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Quantifying fidelity and purity for some quantum states in dissipative channel by virtue of trace rule in the phase space
Optik 124 (2013) 1814–1819
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Quantifying fidelity and purity for some quantum states in dissipative channel by virtue of trace rule in the phase space夽 Xue-xiang Xu a,∗ , Liang-sheng Xiong b , Hong-chun Yuan c , Li-yun Hu a , Zhen Wang d , Shuai Wang d , Hong-yi Fan d a
College of Physics and Communication Electronics, Jiangxi Normal University, Nanchang 330022, China Yichun Vocational and Technical College, Yichun 336000, China c College of Optoelectronic Engineering, Changzhou Institute of Technology, Changzhou 213002, China d Department of Physics, Shanghai Jiao Tong University, Shanghai 200030, China b
a r t i c l e
i n f o
Article history: Received 27 December 2011 Accepted 22 May 2012
PACS: 03.65.Yz 05.30.-d 42.50.Ar
a b s t r a c t By virtue of trace rule in the phase space, we investigate the time evolution of fidelity and purity for some quantum states (including Fock state, coherent state, squeezed state, and thermal state) in dissipative channel. All the relations are calculated by directly applying the Wigner function evolution formula in channel. The method is more convenient and concise than other method such as density operators’ trace. It is found that the dissipation induces all the quantum state eventually to vacuum state (a pure Gaussian state). © 2012 Elsevier GmbH. All rights reserved.
Keywords: Fidelity Purity Dissipative channel Trace rule in the phase space
1. Introduction One of the major topics in Quantum Statistical Mechanics is the evolution of (pure or mixed) quantum states. In recent years, much attention has been paid to the investigation of evolution between pure states and mixed states [1–3]. As we all know, physical systems are never perfectly isolated from their environment. Therefore some evolution, which is described by a master equation, usually happens when a system is immersed in a thermal environment, or a signal (a quantum state) passes through a quantum channel. Many physicists have been carried out research in this field. The evolution of fidelity of generic bosonic fields in noisy channels has been addressed in Ref. [4]. Paris et al. [5] presented a systematic study for Gaussian states of single-mode continuous variable system and considered the dynamics of purity in noisy channels.
夽 This project was supported by the National Natural Science Foundation of China (Nos. 11047133 and 11174114),the Natural Science Foundation of Jiangxi Province of China (No. 2010GQW0027), the Key Programs Foundation of Ministry of Education of China (No. 210115) and the Research Foundation of the Education Department of Jiangxi Province of China (Nos. GJJ12171 and GJJ11390). ∗ Corresponding author. Tel.: +86 791 88120370; fax: +86 791 88120370. E-mail address: [email protected] (X.-x. Xu). 0030-4026/$ – see front matter © 2012 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2012.05.047
They also studied the evolution of purity, entanglement, and total correlations of general two-mode continuous variable Gaussian states in arbitrary uncorrelated Gaussian environments [6]. The evolution of cat-like states in general Gaussian noisy channels were also studied [7]. Alternately, description of evolution of density matrices can be replaced by its Wigner function (WF)’s evolution in phase space [8,9]. Recently, a F-P equation for WF evolution in noisy self-Kerr medium is presented [10]. Moreover, a WF evolution formula in the self-Kerr medium with photon loss is derived [11]. The WF in phase space is very useful in studying quantum dynamics and is a powerful tool for investigating such nonclassical effect. In this paper, we study the decoherence of several initial quantum states and focus our attention on the evolution of the fidelity and purity in dissipative channel. The rate of decoherence is quantified by the evolution of fidelity and purity. The fidelity reflects that how close two states are to each other. While the purity, the degree of mixedness of a quantum state, is fundamental in distinguishing a pure state from a mixed one [12]. When an initial quantum state 0 passes through a dissipative channel and evolutes into (t) after some time, we define the fidelity F(t) = Tr(0 (t)),
(1)
X.-x. Xu et al. / Optik 124 (2013) 1814–1819
and the purity 2
(t) ≡ Tr[ (t)],
(2)
to determine decoherence. Obviously, it is difficult to evaluate operators’ traces directly. However, the introduction of the WF allows to evaluate operators’ traces as integrals in the phase space, i.e. trace rule in the phase space for single-mode case [13]
d2 ˛W1 (˛)W2 (˛),
Tr[1 2 ] =
(3)
which bring much convenience to calculating the fidelity and the purity. That is, the overlap between two operators 1 and 2 can be transformed into the overlap of the corresponding Wigner functions W1 (˛) and W2 (˛). The paper is structured as follows. In Section 2, we briefly review the theory of WF and its evolution formula in dissipative channel, then we define the fidelity and the purity in dissipative channel. In Section 3 we provide a detailed study of time evolution of fidelity and purity for several initial quantum states of major interest such as coherent state, Fock state, squeezed state and thermal state. Some concise analytic expressions have been represented. Through plotting the graph, we discuss in detail their evolution of trends. It is found that all the quantum states in the channel will loss its initial character due to the factor of environment and evolve into vacuum state at last. 2. Basic theory Let us firstly review the theory of WF and its evolution formula in dissipative channel, then we define the fidelity and purity in dissipative channel. All the relations is very important in our following work. 2.1. Wigner function and its evolution formula in dissipative channel The Wigner function (WF) [14,15] was first introduced by Wigner in 1932 to calculate quantum correction to a classic distribution function of a quantum-mechanical system. It now becomes a very popular tool to study the nonclassical properties of quantum states. It is well known that WFs are quasiprobability distributions because it may be negative in phase space. The presence of negativity of the WF for an optical field is a signature of its nonclassicality [16]. For a single-mode quantum system , the WF in the coherent state representation |z is given by [17]
2
2e2|˛| W (˛, ˛ ) = ∗
d2 z ∗ ∗ − z||ze−2(z˛ −z ˛) ,
(4)
√ where ˛ = q + ip/ 2. Using this equation, we can derive the analytic expression for a given quantum state in a good manner. When the quantum state passes through a dissipative channel, the time evolution of its density operator can be described by the following master equation [18,19] d(t) = [2a(t)a† − a† a(t) − (t)a† a], dt
(5)
where a and a† are creation and annihilation operator satisfying [a, a† ] = 1, and is the dissipative coefficient. By virtue of the entangled state representation, the authors Fan and Hu find the density operator (t) for the dissipative channel can be expressed in the Kraus operator sum representation as follows [20,21] (t) =
∞ Tn n=0
n!
