Manipulating entanglement of two qubits in a common environment by ...

PHYSICAL REVIEW A 86, 012325 (2012)

Manipulating entanglement of two qubits in a common environment by means of weak measurements and quantum measurement reversals Zhong-Xiao Man* and Yun-Jie Xia‡ Shandong Provincial Key Laboratory of Laser Polarization and Information Technology, Department of Physics, Qufu Normal University, Qufu 273165, China

Nguyen Ba An† Center for Theoretical Physics, Institute of Physics, 10 Dao Tan, Thu Le, Ba Dinh, Hanoi, Vietnam (Received 18 June 2012; published 23 July 2012) Addressing various types of decoherence is of paramount importance for employing entanglement in quantum information processing and quantum computing. In this paper, we propose two schemes to manipulate entanglement of two qubits stored in a common environment by means of combined weak measurements and quantum measurement reversals. We derive the explicit conditions for the measurement strengths under which at any time and for any initial states the qubits’ entanglement can be protected quite well with a reasonable finite success probability. Moreover, the two schemes enable us in principle to achieve the maximal entanglement or to restore the initial state (entanglement) of the qubits. DOI: 10.1103/PhysRevA.86.012325

PACS number(s): 03.67.Bg, 03.65.Yz, 03.65.Ta

I. INTRODUCTION

Entanglement has attracted more and more attention in recent years since it can serve as a resource to realize various global tasks in quantum information processing and quantum computing [1]. Although countless entanglementbased quantum schemes have been proposed, progress in their realization is slow, because entanglement is very fragile. The unavoidable coupling of a quantum system with its surrounding environment destroys the necessary coherence of the system. The effect of decoherence becomes critically detrimental in cases when the system’s entanglement is terminated abruptly, a phenomenon called finite-time disentanglement or entanglement sudden death [2]. Local decoherences are caused during the process of entanglement distribution by interactions between different subsystems of the entangled system and their own independent environments. However, before entanglement distribution the whole entangled system of interest is usually stored for some time inside a quantum register or memory, giving rise to additional indirect interactions between the subsystems mediated by the common environment. Therefore, protecting entanglement in various realistic scenarios proves to be a precondition for actual application. So far, many methods have been proposed to protect entanglement from decoherence. Entanglement distillation is a method of distilling a pure maximally entangled state from decohered (partially entangled mixed) states [3–7]. This method requires a large number of identically decohered states and will not work if the amount of entanglement contained in these states is too small. If there are several qubits interacting with the same environment and the total Hamiltonian is highly symmetric, there may exist a decoherence-free subspace [8–10], in which, however, only a certain entangled *

[email protected] Corresponding author address: [email protected] ‡ [email protected] †

1050-2947/2012/86(1)/012325(9)

state becomes decoupled from the environment [11,12]. The decoherence process can also be manipulated via the quantum Zeno effect [13], but, to prevent considerable degradation of entanglement, special measurements should be done very frequently at equal time intervals [11,12]. Another method is to introduce redundancy as in quantum error correction. Each physical qubit is encoded onto a logical one composed of several physical qubits [14–17]. After decoherence has taken place a multiqubit measurement is carried out to learn what error possibly occurred so that an appropriate reversal procedure can be applied to correct the error. Yet, as has been shown [18], in some cases this method can indeed delay entanglement degradation but in other cases it brings about sudden disentanglement for states that otherwise disentangle only asymptotically. In most cases, the energy dissipation of individual subsystems of a composite system is responsible for the system’s entanglement degradation. Hence, methods that can prevent the decay of a system’s excited-state population would be applicable. For example, entanglement of two independent atoms can be trapped in photonic-band-gap materials [19–21] if these are structured so as to inhibit spontaneous emission of individual atoms. Quantum interference [22] can also be exploited to quench spontaneous emission in atomic systems [23,24]. One can use driving external fields as well to govern entanglement decay through controlling amplitude-damping decoherence caused by spontaneous emissions. Recently, interesting methods using weak measurements followed by quantum measurement reversals have been put forward to cope with decoherence for single quantum systems [25–27] as well as for bipartite quantum systems [28,29]. A weak measurement [30] contrasts with an ordinary projective one in that it is reversible because of its lack of sharpness (i.e., it does not totally collapse the measured system). Thus, after a weak measurement the measured state could be recovered by a proper quantum measurement reversal operation [31,32]. It was shown that such a procedure can effectively suppress decoherence due to amplitude damping for a single qubit [25,26].

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©2012 American Physical Society

ZHONG-XIAO MAN, YUN-JIE XIA, AND NGUYEN BA AN

PHYSICAL REVIEW A 86, 012325 (2012)

The case with two qubits initially prepared in an entangled state and later on interacting with their own independent noisy channels was also considered [28,29]. In Ref. [28] an attempt to reverse the amplitude damping was made, but the efficiency was not high and entanglement sudden death, if it occurs, cannot be circumvented. For the same damping scenario, the authors of Ref. [29] introduced an essential improvement. Namely, prior to entanglement distribution a weak measurement is done on each qubit; then, when the qubits arrive at the receiving stations after having gone through the decoherence channels, proper quantum measurement reversals are performed. This improved method recovers entanglement much better than that in Ref. [28]. More importantly, by the method in Ref. [29] the qubits’ entanglement can escape sudden death. In this work, we are interested in the situation when it is necessary before actual use to store two entangled qubits inside a common environment, which can be treated as a lossy cavity [11,33,34]. We develop two schemes either to obtain an optimal entanglement or to restore the initial entanglement of the qubits. The two schemes apply the same prior weak measurements on individual qubits before they interact with the common environment, but different post measurements after the decoherence. Namely, to make the entanglement optimal, the post measurements should be either weak measurements or quantum measurement reversals, depending on the prior weak measurement strength and the qubit-environment interaction time. Yet, to restore the initial entanglement, only quantum measurement reversals should be applied as the post measurement. We shall explicitly derive the conditions that the post measurements must satisfy for each purposes. The two schemes are in fact very useful in the sense that the qubits’ entanglement at any time during its evolution can be manipulated so that it is arbitrarily close to the maximal one or to the initial one, even in the case when the uncontrolled entanglement vanishes in a finite time. Our paper is organized as follows. In Sec. II, we describe the model of two qubits interacting with a common environment. Since the qubits may possess two excitations initially, we adopt the pseudomode approach to derive their evolution after the prior weak measurements. In Sec. III, we propose the first scheme to optimize the qubits’ entanglement and, in Sec. IV, we present the second scheme to restore the qubits’ initial state (entanglement as well). For both the schemes we establish the conditions for the post measurements to achieve their goals. Finally, we conclude in Sec. V. II. THE MODEL

