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Effect of two-qutrit entanglement on quantum speed limit time of a bipartite V-type open system
Annals of Physics 378 (2017) 407–417
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Annals of Physics journal homepage: www.elsevier.com/locate/aop
Effect of two-qutrit entanglement on quantum speed limit time of a bipartite V-type open system N. Behzadi a,∗ , B. Ahansaz b , A. Ektesabi b , E. Faizi b a
Research Institute for Fundamental Sciences, University of Tabriz, Iran
b
Physics Department, Azarbaijan Shahid Madani University, Tabriz, Iran
article
abstract
info
Article history: Received 2 September 2016 Accepted 25 January 2017 Available online 31 January 2017 Keywords: Quantum speed limit Two-qutrit entanglement V-type atoms Non-Markovian dynamics
In the present paper, quantum speed limit (QSL) time of a bipartite V-type three-level atomic system under the effect of two-qutrit entanglement is investigated. Each party interacts with own independent reservoir. By considering two local unitarily equivalent Werner states and the Horodecki PPT state, as initial states, the QSL time is evaluated for each of them in the respective entangled regions. It is counterintuitively observed that the effect of entanglement on the QSL time driven from each of the initial Werner states are completely different when the degree of nonMarkovianity is considerable. In addition, it is interesting that the effect of entanglement of the non-equivalent Horodecki state on the calculated QSL time displays an intermediate behavior relative to the cases obtained for the Werner states. © 2017 Elsevier Inc. All rights reserved.
1. Introduction Quantum speed limit (QSL) is of particular interest and has attracted much attention in tremendous areas of quantum physics and quantum information, such as nonequilibrium thermodynamics [1], quantum metrology [2–5], quantum optimal control [6–11], quantum computation [12–14], and quantum communication [1,15]. In these fields, QSL time is a key concept, defined as the minimum
∗
Corresponding author. E-mail address: [email protected] (N. Behzadi).
http://dx.doi.org/10.1016/j.aop.2017.01.026 0003-4916/© 2017 Elsevier Inc. All rights reserved.
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evolution time in which a system evolves from an initial state to a target state. In the other hand, the QSL time sets a bound on the minimal time a system which needs to evolve between two distinguishable states, and it can be understood as a generalization of the time-energy uncertainty principle. For isolated systems, the QSL time under unitary evolution is determined by the Mandelstam–Tamm bound [16] and Margolus–Levitin bound [17]. Because of the inevitable interactions with the environments, quantum systems should be generally regarded as open systems. Therefore, it is necessary to determine a QSL time for open systems. Recently, Taddei et al. developed a method to investigate the QSL problem in open systems described by positive non-unitary maps by using of quantum Fisher information for time estimation [18]. While, for the case of the initial mixed states, a Hermitian operator is required to minimize the Fisher information in the extended system–environment space, which is generally a hard task. Afterwards, Campo et al. exploited the concept of relative purity to derive an analytical and computable QSL time for open systems undergoing a completely positive and trace preserving evolution [19]. It should be noted that relative entropy can perfectly make a distinction between an initial pure state and its target state, however it may fail to distinguish an initial mixed state to its target state. Also, a tight bound on the minimal evolution time of an arbitrarily driven open system has been formulated by Deffner and Lutz [20]. However, their time bound is derived from pure initial states and cannot be directly applied into the mixed initial states. Motivated by the recent studies above, the authors in Ref. [21] employed an alternative fidelity definition different from relative entropy in order to derive an easily computable QSL time bound for open systems whose initial states can be chosen as either pure or mixed states and this QSL time is applicable to either Markovian or non-Markovian dynamics. Alongside these efforts, several attentions have been devoted to investigate the effects of entanglement on the QSL time in multiqubit systems (for instance, see [18,22]). In this paper, we consider a bipartite system composed of two identical V-type three-level atoms as a two-qutrit system where each atom interacts with own independent reservoir. We use the QSL time bound obtained in [21] to evaluate the speed of evolution of the system under the effects of two-qutrit entanglement exhibited in the system. To this aim, we choose two local unitarily equivalent Werner states [23] and the Horodecki PPT (partial positive transpose) state [24] which is not equivalent to the Werner states, as initial states, and obtained the QSL time for each of them in the respective entangled regions. When the degree of non-Markovianity in the system dynamics is considerable, surprisingly distinct behaviors for the QSL time are obtained for the equivalent Werner states in the entangled regions. In other words, under the condition of considerable non-Markovianity, effect of the same two-qutrit entanglement related to equivalent Werner states are completely different on the respective QSL times. It is interesting to note that for the non-equivalent Horodecki state, the calculated QSL time has an intermediate behavior relative to the cases obtained for the Werner states. We see that, although any local unitary operation does not increases the degree of entanglement (in the case of Werner states discussed in the present paper, the introduced local unitary operator does not change the degree of entanglement) but, as illustrated in the text, it is observed that the related QSL time changes under the mentioned local unitary operator effect. The paper is organized as follows: In Section 2, we describe dynamics of a V-type three-level atomic system as an open system along with giving a quantitative description of non-Markovian behavior of that system. Also, we extend the dynamics of the V-type three-level system to a bipartite case. Section 3 is devoted to the obtaining of QSL time for the bipartite system and discussing about the connection between two-qutrit entanglement and the related QSL time. Finally, in Section 4, we present our conclusions. 2. Dynamics of V-type three-level open systems In this section, we study at first, Markovian and non-Markovian dynamics of a V-type three level atom, as a qutrit, coupled to a dissipative environment. Each atom has two excited states each of them spontaneously decays into ground states such that the respective dipole moments of transitions may have interaction with each other. We then extend the scheme into a two-qutrit case which each of them coupled to independent reservoirs as depicted in Fig. 1. For a V-type atom, under the interaction with the same environment, the two upper levels |2⟩ and |1⟩ are coupled with the ground one, |0⟩,
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Fig. 1. A system of two identical V-type three-level atoms each of them interacts with own independent reservoir.
with transition frequency ω2 and ω1 respectively. The Hamiltonian of the total system is given by H = HS + HE + HI = H0 + HI ,
(1)
where H0 =
2
ωl σ+l σ−l +
l =1
ωk bĎk bk ,
(2)
k
is the free Hamiltonian of the system and environment, and HI =
2 (glk σ+l bk + glk∗ σ−l bĎk ), l =l
(3)
k
is the interaction Hamiltonian. σ±l (l = 1, 2) are the raising and lowering operators of the levels |2⟩ and |1⟩ to the ground state |0⟩. The index k labels the different field modes of the environment with Ď frequencies ωk , creation and annihilation operators bk , bk and coupling constants glk . In the interaction picture the total system obeys the Schrödinger equation i
d dt
|ψ(t )⟩ = Hint |ψ(t )⟩,
(4)
where Hint =
2 (glk σ+l bk ei(ωl −ωk )t + glk∗ σ−l bĎk e−i(ωl −ωk )t ), l =1
(5)
k
and
|ψ(t )⟩ =
2
cl (t )|l⟩S ⊗ |0⟩E +
l =0
ck (t )|0⟩S ⊗ |1k ⟩E .
(6)
k
Since Hint |0⟩S ⊗ |0⟩E = 0 then c0 (t ) will be invariant in time, while the amplitudes c1 (t ) and c2 (t ) will not. The time variations of these amplitudes is obtained by solving a system of differential equations which is easily derived from the Schrödinger equation c˙l (t ) = −i
glk ck (t )ei(ωl −ωk )t ,
l = 1, 2,
(7)
k
c˙k (t ) = −i
2 l=1
glk∗ cl (t )e−i(ωl −ωk )t .
(8)
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By assuming ck (0) = 0, i.e. there is no photon in the initial state, the solution of the second equation is inserted into the first one to get a closed equation for cl (t ), c˙l (t ) = −
2
t
flm (t − t ′ )cm (t ′ )dt ′ ,
l = 1, 2.
