common reservoirs. After the evolution, the ancilla is 1050-2947/2009/79共2兲/022306共6兲. measured locally and Alice and Bob communicate via a classical channel. In the following section, we review briefly Bennett’s distillation protocol. In Sec. III, we consider the problem of four particles in the presence of two local common reservoirs in vacuum. The results are presented in Sec. IV and the discussion and summary are given in Sec. V. II. BENNETT’S DISTILLATION PROTOCOL. Bennett’s distillation protocol 关3兴 for a 2 ⫻ 2 entangled system is briefly described in terms of the following algorithm. 共i兲 Consider a mixed state ␳ with fidelity F = 具␺− 兩 ␳ 兩 ␺−典, where the 兩␺−典 is the singlet state, 兩 ␺ −典 =. 1. 冑2 共兩 + − 典 − 兩−. + 典兲.. 共1兲. 共ii兲 Any mixed states can be transformed to a Werner state ␳W using a random bilateral rotation. If the fidelity of the state is F, after the random bilateral rotation the state is transformed to. ␳W = F兩␺−典具␺−兩 +. 1−F + + 共兩␺ 典具␺ 兩 + 兩␾−典具␾−兩 + 兩␾+典具␾+兩兲, 3 共2兲. where 兩 ␺ +典 =. 1. + 典兲,. 共3兲. 1. 冑2 共兩 + − 典 + 兩−. 兩 ␾ −典 =. 冑2 共兩 +. + 典 − 兩− − 典兲,. 共4兲. 兩 ␾ +典 =. 冑2 共兩 +. 1. + 典 + 兩− − 典兲.. 共5兲. and. 共iii兲 After the random bilateral rotation that leaves the system in a Werner state, a unilateral Sy rotation is performed transforming it in the following state:. 022306-1. ©2009 The American Physical Society.

(2) PHYSICAL REVIEW A 79, 022306 共2009兲. D. F. MUNDARAIN AND M. ORSZAG. ⬘ = F兩␾+典具␾+兩 + ␳W. III. TWO EPR PAIRS IN TWO LOCAL COMMON RESERVOIRS. 1−F − − 共兩␾ 典具␾ 兩 + 兩␺+典具␺+兩 + 兩␺−典具␺−兩兲. 3 共6兲. 共iv兲 At this point, Alice and Bob divide the original ensemble into two subensembles: one is used as the source and the other one as the ancilla. Alice and Bob perform a bilateral CNOT using the source as the control and the ancilla as the target. After that, they proceed to measure locally the ancilla along the Z axis. If both obtain the same results, the state of the source after measurements becomes. ␳⬙ =. 1 5 2 F2 + F共1 − F兲 + 共1 − F兲2 3 9. 冋冉. Consider two pairs of two-level particles. One pair 共AB兲 is considered to be the source and the other 共A⬘B⬘兲 the ancilla. We assume that both A and A⬘ are interacting with a common vacuum reservoir. The same is true for B and B⬘. Measurements on the ancilla qubits define what we call the local common reservoir distillation protocol. We start initially with both pairs in the singlet state, so the initial state of the composed system is. 