partially collapsed and we let it evolve. The map of weak measurement on qubit j can be represented as |0j 0| → |0j 0|, |1j 1| → (1 − pj )|1j 1|,   |0j 1| → (1 − pj )|0j 1|, |1j 0| → (1 − pj )|1j 0|, (2) where 0  pj < 1 is called the weak measurement strength, which is the probability of the qubit transition |1j → |0j . For simplicity, we assume pA = pB = p, so after the weak measurements with null outcomes the state (1) becomes (unnormalized) ρAB (p,0) = |α|2 |0,0AB 0,0| + |β|2 (1 − p)2 |1,1AB 1,1| + αβ ∗ (1 − p)|0,0AB 1,1| + α ∗ β(1 − p)|1,1AB 0,0|.

To study the evolution the qubit-environment coupling should be specified. As in Refs. [11,33,34], we model the environment as a multimode electromagnetic field at zero temperature in a lossy cavity characterized by the spectral density J (ω) =

which are stored together in a lossy cavity. To best manipulate the qubits’ entanglement we perform a weak measurement on each qubit right after their preparation. In practice, a weak measurement can be implemented by a device that indirectly monitors a qubit. If the device signals, we know that the qubit transition |1 → |0 occurred and discard the result. If the device does not signal (null outcome), the qubit state was only

2 , 2 2π (ω − ω0 ) + ( /2)2

(4)

with /2 the width of the energy spectrum,  the coupling constant, and ω0 the qubit transition frequency. Let the field be initially in the vacuum state. The total Hamiltonian in the rotating-wave approximation is H = H0 + HI , with H0 and HI being of the forms  † ωk ak ak , (5) H0 = ω0 (σ+A σ−A + σ+B σ−B ) + HI =

(σ+A

+

σ+B )



k

gk ak + H.c.

(6)

k j

In the above equations σ± are the inversion operators † of qubit j ∈ {A,B}, ak (ak ) is the annihilation (creation) operator for the kth mode of the field with frequency ωk , and gk is the coupling constant of a qubit with the mode k. Here, we treat identical qubits equally coupled to the field. Hence, the two-qubit space is effectively spanned by four√basic states |0AB = |0,0AB ,|+AB√= (|0,1AB + |1,0AB )/ 2, |−AB = (|1,0AB − |0,1AB )/ 2, and |2AB = |1,1AB , in terms of which H0 and HI are rewritten as  † ωk ak ak , (7) H0 = 2ω0 |22| + ω0 (|++| + |−−|) + HI =

Consider two qubits A and B initially produced in a Belllike state of the form |ψ(0)AB = α|0,0AB + β|1,1AB , |α|2 + |β|2 = 1, (1)

(3)

k √ 2gk ak (|+0| + |2+|) + H.c.

(8)

k

Clearly from Eq. (8), states |0AB , |+AB , and |2AB form an equally separated three-level ladder system and are coupled to each other via the field, whereas state |−AB does not evolve at all. Moreover, since the component |−AB is absent in the qubits’ state at t = 0, it will be so at any later time t > 0. Thus we need to deal with only the three components |0AB , |+AB , and |2AB for the whole evolution. The exact evolution of the two qubits inside the cavity with the spectral density (4) can be obtained within the pseudomode framework [35] by solving

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MANIPULATING ENTANGLEMENT OF TWO QUBITS IN A . . .

the following master equation: d ρ (p,t) † ρ (p,t) − 2 b ρ (p,t) b† = −i[H, ρ (p,t)] − [ b b dt 2 +ρ (p,t) b† b], (9) with H =

√ 2[ b(|+0| + |2+|) + H.c.].

(10)

