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Dynamical process of quantum dense coding in an exactly solvable model at finite temperature
Physics Letters A 334 (2005) 345–351 www.elsevier.com/locate/pla
Dynamical process of quantum dense coding in an exactly solvable model at finite temperature Sachiko Kitajima Graduate School of Humanities and Sciences, Ochanomizu University, Tokyo 112-8610, Japan Received 13 June 2004; received in revised form 2 October 2004; accepted 22 November 2004 Available online 2 December 2004 Communicated by P.R. Holland
Abstract Dynamical quantum dense coding process is discussed on the basis of an exactly solvable boson detector model. A previous treatment is extended to include finite temperature effect. Time evolution of density matrices is completely determined. Moreover, a quantum mechanical communication channel matrix and Shannon’s mutual information are obtained as functions of time. Quantum and thermal noise effects due to the fluctuating environment are discussed. 2004 Elsevier B.V. All rights reserved. PACS: 03.65.-w; 05.30.-d; 05.70.Ln Keywords: Decoherence; Relaxation; Entangled state; Mutual information; Quantum dense coding
1. Introduction Quantum mechanical dense coding (QDC) [1–4] is one of important method in quantum information processing [5–7]. This utilizes the entangled states importance of which was already recognized in an early stage of development of quantum mechanics [8,9]. In a previous paper [10] the author proposed a new exactly solvable dynamical model of QDC which had been introduced originally as a simplified version of Coleman–Hepp (CH) model [11– 13] of quantum mechanical measurement process [14,15]. We call it the boson detector (BD) model and used both models (CH and BD) to examine decoherence processes [16–22]. In this Letter we extend the previous theory to include finite temperature effect. In contrast to the usual treatment of the damping channel, we can treat dynamical process of QDC even when the system–reservoir interaction is strong and moreover, our theory is valid for all tem-
E-mail address: [email protected] (S. Kitajima). 0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.11.039
346
S. Kitajima / Physics Letters A 334 (2005) 345–351
perature regions from absolute zero to very high temperature. In the following we first determine time evolution of density matrices which give exactly a quantum mechanical channel matrix and Shannon’s entropy.
2. Model Hamiltonian and density matrix In this section, we extend a boson detector (BD) model to include an entangled state. The total Hamiltonian of the model is given by A H = H0 + P+ H1 ,
(1)
where H0 = HS + HD , HS = vA PA + h¯ ωA IAz N HD = h¯ ωl al† al l=1
(2) + h¯ ωB IBz ,
(3) (4)
and 1 h¯ Ωl (XA − xl ) eiωl XA /vA al + e−iωl XA /vA al† . 2 N
H1 =
(5)
l=1
This is composed of a two-particle (A and B) system S and a detector (environment) D. Particles A and B have z z 1/2-spins IA and IB with energies hω ¯ A IA and h¯ ωB IB , respectively. Particle A moves with the velocity vA and PA being the momentum operator. Particle A has the kinetic energy of the form, vA PA , which is the characteristic of, for instance, an ultrarelativistic particle or a one-dimensional Tomonaga–Luttinger particle [23,24]; HS of Eq. (3) represents a total energy of S. Detector D is composed of an array of N harmonic oscillators whose quanta are annihilated by al ’s and created by al† ’s with energy hω ¯ l . These constitute the unperturbed Hamiltonian H0 of Eq. (2). The Hamiltonian H1 of (5) represents the interaction between particle A and each constituent oscillator of D, where XA is the position operator of particle A and the interaction potential around the site position xl (xl+1 − xl = d) is given, for instance, by Ωl · d −(x−xl )2 /2δ 2 Ωl (x − xl ) = √ e , 2πδ 2