冊. 兩␺共0兲典 =. 1 F2 + 共1 − F兲2 兩␾+典具␾+兩 9. 册. 2 5 Pb = F + F共1 − F兲 + 共1 − F兲2 . 3 9. 共8兲. 1 2 5 F2 + F共1 − F兲 + 共1 − F兲2 3 9. 冋冉. 冊. 1 F2 + 共1 − F兲2 兩␺−典具␺−兩 9. 册. 2 2 + F共1 − F兲兩␺+典具␺+兩 + 共1 − F兲2共兩␾+典具␾+兩 + 兩␾−典具␾−兩兲 . 3 9 共9兲. 共11兲. 共12兲. 兩0典 = 兩 + + 典,. 兩1典 = 兩− + 典,. 兩2典 = 兩 + − 典,. 兩3典 = 兩− − 典.. This initial state can also be written in the following form: 1 兩␺共0兲典 = 共兩0典AA⬘兩3典BB⬘ − 兩1典AA⬘兩2典BB⬘ − 兩2典AA⬘兩1典BB⬘ 2 + 兩3典AA⬘兩0典BB⬘兲.. 共13兲. The master equation for this system in the presence of two local common reservoirs in vacuum is. ␳˙ = LAA⬘共␳兲 + LBB⬘共␳兲,. 共14兲. where. ␥ † † † LAA⬘共␳兲 = 共2␴AA⬘␳␴AA⬘ − ␳␴AA⬘␴AA⬘ − ␴AA⬘␴AA⬘␳兲 2. This final state has a fidelity 1 F2 + 共1 − F兲2 9 . F⬘ = 5 2 F2 + F共1 − F兲 + 共1 − F兲2 3 9. 冑2 共兩2典A⬘B⬘ − 兩1典A⬘B⬘兲. where. 共v兲 Finally, a unilateral Sy rotation is performed and the final state is. ␳⬙ =. 1. + 兩1典AB兩1典A⬘B⬘兲,. If Alice and Bob do not obtain the same results, the particles are discarded. The success probability of the process is 2. 冑2. 共兩2典AB − 兩1典AB兲 丢. 1 = 共兩2典AB兩2典A⬘B⬘ − 兩2典AB兩1典A⬘B⬘ − 兩1典AB兩2典A⬘B⬘ 2. 2 2 + F共1 − F兲兩␾−典具␾−兩 + 共1 − F兲2共兩␺−典具␺−兩 + 兩␺+典具␺+兩兲 . 3 9 共7兲. 1. 共15兲 共10兲. If F ⬎ 1 / 2, this new fidelity is larger than the initial one. In their original work, Bennett et al. 关3兴 also suggest performing a final random bilateral rotation. It is not difficult to show that this final suggestion does not modify the entanglement. The final state in expression 共9兲 is a Bell diagonal state. It can be proven that the Werner state obtained via the random bilateral rotation has exactly the same concurrence as the initial Bell diagonal state. This property is not true for other kinds of states. In Sec. IV, we will consider how the entanglement of two qubits in the presence of vacuum is modified if at time t one applies Bennett’s protocol, having prepared initially the system in the singlet state.. and. ␥ † † † LBB⬘共␳兲 = 共2␴BB⬘␳␴BB⬘ − ␳␴BB⬘␴BB⬘ − ␴BB⬘␴BB⬘␳兲 2 共16兲 with. ␴AA⬘ = ␴A 丢 1A⬘ 丢 1B 丢 1B⬘ + 1A 丢 ␴A⬘ 丢 1B 丢 1B⬘ 共17兲 and. ␴BB⬘ = 1A 丢 1A⬘ 丢 ␴B 丢 1B⬘ + 1A 丢 1A⬘ 丢 1B 丢 ␴B⬘ . 共18兲 At this point, it is useful to write the density matrix in the following way:. 022306-2.