†
†
e−ta a an 0 a†n e−ta a ,
(6)
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where T = 1 − e−2t . As the asymptotic case, we find from Eq. (6) that, when t→ ∞, (t) must evolve towards a desired steady state (vacuum state |00|, which is a Gaussian state and is related to the character of channel) for arbitrary initial quantum state. This is also the so-called environment engineering for pure Gaussian state preparation [22]. Obviously, Eq. (6) is an infinite summation in form in itself. If using density operators’ trace to evaluate the fidelity and the purity, it will bring much difficulty. Once the WF at arbitrary time in channel W (˛, ˛∗ ; t) is known, we can convenient to obtain the results by virtue of trace rule in the phase space. By using the thermal field dynamics theory and thermal entangled state representation, Fan and Hu, also the authors, have derived the analytic formula of the time evolution of WF at time t expressed as the following convolution [23] W (˛, ˛∗ ; t) =
2 T
d2 z −2/T |˛−ze−t |2 W (z, z ∗ ; 0), e
(7)
where W(z, z∗ ; 0) is the initial WF. Eq. (7) is just the evolution formula of WF of single mode quantum state in dissipative channel. By observing Eq. (7), we see that when t → 0, T → 0,
2 2 exp − |˛ − ze−t |2 T T
→ ı(˛ − z)ı(˛∗ − z ∗ ),
so W(˛, ˛∗ ; t) → W(˛, ˛∗ ; 0) as expected. Thus the WF at any time can be obtained by performing the integration when the initial WF is known. 2.2. Trace rule calculated by Wigner function in phase space The quantum state 0 evolutes into quantum state (t) after some time in the channel. Now, we face two problems: (1) How to measure how close the initial state and final state are to each other; (2) How to measure the degree of mixedness of the quantum state at any time. 2.2.1. Evolution of fidelity in dissipative channel One widely accepted measure is the fidelity to measure how (t) is close to 0 . The fidelity is defined as F(t) = Tr(0 (t)). When the two states are the same, the fidelity is 1 with a perfect overlap; while when the two states are orthogonal to each other, it is 0. According to trace rule in the phase space, F(t) can be reformed as
W0 (˛, ˛∗ ; 0)W (˛, ˛∗ ; t)d2 ˛.
F(t) =
(8)
as fidelity in dissipative channel. 2.2.2. Evolution of purity in dissipative channel Similarly, the purity is fundamental in distinguishing a pure state from a mixed one and determining the degree of mixedness of the quantum state (t). Thus we define (t) ≡ Tr[2 (t)] as purity in dissipative channel. For continuous variable systems one has 0 < 1. According to trace rule in the phase space, (t) can be rewritten as
(t) =
|W (˛, ˛∗ ; t)|2 d2 ˛.
(9)
It is easy to see that as long as we know W0 (˛, ˛∗ ; 0) and W(˛, ˛∗ ; t), we can evaluate the explicit analytical expressions of fidelity and purity from Eqs. (8) and (9). 3. Several cases for different initial quantum states In this section, we deal with the problem how to determine the fidelity and the purity of some quantum states in dissipative
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channel. These states include Fock state (FS), coherent state (CS), squeezed state (SS), and thermal state (TS). As we will see later, the time evolution of the fidelity and the purity can be expressed as a function of the initial observable parameters of the input state and of the character of the channel.
1.0
m=0
0.8
Ft
3.1. Fock state (FS)
0.4
m=2
0.2
The photon number states, also FSs, are easy to comprehend and they are the basic states of the quantum theory of light. They form a complete set for the states of a single mode and they are easy to manipulate in calculations of their quantum optical properties. The FS form the natural starting point for a treatment of single mode light [24]. Now we consider the case of Fock state (FS) |m as initial quantum state channel. Using Eq. (4) and noticing |z = ∞in dissipative n 2 e−|z| /2 n=0 √z |n, we get the WF of FS
m=5
0.0
m=8
0.0
0.2
2
2(−1)m e2|˛| WFS (˛, ˛ ; 0) = m!
1.0
=
2 2(−1)m e2|˛|
m!
m m
d2 z 2 ∗ ∗ (|z|2 )m e−|z| −2z˛ +2z ˛
∂ ∂ ×
d2 z
2 +(−2˛∗ +)z+(2˛+)z ∗
e−|z|
0.6
0.6
m=8
0.0
0
1
2
(11)
(12)
We retain Eq. (12) in the present form in order to evaluate the following content conveniently. Substituting Eq. (12) into Eq. (7) and treating as above, we get the time evolution of WF WFS (˛, ˛∗ ; t) =
m
2(−1) 2 −t −t ∗ m m ∂ ∂ e−2|˛| +2e ˛−2e ˛ m! ×e
(2e−2t −1)
|==0 ,
(13)
−2t 1 m m m m ∂ ∂ ∂ ∂ e(e −1)f f m!m! i i f f
×e
−e−t f i −e−t f i
|i =i =f =f =0 ,
1 m!m!
(14)
m
∞ (e−2t − 1)j (−e−t )l (−e−t )h
j!l!h!
j,l,h=0 m m
m
j+l j+h
× ∂i ∂i ∂f ∂f hi il f f
|i =i =f =f =0 = m!m!
∞ (e−2t − 1)j (−e−t )l (−e−t )h j,l,h=0
j!l!h!
× ım,j+l ım,j+h ım,l ım,h = e−2mt .
5
as one expected. Since channel damps the initial state to a vacuum state. Substituting Eq. (13) into Eq. (9) and employing the similar procedure above, we obtain the time evolution purity of FS in dissipative channel, FS (t) =
1 −2t −2t m m m m ∂ ∂ ∂ ∂ e(e −1)1 1 +(e −1)2 2 m!m! 1 1 2 2 −2t −e−2t 1 2 2 1
(15)
This is a new concise result. In particular, when t → 0, F(t) → 1, since |m is a pure state. It is easy to see that, as t→ ∞, F(t) → 0,
|1 =1 =2 =2 =0 = m!m!
(e−2t − 1)j+l (−e−2t )h+r ∞
j!l!h!r!
j,l,h,r=0
=
× ım,j+h ım,j+r ım,l+r ım,h+l .
m m!m!(e−2t − 1)2j (e−2t )2m−2j j=0
where the subscript i and f is to differentiate the initial and final states. Furtherexpanding all the exponential items into series and noticing ∂xn xm = m!ım,n (where ım,n denotes Kroncher symx=0 bol), Eq. (14) reduces to FFS (t) =
4
Fig. 1. Evolution of (a) fidelity and (b) purity for Fock state |m in dissipative channel with m = 0, 1, 2, 5, 8.