(6)
A (= 1 + I z ) imposes where Ωl on the right-hand side shows the interaction strength. The projection operator P+ A 2 a condition on the detector that the interaction occurs only when the incident particle spin is in an up state. That is, the interaction becomes effective only when the particle A with up spin component enters into the potential range (width of order δ) around xl . That is, IA does not flip by the interaction and no bit flip error is allowed. Usually, interactions are written in the relative form of l V (X − Ql ) where the lattice coordinate Ql is expanded in the phonon coordinate δQl (Ql = xl + δQl ) and thus the interaction term is given by V (X − xl )δQl . In this derivation, the interaction term is proportional to δQl ∝ al + al† . This usual form of interaction between S and D should be compared with the present Hamiltonian (5) where the exponential operators are inserted to take into account energy (momentum) conservation in a simple but intuitive manner. This is recognized by the properties of the displacement operators eiωl XA /vA and e−iωl XA /vA as follows: √ eiωl XA /vA al |pA ⊗ |nl = nl |pA + h¯ ωl /vA ⊗ |nl − 1, (7) −iωl XA /vA † al |pA ⊗ |nl = nl + 1 |pA − h¯ ωl /vA ⊗ |nl + 1, e (8)
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where PA |pA = pA |pA and al† al |nl = nl |nl . That is, annihilation and creation of quanta give rise to increase and decrease in momenta of particle A. Particle A starts from x0 and passes through D toward particle B outside of D. We regard the extended BD model as a new dynamical system of QDC identifying the spin of particle A with Alice’s qubit. In order to implement the QDC protocol in the BD model, we impose the following initial condition on a density matrix W (t): W (0) = ρI (0) ⊗ ρψ (0) ⊗ ρD (0)
(9)
with ρI (0) = |I I |,
(10)
ρψ (0) = |ψψ|
(11)
where |I is one of Bell’s entangled spin states, AB 1 A B A B Φ 0 ⊗ 0 ± 1 ⊗ 1 , (12) ± = √ 2 AB 1 Ψ = √ 0 A ⊗ 1 B ± 1 A ⊗ 0 B , (13) ± 2 |0 and |1 being the spin-up state and the spin-down state, respectively. The orbital state of the particles is given by |ψ = |ψA ⊗ |ψB = dxA ψA (xA )|xA ⊗ dxB ψB (xB )|xB .
(14) (15)
The initial density matrix for the detector is of the form ρD (0) =
N
ρD,l (0),
(16)
l=1
where ρD,l (0) is a density matrix for the lth harmonic oscillator in thermal equilibrium. This is represented by †
ρD,l (0) =
¯ l al al e−β hω †
Tr e−β h¯ ωl al al 2 0 d zl (A) 0 0∗ ρD,l zl , zl , t = 0 zl0 zl0 , = π
(17) (18)
where |zl0 is a coherent state [25–27] for the lth harmonic oscillator and the antinormal function (P -function) is given by [25,28,29] (A) 0 0∗ ρD,l (19) zl , zl , t = 0 = eβ h¯ ωl − 1 exp − eβ h¯ ωl − 1 zl0∗ zl0 with β = 1/kB T , T the temperature. We encode two-bit classical information (0, 0), (0, 1), (1, 0) and (1, 1) by suitable choice of unitary operators U A ’s to obtain AB AB U A (0, 0)Φ+ (20) = Φ+ , AB AB A U (0, 1)Φ+ = Φ− , (21) AB U A (1, 0)Φ+ (22) = Ψ+AB
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and
AB AB = Ψ − . U A (1, 1)Φ+
(23)
When Alice sends her particle A to Bob who has particle B, particle A moves into D interacting with constituent harmonic oscillators. Bob receives particle A perturbed by the environment. We are thus able to incorporate the QDC protocol into the extended BD model. In the following we solve the dynamical problem of QDC exactly by taking environmental fluctuations at finite temperature into account, irrespective of the interaction strength between S and D. This is contrasted with existing treatments where the system–reservoir interaction is assumed to be weak.