(9)
0
m=1
The kernel in Eq. (9) can be expressed in terms of spectral density J (ω) of the reservoir as follows flm (t − t ′ ) =
t
dωJlm (ω)ei(ωl −ω)t −i(ωm −ω)t . ′
(10)
0
Jlm (ω) is chosen as Lorentzian distribution Jlm (ω) =
γlm λ2 , 2π (ω0 − ∆ − ω)2 + λ2 1
(11)
where ∆ is the detuning of the atomic transition frequency from the central frequency of the reservoir and λ is the spectral width of the coupling. γii = γi are the relaxation rates of the two upper levels, √ and γij = γi γj θ (i ̸= j and |θ | ≤ 1) are responsible for the spontaneously generated interference (SGI) between the two decay channels |2⟩ → |0⟩ and |1⟩ → |0⟩. θ depends on the relative angle between two dipole moment elements related to the mentioned transitions. θ = 0 means that the dipole moments of two transitions are perpendicular to each other and this is corresponding to the case that there is no SGI between two decay channels. On the other hand, θ = ±1 indicating that the two dipole moment are parallel or antiparallel, corresponding to the strongest SGI between two decay channels. If we take Laplace transform from Eq. (9), it becomes
√ λ γ2 − γ1 γ2 θ c˜2 (p) √ , (12) =− c˜1 (p) γ1 γ2 θ γ1 2(p + λ) ∞ where c˜l (p) = L[cl (t )] = 0 cl (t )e−pt dt (l = 1, 2), is the Laplace transform of cl (t ). By considering the subspace spanned by {|2⟩, |1⟩}, and doing the following unitary transformation q + γ1 − γ2 q − γ1 + γ2 − 2q 2q , U= (13) q − γ + γ q + γ1 − γ2 1 2
pc˜2 (p) − c2 (0) pc˜1 (p) − c1 (0)
2q
2q
on Eq. (12) and taking inverse Laplace transform we get c± (t ) = G± (t )c± (0), where c± (0) =
√1 2q
G± (t ) = e
(14)
√
√ c2 (0) q ± γ1 ∓ γ2 ∓ c1 (0) q ∓ γ1 ± γ2 . In Eq. (14),
−λt /2
cosh
d± t 2
+
λ d±
sinh
γ +γ ±q
d± t
2
,
(15)
with d± = λ2 − 2λγ± , γ± = 1 22 and q = (γ1 − γ2 )2 + 4γ1 γ2 θ 2 . It should be noted that the extended unitary transformation on the whole space of the system spanned by {|2⟩, |1⟩, |0⟩}, becomes as
q + γ1 − γ2
2q U = q − γ 1 + γ2 2q 0
q − γ1 + γ2 − 2q q + γ1 − γ2 2q 0
0
. 0 1
(16)
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By these Considerations, the density matrix of a V-type system at time t becomes
|G+ (t )|2 |c+ (0)|2 G∗ (t )c+ (0)∗ G− (t )c− (0) + ϱS (t ) = ∗ G∗+ (t )c+ (0)c0
∗ G+ (t )c+ (0)G∗− (t )c− (0) 2 |G− (t )| |c− (0)|2
G+ (t )c+ (0)c0∗ G− (t )c− (0)c0∗ 2 2 . 1 − |G+ (t )| |c+ (0)| −|G− (t )|2 |c− (0)|2
∗ G∗− (t )c− (0)c0
(17)
Clearly, the density matrix in Eq. (17) is, indeed, the time development of the state ϱS (0) = U |ψ(0)⟩⟨ψ(0)|U Ď in which |ψ(0)⟩ is the state (6) at time t = 0. It is very instructive to write the density matrix (17) in the Krauss representation. It can be easily obtained that 3
ϱS (t ) =
Ď
Ki ϱS (0)Ki ,
(18)
i =1
with
3
i=1
Ď
Ki Ki = I3 , where I3 is identity operator for the Hilbert space of a three-level system and
K1 =
G+ (t ) 0 0
0
0 0 , 1
G− (t ) 0
0 0
K2 =
1 − |G+ (t )|2
0 K3 = 0 0
0 0
0 0 ,
0
0
(19)
0 0
1 − |G− (t )|2
0 0 . 0
Also, the original density matrix is obtained as ρS (t ) = U Ď ϱS (t )U and the respective Krauss operators are Ki = U Ď Ki U (i = 1, 2, 3), with explicit forms
q + γ1 − γ2
q − γ1 + γ2
+ G− G+ 2q 2q K1 = 2 q − (γ1 − γ2 )2 (−G+ + G− )
(−G+ + G− ) G+
2q
q2 − (γ1 − γ2 )2
2q
q − γ1 + γ2 2q
0
K2 =
(1 − |G+ |2 ) 2q
K3 =
(1 − |G− |2 ) 2q
+ G−
q + γ1 − γ2 2q
0
√
0 0
√
0 0
q + γ1 − γ2
q − γ1 + γ2 Ď
√
q + γ1 − γ2
, 0
(20)
1
0 0 , 0
− q − γ1 + γ2 √
0
0 0
0 0
(21)
0 0 , 0
(22) Ď
hence, it is concluded that i=1 Ki Ki = I3 and ρS (t ) = i=1 Ki ρS (0)Ki . It should be noted that the initial state of a V-type atom, i.e. ρS (0), can be generally considered as a mixed state so its time development can also be obtained easily. In this paper, we consider the case in which the two upper atomic levels are degenerated and the atomic transitions are in resonant with the central frequency of the reservoir, i.e. ω1 = ω2 = ω0 and ∆ = 0. Under this consideration, we assume that the relaxation rates of two decay channels are equal, i.e. γ1 = γ2 = γ . Therefore, the statement in Eq. (15), takes a simple form. In order to study the non-Markovian dynamics of the V-type atom, exploiting a non-Markovian measure is inevitable. In Ref. [25], Breuer et al. introduced a measure of non-Markovianity, which is based on the reverse flow
3
3
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Fig. 2. Measure of non-Markovianity N versus γ with θ = 1 (solid line) and θ = 0.6 (dashed line).