(3) PHYSICAL REVIEW A 79, 022306 共2009兲. ENTANGLEMENT DISTILLATION WITH LOCAL COMMON…. At time t, Alice measures the ancilla A⬘ along the Z axis obtaining the ⫹ result. The state of the remaining system 共A , B , B⬘兲 collapses to. 3. ␳=. ⬘ 兺 ␳BB ij 兩i典AA⬘具j兩, i,j=0. 共19兲. ⬘ are 16 operators of the local Hilbert space of where ␳BB ij Bob. Observing that. BB⬘ BB⬘ 共t兲兩 + 典A具+ 兩 + ␳01 共t兲兩 + 典A具− 兩 ␳A,BB⬘共t兲 ⬇ ␳00 BB⬘ BB⬘ 共t兲兩− 典A具− 兩, + ␳10 共t兲兩− 典A具+ 兩 + ␳11. 3. LAA⬘共␳兲 =. ⬘ ␳BB 兺 ij LAA⬘共兩i典AA⬘具j兩兲 i,j=0. with. 共20兲. BB⬘ ␳00 共t兲 =. and 3. LBB⬘共␳兲 =. ⬘ 兺 LBB⬘共␳BB ij 兲兩i典AA⬘具j兩 i,j=0. 共21兲. and using the master equation, one can write a submaster ⬘ equation for each ␳BB ij . There are 16 of these equations that have the following general structure:. BB⬘ ␳01 共t兲 =. 3. BB⬘ ⬘ ␳˙ BB ij = LBB⬘共␳ij 兲 +. ⬘ 兺 Akl␳BB kl ,. 共22兲. 冢. 0 − 0 0. 0 0 0. 0. 0 0 0. 0. 0 0 0. 1 −2␥t 4e. 冣. ,. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 1 −␥t 8 共e. 冢. − e−3␥t兲 − 81 共e−␥t + e−3␥t兲 0. 0 0 0. 0. 0 0 0. 1 −␥t − e−3␥t兲 8 共e − 81 共e−␥t + e−3␥t兲. 0 0 0. 0. 0 0 0. 冣. 冣. ,. ,. and. 0. 0. 1 −2␥t 1 + 16 共1 + e−4␥t兲 8e 1 −4␥t 兲 16 共− 1 + e. 0. 0. BB⬘ ␳10 共t兲 =. where Akl are some constant numerical coefficients. In the Appendix, we write each one of these 16 equations. The solution of the submaster equations requires an initial condition that can be extracted from the initial state of the four particles, as per Eq. 共13兲.. BB⬘ ␳11 共t兲 =. 冢. 冢. 0 0 0. 0. k,l=0. 0. 共23兲. 0. 1 −4␥t 兲 16 共− 1 + e 1 −2␥t 1 −4␥t + 16 共1 + e 兲 8e. 0. 0. − 81 共e−2␥t + 2te−2␥t − e−4␥t兲. 0. 0. 冣. ,. BB⬘ where the matrices ␳11 are written in the 兵兩0典BB⬘ , 兩1典BB⬘ , 兩2典BB⬘ , 兩3典BB⬘其 basis. These expressions were found solving analytically all the 16 submaster equations mentioned above. After this, Bob measures the ancilla B⬘ along the Z axis. If he obtains the opposite result 共⫺兲, the state of the source becomes. AB ␳+− 共t兲 =. 1 ⫻ P+−共t兲. 冢. 0. 0. 0. 0. 1 −2␥t 1 + 16 共1 + e−4␥t兲 8e − 81 共e−␥t + e−3␥t兲. 0. 0. 0. 0. where the normalization function P+− is the probability that the observers obtain the respective results 兵⫹, ⫺其 P+−共t兲 =. 1 1 1 + 共2 + t兲e−2␥t − e−4␥t . 16 4 16. −. 1 −␥t + e−3␥t兲 8 共e 1 −2␥t 4e. 0 0 − 81 共e−2␥t + 2te−2␥t − e−4␥t兲. 冣. ,. will be the same as in the previous case, so the state after AB AB 共t兲 = ␳+− 共t兲 and the measurement for opposite results is ␳−+ success probability is Ps共t兲 = 2P+−共t兲.. 共24兲. If Alice and Bob measure the ancilla and obtain the results 兵⫺, ⫹其, the state of the source and the success probability. 0. IV. RESULTS. In Fig. 1, we plot the concurrence of the source for three cases: 共a兲 The source evolving in the presence of two inde-. 022306-3.