Here, ρ (p,t) is the complete density matrix of the qubitpseudomode system and  b ( b† ) is the annihilation (creation) operator of the pseudomode quantum associated with the spectral density (4), which we call the boson in what follows. Because the field is initially in the vacuum, there are at most only two bosons at a time. Therefore, the basis for the total qubit-pseudomode system consists of six states | 1 ≡ |0AB |0F ,| 2 ≡ |0AB |1F ,| 3 ≡ |0AB |2F ,| 4 ≡   |+AB |0F ,|5 ≡ |+AB |1F , and |6 ≡ |2AB |0F , where |nF denotes the field state containing n bosons. From Eqs. (9) we have the following differential equations for the matrix elements ρ ij =  i| ρ (p,t)| j : ˙ 11 =  ρ  ρ22 , √ ˙ρ 22 = −i 2( ρ42 − ρ 24 ) − ( ρ22 − 2 ρ33 ), ˙ρ 33 = −i2( ρ53 − ρ 35 ) − 2  ρ33 , √ ˙ρ ρ24 − ρ 42 ) +  ρ55 , 44 = −i 2( √ ˙ρ 55 = −i2( ρ35 − ρ 53 ) − i 2( ρ65 − ρ 56 ) −  ρ55 , √ ˙ 66 = −i 2( ρ56 − ρ 65 ), ρ  √ √ ˙ 24 = −i 2( ρ24 − 2 2 ρ44 − ρ 22 ) − ( ρ35 ), ρ  2 √ √ ˙ 42 = −i 2( ρ42 − 2 2 ρ  ρ22 − ρ 44 ) − ( ρ53 ), 2 √ √ √ 3 ˙ 35 = −i 2( 2 ρ 35 , ρ55 − 2 ρ33 − ρ 36 ) − ρ  2 √ √ √ 3 ˙ 53 = −i 2( 2 ρ 53 , ρ33 − 2 ρ55 + ρ 63 ) − ρ  2 √ √ ˙ 36 = −i 2( 2 ρ  ρ56 − ρ 35 ) −  ρ36 , √ √ ˙ 63 = −i 2( ρ53 − 2 ρ65 ) −  ρ63 , ρ  √ √ ˙ 56 = −i 2( 56 , ρ66 + 2 ρ36 − ρ 55 ) − ρ ρ  2 √ √ ˙ 65 = −i 2(− 65 , ρ  ρ66 − 2 ρ63 + ρ 55 ) − ρ 2 ˙ 13 = i2 ρ15 −  ρ13 , ρ  √ √ ˙ 15 = −i 2(− 15 , ρ16 − 2 ρ13 ) − ρ ρ  2 √ ˙ 16 = i 2 ρ  (11) ρ15 . By using Laplace transform and the initial condition ρ (p,0) = ρAB (p,0)|0F , we obtained the matrix elements ρ ij (p,t) of ρ (p,t). Tracing out ρ (p,t) over the pseudomode, we derived the time-dependent reduced density matrix of the qubits as ρAB (p,t) = a(p,t)|0AB 0|+b(p,t)|+AB +|+c(p,t)|2AB 2| + d(p,t)|0AB 2| + d ∗ (p,t)|2AB 0|, (12)

PHYSICAL REVIEW A 86, 012325 (2012)

ij (p,t) are collected in the ρ 16 (p,t). The expressions of ρ Appendix. Although the coefficients in (12) cannot befully written in analytical forms, we may still represent them as a(p,t) = |β|2 (1−p)2 F0 (t)+|α|2 , b(p,t) = |β|2 (1 − p)2 F+ (t)  c(p,t) = |β|2 (1 − p)2 F2 (t), d(p,t) = αβ ∗ (1 − p) F2 (t), (13) where the real positive time-dependent functions F0 (t) and F+ (t) are the probabilities of decoherence-induced transitions |2AB → |0AB and |2AB → |+AB at time t, and F2 (t) = 1 − F0 (t) − F+ (t) is the probability that |2AB remains unchanged. III. THE SCHEME FOR OPTIMIZING ENTANGLEMENT

As time evolves the qubits’ entanglement changes, but we can manipulate it at any time by applying on each qubit a post measurement with the strength 0  prj < 1 (j = A,B), which is either a weak measurement if c(p,t) > a(p,t) or a quantum measurement reversal if c(p,t) < a(p,t). A quantum measurement reversal with strength prj on qubit j affects its state as |0j 0| → (1 − prj )|0j 0|, |1j 1| → |1j 1|,   |0j 1| → 1 − prj |0j 1|, |1j 0| → 1 − prj |1j 0|. (14) In practice, a quantum measurement reversal (14) with strength pr can be implemented by first bit-flipping the qubit, then performing a null-outcome weak measurement (2) with strength p = pr , and finally bit-flipping the qubit back. In some situations, say, when |β|2 > |α|2 , p is small, and the interaction time is short, c(p,t) may be greater than a(p,t). In such situations, we perform weak measurements with strengths prA = prB = pr on the qubits, which in the case of null outcomes transform the state (12) to (1) ρAB (p,t,pr ) =

1 {a(p,t)|0AB 0| + b(p,t)(1 − pr ) P1 (p,t,pr ) × |+AB +| + c(p,t)(1 − pr )2 |2AB 2| + (1−pr )[d(p,t)|0AB 2|+d ∗ (p,t)|2AB 0|]}, (15)

with P1 (p,t,pr ) = a(p,t) + b(p,t)(1 − pr ) + c(p,t)(1 − pr )2 (16) the success probability of both the prior and post measurements with null outcomes. In the other situations when c(p,t) < a(p,t) quantum measurement reversals with prA = prB = pr are to be performed, transforming the state (12) to (2) ρAB (p,t,pr ) =

where a(p,t) = ρ 11 (p,t) + ρ 22 (p,t) + ρ 33 (p,t), b(p,t) = ρ 44 (p,t) + ρ 55 (p,t), c(p,t) = ρ 66 (p,t), and d(p,t) = 012325-3

1 {a(p,t)(1−pr )2 |0AB 0| + b(p,t) P2 (p,t,pr ) × (1 − pr )|+AB +| + c(p,t)|2AB 2| + (1−pr )[d(p,t)|0AB 2|+d ∗ (p,t)|2AB 0|]}, (17)

ZHONG-XIAO MAN, YUN-JIE XIA, AND NGUYEN BA AN

PHYSICAL REVIEW A 86, 012325 (2012)

with P2 (p,t,pr ) = a(p,t)(1 − pr ) + b(p,t)(1 − pr ) + c(p,t) (18) 2

the success probability of the whole process of weak measurements and their reversals. The entanglement quality can be assessed by the concurrence [36] which for states (15) and (17) is formally derived (i) (p,t,pr ) = 2 max{0,(p,t,pr ), (p,t,p as CAB r )}/Pi (p,t), √ with (p,t,pr ) = (1 − pr )[b(p,t)/2 − a(p,t)c(p,t) and (p,t,pr ) = (1 − pr ) [|d (p, t)| − b (p, t)/2 ]. Actually, (p,t,pr ) is always negative, so we are left with (i) CAB (p,t,pr ) =