× e−e
It is clear from Eq. (13) that when t→ ∞, WFS (˛, ˛∗ ; ∞) → 2 (2/)e−2|˛| , i.e. corresponding to vacuum state. Substituting Eqs. (12) and (13) into Eq. (8), we derive the time evolution fidelity of FS in dissipative channel yielding FFS (t) =
3 t
it leads to 2(−1)m m m −2|˛|2 +2˛−2˛∗ + WFS (˛, ˛ ; 0) = ∂ ∂ e |==0 . m!
1.4
(b)
After employing the integration formula [25],
∗
1.2
0.2
|==0 ,
1 −
d2 z |z|2 + z+ z∗ = − e , Re () < 0, e
1.0
m=1 m=2 m=5
0.4
(10)
0.8
m=0
0.8 t
0.4
t
n!
∗
(a)
m=1
0.6
= e−4mt
j!j!(m − j)!(m − j)! 2 F1 (−m, −m; 1; (e
2t
− 1)2 )
(16)
where 2 F1 (− m, − m ; 1 ; (e2t − 1)2 ) is a hypergeometric function[26]. In particular, when t → 0, (t) → 1. While t→ ∞, there is also (t) → 1. Eqs. (15) and (16) reflect that FFS (t) and FS (t) are only relative to the initial observable parameters m of the input state and of the character of the channel. The results of the numerical analysis of the evolution of fidelity and purity of FSs in dissipative channel for different number m are reported in Fig. 1. The fidelity is a monotonically decreasing function of t and tends to zero as the time increase. Moreover, the higher the number m, the faster the decline (see Fig. 1(a)). But for purity, it decreases from the maximum 1 to a minimum, then rose to the maximum 1 except m = 0. Moreover, the higher the number m, the smaller the minimum (see Fig. 1(b)). 3.2. Coherent state (CS) There is a wide variety of possible superposition states of FS. But one kind, the CS is of particular importance in the practical
X.-x. Xu et al. / Optik 124 (2013) 1814–1819
(a)
1.0
1.0
0.8
0.6
Ft
Ft
0.8
0.6 0.4
0.4
(a)
0.2
0.2 0.0
1817
0.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0
0.1
0.2
0.3
t
(b)
0.8
t
0.8 t
0.5
1.0
1.0
0.6
0.6 0.4
0.4
(b)
0.2
0.2 0.0
0.4
t
0.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
t
t Fig. 2. Evolution of (a) fidelity and (b) purity for coherent state |ˇ in dissipative channel with |ˇ| = 0, 0.5, 0.8, 1, 2 (from top to bottom).
Fig. 3. Evolution of (a) fidelity and (b) purity for squeezed state S(r)|0 in dissipative channel with r = 0, 0.3, 0.8, 1, 2 (from top to bottom).
3.3. Squeezed state (SS) applications of the quantum theory of light [27]. Here we consider the case of coherent state |ˇ in dissipative channel. From Eq. (4), we readily see WCS (˛, ˛∗ ; 0) =
2 exp(−2|˛ − ˇ|2 ),
(17)
Here, we discuss the case of SS S(r)|0 [28] in dissipative channel, †2
r
2
where S(r) = e 2 (a −a ) is a squeezed operator with the squeezing parameter r. Employing the similar procedure, we have 2 −2|˛ cosh r+˛∗ sinh r|2 e ,
WSS (˛, ˛∗ ; 0) =
(21)
and Then substituting Eq. (17) into Eq. (7), the time evolution of WF is obtained, i.e. WCS (˛, ˛∗ ; t) =
2 exp(−2|˛ − e−t ˇ|2 ).
2 −2t (cosh 2r−1))/M|˛|2 √ e−(2+2e M
WSS (˛, ˛∗ ; t) =
× e−((e
(18)
M = (e−2t
Comparing Eq. (18) with Eq. (17), we find that the coherent state |ˇ remains as a pure coherent state |e−t ˇ only having an exponential damping in the amplitude. Moreover, we easily know 2 WCS (˛, ˛∗ ; ∞) = 2 e−2|˛| , which is just the WF of vacuum state. Substituting Eqs. (17) and (18) into Eq. (8), we obtain the time evolution fidelity of CS in dissipative channel
−2t
sinh 2r)/M)(˛2 +˛∗2 )
+ T cosh 2r)2
− T2
,
(22)
sinh 2 2r .
with When t→ ∞, 2 WSS (˛, ˛∗ ; ∞) → (2/)e−2|˛| , the WF of vacuum state. Substituting Eqs. (21) and (22) into Eq. (8), we have the time evolution fidelity of SS in dissipative channel FSS (t) =
2 M(A2 − B2 )
,
(23)
The result is also identical to that calculated by. Similarly, we have the time evolution purity of CS
with A = (1 + e−2t (cosh 2r − 1))/M + cosh 2r and B = ((e−2t /M) + 1) sinh 2r. When t→ ∞, M → 1, A → 1 + cosh 2r, B → sinh 2r, it leads to FSS (t) → 2/ (cosh 2r + 1). Substituting Eq. (22) into Eq. (9), we obtain the time evolution purity of SS in dissipative channel
CS (t) = 1,
SS (t) =
FCS (t) = exp[−(e−t − 1)2 |ˇ|2 ].
(19)
(20)
which means that the coherent state remains the pure character for all time in dissipative channel. In Fig. 2, the behavior of fidelity and purity for CS in dissipative channel are presented in detail. Obviously, when t→ ∞, 2 FCS (t) → e−|ˇ| . For different amplification |ˇ| of CS, the fidelity 2 decreases and eventually reach a steady value e−|ˇ| as the time increase. Moreover, the bigger |ˇ|, the smaller the fidelity. But the purity always remains 1 for any time.