3. Dynamics of quantum dense coding process at finite temperature In the following, we will determine time-evolution of the total density matrix given by W (t) = e−iHt /h¯ W (0)eiHt /h¯ . This is done with the use of the relation, A A A A = e−i(H0 +H1 )t /h¯ P+ + P− + e−iH0 t /h¯ P− e−iHt /h¯ = e−iHt /h¯ P+ −iH t /h¯ † A B S VA (t)P+ DA {ωl X/vA } , = DA {ωl X/vA } e + P+
(24)
(25)
where N N † DA {φl } = Dl {φl } = e−iφl al al , l=1
(26)
l=1
N i VA (t) = exp − Θl (XA ; t) al† + al 2
(27)
l=1
and t Θl (x; t) =
dt Ωl (x + vA t − xl )
(28)
0 A + P A = 1. with P+ − After determining the time evolution of the total density matrix W (t), we eliminate other variables than spins of particles A and B. Then we find reduced density matrices for respective initial Bell states (20)–(23). When the AB , we obtain the reduced density matrices of the form initial encoded quantum state is |I = |Φ± 1 A A B B A A B B Φ 0 0 ⊗ 0 0 + 1 1 ⊗ 1 1 ρ± (t) = 2 2 −iω(+) t A A B B ± dxA ψA (xA ) e (29) C(N, T ; xA, t) 0 1 ⊗ 0 1 + h.c. ,
while for |I = |Ψ±AB , we obtain 1 A A B B A A B B Ψ 0 0 ⊗ 1 1 + 1 1 ⊗ 0 0 ρ± (t) = 2 2 −iω(−) t A A B B ± dxA ψA (xA ) e C(N, T ; xA , t) 0 1 ⊗ 1 0 + h.c. ,
(30)
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349
where C(N, T ; xA, t) =
N
exp − coth(β h¯ ωl /2)Θl (xA ; t)2 /8 ,
(31)
l=1
and t Θl (x; t) =
dt Ωl (x + vt − xl )
(32)
0
with ω(±) ≡ ωA ± ωB . Once the reduced density matrices (29) and (30) are obtained, we find immediately a quantum mechanical communication channel matrix P (t): 1 (+) , t) 1 − R(ω(+) , t) 0 0 2 + R(ω 2 1 − R(ω(+) , t) 1 + R(ω(+) , t) 0 0 2 P (t) = 2 (33) , 1 (−) , t) 1 − R(ω(−) , t) 0 0 + R(ω 2 2 0
0
1 2
− R(ω(−) , t)
1 2
+ R(ω(−) , t)
where
2 1 −iωt e dxA ψA (xA ) C(N, T ; xA, t). + eiωt 4 In Eq. (33) the matrix elements of P (t) are defined by Pf,i (t) = Tr ρf ρi (t) = f |ρi (t)|f R(ω, t) =
≡ P (f, t|i, 0),
(34)
(35) (36)
where ρi (t) is a reduced density matrix with an initial state |i numbered from 1 to 4 according to the order (12), Φ (t) and ρ (t) = ρ Φ (t). The density matrix ρ = |f f | specifies the final Bell measurement (13), e.g., ρ1 (t) = ρ+ 2 f − state |f : ρ1 = |Φ+ Φ+ |, ρ2 = |Φ− Φ− | and so on. From the channel matrix P (t), we find Shannon’s mutual information, IS (out : in) = H (in) + H (out) − H (out, in) P (f, t|i, 0)πi log P (f, t|i, 0) − log P (f, t) , =
(37) (38)
f,i
where H (in) = −
πi log πi ,
(39)
i
H (out) = −
P (f, t) log P (f, t),
(40)
f
and H (out, in) = −
P (f, t|i, 0)πi log P (f, t|i, 0)πi .
(41)
f,i
The probability for determining the final state |f at time t is given by P (f, t) = P (f, t|i, 0)πi . i
(42)
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Fig. 1. Time evolution of Shannon’s mutual information, IS (out : in) as a function of tˆ = vA t/d varying the temperature T¯ = kB T /hω ¯ l for N = 5 and Ωˆ l = Ωl d/vA = 0.2. The thick broken, thick solid, thin broken and thin solid lines correspond to T¯ = 0.1, 1, 10 and 100, respectively.