2 max{0, (p,t,pr )}. Pi (p,t,pr )

∂ 2 C (i)

the conditions ∂pABr |pr =pr(i) = 0 and ∂pAB 2 |p =p (i) < 0, yielding r r r for the case of c(p,t) > a(p,t)  a(p,t) (1) pr = 1 − , (20) c(p,t) and for the case of c(p,t) < a(p,t)  c(p,t) pr(2) = 1 − . a(p,t)

(21)

The concurrences corresponding to the two optimal conditions (20) and (21) have the same form, opt

CAB (p,t) = max{0, opt (p,t)}

(22)

with   opt (p,t) ≡ p,t,pr(i) =

2|d(p,t)| − b(p,t) , √ b(p,t) + 2 a(p,t)c(p,t)

(23)

but the corresponding success probabilities P1 and P2 are different:    a(p,t) opt (24) P1 (p,t) ≡ P1 p,t,pr(1) = 2a(p,t) + b(p,t) c(p,t) and opt P2 (p,t)

   c(p,t) (2) . (25) ≡ P2 p,t,pr = 2c(p,t) + b(p,t) a(p,t) opt

opt (p,t)

√ 2|α| F2 (t) − |β|(1 − p)F+ (t)  = √ 2 F2 (t) |β|2 (1 − p)2 F0 (t) + |α|2 + |β|(1 − p)F+ (t) (26)

and

We observe that for p → 1 the concurrence CAB tends to1 (i.e., maximal entanglement could in principle be achieved at any time, irrespective of the qubits’ initial state), at the expense that the success probability tends to 0. These results can be proved analytically by noting that in the limit p → 1 we always have c(p,t) < a(p,t) so the success probability opt is P opt (p,t) ≡ P2 (p,t) and by using the general forms of

opt P2 (p,t)

= |β| (1 − p) 2

 ×

(19)

In view of the above procedure we have at hand two control parameters p and pr , which we can manage to manipulate the qubits’ entanglement for a certain purpose at any time during the evolution. To have as much entanglement as possible at a (i) (p,t,pr ) does not given time t we note that the concurrence CAB vary monotonically with pr for a fixed p. The optimal pr = pr(i) that maximizes the concurrence can be determined from ∂C (i)

a(p,t), b(p,t), c(p,t), and d(p,t) in Eq. (13) to reexpress opt opt (p,t) (23) and P2 (p,t) (25) as

2

2F2 (t) + |β|(1 − p)F+ (t)

F2 (t) , |β|2 (1 − p)2 F0 (t) + |α|2

(27)

from which it is clear that limp→1 opt (p,t) = 1 and opt limp→1 P2 (p,t) = 0. opt In Figs. 1(a) and 1(b) we show the evolution of CAB , Eq. (22), as influenced √bypin the weak-coupling regime √ ( /   4) for α = 1/ 2 [Fig. 1(a)] and α = 1/ 15 [Fig. 1(b)]. We see that at any time t the concurrence increases with increasing p and in principle could be greater than its initial value or even become arbitrarily close to 1 for any α if p is chosen close enough to 1. Particularly in the small-|α| case [Fig. 1(b)] the concurrence vanishes within a time window t ∈ [1,2.2] if p = 0. However, with increasing p the zero-concurrence time window is shrinking and will disappear completely starting from p ≈ 0.2. The same dependences as in Figs. 1(a) and 1(b), but in the strong-coupling regime ( /√ < 4) with an experimentally achievable value of /  = 0.8 [37], are shown in Figs. 1(c) and 1(d). Despite oscillating in time, the concurrence at any time also improves with p for both large- and small-|α| cases, opt and limp→1 CAB = 1, as in the weak-coupling regime. Note that, in the small-|α| case [Fig. 1(d)] there may exist several time windows of zero concurrence if p = 0 [33]. To get rid of all such zero-concurrence windows (i.e., for all times) one should apply weak measurements with a quite large p. As an example, for the parameters used in Fig. 1(d), the concurrence will never vanish if p > 0.5. Compared with Fig. 1(b), where p > 0.2 is required to make the concurrence always nonzero, we may say that our method for optimizing entanglement of qubits coupled to a common environment is more efficient in the weak-coupling than in the strong-coupling regime. opt It is worth noting that the 3D surfaces of CAB in Fig. 1 have (i) pr = pr = 0 even when p = 0 (i.e., the post measurements are still performed even in the absence of the prior measurements). To highlight the advantage of our method as compared to the case without any entanglement manipulation, we also display, as 2D dashed curves in the p = 0 plane in Fig. 1, the concurrence with p = pr = 0. As can be seen, in the absence of the prior measurements the post measurements alone can slightly enhance the entanglement but entanglement sudden death, if any, cannot be fully avoided, somewhat similarly to the case when the qubits suffer from independent decoherences due to local amplitude dampings [28]. opt opt The total success probabilities P opt = P1 or P opt = P2 corresponding to the two types of post measurements are plotted in Fig. 2 versus t and p. In Figs. 2(a) and 2(c)

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PHYSICAL REVIEW A 86, 012325 (2012)

a

b

1.0 opt CAB

1.0 1.0

0.5 0.0 0

opt CAB

0.0 0

0.5p

1 t

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1

2 3 4

t

0.0

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d

1.0 opt CAB

0.0

1.0 1.0

0.5 0.0 0

opt

CAB 0.5

1.0

0.0 0

0.5p

0.5p

2

2 t

4 6

t

0.0

4 6

0.0

opt

FIG. 1. (Color online) The optimized concurrence CAB , Eq. (22), as functions of rescaled time √ t and prior weak measurement strength p 0.8 (c),(d). The initial states of the qubits in the weak-coupling regime with /  = 6 (a),(b) and the strong-coupling regime with /  = √ √ are prepared with α = 1/ 2 (a),(c) and α = 1/ 15 (b),(d). The two-dimensional (2D) dashed curves in the p = 0 plane show the concurrence without any manipulation (i.e., with p = pr = 0).