1 C2
− e−4t sinh2 2r
(24)
with C = 1 + e−2t (cosh 2r − 1). Eqs. (23) and (24) reflect that FSS (t) and SS (t) are only relative to the initial squeezing parameters r of the SS and of the character of the channel. The behavior of fidelity and purity for SS in dissipative channel are presented in Fig. 3. The fidelity of SS is a monotonically decreasing function of t and tends to a steady 2/ (cosh 2r + 1) for different r as the time increase. Morevalue over, the bigger the squeezing parameter r, the faster the decline
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X.-x. Xu et al. / Optik 124 (2013) 1814–1819
Then substituting Eq. (27) into Eq. (7) yields the Wigner function
1.0
∗
WTS (˛, ˛ ; t) =
0.8
Ft
0.6
(a)
0.0 0.0
0.5
1.0
1.5
2.0
2.5
FTS (t) =
3.0
t
(b)
t
0.2 0.0
0.5
1.0
1.5
2.0
2.5
3.0
t Fig. 4. Evolution of (a) fidelity and (b) purity for thermal state in dissipative channel with n = 0, 0.1, 0.2, 0.5, 1 (from top to bottom).
(see Fig. 3(a)). But for purity, it decreases from the maximum 1 to a minimum, then rose to the maximum 1 except r = 0. Moreover, the bigger the squeezing parameter r, the smaller the minimum (see Fig. 3(b)). 3.4. Thermal state (TS) As a special case, we discuss a mixed state – TS th [1–3,24] in dissipative channel. TS is the equilibrium state for a field coupled to a reservoir at finite temperature . This state is commonly discussed in statistical and thermal physics courses because it is used to derive the Planck radiation law. The distribution function which describes a thermal state’s statistics is similar to the Maxwell–Boltzmann dis† tribution th = (1 − e−ω/k )e−ωa a/k (k is Boltzmann constant). The average photon number n = Tr(tha† a) of TS is a well-known result which is often termed the Planck distribution function, i.e. n = (1 − eω/k )−1 . Alternately, the density operator th of thermal state in Fock representation is l
n
(n + 1)
l+1
|ll|
(25)
√ where |l is the Fock state. Due to |l = a†l / l!|0 and |00| =: † e−a a :(the symbol : :denotes normally ordering), Eq. (25) can be rewritten as 1 1 † † : e(n/(n+1)−1)a a := ea a ln n/(n+1) , n+1 n+1
where we use the following operator exp [− a† a] = : exp [− (1 − e− )a† a]: [29,30]. Substituting Eq. (26) into Eq. (4), we have 2 WTS (˛, ˛ ; 0) = exp (2n + 1) ∗
,
(28)
2|˛|2 − 2n + 1
(29)
identity
(26) formula
,
(30)
In particular, when t → 0, (t) → 1/(2n + 1). While t→ ∞, there is also (t) → 1. Eqs. (29) and (30) reflect that FTS (t) and TS (t) are only relative to the initial average photon number n of the TS and of the character of the channel. The behavior of fidelity and purity for SS in dissipative channel are presented in Fig. 4. From Fig. 4, we find that for TS with different n, the initial fidelity and purity is different and all equal to 1/ (2n + 1). As time increase, the fidelity reach a fixed value 1/ (n + 1) , while the purity tend to the same value 1.
0.6 0.4
th =
1 n(1 + e−2t ) + 1
TS (t) = 1/(2ne−2t + 1).
0.8
l=0
In particular, when t → 0, F(t) → 1/(2n + 1). While t→ ∞, there is also F(t) → 1/(n + 1). Substituting Eq. (28) into Eq. (9), we have the time evolution purity of TS in dissipative channel
1.0
∞
exp
2|˛|2 − 2ne−2t + 1
2
0.2
th =
(2ne−2t + 1)
Obviously when t→ ∞, WTS (˛, ˛∗ ; ∞) → (2/)e−2|˛| , the WF of vacuum state. Substituting Eqs. (27) and (28) into Eq. (8), we obtain the time evolution fidelity of TS in dissipative channel
0.4
0.0
2
(27)
4. Conclusions In summary, we have studied in detail the fidelity and purity for some quantum states in dissipative channel. These states include Fock state, coherent state, squeezed state, and thermal state. We derived the exact general relations of fidelity and purity. Such analytical expressions, supported by direct numerical analysis, clearly show the quantum dynamics of decoherence. Our skill lies in using trace rule in the phase space instead of operators’ traces. Firstly, we calculate the initial and final WF and then obtain fidelity and purity. The method is effective and convenient. The results of our analysis, together with the above discussions, naturally lead us to formulate the following general conjecture: for any quantum state, either pure or mixed, will be induced into pure Gaussian state (vacuum state) at last in the dissipative channel. This is also the basic idea of the so-called environment engineering for pure Gaussian state preparation. References [1] W.H. Louisell, Quantum Statistical Properties of Radiation, Wiley, New York, 1973. [2] H.J. Carmichael, Statistical Methods in Quantum Optics 1: Master Equations and Fokker–Planck Equations, Springer-Verlag, Berlin, 1999. [3] H.J. Carmichael, Statistical Methods in Quantum Optics 2: Non-classical Fields, Springer-Verlag, Berlin, 2008. [4] L.M. Duan, G.C. Guo, Influence of noise on the fidelity and the entanglement fidelity of states, Quantum Semiclass. Opt. 9 (1997) 953–959. [5] M.G.A. Paris, F. Illuminati, A. Serafini, S.D. Siena, Purity of Gaussian states: measurement schemes and time evolution in noisy channels, Phys. Rev. A 68 (2003) 012314. [6] A. Serafini, F. Illuminati, M.G.A. Paris, S.D. Siena, Entanglement and purity of two-mode Gaussian states in noisy channels, Phys. Rev. A 69 (2004) 022318. [7] A. Serafini, S.D. Siena, F. Illuminati1, M.G.A. Paris, Minimum decoherence catlike states in Gaussian noisy channels, J. Opt. B: Quantum Semiclass. Opt. 6 (2004) S591–S596. [8] W.P. Schleich, Quantum Optics in Phase Space, Wiley-VCH, Berlin, 2001. [9] M. Hillery, R.F. O’Connell, M.O. Scully, E.P. Wigner, Distribution functions in physics: fundamentals, Phys. Rep. 106 (1984) 121–167. [10] M. Stobinska, G.J. Milburn, K. Wodkiewicz, Wigner function evolution of quantum states in the presence of self-Kerr interaction, Phys. Rev. A 78 (2008) 013810. [11] L.Y. Hu, Z.L. Duan, X.X. Xu, Z.S. Wang, Wigner function evolution in a self-Kerr medium derived by entangled state representation, J. Phys. A: Math. Theor. 44 (2011) 195304.