of information from the reservoir back to the system as follows
N = max
ρ1,2 (0) η>0
η[t , ρ1,2 (0)]dt ,
(23)
where
η[t , ρ1,2 (0)] =
d dt
D[ρ1 (t ), ρ2 (t )],
(24)
indicates the changing rate of the trace distance of a pair of states denoted by√D[ρ1 (t ), ρ2 (t )] = 1 Tr |ρ1 (t )−ρ2 (t )| with the trace norm definition for an operator such as A as |A| = AĎ A. In general, in 2 order to calculate the N in Eq. (23), it is required to cover any pair of initial states. However, in Ref. [26] the optimized pair of initial states for a V-type three-level atom is obtained analytically. Fig. 2, shows the non-Markovian behavior of the system in terms of γ , λ and θ . When θ (|θ|) increases the SGI becomes more and more strong and this leads to the improvement of the non-Markovian behavior. Similar process takes place by increasing the system–environment coupling γ . But the non-Markovian measure decreases in terms of the spectral width of the coupling λ. Extending above method for dynamics of a system consists of two identical V-type atoms each of them independently interact with a Lorentzian environment is a trivial task. Let ρS (0) be defined as a density matrix of a two-qutrit V-type system on the 3 ⊗ 3 Hilbert space. So in this way, its time development at time t becomes
ρS (t ) =
3 k,l=1
Ď
Kk,l ρS (0)Kk,l ,
3 k,l=1
Ď
Kk,l Kk,l = I3 ⊗ I3 ,
(25)
where Kk,l = Kk ⊗ Kl . In the next section, we consider possible types of entanglement for the twoqutrit V-type system at initial time, and investigate their effects on the related QSL time throughout various non-Markovian dynamics. 3. Quantum speed limit time for a two-qutrit V-type system In this section, we introduce an easily computable QSL time for a two-qutrit V-type open system described in the previous section, whose initial states can be chosen as either pure or mixed. To
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this aim, we employ an alternative fidelity definition, as a distance measure of two quantum states introduced in [27] as F (ρ0 , ρt ) =
Tr (ρ0 , ρt ) Tr (ρ02 )Tr (ρt2 )
,
(26)
to calculate the QSL time bound, as derived in [21] (note that ρ(t ) ≡ ρt ). According to Ref. [21], the derived QSL time bound which is applicable to either Markovian and non-Markovian dynamics, is as the following form
|1 − F (ρ0 , ρτ )| , X (τ )
τ ≥ τQSL =
(27)
where X (τ ) =
2
τ
τ
Tr (ρ˙ t2 ) Tr (ρt2 )
0
dt ,
(28)
by denoting that ρ˙ t is the time derivative of the state ρt and τ is the actual driving time. At the first step, we consider the initial state of the two-qutrit system to be the following Werner state [23]
ρ0p (0) = (1 − p)
I3 ⊗ I3 9
+ p|ψ0 ⟩⟨ψ0 |,
(29)
where p ∈ [0, 1] and |ψ0 ⟩ = √1 (|00⟩ + |11⟩ + |22⟩). This state for p ≤ 1/4, is separable whereas for 3
p > 1/4 entangled. In fact, since it violate reduction criterion of separability so such Werner state is distillable [28]. By this consideration, in Figs. 3(a), 4(a) and 5(a), we have shown the τQSL (QSL time) in terms of γ for parameter values λ = 0.1, λ = 1 and λ = 10 (for fixed θ = 1) respectively. In Fig. 3(a) (solid red lines) for λ = 0.1, as the non-Markovianity of the system grows up in terms of γ , τQSL suddenly decreases. It is observed that the decrement rate of τQSL is proportional to the degree of entanglement of Werner state. In other words, more entanglement leads to evolution with more speed. In Fig. 4(a) (solid red lines) for λ = 1, τQSL fluctuationally decreases and the fluctuations amplitude is proportional to the degree of entanglement. Fig. 5(a) (solid red lines), for λ = 10, shows a smooth decreasing of τQSL in terms of γ and the speed of evolution of the system is always inversely proportional to the degree of entanglement. If we perform the following local unitary shift operator as 1 0 0
0 1 0
U=
0 0 1
⊗
0 0 1
1 0 0
0 1 , 0
(30)
on the state (29), we obtain the other Werner state as our another initial state as follows
ρ1p (0) = (1 − p)
I3 ⊗ I3 9
+ p|ψ1 ⟩⟨ψ1 |,
(31)
where |ψ1 ⟩ = √1 (|01⟩+|12⟩+|20⟩). Since the local shift operator (30) does not change the amount of 3
entanglement of the state (29) but we obtain a surprisingly different result for the QSL time when the initial state is chosen as (31). As observed from dashed blue lines in Fig. 3(a), QSL time for λ = 0.1 does not generally suffer sudden decrement in comparison to the previous one as shown by solid red lines in Fig. 3(a). In other words, in contrast to the result of the Werner state in (29), the decrement rate of τQSL is inversely proportional to the degree of entanglement of the Werner state in (31). Consequently, more entanglement leads to evolution with less speed. In Fig. 4(a) (dashed blue lines) for λ = 1, τQSL fluctuationally decreases with less amplitude in comparison to Fig. 4(a) (solid red lines) and the fluctuations amplitude is also proportional to the degree of entanglement. Fig. 5(a) (dashed blue lines),
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Fig. 3. QSL time versus γ with θ = 1 and λ = 0.1. (a) The Werner state in (29) (solid red line) and the other Werner state in (31) (dashed blue line) are the initial states where p is chosen in the entangled region. As the non-Markovianity of the system grows up in terms of γ , the decrement rate of QSL time for the Werner state in (29) is proportional to the entanglement, while it is inversely proportional to the entanglement for the Werner state in (31). Also, the QSL time for the Werner state in (31) is always greater than the corresponding case for the Werner state in (29) in the entangled region. (b) The Horodecki state in (32) is the initial state where α is chosen in the entangled region (free and PPT entangled region). The QSL time for this state relies between the corresponding cases of Werner states presented in the left panel. In both panels (a) and (b), the actual driving time is τ = 1. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 4. QSL time versus γ with θ = 1 and λ = 1. (a) The Werner state in (29) (solid red line) and the Werner state in (31) (dashed blue line) are the initial states where p is chosen in the entangled region. By considering the Werner state in (31) as the initial state, the QSL time fluctuationally decreases with less amplitude than the respective case in (29) and the fluctuations amplitude is also proportional to the degree of entanglement for both of them. Also, the QSL time for the Werner state in (31) is always greater than the corresponding case for the Werner state in (29) in the entangled region. (b) The Horodecki state in (32) is the initial state where α is chosen in the entangled region (free and PPT entangled region). As the previous case, the QSL time for this state relies between the corresponding cases of Werner states presented in the left panel. In both panels (a) and (b), the actual driving time is τ = 1. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
for λ = 10, shows that the effect of two-qutrit entanglement on the QSL time for the initial state (31) is similar to the case presented in Fig. 5(a) (solid red lines). It is concluded that as the degree of nonMarkovianity is considerable in the system (see Fig. 2), the effects of two-qutrit entanglement of two local unitarily equivalent Werner states on the corresponding QSL times will be completely different.
N. Behzadi et al. / Annals of Physics 378 (2017) 407–417
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Fig. 5. QSL time versus γ with θ = 1 and λ = 10. (a) The Werner state in (29) (solid red line) and the Werner state in (31) (dashed blue line) are the initial states where p is chosen in the entangled region. Speed of evolution of the system is inversely proportional to the degree of entanglement for the Werner states in (29) and (31). Also, the QSL time for the Werner state in (31) is always greater than the corresponding case for the Werner state in (29) in the entangled region. (b) The Horodecki state in (32) is the initial state where α is chosen in the entangled region (free and PPT entangled region). The QSL time for this state relies also between the corresponding cases of Werner states presented in the left panel. In both panels (a) and (b), the actual driving time is τ = 1. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
In the next step, we take the well-known Horodecki state [24] as an another initial state for the two-qutrit V-type system as follows
ρ α (0) =
2 7
|ψ0 ⟩⟨ψ0 | +
α 7
σ+ +
5−α 7
σ− ,
(32)
where
σ+ = σ− =
1 3 1 3
(|01⟩⟨01| + |12⟩⟨12| + |20⟩⟨20|), (33)
(|10⟩⟨10| + |21⟩⟨21| + |02⟩⟨02|),