(4) PHYSICAL REVIEW A 79, 022306 共2009兲. D. F. MUNDARAIN AND M. ORSZAG 1 1. Concurrence. Success Probability. Free Concurrence Common distillation Bennett’s distillation. 0.8. 0.6. 0.4. 0.2. 0.8. 0.6. 0.4. 0.2. 0. 0 0. 1. 2. 3. 4. 0. 5. 1. 2. 3. 4. 5. γt. γt. pendent vacuum reservoirs without any interaction with the ancilla. 共b兲 Both the source and the ancilla, evolving in the presence of two respective independent vacuum reservoirs. At time t, Bennett’s distillation protocol is applied. 共c兲 The source and the ancilla evolving in two local common vacuum reservoirs. At time t, local measurements on the ancilla are performed. As we can observe, Bennett’s protocol destroys the entaglement at a finite time. This effect is a consequence of two facts: in the first place, at any time the random bilateral rotation partially destroys the entanglement of the system. The gain in concurrence as a result of the bilateral operations applied after random bilateral rotation is smaller than the initial loss. On the other hand, although the system is always entangled, there is a finite time for which the fidelity of the state becomes smaller than one-half, in which case the random bilateral rotation leaves the system in a separable state. We also observe from Fig. 1 that a local common distillation leaves, at the beginning, the concurrence invariant, with respect to the free evolution. However, for a wide time range, one observes a significant increase of the entanglement. Also, it is important to emphasize that in this range, the success probability is very high, as we show in Fig. 2, where we plot the evolution of this quantity. Finally, in Fig. 3 we plot the fidelity versus time, when the local common reservoir distillation protocol is applied at time t. We compare this evolution with one without any distillation protocol. We do not plot Bennett’s distilled fidelity because in that case the gain is very small and Bennett’s protocol does not modify substantially the system’s fidelity evolution. The most important feature one observes in this figure is the fact that at any time after the distillation procedure, the system is left in a state with fidelity higher than one-half 共F ⬎ 1 / 2兲, so other distillation procedures, like the Bennett scheme, which requires F ⬎ 1 / 2 for optimal operation, can be applied after the local common reservoir procedure in order to increase the entanglement in the system. V. DISCUSSION AND SUMMARY. The general idea of distillation is to extract from a large set of bipartite systems, in either pure or mixed states, a. FIG. 2. Success probability vs time for a system distilled at time t with local common reservoir protocol.. smaller set of pure or mixed state, but with a higher fidelity with respect to a given Bell state. In this process, we increase the entanglement since we approach a maximally entangled state. In the present work, we considered two pairs of two-level atoms. One of the pairs is called the source and labeled 共AB兲, and the second pair is an ancilla labeled 共A⬘B⬘兲. Since the communication between Alice and Bob may take place between two remote locations, it would be unreasonable to assume that they would share a common reservoir. However, A and A⬘ as well as B and B⬘ may share common vacuum reservoirs. Intuitively, this would greatly enhance the correlation between the two pairs, so that, following Bennett’s idea to perform measurements on the ancilla, and thus indirectly enhance the entanglement of the source pair, the effect of such measurements with these common reservoirs would have a larger effect on the source pair. In summary, in this work we studied the composed system of two source and two ancilla qubits. We found that if the two Alice’s 共source and ancilla兲 share a common vacuum reservoir, and the same is true for the two Bob’s, and if 1.2. Free Fidelity Distilled Fidelity. 1. 0.8 Fidelity. FIG. 1. Concurrence of source state. 共a兲 Solid line: distillation free evolution. 共b兲 Line with squares: Bennett’s distillation at time t. 共c兲 Line with circles: distillation with local common reservoirs.. 0.6. 0.4. 0.2. 0 0. 2. 4. 6. 8. 10. γt. FIG. 3. Fidelity vs time. 共a兲 Solid line: evolution of distilled free fidelity. 共b兲 Line with circles: evolution of fidelity for a system distilled at time t with the local common reservoir protocol.. 022306-4.