X.-x. Xu et al. / Optik 124 (2013) 1814–1819 [12] M.S. Kim, Recent developments in photon-level operations on traveling light fields, J. Phys. B: At. Mol. Opt. Phys. 41 (2008) 133001. [13] A. Ferraro, S. Olivares, M.G.A. Paris, Gaussian states in quantum information, Bibliopolis, Napoli, 2005. [14] E. Wigner, On the quantum correction for thermodynamic equilibrium, Phys. Rev. 40 (1932) 749–759. [15] X.X. Xu, H.C. Yuan, L.Y. Hu, H.Y. Fan, Non-Gaussianity of photon-added-thensubtracted squeezed vacuum state, Optik 123 (2012) 16–20. [16] A. Kenfack, K. Zyczkowski, Negativity of the Wigner function as an indicator of nonclassicality, J. Opt. B: Quantum Semiclass. Opt. 6 (2004) 396–404. [17] H.Y. Fan, H.R. Zaidi, Application of IWOP technique to the generalized Weyl correspondence, Phys. Lett. A 124 (1987) 303–307. [18] C. Garder, P. Zoller, Quantum Noise, Springer, Berlin, 2000. [19] M.O. Scully, M.S. Zubairy, Quantum Optics, Cambridge University Press, 1997. [20] L.Y. Hu, H.Y. Fan, Infinite-dimensional Kraus operators for describing amplitude-damping channel and laser process, Opt. Commun. 282 (2009) 932–935. [21] H.Y. Fan, L.Y. Hu, Operator-sum representation of density operators as solutions to master equations obtained via the entangled state approach, Mod. Phys. Lett. B 22 (2008) 2435–2468.
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[22] K. Koga, N. Yamamoto, Dissipation-induced pure Gaussian state, Phys. Rev. A 85 (2012) 022103. [23] L.Y. Hu, H.Y. Fan, Time evolution of Wigner function in laser process derived by entangled state representation, Opt. Commun. 282 (2009) 4379–4383. [24] R. Loudon, The Quantum Theory of Light, Oxford University Press, 2001. [25] R.R. Puri, Mathematical Methods of Quantum Optics, Springer Verlag, Berlin, 2001. [26] E.D. Rainville, Special Function, MacMillan, New York, 1960. [27] R.J. Glauber, Coherent and Incoherent States of the Radiation Field, Phys. Rev. 131 (1963) 2766–2788. [28] J.C. Garrison, R.Y. Chiao, Quantum Optic, Oxford University Press, 2008. [29] H.Y. Fan, H.L. Lu, Y. Fan, Newton–Leibniz integration for ket-bra operators in quantum mechanics and derivation of entangled state representations, Ann. Phys. 321 (2006) 480–494. [30] H.Y. Fan, Newton-Leibniz integration for ket-bra operators in quantum mechanics (V)—deriving normally ordered bivariate-normal-distribution form of density operators and developing their phase space formalism, Ann. Phys. 323 (2008) 1502–1528.
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Optik journal homepage: www.elsevier.de/ijleo
Quantifying fidelity and purity for some quantum states in dissipative channel by virtue of trace rule in the phase space夽 Xue-xiang Xu a,∗ , Liang-sheng Xiong b , Hong-chun Yuan c , Li-yun Hu a , Zhen Wang d , Shuai Wang d , Hong-yi Fan d a
College of Physics and Communication Electronics, Jiangxi Normal University, Nanchang 330022, China Yichun Vocational and Technical College, Yichun 336000, China c College of Optoelectronic Engineering, Changzhou Institute of Technology, Changzhou 213002, China d Department of Physics, Shanghai Jiao Tong University, Shanghai 200030, China b
a r t i c l e
i n f o
Article history: Received 27 December 2011 Accepted 22 May 2012
PACS: 03.65.Yz 05.30.-d 42.50.Ar
a b s t r a c t By virtue of trace rule in the phase space, we investigate the time evolution of fidelity and purity for some quantum states (including Fock state, coherent state, squeezed state, and thermal state) in dissipative channel. All the relations are calculated by directly applying the Wigner function evolution formula in channel. The method is more convenient and concise than other method such as density operators’ trace. It is found that the dissipation induces all the quantum state eventually to vacuum state (a pure Gaussian state). © 2012 Elsevier GmbH. All rights reserved.
Keywords: Fidelity Purity Dissipative channel Trace rule in the phase space
1. Introduction One of the major topics in Quantum Statistical Mechanics is the evolution of (pure or mixed) quantum states. In recent years, much attention has been paid to the investigation of evolution between pure states and mixed states [1–3]. As we all know, physical systems are never perfectly isolated from their environment. Therefore some evolution, which is described by a master equation, usually happens when a system is immersed in a thermal environment, or a signal (a quantum state) passes through a quantum channel. Many physicists have been carried out research in this field. The evolution of fidelity of generic bosonic fields in noisy channels has been addressed in Ref. [4]. Paris et al. [5] presented a systematic study for Gaussian states of single-mode continuous variable system and considered the dynamics of purity in noisy channels.