with 0 ≤ α ≤ 5. ρ α (0) is free entangled for 0 ≤ α < 1, bound entangled for 1 ≤ α < 2 and is separable for 2 ≤ α ≤ 3. Notice that ρ α is invariant under the simultaneous change of α −→ 5 − α and interchange of the parties so by these operations, ρ α is free entangled for 4 < α ≤ 5, bound entangled for 3 < α ≤ 4 and is separable for 2 ≤ α ≤ 3. It was shown in [29] that the entanglement of ρ α is decreased, in terms of parameter α , in the interval α ∈ [0, 2] and increased in the interval α ∈ [3, 5] symmetrically. Therefore, we only analysis effect of entanglement of ρ α (0) on the related QSL for α ∈ [0, 2]. In similar way discussed above, for the Horodecki state (32), we have shown the behaviors of τQSL in terms of γ for parameter values λ = 0.1, λ = 1 and λ = 10 (for fixed θ = 1) throughout the Figs. 3(b), 4(b) and 5(b) respectively. It is evident that the Horodecki state (32) is completely different from the previous Werner states such that we cannot obtain it from the Werner states by any local unitary operations. However, we observe that the effect of entanglement (free and PPT entanglement) of Horodecki state on the related QSL time has an intermediate behavior in comparison to the effect of entanglement of Werner states (29) and (31). Another point which should be noted in this paper is that though two Werner states (29) and (31) are local unitarily equivalent but the QSL time of (31) is always greater than the case of (29) (compare the solid red lines in Figs. 3(a), 4(a) and 5(a) with dashed blue lines in Figs. 3(a), 4(a) and 5(a) respectively). In other words, performing the local unitary operation (30) on the Werner state
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(29) leads to reduce the speed of evolution of the system. Also, the QSL time for the non-equivalent Horodecki state (32) relies between the QSL times obtained from the Werner states. At the end, by interchanging the atoms (atom1 atom2) in (31), another Werner state is obtained. By considering this state as an initial state, the obtained results are the same as the results of the state (31). To give a more clear physical reason for our results in this paper, we remember that the τQSL is the minimum time that it takes, under a certain Hamiltonian, for the initial state ρ0 to evolve to a perfectly distinguishable state ρτ . According to Ref. [30], the function 1/τQSL defines a natural notion of speed of evolution. In dynamics of open systems, in general, the amount of speed of evolution 1/τQSL (and so τQSL ) depends on the shape of the interaction between the system of concern and the corresponding environment, and also on the system’s initial state (the environment is in its corresponding vacuum state). We illustrate these points by examples from a typical single qubit open system. Consider that Ď ∗ the interaction of the qubit with its surrendering environment be as HI = k (gk σ+ bk + gk σ− bk ) with the system’s initial state as ρ0 = |0⟩⟨0| (the ground state of the qubit without excitation). It is clear from Ref. [25] that ρτ = ρ0 , i.e. the state ρ0 does not evolve so its speed of evolution is zero (or τQSL = ∞). Evidently, if ρ0 is not the ground state then the speed of evolution for this state is a non-zero value (so τQSL is finite) depending on the amount of excitation. Suppose that the interaction ∗ Ď between the qubit and the environment turns on to HI = k σz (gk bk + gk bk ). According to Ref. [31], D any diagonal state ρ0 has no speed of evolution and τQSL = ∞. So the states with non-zero off-diagonal terms (or non-zero coherence) has a non-zero speed of evolution and a finite τQSL . Now let us consider the two-qutrit V-type open system discussed in this paper. Let the state ρ0 = |00⟩⟨00| be the initial state (without excitation). Eq. (25) (which originates from the interaction Hamiltonian (5)) shows for this case that ρτ = ρ0 , i.e. there is no speed of evolution for this state and consequently, any initial state with non-zero excitation has a non-zero speed of evolution. Roughly speaking, by this argument the initial state entanglement also affects the speed of evolution as is clear, for example, from the dashed blue lines in Fig. 3(a) which shows that more entanglement leads to evolution with less speed. In other words, uncorrelated environments more speedup the evolution of a bipartite two-qutrit system whose given initial state has less amount of quantum correlation (entanglement). The other observation which relates non-trivially to the system–environment interaction and the system’s initial state is that the speed of evolution for the Werner states with the same degree of entanglement are different (compare the solid red lines in Figs. 3(a), 4(a) and 5(a) with dashed blue lines in Figs. 3(a), 4(a) and 5(a) respectively). Similar observation was reported for pure two-qubit entangled Bell states in Ref. [32]. The same reason also determines that for the Horodecki state which is not local unitarily equivalent with the Werner states, the respective speed of evolution or τQSL relies intermediately between those of the corresponding Werner states. In the next step, τQSL depends also on the effective spectral density J (ω) which is related to the spectral distribution of the environment and the system–environment coupling strength. In fact, Xu et al. showed that the mechanism of speedup of quantum evolution of an open system not only depends on the non-Markovianity (memory) effect but also on the population of initial state excitation, under a given driving time [33]. In this paper, we utilized the Lorentzian spectral density for J (ω) and observe that how non-Markovian effect speedups the evolution of the system. If J (ω) is taken to be Ohmic instead of Lorentzian, different results may be obtained in this way which can be considered as the subject of future research. 4. Conclusions In summary, we investigated effect of two-qutrit entanglement on the QSL time of a bipartite V-type atomic open system. It was shown that as the non-Markovianity in the system dynamics is considerable, the entanglement effect of two local unitarily equivalent Werner states are completely different. Also, we found out that the entanglement effect of non-equivalent Horodecki state on the related QSL time has an intermediate behavior relative to the previous cases. We observed that although the local unitary shift operator (30) does not change the two-qutrit entanglement but,
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in turn, may affect speed of evolution of the system considerably. Generally speaking, this point encourage us to examine the effect of other local unitary operators (local quantum gates) on the speed of evolution of systems in the future studies, which is of essential importance in quantum information processing. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33]
S. Deffner, E. Lutz, Phys. Rev. Lett. 105 (2010) 170402. V. Giovannetti, S. Lloyd, L. Maccone, Nat. Photonics 5 (2011) 222. S. Alipour, M. Mehboudi, A.T. Rezakhani, Phys. Rev. Lett. 112 (2014) 120405. A.W. Chin, S.F. Huelga, M.B. Plenio, Phys. Rev. Lett. 109 (2012) 233601. M. Tsang, New J. Phys. 15 (2013) 073005. T. Caneva, M. Murphy, T. Calarco, R. Fazio, S. Montangero, V. Giovannetti, G.E. Santoro, Phys. Rev. Lett. 103 (2009) 240501. G.C. Hegerfeldt, Phys. Rev. Lett. 111 (2013) 260501. G.C. Hegerfeldt, Phys. Rev. A 90 (2014) 032110. S. Lloyd, S. Montangero, Phys. Rev. Lett. 113 (2014) 010502. M. Gajdacz, K.K. Das, J. Arlt, J.F. Sherson, T. Opatrny, Phys. Rev. A 92 (2015) 062106. V. Mukherjee, A. Carlini, A. Mari, T. Caneva, S. Montangero, T. Calarco, R. Fazio, V. Giovannetti, Phys. Rev. A 88 (2013) 062326. J.D. Bekenstein, Phys. Rev. Lett. 46 (1981) 623. S. Lloyd, Phys. Rev. Lett. 88 (2002) 237901. L.B. Levitin, Internat. J. Theoret. Phys. 21 (1982) 299. M.-H. Yung, Phys. Rev. A 74 (2006) 030303. L. Mandelstam, I. Tamm, J. Phys. USSR 9 (1945) 249. N. Margolus, L.B. Levitin, Physica D 120 (1998) 188. M.M. Taddei, B.M. Escher, L. Davidovich, R.L. de Matos Filho, Phys. Rev. Lett. 110 (2013) 050402. A. del Campo, I.L. Egusquiza, M.B. Plenio, S.F. Huelga, Phys. Rev. Lett. 110 (2013) 050403. S. Deffner, E. Lutz, Phys. Rev. Lett. 111 (2013) 010402. Z. Sun, J. Liu, J. Ma, X. Wang, Sci. Rep. 5 (2015) 8444. Chen Liu, Zhen-Yu Xu, Shiqun Zhu, Phys. Rev. A 91 (2015) 022102. M. Horodecki, P. Horodecki, R. Horodecki, Phys. Rev. A 60 (1999) 1888. P. Horodecki, M. Horodecki, R. Horodecki, Phys. Rev. Lett. 82 (1999) 1056. H.P. Breuer, E.M. Laine, J. Piilo, Phys. Rev. Lett. 103 (2009) 210401. Wen-ju Gu, Gao-xiang Li, Phys. Rev. A 85 (2012) 014101. Xiaoguang Wang, Chang-Shui Yu, X.X. Yi, Phys. Lett. A 373 (2008) 58. M. Horodecki, P. Horodecki, Phys. Rev. A 59 (1999) 4206. M.A. Jafarizadeh, N. Behzadi, Y. Akbari, Eur. Phys. J. D 55 (2009) 197. I. Marvian, R.W. Spekkens, P. Zanardi, Phys. Rev. A 93 (2016) 052331. B. Bylicka, D. Chruscinski, S. Maniscalco, Sci. Rep. 4 (2014) 5720. C. Liu, Z.-Y. Xu, S. Zhu, Phys. Rev. A 91 (2015) 022102. Z.-Y. Xu, S. Luo, W.L. Yang, C. Liu, S. Zhu, Phys. Rev. A 89 (2014) 012307.
