(2) PHYSICAL REVIEW A 79, 052333 共2009兲. D. MUNDARAIN AND M. ORSZAG. independent thermal baths. The states of this family are characterized by the following real parametrization of the density matrix:. C4⫻4 =. 冢. 1. 0. 0 d1. 0 a1 0 0. 0. 0 b1 0. d1 0. 0. c1. 冣. 共5兲. .. = a⬘兩+典具+兩 + b⬘兩−典具−兩 + c⬘兩+典具+兩 + d⬘兩−典具−兩,. For these kinds of states the optimal filtered states are Bell diagonal states with the following entries: r1,r2 =. a1,b1 , ␤2 + 2␣␤d1 + ␣2c1. 共11兲 where. ␣ + 2␣␤d1 + ␤ c1 , ␤2 + 2␣␤d1 + ␣2c1 2. r3 =. 共3兲 Alice and Bob divide the total ensemble into two subensembles, one as a source and the other as ancilla. Alice and Bob perform a bilateral controlled NOT 共C-NOT兲 operation using the source as control and the ancilla as target. After this, a local measurement over the ancilla is done. If Alice and Bob obtain the same results, the final state of the source, which has not been measured, becomes. 2. a⬘ =. a2 + b2 , 共a + b兲2 + 共c + d兲2. b⬘ =. 2cd , 共a + b兲 + 共c + d兲2. c⬘ =. c2 + d2 , 共a + b兲2 + 共c + d兲2. d⬘ =. 2ab . 共12兲 共a + b兲 + 共c + d兲2. 共6兲 where. ␣2 = −. 1 1 + 2 2. 冑. 1−. 4d21 4d21. − 共1 + c1兲2. ,. ␤ = 冑1 + ␣2 . 共7兲. If the evolution preserves the structure of the real parametrization, one can obtain the optimal state at any time, using the previous expressions. The maximal achievable entanglement increase obtained by LOCC operations acting independently over each member of the ensemble describing the system is determined by the optimal filtering operation and the optimal state is a Bell diagonal state. One can relax this construction, allowing the observers to execute collective operations over two members of the same ensemble. One can separate the ensemble into two parts. The first part is considered as a source 共control for some bilateral operations兲 and the second as the ancilla 共target兲. After some bilateral operations that involve source and ancilla, the ancilla is measured. For some specific results in the measurement, the source, which initially is in an entangled state, is left in a more entangled state. The ancilla is always left in a separable state and the failure cases are rejected since in those cases, the source is left in a separable state. The next distillation protocol was proposed by Deutsch et al. 关11兴 following the above recipe.. 2. 2. If Alice and Bob obtain different results, the particles are rejected since in that case after the measurement the source is left in a separable state. At the end of the process all particles in the ancilla subensemble are in a separable states. Finally Alice and Bob perform a unilateral Sy rotation. The state after this is. = a⬘兩−典具−兩 + b⬘兩+典具+兩 + c⬘兩−典具−兩 + d⬘兩+典具+兩. 共13兲 The initial and final states are both Bell diagonal states, so one can write the final real parametrization in terms of the initial one: r1⬘ = −. r21 + r23 1+. r22. ,. r2⬘ = −. 2r1r3 1+. , r22. r3⬘ =. 2r2 1 + r22. ,. 共14兲. and the success probability is Pd = 21 共1 + r22兲.. 共15兲. III. ENTANGLEMENT PRESERVATION AND THE MASTER EQUATION. II. DEUTSCH PROTOCOL. 共1兲 Consider the following Bell diagonal state:. = a兩−典具−兩 + b兩+典具+兩 + c兩−典具−兩 + d兩+典具+兩,. 共8兲. where the ’s and ’s are the usual Bell states: 兩 ⫾典 =. 1. 冑2. 共兩00典 ⫾ 兩11典兲,. 兩 ⫾典 =. 1. 冑2 共兩01典 ⫾ 兩10典兲.. Next, we consider a sequence of both filtering operations over a system in the presence of thermal independent bath and a sequence of a combination of filtering plus the Deutsch distillation protocol. The master equation for two qubits in presence of two independent thermal baths is. 共9兲. In terms of the real parametrization of density matrix, the probabilities 兵a , b , c , d其 can be written as a = 共1 − r1 − r2 − r3兲 / 4, b = 共1 + r1 − r2 + r3兲 / 4, c = 共1 − r1 + r2 + r3兲 / 4, and d = 共1 + r1 + r2 − r3兲 / 4. 共2兲 After a couple of rotations performed by Alice and Bob on each copy, the state of the system becomes. = a兩+典具+兩 + d兩−典具−兩 + c兩+典具+兩 + b兩−典具−兩. 共10兲 052333-2. ␥ ˙ = 共N + 1兲共2a†a − †aa − †aa兲 2 ␥ + N共2†aa − a†a − a†a兲 2 ␥ + 共N + 1兲共2b†b − †bb − †bb兲 2 ␥ + N共2†bb − b†b − b†b兲, 2. 共16兲.