夽 This project was supported by the National Natural Science Foundation of China (Nos. 11047133 and 11174114),the Natural Science Foundation of Jiangxi Province of China (No. 2010GQW0027), the Key Programs Foundation of Ministry of Education of China (No. 210115) and the Research Foundation of the Education Department of Jiangxi Province of China (Nos. GJJ12171 and GJJ11390). ∗ Corresponding author. Tel.: +86 791 88120370; fax: +86 791 88120370. E-mail address: [email protected] (X.-x. Xu). 0030-4026/$ – see front matter © 2012 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2012.05.047
They also studied the evolution of purity, entanglement, and total correlations of general two-mode continuous variable Gaussian states in arbitrary uncorrelated Gaussian environments [6]. The evolution of cat-like states in general Gaussian noisy channels were also studied [7]. Alternately, description of evolution of density matrices can be replaced by its Wigner function (WF)’s evolution in phase space [8,9]. Recently, a F-P equation for WF evolution in noisy self-Kerr medium is presented [10]. Moreover, a WF evolution formula in the self-Kerr medium with photon loss is derived [11]. The WF in phase space is very useful in studying quantum dynamics and is a powerful tool for investigating such nonclassical effect. In this paper, we study the decoherence of several initial quantum states and focus our attention on the evolution of the fidelity and purity in dissipative channel. The rate of decoherence is quantified by the evolution of fidelity and purity. The fidelity reflects that how close two states are to each other. While the purity, the degree of mixedness of a quantum state, is fundamental in distinguishing a pure state from a mixed one [12]. When an initial quantum state 0 passes through a dissipative channel and evolutes into (t) after some time, we define the fidelity F(t) = Tr(0 (t)),
(1)
X.-x. Xu et al. / Optik 124 (2013) 1814–1819
and the purity 2
(t) ≡ Tr[ (t)],
(2)
to determine decoherence. Obviously, it is difficult to evaluate operators’ traces directly. However, the introduction of the WF allows to evaluate operators’ traces as integrals in the phase space, i.e. trace rule in the phase space for single-mode case [13]
d2 ˛W1 (˛)W2 (˛),
Tr[1 2 ] =
(3)
which bring much convenience to calculating the fidelity and the purity. That is, the overlap between two operators 1 and 2 can be transformed into the overlap of the corresponding Wigner functions W1 (˛) and W2 (˛). The paper is structured as follows. In Section 2, we briefly review the theory of WF and its evolution formula in dissipative channel, then we define the fidelity and the purity in dissipative channel. In Section 3 we provide a detailed study of time evolution of fidelity and purity for several initial quantum states of major interest such as coherent state, Fock state, squeezed state and thermal state. Some concise analytic expressions have been represented. Through plotting the graph, we discuss in detail their evolution of trends. It is found that all the quantum states in the channel will loss its initial character due to the factor of environment and evolve into vacuum state at last. 2. Basic theory Let us firstly review the theory of WF and its evolution formula in dissipative channel, then we define the fidelity and purity in dissipative channel. All the relations is very important in our following work. 2.1. Wigner function and its evolution formula in dissipative channel The Wigner function (WF) [14,15] was first introduced by Wigner in 1932 to calculate quantum correction to a classic distribution function of a quantum-mechanical system. It now becomes a very popular tool to study the nonclassical properties of quantum states. It is well known that WFs are quasiprobability distributions because it may be negative in phase space. The presence of negativity of the WF for an optical field is a signature of its nonclassicality [16]. For a single-mode quantum system , the WF in the coherent state representation |z is given by [17]
2
2e2|˛| W (˛, ˛ ) = ∗
d2 z ∗ ∗ − z||ze−2(z˛ −z ˛) ,
(4)
√ where ˛ = q + ip/ 2. Using this equation, we can derive the analytic expression for a given quantum state in a good manner. When the quantum state passes through a dissipative channel, the time evolution of its density operator can be described by the following master equation [18,19] d(t) = [2a(t)a† − a† a(t) − (t)a† a], dt
(5)
where a and a† are creation and annihilation operator satisfying [a, a† ] = 1, and is the dissipative coefficient. By virtue of the entangled state representation, the authors Fan and Hu find the density operator (t) for the dissipative channel can be expressed in the Kraus operator sum representation as follows [20,21] (t) =
∞ Tn n=0
n!
†
†
e−ta a an 0 a†n e−ta a ,
(6)
1815
where T = 1 − e−2t . As the asymptotic case, we find from Eq. (6) that, when t→ ∞, (t) must evolve towards a desired steady state (vacuum state |00|, which is a Gaussian state and is related to the character of channel) for arbitrary initial quantum state. This is also the so-called environment engineering for pure Gaussian state preparation [22]. Obviously, Eq. (6) is an infinite summation in form in itself. If using density operators’ trace to evaluate the fidelity and the purity, it will bring much difficulty. Once the WF at arbitrary time in channel W (˛, ˛∗ ; t) is known, we can convenient to obtain the results by virtue of trace rule in the phase space. By using the thermal field dynamics theory and thermal entangled state representation, Fan and Hu, also the authors, have derived the analytic formula of the time evolution of WF at time t expressed as the following convolution [23] W (˛, ˛∗ ; t) =
2 T
d2 z −2/T |˛−ze−t |2 W (z, z ∗ ; 0), e
(7)
where W(z, z∗ ; 0) is the initial WF. Eq. (7) is just the evolution formula of WF of single mode quantum state in dissipative channel. By observing Eq. (7), we see that when t → 0, T → 0,
2 2 exp − |˛ − ze−t |2 T T
→ ı(˛ − z)ı(˛∗ − z ∗ ),
so W(˛, ˛∗ ; t) → W(˛, ˛∗ ; 0) as expected. Thus the WF at any time can be obtained by performing the integration when the initial WF is known. 2.2. Trace rule calculated by Wigner function in phase space The quantum state 0 evolutes into quantum state (t) after some time in the channel. Now, we face two problems: (1) How to measure how close the initial state and final state are to each other; (2) How to measure the degree of mixedness of the quantum state at any time. 2.2.1. Evolution of fidelity in dissipative channel One widely accepted measure is the fidelity to measure how (t) is close to 0 . The fidelity is defined as F(t) = Tr(0 (t)). When the two states are the same, the fidelity is 1 with a perfect overlap; while when the two states are orthogonal to each other, it is 0. According to trace rule in the phase space, F(t) can be reformed as
W0 (˛, ˛∗ ; 0)W (˛, ˛∗ ; t)d2 ˛.
F(t) =
(8)
as fidelity in dissipative channel. 2.2.2. Evolution of purity in dissipative channel Similarly, the purity is fundamental in distinguishing a pure state from a mixed one and determining the degree of mixedness of the quantum state (t). Thus we define (t) ≡ Tr[2 (t)] as purity in dissipative channel. For continuous variable systems one has 0 < 1. According to trace rule in the phase space, (t) can be rewritten as
(t) =
|W (˛, ˛∗ ; t)|2 d2 ˛.
(9)
It is easy to see that as long as we know W0 (˛, ˛∗ ; 0) and W(˛, ˛∗ ; t), we can evaluate the explicit analytical expressions of fidelity and purity from Eqs. (8) and (9). 3. Several cases for different initial quantum states In this section, we deal with the problem how to determine the fidelity and the purity of some quantum states in dissipative
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channel. These states include Fock state (FS), coherent state (CS), squeezed state (SS), and thermal state (TS). As we will see later, the time evolution of the fidelity and the purity can be expressed as a function of the initial observable parameters of the input state and of the character of the channel.