√ we choose α = 1/2, for which the inequality c(p,t) < a(p,t) holds for all t and p. Therefore, only the optimized quantum measurement reversals, as the post measurements, are performed and the corresponding total success probability opt opt is P opt = P2 . In the weak-coupling regime [Fig. 2(a)] P2 decreases monotonically with both t and p. By contrast, opt in the strong-coupling regime [Fig. 2(c)] P2 undergoes damped oscillations in time. The underlying reason can be explained by examining the success probability of the quantum measurement reversals on the qubits. Actually, the larger the weight of the qubits’ component |11AB , the bigger the success probability for the quantum measurement reversals. Owning to the back action of the structured cavity and the pseudomode-mediated qubit-qubit interaction in the strongcoupling regime, the component |11AB can revive after a opt decay. As a consequence,  P2 oscillates correspondingly. The small-|α| case (say, α = 1/15) is considered in Figs. 2(b) and 2(d). As discussed previously, when |α|2 < |β|2 ,there exists a domain in the t-p space in which c(p,t) > a(p,t). Inside this domain [which is surrounded by the bold black curve in Figs. 2(b) and 2(d)] the post measurements are again weak measurements with optimized strengths, and the opt corresponding total success probability is P1 . As can be seen, opt opt P1 (i.e., P inside the surrounded domain) increases with t for a fixed p but decreases with p for a fixed t. Outside the domain [i.e., when c(p,t) < a(p,t) and the post measurements are quantum measurement reversals with optimized strengths]

opt

P opt = P2 decreases monotonically with respect to both t and p in the weak-coupling regime [Fig. 2(b)] and exhibits damped oscillations in the strong-coupling regime [Fig. 2(d)]. However, in both the weak- and strong-coupling regimes limp→1 P opt = 0, which is the price to pay for an attempt to obtain a nearly maximal entanglement. IV. THE SCHEME FOR RESTORING THE INITIAL ENTANGLEMENT

In this section we present a scheme to restore the initial qubits’ state and entanglement. The procedure for that purpose is similar to the scheme for optimizing entanglement when c(p,t) < a(p,t), i.e., weak measurements with strengths p are followed by quantum measurement reversals with properly chosen strengths pr = prres . The reason why the post measurements should not be weak measurements again is because both the prior weak measurements and the decoherence have already reduced the weight of the qubits’ component |11AB . To determine prres it is convenient to resort to the mathematical technique of the quantum trajectory dealing with pure states associated with fictitious “jump” and “no-jump” scenarios, as adopted in Ref. [25] for the single-qubit case. Here, the two-qubit state (3) after the null-outcome weak measurements with strengths p remains in fact a pure state of the form

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|ψ(p,0)AB =

α|0AB + β(1 − p)|2AB  . |α|2 + |β|2 (1 − p)2

(28)

ZHONG-XIAO MAN, YUN-JIE XIA, AND NGUYEN BA AN

PHYSICAL REVIEW A 86, 012325 (2012)

a

b

1.0 opt

P

1.0 1.0

0.5 0.0 0

opt

P

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0.5 0.0 0

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0.5p

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1 2

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t 3

0.0

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1.0

1.0 Popt 0.5

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opt

0.0 0

0.5 p

0.5p

1

1

t

2

t

1.0

0.5

3

0.0

2 3 4

0.0

FIG. 2. (Color online) The success probability of the entanglement optimization versus t and with √ √ p in the weak-coupling regime √ /  = 6 (a),(b) and the strong-coupling regime with /  = 0.8 (c),(d). In (a), and (c) α = 1/ 2 and in (b) and (d) α = 1/ 15. In the domains surrounded by the bold black curve in (b) and (d) c(p,t) > a(p,t) and outside these domains c(p,t) < a(p,t).

While being coupled to the same environment via the interaction Hamiltonian (8), the state (28) might jump into state |+AB or into state |0AB . In our model we learn from Eqs. (12) and (13) that the jumps into state |+AB or state |0AB happen respectively with the probabilities P|+ (p,t) = b(p,t) = |β|2 (1 − p)2 F+ (t) or P|0 (p,t) = a(p,t) − |α|2 = |β|2 (1 − p)2 F0 (t). Otherwise, if there are no jumps at all, the qubits’ state (28) at time t > 0 just evolves into 1 |ψno jump (p,t)AB =  [α|0AB Pno jump (p,t)  + β(1 − p) F2 (t)|2AB ],

With such a pr = prres the probabilities concerned become   res res Pno = (1 − p)2 F2 (t), (32) jump (p,t) ≡ Pno jump p,t,pr    res P|+ (p,t) ≡ P|+ p,t,prres = |β|2 (1 − p)3 F+ (t) F2 (t), (33) and

(29)

with the no-jump probability Pno jump (p,t) = |α| + c(p,t) = |α|2 + |β|2 (1 − p)2 F2 (t). Then, the reversing quantum measurements (14) with strengths pr performed at time t bring the no-jump state (29) to 2

|ψno jump (p,t,pr )AB = 

that would exactly restore the initial qubits’ state [i.e., |ψno jump (p,t,prres )AB ≡ |ψ(0)AB ] satisfies the condition  prres (p,t) = 1 − (1 − p) F2 (t). (31)

1

[α(1 − pr )|0AB Pno jump (p,t,pr )  + β(1 − p) F2 (t)|2AB ], (30)

with the probability Pno jump (p,t,pr ) = |α|2 (1 − pr )2 + c(p,t) = |α|2 (1−pr )2 +|β|2 (1−p)2 F2 (t), while states |+AB and |0AB remain themselves with the probabilities P|+ (p, t, pr ) = P|+ (p, t) (1 − pr ) and P|0 (p, t, pr ) = P|0 (p,t)(1 − pr )2 , respectively. From Eq. (30), it is simple to find that the reversing measurement strength pr = prres

  res P|0 (p,t) ≡ P|0 p,t,prres = |β|2 (1 − p)4 F0 (t)F2 (t).