(3) PHYSICAL REVIEW A 79, 052333 共2009兲. ENTANGLEMENT PRESERVATION BY CONTINUOUS…. 1. 1 Without filtering With filtering. 0.8 Concurrence. Concurrence. 0.8. Without distillation With distillation. 0.6. 0.4. 0.2. 0.6. 0.4. 0.2. 0. 0 0. 1. 2. 3. 4. 5. 0. 1. 2. γt. 3. 4. 5. γt. FIG. 1. Concurrence vs time for two qubits in two independent thermal baths: without continuous filtering 共solid line兲, and with continuous filtering 共circles兲, N = 0.001. The initial state is the singlet −.. FIG. 2. Concurrence vs time for two qubits in two independent thermal baths: without continuous filtering and distillation 共solid line兲, and with continuous filtering and distillation 共circles兲, N = 0.001. The initial state is the singlet −.. where we have assumed that the two baths have the same number of thermal photons. If one has an initial state which is Bell diagonal, the real parametrization of the matrix for a short time, C4⫻4共⌬t兲, is of the form given in Eq. 共5兲 with parameters a1 = r1共1 − ␥共2N + 1兲⌬t兲, b1 = r2共1 − ␥共2N + 1兲⌬t兲, c1 = r3共1 − 2␥共2N + 1兲⌬t兲, and d1 = −␥⌬t, where 兵r1 , r2 , r3其 are the nonzero entries of the real parametrization of the initial Bell diagonal state. Since the state of the system after the evolution belongs to the family considered by Isasi et al. 关13兴, one can apply the results for filtering operation. After this filtering operation the system is left again in a Bell diagonal state and one can repeat the process many times.. diagonal state, in the domain of the singlet fixed point, and get a dramatic increase in the concurrence to end up approximately in the singlet state. In the process of filtering the success probability is P f and that for the distillation is Pd. In each iteration more than half of particles are discarded, all the ancilla particles and some of the source particles that do not verify the success criteria. The overall success probability of the process is a time-dependent quantity that combines the success probabilities in all the previous iterations. After a rather long algebraic work, one finds an expression for the overall success probability for the case of very frequent filtering and distillation and the singlet as the initial state:. IV. RESULTS AND DISCUSSION. P共t兲 = exp兵− ␣共N兲␥t其, where. 1. 0.8 Concurrence. In Fig. 1 we plot the time evolution of concurrence using this iterative method. The first iteration begins with the singlet − as initial state. In the same figure we plot the evolution of concurrence without frequent filtering. As one can observe the continuous filtering of the system accelerates the entanglement loss rate. The results are completely different if one distillates after each step of filtering. Obviously more particles must be discarded in this case. In Figs. 2 and 3, we show the main results of this paper. Figure 2 shows the time evolution of the concurrence by iteration, using, in each step, filtering and distillation. These results imply that there is a small fraction of particles that preserve its initial entanglement. In this case, we started from −, which is a fixed point of the distillation map. If we start in any state in the neighborhood of this state, after the evolution, we end up approximately in the singlet state. However, our map has also a fixed point at +. On the other hand, if we do not perform the unilateral Sy rotation, the fixed points are then located in + and −. In other words, if our system is originally in the neighborhood of any of the Bell states, following the present procedure, the dynamics is forced to end up approximately in the corresponding Bell state. This is shown in Fig. 3, where we start from a Bell. 共17兲. Without distillation With distillation. 0.6. 0.4. 0.2. 0 0. 0.1. 0.2. 0.3. 0.4. 0.5. γt. FIG. 3. Concurrence vs time for two qubits in two independent thermal baths: without continuous filtering and distillation 共solid line兲, and with continuous filtering and distillation 共circles兲, N = 0.1. The initial state is a Bell diagonal state with r1共0兲 = r2共0兲 = r3共0兲 = −0.5.. 052333-3.