1.0
m=0
0.8
Ft
3.1. Fock state (FS)
0.4
m=2
0.2
The photon number states, also FSs, are easy to comprehend and they are the basic states of the quantum theory of light. They form a complete set for the states of a single mode and they are easy to manipulate in calculations of their quantum optical properties. The FS form the natural starting point for a treatment of single mode light [24]. Now we consider the case of Fock state (FS) |m as initial quantum state channel. Using Eq. (4) and noticing |z = ∞in dissipative n 2 e−|z| /2 n=0 √z |n, we get the WF of FS
m=5
0.0
m=8
0.0
0.2
2
2(−1)m e2|˛| WFS (˛, ˛ ; 0) = m!
1.0
=
2 2(−1)m e2|˛|
m!
m m
d2 z 2 ∗ ∗ (|z|2 )m e−|z| −2z˛ +2z ˛
∂ ∂ ×
d2 z
2 +(−2˛∗ +)z+(2˛+)z ∗
e−|z|
0.6
0.6
m=8
0.0
0
1
2
(11)
(12)
We retain Eq. (12) in the present form in order to evaluate the following content conveniently. Substituting Eq. (12) into Eq. (7) and treating as above, we get the time evolution of WF WFS (˛, ˛∗ ; t) =
m
2(−1) 2 −t −t ∗ m m ∂ ∂ e−2|˛| +2e ˛−2e ˛ m! ×e
(2e−2t −1)
|==0 ,
(13)
−2t 1 m m m m ∂ ∂ ∂ ∂ e(e −1)f f m!m! i i f f
×e
−e−t f i −e−t f i
|i =i =f =f =0 ,
1 m!m!
(14)
m
∞ (e−2t − 1)j (−e−t )l (−e−t )h
j!l!h!
j,l,h=0 m m
m
j+l j+h
× ∂i ∂i ∂f ∂f hi il f f
|i =i =f =f =0 = m!m!
∞ (e−2t − 1)j (−e−t )l (−e−t )h j,l,h=0
j!l!h!
× ım,j+l ım,j+h ım,l ım,h = e−2mt .
5
as one expected. Since channel damps the initial state to a vacuum state. Substituting Eq. (13) into Eq. (9) and employing the similar procedure above, we obtain the time evolution purity of FS in dissipative channel, FS (t) =
1 −2t −2t m m m m ∂ ∂ ∂ ∂ e(e −1)1 1 +(e −1)2 2 m!m! 1 1 2 2 −2t −e−2t 1 2 2 1
(15)
This is a new concise result. In particular, when t → 0, F(t) → 1, since |m is a pure state. It is easy to see that, as t→ ∞, F(t) → 0,
|1 =1 =2 =2 =0 = m!m!
(e−2t − 1)j+l (−e−2t )h+r ∞
j!l!h!r!
j,l,h,r=0
=
× ım,j+h ım,j+r ım,l+r ım,h+l .
m m!m!(e−2t − 1)2j (e−2t )2m−2j j=0
where the subscript i and f is to differentiate the initial and final states. Furtherexpanding all the exponential items into series and noticing ∂xn xm = m!ım,n (where ım,n denotes Kroncher symx=0 bol), Eq. (14) reduces to FFS (t) =
4
Fig. 1. Evolution of (a) fidelity and (b) purity for Fock state |m in dissipative channel with m = 0, 1, 2, 5, 8.
× e−e
It is clear from Eq. (13) that when t→ ∞, WFS (˛, ˛∗ ; ∞) → 2 (2/)e−2|˛| , i.e. corresponding to vacuum state. Substituting Eqs. (12) and (13) into Eq. (8), we derive the time evolution fidelity of FS in dissipative channel yielding FFS (t) =
3 t
it leads to 2(−1)m m m −2|˛|2 +2˛−2˛∗ + WFS (˛, ˛ ; 0) = ∂ ∂ e |==0 . m!
1.4
(b)
After employing the integration formula [25],
∗
1.2
0.2
|==0 ,
1 −
d2 z |z|2 + z+ z∗ = − e , Re () < 0, e
1.0
m=1 m=2 m=5
0.4
(10)
0.8
m=0
0.8 t
0.4
t
n!
∗
(a)
m=1
0.6
= e−4mt
j!j!(m − j)!(m − j)! 2 F1 (−m, −m; 1; (e
2t
− 1)2 )
(16)
where 2 F1 (− m, − m ; 1 ; (e2t − 1)2 ) is a hypergeometric function[26]. In particular, when t → 0, (t) → 1. While t→ ∞, there is also (t) → 1. Eqs. (15) and (16) reflect that FFS (t) and FS (t) are only relative to the initial observable parameters m of the input state and of the character of the channel. The results of the numerical analysis of the evolution of fidelity and purity of FSs in dissipative channel for different number m are reported in Fig. 1. The fidelity is a monotonically decreasing function of t and tends to zero as the time increase. Moreover, the higher the number m, the faster the decline (see Fig. 1(a)). But for purity, it decreases from the maximum 1 to a minimum, then rose to the maximum 1 except m = 0. Moreover, the higher the number m, the smaller the minimum (see Fig. 1(b)). 3.2. Coherent state (CS) There is a wide variety of possible superposition states of FS. But one kind, the CS is of particular importance in the practical
X.-x. Xu et al. / Optik 124 (2013) 1814–1819
(a)
1.0
1.0
0.8
0.6
Ft
Ft
0.8
0.6 0.4
0.4
(a)
0.2
0.2 0.0
1817
0.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0
0.1
0.2
0.3
t
(b)
0.8
t
0.8 t
0.5
1.0
1.0
0.6
0.6 0.4
0.4
(b)
0.2
0.2 0.0
0.4
t
0.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
t
t Fig. 2. Evolution of (a) fidelity and (b) purity for coherent state |ˇ in dissipative channel with |ˇ| = 0, 0.5, 0.8, 1, 2 (from top to bottom).
Fig. 3. Evolution of (a) fidelity and (b) purity for squeezed state S(r)|0 in dissipative channel with r = 0, 0.3, 0.8, 1, 2 (from top to bottom).
3.3. Squeezed state (SS) applications of the quantum theory of light [27]. Here we consider the case of coherent state |ˇ in dissipative channel. From Eq. (4), we readily see WCS (˛, ˛∗ ; 0) =
2 exp(−2|˛ − ˇ|2 ),
(17)
Here, we discuss the case of SS S(r)|0 [28] in dissipative channel, †2
r
2
where S(r) = e 2 (a −a ) is a squeezed operator with the squeezing parameter r. Employing the similar procedure, we have 2 −2|˛ cosh r+˛∗ sinh r|2 e ,
WSS (˛, ˛∗ ; 0) =
(21)
and Then substituting Eq. (17) into Eq. (7), the time evolution of WF is obtained, i.e. WCS (˛, ˛∗ ; t) =
2 exp(−2|˛ − e−t ˇ|2 ).