(34)

The above jump and no-jump technique yields the total probability of a successful restoration process as P res = res res res Pno jump + P|+ + P|0 and the resulting density matrix of the two qubits reads   1  res res ρAB (p,t) ≡ ρAB p,t,prres = res Pno jump |ψ(0)AB ψ(0)| P res res + P|+ |+AB +| + P|0 |0AB 0| . (35) res It is worth noticing from Eqs. (32)–(34) that Pno jump scales res res as (1 − p)2 , P|+ as (1 − p)3 , and P|0 as (1 − p)4 . Hence, res ρAB (p,t) in Eq. (35), for any α,β can be made arbitrarily close to the initial state ρAB (0,0) = |ψ(0)AB ψ(0)| in the limit of p → 1 (i.e., by choosing p close enough to 1), despite a possibly strong decoherence effect. In other words, decoherence due to the qubits’ interaction with a common

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a

res CAB

b

1.0 0.8 0.6 0.4 0

1.0

res CAB

t

4

0.5p

1

2 3

1.0

0.2 0.0 0

0.5p

1

0.4

t

0.0

2 3 4

0.0

res FIG. 3. (Color online) The concurrence CAB , Eq. (36), √ under the restoration condition (31) as a function of t and of p in the weak-coupling √ regime with /  = 6 for α = 1/ 2 (a) and α = 1/ 15 (b).

environment can be suppressed by our method described above. To see the entanglement restoration explicitly we express the concurrence (19) (with i = 2) for pr = prres as   res CAB (p,t) ≡ CAB p,t,prres = max{0, res (p,t)}, (36) where res (p,t)   ≡ p,t,prres   1 − prres [2|d(p,t)| − b(p,t)] =   2  a(p,t) 1 − prres + b(p,t) 1 − prres + c(p,t) √ 2|αβ| F2 (t) − (1 − p)|β|2 F+ (t) =√ , √ F2 (t) + (1 − p)|β|2 [(1 − p)F0 (t) F2 (t)+F+ (t)] (37) which in the limit p → 1 tends to 2|αβ|, the concurrence of the initial state |ψ(0)AB , independent of the decoherence governed by F0 (t), F+ (t), and F2 (t). This is, of course, not for free. As seen from Eqs. (32)–(34), the restoration probability

res res res P res = Pno jump + P|+ + P|0 becomes vanishingly small if one wishes an exact entanglement restoration. res In Fig. 3 we show how the concurrence CAB , Eq. (36), under the condition (31) for the post-quantum-measurement reversals with strength prres depends on the scaled interaction time t and the prior weak measurement strength p. Visibly, at any evolution time when p is approaching1 the concurrence 2 is tending √ to the initial value 2|αβ|√= 2|α| 1 − |α| for both α  1/ 2 [Fig. 3(a)] and α < 1/ 2 [Fig. 3(b)]. Remarkable from Fig. 3 is the fact that by our scheme in this section the qubits’ entanglement is still considerably improved and, as in the scheme in Sec. III, entanglement sudden death (if any) can be circumvented even for a moderate value of p < 1. To have a simultaneous view of both schemes for optimizing and restoring entanglement we plot in Fig. 4 opt res and P opt ,P res versus p for a couple of moments CAB ,CAB during the evolution. It is clear from Fig. 4(a) that in the opt scheme for optimizing entanglement CAB is tending to 1, while in the scheme for restoring the initial entanglement res CAB is regaining its initial √ value [here CAB (0,0)  0.499 for the value used of α = 1/ 15] when p is approaching 1, regardless of the interaction time (i.e., for any fixed value of t). However, the cost to pay is that both the success probabilities for entanglement optimization and entanglement restoration become vanishingly small in the limit of p → 1, as illustrated in Fig. 4(b).

V. CONCLUSION

FIG. 4. The concurrences (a) and the success probabilities (b) of both the entanglement optimization scheme (solid lines) and the entanglement restoration scheme (dashed lines) as functions of p at different moments of evolution (t = 0.5 and t = 1.5 are indicated near the relevant curves). The initial state of the qubits is prepared  with α = 1/15; thereby their initial concurrence is about 0.499. In this figure /  = 6 is used.

We have described methods to manipulate the entanglement of two qubits stored in a common environment modeled as a lossy cavity with Lorentzian spectral density, by means of prior and post measurements. Two schemes are proposed to cope with the decoherence. The first scheme aims at obtaining an optimal (i.e., as much as possible) entanglement, while the second one aims at restoring the initial entanglement. For both schemes the prior measurements are weak measurements with a controllable strength p. However, the post measurements are scheme dependent. In restoring the initial entanglement the post measurements are quantum measurement reversals with a strength prres . In optimizing entanglement the post measurements are, however, case sensitive: they may be again weak