(5) PHYSICAL REVIEW A 79, 022306 共2009兲. ENTANGLEMENT DISTILLATION WITH LOCAL COMMON…. ␳˙ = LAA⬘共␳兲 + LBB⬘共␳兲,. initially both pairs start from the singlet state, then if at time t local measurements on the ancilla are performed, a dramatic increase in the concurrence as well as in the fidelity is observed on the source pair. The fact that the reservoir is common for both Alice and Bob is crucial for the present results, since in this case they are connected to independent reservoirs, and if a distillation procedure is applied, the concurrence not only does not increase, it actually becomes worse as compared with the free evolution case. Finally, we show that this spectacular increase of the concurrence 共and fidelity兲, for the common reservoir case, has a large success probability, as shown in Fig. 2.. 共A1兲. where. ␥ † † † LAA⬘共␳兲 = 共2␴AA⬘␳␴AA⬘ − ␳␴AA⬘␴AA⬘ − ␴AA⬘␴AA⬘␳兲 2 共A2兲 and. ␥ † † † LBB⬘共␳兲 = 共2␴BB⬘␳␴BB⬘ − ␳␴BB⬘␴BB⬘ − ␴BB⬘␴BB⬘␳兲, 2 共A3兲 with. ACKNOWLEDGMENTS. ␴AA⬘ = ␴A 丢 1A⬘ 丢 1B 丢 1B⬘ + 1A 丢 ␴A⬘ 丢 1B 丢 1B⬘. D.M. was supported by Did-Usb Grant No. Gid-30 and by Fonacit Grant No. G-2001000712. M.O was supported by Fondecyt No. 1051062.. 共A4兲 and. ␴BB⬘ = 1A 丢 1A⬘ 丢 ␴B 丢 1B⬘ + 1A 丢 1A⬘ 丢 1B 丢 ␴B⬘ , 共A5兲. APPENDIX. If we write. it is simple to show, via matrix multiplication, that. LAA⬘共␳兲 = −. 冤. BB⬘ 4␳00 BB⬘ BB⬘ 共3␳10 + ␳20 兲. ␥ BB⬘ BB⬘ 2 共␳10 + 3␳20 兲 BB⬘ 2␳30. 冋 冋 冋. BB⬘ BB⬘ 共3␳01 + ␳02 兲 BB⬘ BB⬘ 2␳11 + ␳12 BB⬘ BB⬘ + ␳21 − 2␳00. +. BB⬘ BB⬘ ␳11 + ␳22 BB⬘ 2␳21. −. 册冋 册冋 册冋. BB⬘ 2␳00. BB⬘ BB⬘ ␳31 + ␳32. BB⬘ BB⬘ − 2共␳10 + ␳20 兲. BB⬘ BB⬘ ␳11 + ␳22. 册冋 册冋 册冋. BB⬘ 2␳03 BB⬘ BB⬘ ␳13 + ␳23. BB⬘ BB⬘ + 2␳12 − 2␳00. BB⬘ BB⬘ − 2共␳01 + ␳02 兲. BB⬘ BB⬘ 2␳22 + ␳12. BB⬘ BB⬘ ␳13 + ␳23. +. BB⬘ ␳21. −. BB⬘ 2␳00. BB⬘ BB⬘ ␳31 + ␳32. BB⬘ BB⬘ − 2共␳10 + ␳20 兲. −. BB⬘ 2共␳01. +. BB⬘ ␳02 兲. BB⬘ BB⬘ − 2␳11 − 2␳22 BB⬘ BB⬘ − 2共␳12 + ␳21 兲. 册 册 册. 冥. ,. ␥ BB⬘ BB⬘ BB⬘ BB⬘ BB⬘ BB⬘ ␳˙ 12 = LBB⬘共␳12 兲 − 共␳11 + ␳22 + 2␳12 − 2␳00 兲, 2. so that the 16 differential equations are BB⬘ BB⬘ BB⬘ ␳˙ 00 = LBB⬘共␳00 兲 − 2␥␳00 ,. ␥ BB⬘ BB⬘ BB⬘ BB⬘ BB⬘ BB⬘ ␳˙ 13 = LBB⬘共␳13 兲 − 关␳13 + ␳23 − 2共␳01 + ␳02 兲兴, 2. ␥ BB⬘ BB⬘ BB⬘ BB⬘ ␳˙ 01 = LBB⬘共␳01 + ␳02 兲, 兲 − 共3␳01 2. ␥ BB⬘ BB⬘ BB⬘ BB⬘ BB⬘ BB⬘ ␳˙ 22 = LBB⬘共␳22 兲 − 共2␳22 + ␳21 + ␳12 − 2␳00 兲, 2. ␥ BB⬘ BB⬘ BB⬘ BB⬘ ␳˙ 02 = LBB⬘共␳02 兲 − 共␳01 + 3␳02 兲, 2. ␥ BB⬘ BB⬘ BB⬘ BB⬘ BB⬘ BB⬘ + ␳02 兲兴, ␳˙ 23 = LBB⬘共␳23 兲 − 关␳13 + ␳23 − 2共␳01 2. BB⬘ BB⬘ BB⬘ ␳˙ 03 = LBB⬘共␳03 兲 − ␥␳03,. ␥ BB⬘ BB⬘ BB⬘ BB⬘ BB⬘ BB⬘ 兲, ␳˙ 11 = LBB⬘共␳11 兲 − 共2␳11 + ␳12 + ␳21 − 2␳00 2. BB⬘ BB⬘ 共␳01 + 3␳02 兲. ␥ BB⬘ BB⬘ BB⬘ BB⬘ BB⬘ BB⬘ 兲兴. 兲 − 关− 2␳11 − 2␳22 − 2共␳12 + ␳21 ␳˙ 33 = LBB⬘共␳33 2 The initial conditions for the above equations can be calculated from 022306-5.