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Annals of Physics journal homepage: www.elsevier.com/locate/aop
Effect of two-qutrit entanglement on quantum speed limit time of a bipartite V-type open system N. Behzadi a,∗ , B. Ahansaz b , A. Ektesabi b , E. Faizi b a
Research Institute for Fundamental Sciences, University of Tabriz, Iran
b
Physics Department, Azarbaijan Shahid Madani University, Tabriz, Iran
article
abstract
info
Article history: Received 2 September 2016 Accepted 25 January 2017 Available online 31 January 2017 Keywords: Quantum speed limit Two-qutrit entanglement V-type atoms Non-Markovian dynamics
In the present paper, quantum speed limit (QSL) time of a bipartite V-type three-level atomic system under the effect of two-qutrit entanglement is investigated. Each party interacts with own independent reservoir. By considering two local unitarily equivalent Werner states and the Horodecki PPT state, as initial states, the QSL time is evaluated for each of them in the respective entangled regions. It is counterintuitively observed that the effect of entanglement on the QSL time driven from each of the initial Werner states are completely different when the degree of nonMarkovianity is considerable. In addition, it is interesting that the effect of entanglement of the non-equivalent Horodecki state on the calculated QSL time displays an intermediate behavior relative to the cases obtained for the Werner states. © 2017 Elsevier Inc. All rights reserved.
1. Introduction Quantum speed limit (QSL) is of particular interest and has attracted much attention in tremendous areas of quantum physics and quantum information, such as nonequilibrium thermodynamics [1], quantum metrology [2–5], quantum optimal control [6–11], quantum computation [12–14], and quantum communication [1,15]. In these fields, QSL time is a key concept, defined as the minimum
∗
Corresponding author. E-mail address: [email protected] (N. Behzadi).
http://dx.doi.org/10.1016/j.aop.2017.01.026 0003-4916/© 2017 Elsevier Inc. All rights reserved.
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evolution time in which a system evolves from an initial state to a target state. In the other hand, the QSL time sets a bound on the minimal time a system which needs to evolve between two distinguishable states, and it can be understood as a generalization of the time-energy uncertainty principle. For isolated systems, the QSL time under unitary evolution is determined by the Mandelstam–Tamm bound [16] and Margolus–Levitin bound [17]. Because of the inevitable interactions with the environments, quantum systems should be generally regarded as open systems. Therefore, it is necessary to determine a QSL time for open systems. Recently, Taddei et al. developed a method to investigate the QSL problem in open systems described by positive non-unitary maps by using of quantum Fisher information for time estimation [18]. While, for the case of the initial mixed states, a Hermitian operator is required to minimize the Fisher information in the extended system–environment space, which is generally a hard task. Afterwards, Campo et al. exploited the concept of relative purity to derive an analytical and computable QSL time for open systems undergoing a completely positive and trace preserving evolution [19]. It should be noted that relative entropy can perfectly make a distinction between an initial pure state and its target state, however it may fail to distinguish an initial mixed state to its target state. Also, a tight bound on the minimal evolution time of an arbitrarily driven open system has been formulated by Deffner and Lutz [20]. However, their time bound is derived from pure initial states and cannot be directly applied into the mixed initial states. Motivated by the recent studies above, the authors in Ref. [21] employed an alternative fidelity definition different from relative entropy in order to derive an easily computable QSL time bound for open systems whose initial states can be chosen as either pure or mixed states and this QSL time is applicable to either Markovian or non-Markovian dynamics. Alongside these efforts, several attentions have been devoted to investigate the effects of entanglement on the QSL time in multiqubit systems (for instance, see [18,22]). In this paper, we consider a bipartite system composed of two identical V-type three-level atoms as a two-qutrit system where each atom interacts with own independent reservoir. We use the QSL time bound obtained in [21] to evaluate the speed of evolution of the system under the effects of two-qutrit entanglement exhibited in the system. To this aim, we choose two local unitarily equivalent Werner states [23] and the Horodecki PPT (partial positive transpose) state [24] which is not equivalent to the Werner states, as initial states, and obtained the QSL time for each of them in the respective entangled regions. When the degree of non-Markovianity in the system dynamics is considerable, surprisingly distinct behaviors for the QSL time are obtained for the equivalent Werner states in the entangled regions. In other words, under the condition of considerable non-Markovianity, effect of the same two-qutrit entanglement related to equivalent Werner states are completely different on the respective QSL times. It is interesting to note that for the non-equivalent Horodecki state, the calculated QSL time has an intermediate behavior relative to the cases obtained for the Werner states. We see that, although any local unitary operation does not increases the degree of entanglement (in the case of Werner states discussed in the present paper, the introduced local unitary operator does not change the degree of entanglement) but, as illustrated in the text, it is observed that the related QSL time changes under the mentioned local unitary operator effect. The paper is organized as follows: In Section 2, we describe dynamics of a V-type three-level atomic system as an open system along with giving a quantitative description of non-Markovian behavior of that system. Also, we extend the dynamics of the V-type three-level system to a bipartite case. Section 3 is devoted to the obtaining of QSL time for the bipartite system and discussing about the connection between two-qutrit entanglement and the related QSL time. Finally, in Section 4, we present our conclusions. 2. Dynamics of V-type three-level open systems In this section, we study at first, Markovian and non-Markovian dynamics of a V-type three level atom, as a qutrit, coupled to a dissipative environment. Each atom has two excited states each of them spontaneously decays into ground states such that the respective dipole moments of transitions may have interaction with each other. We then extend the scheme into a two-qutrit case which each of them coupled to independent reservoirs as depicted in Fig. 1. For a V-type atom, under the interaction with the same environment, the two upper levels |2⟩ and |1⟩ are coupled with the ground one, |0⟩,
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Fig. 1. A system of two identical V-type three-level atoms each of them interacts with own independent reservoir.
with transition frequency ω2 and ω1 respectively. The Hamiltonian of the total system is given by H = HS + HE + HI = H0 + HI ,
(1)
where H0 =
2
ωl σ+l σ−l +
l =1
ωk bĎk bk ,
(2)
k
is the free Hamiltonian of the system and environment, and HI =
2 (glk σ+l bk + glk∗ σ−l bĎk ), l =l
(3)
k
is the interaction Hamiltonian. σ±l (l = 1, 2) are the raising and lowering operators of the levels |2⟩ and |1⟩ to the ground state |0⟩. The index k labels the different field modes of the environment with Ď frequencies ωk , creation and annihilation operators bk , bk and coupling constants glk . In the interaction picture the total system obeys the Schrödinger equation i
d dt
|ψ(t )⟩ = Hint |ψ(t )⟩,
(4)
where Hint =
2 (glk σ+l bk ei(ωl −ωk )t + glk∗ σ−l bĎk e−i(ωl −ωk )t ), l =1
(5)
k
and
|ψ(t )⟩ =
2
cl (t )|l⟩S ⊗ |0⟩E +
l =0
ck (t )|0⟩S ⊗ |1k ⟩E .
(6)
k
Since Hint |0⟩S ⊗ |0⟩E = 0 then c0 (t ) will be invariant in time, while the amplitudes c1 (t ) and c2 (t ) will not. The time variations of these amplitudes is obtained by solving a system of differential equations which is easily derived from the Schrödinger equation c˙l (t ) = −i
glk ck (t )ei(ωl −ωk )t ,
l = 1, 2,
(7)
k
c˙k (t ) = −i
2 l=1
glk∗ cl (t )e−i(ωl −ωk )t .
(8)
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By assuming ck (0) = 0, i.e. there is no photon in the initial state, the solution of the second equation is inserted into the first one to get a closed equation for cl (t ), c˙l (t ) = −
2
t
flm (t − t ′ )cm (t ′ )dt ′ ,
l = 1, 2.