2 −2t (cosh 2r−1))/M|˛|2 √ e−(2+2e M
WSS (˛, ˛∗ ; t) =
× e−((e
(18)
M = (e−2t
Comparing Eq. (18) with Eq. (17), we find that the coherent state |ˇ remains as a pure coherent state |e−t ˇ only having an exponential damping in the amplitude. Moreover, we easily know 2 WCS (˛, ˛∗ ; ∞) = 2 e−2|˛| , which is just the WF of vacuum state. Substituting Eqs. (17) and (18) into Eq. (8), we obtain the time evolution fidelity of CS in dissipative channel
−2t
sinh 2r)/M)(˛2 +˛∗2 )
+ T cosh 2r)2
− T2
,
(22)
sinh 2 2r .
with When t→ ∞, 2 WSS (˛, ˛∗ ; ∞) → (2/)e−2|˛| , the WF of vacuum state. Substituting Eqs. (21) and (22) into Eq. (8), we have the time evolution fidelity of SS in dissipative channel FSS (t) =
2 M(A2 − B2 )
,
(23)
The result is also identical to that calculated by. Similarly, we have the time evolution purity of CS
with A = (1 + e−2t (cosh 2r − 1))/M + cosh 2r and B = ((e−2t /M) + 1) sinh 2r. When t→ ∞, M → 1, A → 1 + cosh 2r, B → sinh 2r, it leads to FSS (t) → 2/ (cosh 2r + 1). Substituting Eq. (22) into Eq. (9), we obtain the time evolution purity of SS in dissipative channel
CS (t) = 1,
SS (t) =
FCS (t) = exp[−(e−t − 1)2 |ˇ|2 ].
(19)
(20)
which means that the coherent state remains the pure character for all time in dissipative channel. In Fig. 2, the behavior of fidelity and purity for CS in dissipative channel are presented in detail. Obviously, when t→ ∞, 2 FCS (t) → e−|ˇ| . For different amplification |ˇ| of CS, the fidelity 2 decreases and eventually reach a steady value e−|ˇ| as the time increase. Moreover, the bigger |ˇ|, the smaller the fidelity. But the purity always remains 1 for any time.
1 C2
− e−4t sinh2 2r
(24)
with C = 1 + e−2t (cosh 2r − 1). Eqs. (23) and (24) reflect that FSS (t) and SS (t) are only relative to the initial squeezing parameters r of the SS and of the character of the channel. The behavior of fidelity and purity for SS in dissipative channel are presented in Fig. 3. The fidelity of SS is a monotonically decreasing function of t and tends to a steady 2/ (cosh 2r + 1) for different r as the time increase. Morevalue over, the bigger the squeezing parameter r, the faster the decline
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X.-x. Xu et al. / Optik 124 (2013) 1814–1819
Then substituting Eq. (27) into Eq. (7) yields the Wigner function
1.0
∗
WTS (˛, ˛ ; t) =
0.8
Ft
0.6
(a)
0.0 0.0
0.5
1.0
1.5
2.0
2.5
FTS (t) =
3.0
t
(b)
t
0.2 0.0
0.5
1.0
1.5
2.0
2.5
3.0
t Fig. 4. Evolution of (a) fidelity and (b) purity for thermal state in dissipative channel with n = 0, 0.1, 0.2, 0.5, 1 (from top to bottom).
(see Fig. 3(a)). But for purity, it decreases from the maximum 1 to a minimum, then rose to the maximum 1 except r = 0. Moreover, the bigger the squeezing parameter r, the smaller the minimum (see Fig. 3(b)). 3.4. Thermal state (TS) As a special case, we discuss a mixed state – TS th [1–3,24] in dissipative channel. TS is the equilibrium state for a field coupled to a reservoir at finite temperature . This state is commonly discussed in statistical and thermal physics courses because it is used to derive the Planck radiation law. The distribution function which describes a thermal state’s statistics is similar to the Maxwell–Boltzmann dis† tribution th = (1 − e−ω/k )e−ωa a/k (k is Boltzmann constant). The average photon number n = Tr(tha† a) of TS is a well-known result which is often termed the Planck distribution function, i.e. n = (1 − eω/k )−1 . Alternately, the density operator th of thermal state in Fock representation is l
n
(n + 1)
l+1
|ll|
(25)
√ where |l is the Fock state. Due to |l = a†l / l!|0 and |00| =: † e−a a :(the symbol : :denotes normally ordering), Eq. (25) can be rewritten as 1 1 † † : e(n/(n+1)−1)a a := ea a ln n/(n+1) , n+1 n+1
where we use the following operator exp [− a† a] = : exp [− (1 − e− )a† a]: [29,30]. Substituting Eq. (26) into Eq. (4), we have 2 WTS (˛, ˛ ; 0) = exp (2n + 1) ∗
,
(28)
2|˛|2 − 2n + 1
(29)
identity
(26) formula
,
(30)
In particular, when t → 0, (t) → 1/(2n + 1). While t→ ∞, there is also (t) → 1. Eqs. (29) and (30) reflect that FTS (t) and TS (t) are only relative to the initial average photon number n of the TS and of the character of the channel. The behavior of fidelity and purity for SS in dissipative channel are presented in Fig. 4. From Fig. 4, we find that for TS with different n, the initial fidelity and purity is different and all equal to 1/ (2n + 1). As time increase, the fidelity reach a fixed value 1/ (n + 1) , while the purity tend to the same value 1.
0.6 0.4
th =
1 n(1 + e−2t ) + 1
TS (t) = 1/(2ne−2t + 1).
0.8
l=0
In particular, when t → 0, F(t) → 1/(2n + 1). While t→ ∞, there is also F(t) → 1/(n + 1). Substituting Eq. (28) into Eq. (9), we have the time evolution purity of TS in dissipative channel
1.0
∞
exp
2|˛|2 − 2ne−2t + 1
2
0.2
th =
(2ne−2t + 1)
Obviously when t→ ∞, WTS (˛, ˛∗ ; ∞) → (2/)e−2|˛| , the WF of vacuum state. Substituting Eqs. (27) and (28) into Eq. (8), we obtain the time evolution fidelity of TS in dissipative channel
0.4
0.0
2
(27)
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