(6) PHYSICAL REVIEW A 79, 022306 共2009兲. D. F. MUNDARAIN AND M. ORSZAG. ␳共0兲 = 兩␺共0兲典具␺共0兲兩, which in the 兵兩0典BB⬘ , 兩1典BB⬘ , 兩2典BB⬘ , 兩3典BB⬘其 basis can be written as. 冤 冥 冤 冥. 冤. 0 0 0 0. 1 0 0 0 0 BB⬘ ␳00 共0兲 = , 4 0 0 0 0 0 0 0 1 0. BB⬘ ␳02 共0兲 =. 0. 1 0 0 0 0 , 4 0 0 0 0 0 −1 0 0. 冥 冤 冥. 0 0. 0. BB⬘ ␳11 共0兲 =. 0. 1 0 0 0 0 BB⬘ ␳01 , 共0兲 = 4 0 0 0 0 0 0 −1 0. 0 0. 1 0 0 0 0 , 4 0 0 0 0 1 0 0 0. 关1兴 C. H. Bennett, G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W. K. Wootters, Phys. Rev. Lett. 70, 1895 共1993兲. 关2兴 A. K. Ekert, Phys. Rev. Lett. 67, 661 共1991兲. 关3兴 C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, Phys. Rev. Lett. 76, 722 共1996兲. 关4兴 M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Rev. Lett. 78, 574 共1997兲. 关5兴 D. Deutsch, A. Ekert, R. Jozsa, C. Macchiavello, S. Popescu, and A. Sanpera, Phys. Rev. Lett. 77, 2818 共1996兲.. 1 0 0 0 0 , 4 0 0 1 0 0 0 0 0 0. BB⬘ ␳13 共0兲 =. BB⬘ ␳12 共0兲 =. 1 0 1 0 0 , 4 0 0 0 0 0 0 0 0. 1 0 0 0 0 , 4 0 1 0 0 0 0 0 0 0. BB⬘ ␳23 共0兲 =. 0 0 0 0. BB⬘ ␳22 共0兲 =. 冤 冥 冤 冥 冤 冥 0 0 0 0. 0 0 0. 1 0 0 0 0 , 4 −1 0 0 0 0 0 0 0. 0 0 0 0. BB⬘ ␳03 共0兲 =. 冤 冥 冤 冥 冤 冥 0 0 0 0. 0 0 0. 1 −1 0 0 0 , 4 0 0 0 0 0 0 0 0 1 0 0 0. BB⬘ ␳33 共0兲 =. 1 0 0 0 0 . 4 0 0 0 0 0 0 0 0. 关6兴 J. W. Pan, C. Simon, C. Brukner, and A. Zeilinger, Nature 共London兲 423, 417 共2003兲; T. Yamamoto, M. Koashi, S. Ozdemir, and N. Imoto, ibid. 421, 343 共2003兲; P. G. Kwiat, B. Barraza-Lopez, A. Stefanov, and N. Gisin, ibid. 409, 1014 共2001兲. 关7兴 R. Rechle, D. Leibfried, E. Knill, J. Britton, R. B. Blackstead, J. D. Jost, C. Langer, R. Ozeri, S. Seidelin, and D. J. Wineland, Nature 共London兲 443, 838 共2006兲.. 022306-6.

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