(4) PHYSICAL REVIEW A 79, 052333 共2009兲. D. MUNDARAIN AND M. ORSZAG. Success probability. 1 N=0.001 N=0.01 N=0.1. 0.8. 0.6. 0.4. 0.2. 0 0. 1. 2. 3. 4. 5. γt. FIG. 4. Success probability for continuous filtering and distillation vs time for two qubits in two independent thermal baths: N = 0.001 共solid line兲, N = 0.01 共circles兲, and N = 0.1 共squares兲. The initial state in the three cases is the singlet −.. ␣共N兲 = 1 + 2N + 2N −. 冑. 冑. 共2N + 1兲2 + N共N + 1兲. 共2N + 1兲2 − 4. N共N + 1兲. 冑. 共2N + 1兲2 N共N + 1兲 共18兲. may have the opposite effect. The loss rate of entanglement in the presence of baths depends strongly on the state of the system and obviously on the nature of the reservoirs. When one applies a local operation, it is possible that after the operation, the final state gets a higher concurrence, but also a higher loss rate of entanglement, which is what happens in the first case 共Fig. 1兲. Filtering not only increases the concurrence, but it increases the loss rate too, so eventually in each iteration of the protocol, entanglement is lost quickly. However, the filtering plus the distillation operations leave the system in a state that has more concurrence and a smaller loss rate of entanglement, so entanglement is preserved in each iteration. In this case, there is a stationary state, in which what is lost in the interaction with the bath is fully recovered by the local operations. The state after the local operation is the same as the state before the evolution in contact with the bath. One alternative to preserve the entanglement is via modifying the evolution of the entanglement using only unitary local transformations. These operations could modify the loss rate, preserving the entanglement, but cannot increase the entanglement of a fraction of particles, as one observes in our protocol. In summary, we show that although individually the optimal filtering or distillation procedure may result in an accelerated loss rate of entanglement, the combination of filtering with maximal extractable entanglement and the distillation procedure produces a surprising and dramatic increase in the concurrence. The effect reported in this paper could be also understood as a kind of “Zeno-type” effect, since after repeated measurement, in each step of the time evolution of the master equation, and discarding of particles in this combination of optimal filtering and distillation, we are able to stop the damaging effect of the thermal bath on the entanglement of two qubits.. In Fig. 4 we plot the success probabilities for three different thermal average numbers of photons. Experimentally, distillation and entanglement concentration have been realized using partial polarizers 关17,18兴. In addition, there has been an interesting experimental realization of distillation of two qubits under local filtering operations, using pairs of photons from spontaneous parametric down-conversion, using nonmaximally entangled polarization states and optical elements for the local operations 关19兴. In principle, the present proposal could be also achieved, with the added distillation operation 共Deutsch兲, at a given instant of the time evolution of the system. The behavior of the entanglement in the various cases presented here is associated with two opposite effects. The bath destroys the entanglement, while the local operations. D.M. was supported by Did-Usb Grant No. Gid-30 and by Fonacit Grant No. G-2001000712. M.O. was supported by Fondecyt Grant No. 1051062.. 关1兴 A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 共1935兲. 关2兴 E. Schrödinger, Proc. Cambridge Philos. Soc. 31, 555 共1935兲. 关3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4兴 A. K. Ekert, Phys. Rev. Lett. 67, 661 共1991兲. 关5兴 K. Zyczkowski, P. Horodecki, M. Horodecki, and R. Horodecki, Phys. Rev. A 65, 012101 共2001兲. 关6兴 L. Diosi, Lect. Notes Phys. 622, 157 共2003兲. 关7兴 T. Yu and J. H. Eberly, Phys. Rev. Lett. 93, 140404 共2004兲. 关8兴 T. Yu and J. H. Eberly, Phys. Rev. Lett. 97, 140403 共2006兲. 关9兴 C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, Phys. Rev. A 53, 2046 共1996兲.. 关10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11兴 D. Deutsch, A. Ekert, R. Jozsa, C. Macchiavello, S. Popescu, and A. Sanpera, Phys. Rev. Lett. 77, 2818 共1996兲. 关12兴 A. Kent, N. Linden, and S. Massar, Phys. Rev. Lett. 83, 2656 共1999兲. 关13兴 E. Isasi, D. Mundarain, and J. Stephany, J. Phys. B 41, 235504 共2008兲. 关14兴 J. M. Leinaas, J. Myrheim, and E. Ovrum, Phys. Rev. A 74, 012313 共2006兲. 关15兴 F. Verstraete, J. Dehaene, and B. der Moor, Phys. Rev. 64, 010101 共2001兲.. ACKNOWLEDGMENTS. 052333-4.
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