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measurements with a strength pr(1) or quantum measurement reversals with a strength pr(2) , depending on the evolution time as well as on the parameters of the initial state and the prior measurements. We have established the explicit conditions for prres , pr(1) , and pr(2) so that at any time and for any initial state the qubits’ concurrence can be made arbitrarily close to 1 (i.e., maximal entanglement is achieved) or arbitrarily close to the initial value (i.e., exact entanglement restoration is realized) by choosing the strength p of the prior weak measurements close enough to 1, at the expense of vanishingly small success probabilities. Yet our methods prove to be good enough to protect entanglement, with a reasonable finite success probability, from decoherence effects, including entanglement sudden death, by using a moderate value of p < 1. There exists a special case when the qubits’ initial state √ is maximally entangled (|α| = |β| = 1/ 2). In this case both the presented schemes can be applied with the post measurements being quantum measurement reversals with different measurement strengths. For the first scheme, the strength pr(2) should satisfy the condition (21), while for the second scheme, the strength prres should satisfy the condition (31). The question then arises as to which scheme is more efficient (i.e., which choice of pr , pr = pr(2) or pr = prres , is better). We have examined this issue and found that for the same values of p and t the amounts of evolved entanglement are almost the same, but the success probability of the second scheme is higher than that of the√first scheme. Thus, for the initial state with |α| = |β| = 1/ 2, one would prefer the second to the first scheme. Finally, we would like to discuss briefly the possible experimental realization of our schemes. The decoherence suppression of a single qubit via weak measurement and quantum measurement reversal was experimentally demonstrated by using all-optical apparatuses [26]. The protection of entanglement

of two qubits from independent decoherences via the same measurement procedures was realized also by exploiting all-optical apparatuses [29]. However, in our schemes the two qubits interact with the same environment; therefore the contexts of cavity QED experiments with trapped ions and circuit QED experiments would be more suitable. Entanglement between two remotely located trapped atomic ions was demonstrated in Ref. [38,39], and via quantum state tomography multiqubit entangled states can be generated and fully characterized [39]. In addition, field coupling and coherent quantum state storage between two Josephson phase qubits were realized via a microwave cavity on a chip [37,40]. Therefore, within current technologies our schemes could be realized, say, for solid-state or superconducting phase qubits [32]. ACKNOWLEDGMENTS

In this work Z.X.M. and Y.J.X. were supported by the National Natural Science Foundation of China under Grants No. 10947006 and No. 61178012, the Specialized Research Fund for the Doctoral Program of Higher Education under Grant No. 20093705110001, and the Scientific Research Foundation of Qufu Normal University for Doctors, while N.B.A. was funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 103.99-2011.26. APPENDIX: EXPRESSIONS FOR THE MATRIX ELEMENTS

This Appendix provides expressions for the matrix elements ρ ij (p,t) which are solutions of the differential equations (11) with ρ (p,0) = ρAB (p,0)|0F as the initial condition.

γ +i∞ 1 st In the following L−1 L(s) = 2πi γ −i∞ L(s)e ds denotes the inverse Laplace transform of L(s).

ρ 11 (p,t) = L−1 {β 2 (1 − p)2 64 2 4 [20s 3 + 45s 2 + 31s 2 + 6 3 + 60s2 + 56 2 ] + α 2 (2s + )[4s 8 + 28s 7 + 128 2 4 (3 2 + 282 ) + s 6 (79 2 + 1522 ) + s 5 (115 3 + 764 2 + 96s 2 ( 4 + 30 2 2 + 884 ) + s 4 (91 4 + 1480 2 2 + 15364 ) + s 3 (37 5 + 1372 3 2 + 5248 ) + s 2 (6 6 + 600 4 2 + 6208 2 4 + 46086 )]} × {s(2s + )[s(s + ) + 82 ][s(s + )(s + 2 ) + 8(3s + 2 )] × [4s 3 + 12s 2 + 11s 2 + 3 3 + 4(6s + 7 )−1 ]} = L−1 {β 2 (1 − p)2 64 2 4 [20s 3 + 45s 2 + 31s 2 + 6 3 + 60s2 + 56 2 ]} × {s(2s + )[s(s + ) + 82 ][s(s + )(s + 2 ) + 8(3s + 2 )2 ] × [4s 3 + 12s 2 + 11s 2 + 3 3 + 4(6s + 7 )2 ]}−1  + α 2 ;

(A1)

ρ 22 (p,t) = L−1 {β 2 (1 − p)2 64 4 (20s 3 + 45s 2 + 31s 2 + 6 3 + 60s2 + 56 2 )} × {(2s + )[s(s + ) + 82 ][s(s + )(s + 2 ) + 8(3s + 2 )2 ] × [4s 3 + 12s 2 + 11s 2 + 3 3 + 4(6s + 7 )2 ]}−1 ;

(A2)

ρ 33 (p,t) = L−1 {β 2 (1 − p)2 192(s + )4 }/{[s(s + )(s + 2 ) + 8(3s + 2 )2 ] × [4s 3 + 12s 2 + 11s 2 + 3 3 + 4(6s + 7 )2 ]}; 012325-8

(A3)

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ρ 44 (p,t) = L−1 {β 2 (1 − p)2 8 2 [(s + )2 (2s + )(s + 2 )(2s + 3 ) − 4(s + )(2s 2 + 15s + 10 2 )2 + 32(15s + 14 )4 ]}{(2s + )[s(s + ) + 82 ][s(s + )(s + 2 ) + 8(3s + 2 )2 ] × [4s 3 + 12s 2 + 11s 2 + 3 3 + 4(6s + 7 )2 ]}−1 ;

(A4)

ρ 55 (p,t) = L−1 {β 2 (1 − p)2 82 [2s 3 + 9s 2 + 13s 2 + 6 3 + 4(3s + 2 )2 ]} × {[s(s + )(s + 2 ) + 8(3s + 2 )2 ][4s 3 + 12s 2 + 11s 2 + 3 3 + 4(6s + 7 )2 ]}−1 ;

(A5)

ρ 66 (p,t) = L−1 {β 2 (1 − p)2 [(s + )2 (2s + )(s + 2 )(2s + 3 ) + 4(s + )(26s 2 + 65s + 38 2 )2 + 32(9s + 10 )4 ]} × {[s(s + )(s + 2 ) + 8(3s + 2 )2 ][4s 3 + 12s 2 + 11s 2 + 3 3 + 4(6s + 7 )2 ]}−1 ; (A6) ρ 16 (p,t) = L−1 {αβ(1 − p)[(s + )(2s + ) + 82 ]}/{s(s + )(2s + ) + 4(3s + )2 }.