(9)
0
m=1
The kernel in Eq. (9) can be expressed in terms of spectral density J (ω) of the reservoir as follows flm (t − t ′ ) =
t
dωJlm (ω)ei(ωl −ω)t −i(ωm −ω)t . ′
(10)
0
Jlm (ω) is chosen as Lorentzian distribution Jlm (ω) =
γlm λ2 , 2π (ω0 − ∆ − ω)2 + λ2 1
(11)
where ∆ is the detuning of the atomic transition frequency from the central frequency of the reservoir and λ is the spectral width of the coupling. γii = γi are the relaxation rates of the two upper levels, √ and γij = γi γj θ (i ̸= j and |θ | ≤ 1) are responsible for the spontaneously generated interference (SGI) between the two decay channels |2⟩ → |0⟩ and |1⟩ → |0⟩. θ depends on the relative angle between two dipole moment elements related to the mentioned transitions. θ = 0 means that the dipole moments of two transitions are perpendicular to each other and this is corresponding to the case that there is no SGI between two decay channels. On the other hand, θ = ±1 indicating that the two dipole moment are parallel or antiparallel, corresponding to the strongest SGI between two decay channels. If we take Laplace transform from Eq. (9), it becomes
√ λ γ2 − γ1 γ2 θ c˜2 (p) √ , (12) =− c˜1 (p) γ1 γ2 θ γ1 2(p + λ) ∞ where c˜l (p) = L[cl (t )] = 0 cl (t )e−pt dt (l = 1, 2), is the Laplace transform of cl (t ). By considering the subspace spanned by {|2⟩, |1⟩}, and doing the following unitary transformation q + γ1 − γ2 q − γ1 + γ2 − 2q 2q , U= (13) q − γ + γ q + γ1 − γ2 1 2
pc˜2 (p) − c2 (0) pc˜1 (p) − c1 (0)
2q
2q
on Eq. (12) and taking inverse Laplace transform we get c± (t ) = G± (t )c± (0), where c± (0) =
√1 2q
G± (t ) = e
(14)
√
√ c2 (0) q ± γ1 ∓ γ2 ∓ c1 (0) q ∓ γ1 ± γ2 . In Eq. (14),
−λt /2
cosh
d± t 2
+
λ d±
sinh
γ +γ ±q
d± t
2
,
(15)
with d± = λ2 − 2λγ± , γ± = 1 22 and q = (γ1 − γ2 )2 + 4γ1 γ2 θ 2 . It should be noted that the extended unitary transformation on the whole space of the system spanned by {|2⟩, |1⟩, |0⟩}, becomes as
q + γ1 − γ2
2q U = q − γ 1 + γ2 2q 0
q − γ1 + γ2 − 2q q + γ1 − γ2 2q 0
0
. 0 1
(16)
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By these Considerations, the density matrix of a V-type system at time t becomes
|G+ (t )|2 |c+ (0)|2 G∗ (t )c+ (0)∗ G− (t )c− (0) + ϱS (t ) = ∗ G∗+ (t )c+ (0)c0
∗ G+ (t )c+ (0)G∗− (t )c− (0) 2 |G− (t )| |c− (0)|2
G+ (t )c+ (0)c0∗ G− (t )c− (0)c0∗ 2 2 . 1 − |G+ (t )| |c+ (0)| −|G− (t )|2 |c− (0)|2
∗ G∗− (t )c− (0)c0
(17)
Clearly, the density matrix in Eq. (17) is, indeed, the time development of the state ϱS (0) = U |ψ(0)⟩⟨ψ(0)|U Ď in which |ψ(0)⟩ is the state (6) at time t = 0. It is very instructive to write the density matrix (17) in the Krauss representation. It can be easily obtained that 3
ϱS (t ) =
Ď
Ki ϱS (0)Ki ,
(18)
i =1
with
3
i=1
Ď
Ki Ki = I3 , where I3 is identity operator for the Hilbert space of a three-level system and
K1 =
G+ (t ) 0 0
0
0 0 , 1
G− (t ) 0
0 0
K2 =
1 − |G+ (t )|2
0 K3 = 0 0
0 0
0 0 ,
0
0
(19)
0 0
1 − |G− (t )|2
0 0 . 0
Also, the original density matrix is obtained as ρS (t ) = U Ď ϱS (t )U and the respective Krauss operators are Ki = U Ď Ki U (i = 1, 2, 3), with explicit forms
q + γ1 − γ2
q − γ1 + γ2
+ G− G+ 2q 2q K1 = 2 q − (γ1 − γ2 )2 (−G+ + G− )
(−G+ + G− ) G+
2q
q2 − (γ1 − γ2 )2
2q
q − γ1 + γ2 2q
0
K2 =
(1 − |G+ |2 ) 2q
K3 =
(1 − |G− |2 ) 2q
+ G−
q + γ1 − γ2 2q
0
√
0 0
√
0 0
q + γ1 − γ2
q − γ1 + γ2 Ď
√
q + γ1 − γ2
, 0
(20)
1
0 0 , 0
− q − γ1 + γ2 √
0
0 0
0 0
(21)
0 0 , 0
(22) Ď
hence, it is concluded that i=1 Ki Ki = I3 and ρS (t ) = i=1 Ki ρS (0)Ki . It should be noted that the initial state of a V-type atom, i.e. ρS (0), can be generally considered as a mixed state so its time development can also be obtained easily. In this paper, we consider the case in which the two upper atomic levels are degenerated and the atomic transitions are in resonant with the central frequency of the reservoir, i.e. ω1 = ω2 = ω0 and ∆ = 0. Under this consideration, we assume that the relaxation rates of two decay channels are equal, i.e. γ1 = γ2 = γ . Therefore, the statement in Eq. (15), takes a simple form. In order to study the non-Markovian dynamics of the V-type atom, exploiting a non-Markovian measure is inevitable. In Ref. [25], Breuer et al. introduced a measure of non-Markovianity, which is based on the reverse flow
3
3
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N. Behzadi et al. / Annals of Physics 378 (2017) 407–417
Fig. 2. Measure of non-Markovianity N versus γ with θ = 1 (solid line) and θ = 0.6 (dashed line).