[1] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, England, 2000). [2] T. Yu and J. H. Eberly, Phys. Rev. Lett. 93, 140404 (2004); 97, 140403 (2006); J. H. Eberly and T. Yu, Science 316, 555 (2007); T. Yu and J. H. Eberly, ibid. 323, 598 (2009); M. P. Almeida et al., ibid. 316, 579 (2007). [3] C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin, and W. K. Wootters, Phys. Rev. Lett. 76, 722 (1996). [4] C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, Phys. Rev. A 53, 2046 (1996). [5] J. W. Pan, S. Gasparoni, R. Ursin, G. Weihs, and A. Zeilinger, Nature (London) 423, 417 (2003). [6] P. G. Kwiat, S. Barraza-Lopez, A. Stefanov, and N. Gisin, Nature (London) 409, 1014 (2001). [7] R. Dong et al., Nat. Phys. 4, 919 (2008). [8] P. Zanardi and M. Rasetti, Phys. Rev. Lett. 79, 3306 (1997). [9] D. A. Lidar, I. L. Chuang, and K. B. Whaley, Phys. Rev. Lett. 81, 2594 (1998). [10] P. G. Kwiat, A. J. Berglund, J. B. Alterpeter, and A. G. White, Science 290, 498 (2000). [11] S. Maniscalco, F. Francica, R. L. Zaffino, N. Lo Gullo, and F. Plastina, Phys. Rev. Lett. 100, 090503 (2008). [12] N. B. An, J. Kim, and K. Kim, Phys. Rev. A 82, 032316 (2010). [13] P. Facchi, D. A. Lidar, and S. Pascazio, Phys. Rev. A 69, 032314 (2004). [14] P. W. Shor, Phys. Rev. A 52, 2493(R) (1995). [15] A. M. Steane, Phys. Rev. Lett. 77, 793 (1996). [16] A. R. Calderbank and P. W. Shor, Phys. Rev. A 54, 1098 (1996). [17] A. M. Steane, Proc. R. Soc. London, Ser. A 452, 2551 (1996). [18] I. Sainz and G. Bjork, Phys. Rev. A 77, 052307 (2008). [19] J. S. Zhang, J. B. Xu, and Q. Lin, Eur. Phys. J. D 51, 283 (2009); J. Zhang, R. B. Wu, C. W. Li, and T. J. Tarn, J. Phys. A 42, 035304 (2009). [20] B. Bellomo, R. LoFranco, S. Maniscalco, and G. Compagno, Phys. Rev. A 78, 060302 (2008). [21] P. Lodahl et al., Nature (London) 430, 654 (2004). [22] S. Das and G. S. Agarwal, Phys. Rev. A 81, 052341 (2010). [23] S. Y. Zhu and M. O. Scully, Phys. Rev. Lett. 76, 388 (1996).

(A7)

[24] M. O. Scully and S. Y. Zhu, Science 281, 1973 (1998). [25] A. N. Korotkov and K. Keane, Phys. Rev. A 81, 040103(R) (2010). [26] J. C. Lee, Y. C. Jeong, Y. S. Kim, and Y. H. Kim, Opt. Express 19, 16309 (2011). [27] Q. Sun, M. Al-Amri, and M. S. Zubairy, Phys. Rev. A 80, 033838 (2009). [28] Q. Sun, M. Al-Amri, L. Davidovich, and M. S. Zubairy, Phys. Rev. A 82, 052323 (2010). [29] Y. S. Kim, J. C. Lee, O. Kwon, and Y. H. Kim, Nat. Phys. 8, 117 (2012). [30] A. N. Korotkov, Phys. Rev. B 60, 5737 (1999). [31] H. Mabuchi and P. Zoller, Phys. Rev. Lett. 76, 3108 (1996); M. A. Nielsen and C. M. Caves, Phys. Rev. A 55, 2547 (1997). [32] A. N. Korotkov and A. N. Jordan, Phys. Rev. Lett. 97, 166805 (2006); N. Katz, M. Neeley, M. Ansmann, R. C. Bialczak, M. Hofheinz, E. Lucero, A. OConnell, H. Wang, A. N. Cleland, J. M. Martinis, and A. N. Korotkov, ibid. 101, 200401 (2008). [33] L. Mazzola, S. Maniscalco, J. Piilo, K.-A. Suominen, and B. M. Garraway, Phys. Rev. A 79, 042302 (2009); L. Mazzola, S. Maniscalco, J. Piilo, and K. A. Suominen, J. Phys. B 43, 085505 (2010). [34] Y. Li, J. Zhou, and H. Guo, Phys. Rev. A 79, 012309 (2009). [35] B. M. Garraway, Phys. Rev. A 55, 2290 (1997). [36] W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998). [37] M. A. Sillanpaa, J. I. Park, and R. W. Simmonds, Nature (London) 449, 438 (2007). [38] D. L. Moehring, P. Maunz, S. Olmschenk, K. C. Younge, D. N. Matsukevich, L.-M. Duan, and C. Monroe, Nature (London) 449, 68 (2007). [39] H. H¨afner, W. H¨ansel, C. F. Roos, J. Benhelm, D. Chek-al-kar, M. Chwalla, T. K¨orber, U. D. Rapol, M. Riebe, P. O. Schmidt, C. Becher, O. G¨uhne, W. D¨ur, and R. Blatt, Nature (London) 438, 643 (2005). [40] J. Majer, J. M. Chow, J. M. Gambetta, J. Koch, B. R. Johnson, J. A. Schreier, L. Frunzio, D. I. Schuster, A. A. Houck, A. Wallraff, A. Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Nature (London) 449, 443 (2007).

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