of information from the reservoir back to the system as follows
N = max
ρ1,2 (0) η>0
η[t , ρ1,2 (0)]dt ,
(23)
where
η[t , ρ1,2 (0)] =
d dt
D[ρ1 (t ), ρ2 (t )],
(24)
indicates the changing rate of the trace distance of a pair of states denoted by√D[ρ1 (t ), ρ2 (t )] = 1 Tr |ρ1 (t )−ρ2 (t )| with the trace norm definition for an operator such as A as |A| = AĎ A. In general, in 2 order to calculate the N in Eq. (23), it is required to cover any pair of initial states. However, in Ref. [26] the optimized pair of initial states for a V-type three-level atom is obtained analytically. Fig. 2, shows the non-Markovian behavior of the system in terms of γ , λ and θ . When θ (|θ|) increases the SGI becomes more and more strong and this leads to the improvement of the non-Markovian behavior. Similar process takes place by increasing the system–environment coupling γ . But the non-Markovian measure decreases in terms of the spectral width of the coupling λ. Extending above method for dynamics of a system consists of two identical V-type atoms each of them independently interact with a Lorentzian environment is a trivial task. Let ρS (0) be defined as a density matrix of a two-qutrit V-type system on the 3 ⊗ 3 Hilbert space. So in this way, its time development at time t becomes
ρS (t ) =
3 k,l=1
Ď
Kk,l ρS (0)Kk,l ,
3 k,l=1
Ď
Kk,l Kk,l = I3 ⊗ I3 ,
(25)
where Kk,l = Kk ⊗ Kl . In the next section, we consider possible types of entanglement for the twoqutrit V-type system at initial time, and investigate their effects on the related QSL time throughout various non-Markovian dynamics. 3. Quantum speed limit time for a two-qutrit V-type system In this section, we introduce an easily computable QSL time for a two-qutrit V-type open system described in the previous section, whose initial states can be chosen as either pure or mixed. To
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this aim, we employ an alternative fidelity definition, as a distance measure of two quantum states introduced in [27] as F (ρ0 , ρt ) =
Tr (ρ0 , ρt ) Tr (ρ02 )Tr (ρt2 )
,
(26)
to calculate the QSL time bound, as derived in [21] (note that ρ(t ) ≡ ρt ). According to Ref. [21], the derived QSL time bound which is applicable to either Markovian and non-Markovian dynamics, is as the following form
|1 − F (ρ0 , ρτ )| , X (τ )
τ ≥ τQSL =
(27)
where X (τ ) =
2
τ
τ
Tr (ρ˙ t2 ) Tr (ρt2 )
0
dt ,
(28)
by denoting that ρ˙ t is the time derivative of the state ρt and τ is the actual driving time. At the first step, we consider the initial state of the two-qutrit system to be the following Werner state [23]
ρ0p (0) = (1 − p)
I3 ⊗ I3 9
+ p|ψ0 ⟩⟨ψ0 |,
(29)
where p ∈ [0, 1] and |ψ0 ⟩ = √1 (|00⟩ + |11⟩ + |22⟩). This state for p ≤ 1/4, is separable whereas for 3
p > 1/4 entangled. In fact, since it violate reduction criterion of separability so such Werner state is distillable [28]. By this consideration, in Figs. 3(a), 4(a) and 5(a), we have shown the τQSL (QSL time) in terms of γ for parameter values λ = 0.1, λ = 1 and λ = 10 (for fixed θ = 1) respectively. In Fig. 3(a) (solid red lines) for λ = 0.1, as the non-Markovianity of the system grows up in terms of γ , τQSL suddenly decreases. It is observed that the decrement rate of τQSL is proportional to the degree of entanglement of Werner state. In other words, more entanglement leads to evolution with more speed. In Fig. 4(a) (solid red lines) for λ = 1, τQSL fluctuationally decreases and the fluctuations amplitude is proportional to the degree of entanglement. Fig. 5(a) (solid red lines), for λ = 10, shows a smooth decreasing of τQSL in terms of γ and the speed of evolution of the system is always inversely proportional to the degree of entanglement. If we perform the following local unitary shift operator as 1 0 0
0 1 0
U=
0 0 1
⊗
0 0 1
1 0 0
0 1 , 0
(30)
on the state (29), we obtain the other Werner state as our another initial state as follows
ρ1p (0) = (1 − p)
I3 ⊗ I3 9
+ p|ψ1 ⟩⟨ψ1 |,
(31)
where |ψ1 ⟩ = √1 (|01⟩+|12⟩+|20⟩). Since the local shift operator (30) does not change the amount of 3
entanglement of the state (29) but we obtain a surprisingly different result for the QSL time when the initial state is chosen as (31). As observed from dashed blue lines in Fig. 3(a), QSL time for λ = 0.1 does not generally suffer sudden decrement in comparison to the previous one as shown by solid red lines in Fig. 3(a). In other words, in contrast to the result of the Werner state in (29), the decrement rate of τQSL is inversely proportional to the degree of entanglement of the Werner state in (31). Consequently, more entanglement leads to evolution with less speed. In Fig. 4(a) (dashed blue lines) for λ = 1, τQSL fluctuationally decreases with less amplitude in comparison to Fig. 4(a) (solid red lines) and the fluctuations amplitude is also proportional to the degree of entanglement. Fig. 5(a) (dashed blue lines),
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Fig. 3. QSL time versus γ with θ = 1 and λ = 0.1. (a) The Werner state in (29) (solid red line) and the other Werner state in (31) (dashed blue line) are the initial states where p is chosen in the entangled region. As the non-Markovianity of the system grows up in terms of γ , the decrement rate of QSL time for the Werner state in (29) is proportional to the entanglement, while it is inversely proportional to the entanglement for the Werner state in (31). Also, the QSL time for the Werner state in (31) is always greater than the corresponding case for the Werner state in (29) in the entangled region. (b) The Horodecki state in (32) is the initial state where α is chosen in the entangled region (free and PPT entangled region). The QSL time for this state relies between the corresponding cases of Werner states presented in the left panel. In both panels (a) and (b), the actual driving time is τ = 1. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 4. QSL time versus γ with θ = 1 and λ = 1. (a) The Werner state in (29) (solid red line) and the Werner state in (31) (dashed blue line) are the initial states where p is chosen in the entangled region. By considering the Werner state in (31) as the initial state, the QSL time fluctuationally decreases with less amplitude than the respective case in (29) and the fluctuations amplitude is also proportional to the degree of entanglement for both of them. Also, the QSL time for the Werner state in (31) is always greater than the corresponding case for the Werner state in (29) in the entangled region. (b) The Horodecki state in (32) is the initial state where α is chosen in the entangled region (free and PPT entangled region). As the previous case, the QSL time for this state relies between the corresponding cases of Werner states presented in the left panel. In both panels (a) and (b), the actual driving time is τ = 1. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
for λ = 10, shows that the effect of two-qutrit entanglement on the QSL time for the initial state (31) is similar to the case presented in Fig. 5(a) (solid red lines). It is concluded that as the degree of nonMarkovianity is considerable in the system (see Fig. 2), the effects of two-qutrit entanglement of two local unitarily equivalent Werner states on the corresponding QSL times will be completely different.
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Fig. 5. QSL time versus γ with θ = 1 and λ = 10. (a) The Werner state in (29) (solid red line) and the Werner state in (31) (dashed blue line) are the initial states where p is chosen in the entangled region. Speed of evolution of the system is inversely proportional to the degree of entanglement for the Werner states in (29) and (31). Also, the QSL time for the Werner state in (31) is always greater than the corresponding case for the Werner state in (29) in the entangled region. (b) The Horodecki state in (32) is the initial state where α is chosen in the entangled region (free and PPT entangled region). The QSL time for this state relies also between the corresponding cases of Werner states presented in the left panel. In both panels (a) and (b), the actual driving time is τ = 1. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
In the next step, we take the well-known Horodecki state [24] as an another initial state for the two-qutrit V-type system as follows
ρ α (0) =
2 7
|ψ0 ⟩⟨ψ0 | +
α 7
σ+ +
5−α 7
σ− ,
(32)
where
σ+ = σ− =
1 3 1 3
(|01⟩⟨01| + |12⟩⟨12| + |20⟩⟨20|), (33)
(|10⟩⟨10| + |21⟩⟨21| + |02⟩⟨02|),
with 0 ≤ α ≤ 5. ρ α (0) is free entangled for 0 ≤ α < 1, bound entangled for 1 ≤ α < 2 and is separable for 2 ≤ α ≤ 3. Notice that ρ α is invariant under the simultaneous change of α −→ 5 − α and interchange of the parties so by these operations, ρ α is free entangled for 4 < α ≤ 5, bound entangled for 3 < α ≤ 4 and is separable for 2 ≤ α ≤ 3. It was shown in [29] that the entanglement of ρ α is decreased, in terms of parameter α , in the interval α ∈ [0, 2] and increased in the interval α ∈ [3, 5] symmetrically. Therefore, we only analysis effect of entanglement of ρ α (0) on the related QSL for α ∈ [0, 2]. In similar way discussed above, for the Horodecki state (32), we have shown the behaviors of τQSL in terms of γ for parameter values λ = 0.1, λ = 1 and λ = 10 (for fixed θ = 1) throughout the Figs. 3(b), 4(b) and 5(b) respectively. It is evident that the Horodecki state (32) is completely different from the previous Werner states such that we cannot obtain it from the Werner states by any local unitary operations. However, we observe that the effect of entanglement (free and PPT entanglement) of Horodecki state on the related QSL time has an intermediate behavior in comparison to the effect of entanglement of Werner states (29) and (31). Another point which should be noted in this paper is that though two Werner states (29) and (31) are local unitarily equivalent but the QSL time of (31) is always greater than the case of (29) (compare the solid red lines in Figs. 3(a), 4(a) and 5(a) with dashed blue lines in Figs. 3(a), 4(a) and 5(a) respectively). In other words, performing the local unitary operation (30) on the Werner state
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(29) leads to reduce the speed of evolution of the system. Also, the QSL time for the non-equivalent Horodecki state (32) relies between the QSL times obtained from the Werner states. At the end, by interchanging the atoms (atom1 atom2) in (31), another Werner state is obtained. By considering this state as an initial state, the obtained results are the same as the results of the state (31). To give a more clear physical reason for our results in this paper, we remember that the τQSL is the minimum time that it takes, under a certain Hamiltonian, for the initial state ρ0 to evolve to a perfectly distinguishable state ρτ . According to Ref. [30], the function 1/τQSL defines a natural notion of speed of evolution. In dynamics of open systems, in general, the amount of speed of evolution 1/τQSL (and so τQSL ) depends on the shape of the interaction between the system of concern and the corresponding environment, and also on the system’s initial state (the environment is in its corresponding vacuum state). We illustrate these points by examples from a typical single qubit open system. Consider that Ď ∗ the interaction of the qubit with its surrendering environment be as HI = k (gk σ+ bk + gk σ− bk ) with the system’s initial state as ρ0 = |0⟩⟨0| (the ground state of the qubit without excitation). It is clear from Ref. [25] that ρτ = ρ0 , i.e. the state ρ0 does not evolve so its speed of evolution is zero (or τQSL = ∞). Evidently, if ρ0 is not the ground state then the speed of evolution for this state is a non-zero value (so τQSL is finite) depending on the amount of excitation. Suppose that the interaction ∗ Ď between the qubit and the environment turns on to HI = k σz (gk bk + gk bk ). According to Ref. [31], D any diagonal state ρ0 has no speed of evolution and τQSL = ∞. So the states with non-zero off-diagonal terms (or non-zero coherence) has a non-zero speed of evolution and a finite τQSL . Now let us consider the two-qutrit V-type open system discussed in this paper. Let the state ρ0 = |00⟩⟨00| be the initial state (without excitation). Eq. (25) (which originates from the interaction Hamiltonian (5)) shows for this case that ρτ = ρ0 , i.e. there is no speed of evolution for this state and consequently, any initial state with non-zero excitation has a non-zero speed of evolution. Roughly speaking, by this argument the initial state entanglement also affects the speed of evolution as is clear, for example, from the dashed blue lines in Fig. 3(a) which shows that more entanglement leads to evolution with less speed. In other words, uncorrelated environments more speedup the evolution of a bipartite two-qutrit system whose given initial state has less amount of quantum correlation (entanglement). The other observation which relates non-trivially to the system–environment interaction and the system’s initial state is that the speed of evolution for the Werner states with the same degree of entanglement are different (compare the solid red lines in Figs. 3(a), 4(a) and 5(a) with dashed blue lines in Figs. 3(a), 4(a) and 5(a) respectively). Similar observation was reported for pure two-qubit entangled Bell states in Ref. [32]. The same reason also determines that for the Horodecki state which is not local unitarily equivalent with the Werner states, the respective speed of evolution or τQSL relies intermediately between those of the corresponding Werner states. In the next step, τQSL depends also on the effective spectral density J (ω) which is related to the spectral distribution of the environment and the system–environment coupling strength. In fact, Xu et al. showed that the mechanism of speedup of quantum evolution of an open system not only depends on the non-Markovianity (memory) effect but also on the population of initial state excitation, under a given driving time [33]. In this paper, we utilized the Lorentzian spectral density for J (ω) and observe that how non-Markovian effect speedups the evolution of the system. If J (ω) is taken to be Ohmic instead of Lorentzian, different results may be obtained in this way which can be considered as the subject of future research. 4. Conclusions In summary, we investigated effect of two-qutrit entanglement on the QSL time of a bipartite V-type atomic open system. It was shown that as the non-Markovianity in the system dynamics is considerable, the entanglement effect of two local unitarily equivalent Werner states are completely different. Also, we found out that the entanglement effect of non-equivalent Horodecki state on the related QSL time has an intermediate behavior relative to the previous cases. We observed that although the local unitary shift operator (30) does not change the two-qutrit entanglement but,
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in turn, may affect speed of evolution of the system considerably. Generally speaking, this point encourage us to examine the effect of other local unitary operators (local quantum gates) on the speed of evolution of systems in the future studies, which is of essential importance in quantum information processing. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33]
S. Deffner, E. Lutz, Phys. Rev. Lett. 105 (2010) 170402. V. Giovannetti, S. Lloyd, L. Maccone, Nat. Photonics 5 (2011) 222. S. Alipour, M. Mehboudi, A.T. Rezakhani, Phys. Rev. Lett. 112 (2014) 120405. A.W. Chin, S.F. Huelga, M.B. Plenio, Phys. Rev. Lett. 109 (2012) 233601. M. Tsang, New J. Phys. 15 (2013) 073005. T. Caneva, M. Murphy, T. Calarco, R. Fazio, S. Montangero, V. Giovannetti, G.E. Santoro, Phys. Rev. Lett. 103 (2009) 240501. G.C. Hegerfeldt, Phys. Rev. Lett. 111 (2013) 260501. G.C. Hegerfeldt, Phys. Rev. A 90 (2014) 032110. S. Lloyd, S. Montangero, Phys. Rev. Lett. 113 (2014) 010502. M. Gajdacz, K.K. Das, J. Arlt, J.F. Sherson, T. Opatrny, Phys. Rev. A 92 (2015) 062106. V. Mukherjee, A. Carlini, A. Mari, T. Caneva, S. Montangero, T. Calarco, R. Fazio, V. Giovannetti, Phys. Rev. A 88 (2013) 062326. J.D. Bekenstein, Phys. Rev. Lett. 46 (1981) 623. S. Lloyd, Phys. Rev. Lett. 88 (2002) 237901. L.B. Levitin, Internat. J. Theoret. Phys. 21 (1982) 299. M.-H. Yung, Phys. Rev. A 74 (2006) 030303. L. Mandelstam, I. Tamm, J. Phys. USSR 9 (1945) 249. N. Margolus, L.B. Levitin, Physica D 120 (1998) 188. M.M. Taddei, B.M. Escher, L. Davidovich, R.L. de Matos Filho, Phys. Rev. Lett. 110 (2013) 050402. A. del Campo, I.L. Egusquiza, M.B. Plenio, S.F. Huelga, Phys. Rev. Lett. 110 (2013) 050403. S. Deffner, E. Lutz, Phys. Rev. Lett. 111 (2013) 010402. Z. Sun, J. Liu, J. Ma, X. Wang, Sci. Rep. 5 (2015) 8444. Chen Liu, Zhen-Yu Xu, Shiqun Zhu, Phys. Rev. A 91 (2015) 022102. M. Horodecki, P. Horodecki, R. Horodecki, Phys. Rev. A 60 (1999) 1888. P. Horodecki, M. Horodecki, R. Horodecki, Phys. Rev. Lett. 82 (1999) 1056. H.P. Breuer, E.M. Laine, J. Piilo, Phys. Rev. Lett. 103 (2009) 210401. Wen-ju Gu, Gao-xiang Li, Phys. Rev. A 85 (2012) 014101. Xiaoguang Wang, Chang-Shui Yu, X.X. Yi, Phys. Lett. A 373 (2008) 58. M. Horodecki, P. Horodecki, Phys. Rev. A 59 (1999) 4206. M.A. Jafarizadeh, N. Behzadi, Y. Akbari, Eur. Phys. J. D 55 (2009) 197. I. Marvian, R.W. Spekkens, P. Zanardi, Phys. Rev. A 93 (2016) 052331. B. Bylicka, D. Chruscinski, S. Maniscalco, Sci. Rep. 4 (2014) 5720. C. Liu, Z.-Y. Xu, S. Zhu, Phys. Rev. A 91 (2015) 022102. Z.-Y. Xu, S. Luo, W.L. Yang, C. Liu, S. Zhu, Phys. Rev. A 89 (2014) 012307.