When the prior probabilities πi ’s are the same, πi = 1/4 (i = 1–4), we have Shannon’s mutual information for the BD model: 1 1 1 1 1 + R ω(+) , t log + R ω(+) , t + − R ω(+) , t log − R ω(+) , t IS (out : in) = 2 + 2 2 2 2 2
(−) (−) (−) 1 1 1 1 + R ω , t log + R ω , t + − R ω , t log − R ω(−) , t . (43) + 2 2 2 2 In Fig. 1, we show the time evolution of Shannon’s mutual information for N = 5, Ωˆ l = Ωl d/vA = 0.2, ωˆ A ≡ ωA d/vA = 1 and ωA = ωB . We used the interaction function of the form Ωl · d −(x−xl )2 /2δ 2 Ωl (x − xl ) = √ (44) e , 2πδ 2 where Ωl is the interaction strength and δ the potential range. In the figure we changed the temperature parameter T¯ ≡ kB T /h¯ ωl from 0.1 to 100. At the low temperature the oscillatory behavior is found in IS (out : in). This oscillation is due to the mechanical origin and the interference effect occurs in the constituent terms in Eq. (43). That is, in Fig. 1 we show the result with ωA = ωB giving ω(+) = 2ωA , ω(−) = 0. And thus the oscillation comes from R(ω(+) , t) which is written as a product of the mechanical factor cos(ω(+) t) and the damping factor. We note the small decay even for T¯ = 0.1 due to the quantum fluctuation of D. With increasing value of T¯ , the thermal fluctuation becomes dominant yielding the large decay. Especially at T¯ = 100, the information suffers sudden strong perturbation from the fluctuating detector. In addition, the particle spin suffers strong decoherence effect increasing interaction strength as is clearly seen from (31).
4. Concluding remarks We introduced a new dynamical model of QDC which is valid even for finite temperature. This theoretical model was exactly solved to give time evolution of Shannon’s mutual information as a function of time. We could discuss effects due to the thermal noise as well as the quantum noise. Our model can describe various aspects of decoherence in the quantum mechanical channel by changing the interaction strength, the number of constituent
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oscillators, and the temperature of D. We showed in this Letter only the temperature effect and in a future publication another aspects of the model will be examined further in details including the quantum mechanical information and Holevo capacity.
Acknowledgements The author is grateful to Professor F. Shibata for valuable discussions.
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]
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Dynamical process of quantum dense coding in an exactly solvable model at finite temperature Sachiko Kitajima Graduate School of Humanities and Sciences, Ochanomizu University, Tokyo 112-8610, Japan Received 13 June 2004; received in revised form 2 October 2004; accepted 22 November 2004 Available online 2 December 2004 Communicated by P.R. Holland
Abstract Dynamical quantum dense coding process is discussed on the basis of an exactly solvable boson detector model. A previous treatment is extended to include finite temperature effect. Time evolution of density matrices is completely determined. Moreover, a quantum mechanical communication channel matrix and Shannon’s mutual information are obtained as functions of time. Quantum and thermal noise effects due to the fluctuating environment are discussed. 2004 Elsevier B.V. All rights reserved. PACS: 03.65.-w; 05.30.-d; 05.70.Ln Keywords: Decoherence; Relaxation; Entangled state; Mutual information; Quantum dense coding
1. Introduction Quantum mechanical dense coding (QDC) [1–4] is one of important method in quantum information processing [5–7]. This utilizes the entangled states importance of which was already recognized in an early stage of development of quantum mechanics [8,9]. In a previous paper [10] the author proposed a new exactly solvable dynamical model of QDC which had been introduced originally as a simplified version of Coleman–Hepp (CH) model [11– 13] of quantum mechanical measurement process [14,15]. We call it the boson detector (BD) model and used both models (CH and BD) to examine decoherence processes [16–22]. In this Letter we extend the previous theory to include finite temperature effect. In contrast to the usual treatment of the damping channel, we can treat dynamical process of QDC even when the system–reservoir interaction is strong and moreover, our theory is valid for all tem-
E-mail address: [email protected] (S. Kitajima). 0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.11.039
346
S. Kitajima / Physics Letters A 334 (2005) 345–351
perature regions from absolute zero to very high temperature. In the following we first determine time evolution of density matrices which give exactly a quantum mechanical channel matrix and Shannon’s entropy.
2. Model Hamiltonian and density matrix In this section, we extend a boson detector (BD) model to include an entangled state. The total Hamiltonian of the model is given by A H = H0 + P+ H1 ,
(1)
where H0 = HS + HD , HS = vA PA + h¯ ωA IAz N HD = h¯ ωl al† al l=1
(2) + h¯ ωB IBz ,
(3) (4)
and 1 h¯ Ωl (XA − xl ) eiωl XA /vA al + e−iωl XA /vA al† . 2 N
H1 =
(5)
l=1
This is composed of a two-particle (A and B) system S and a detector (environment) D. Particles A and B have z z 1/2-spins IA and IB with energies hω ¯ A IA and h¯ ωB IB , respectively. Particle A moves with the velocity vA and PA being the momentum operator. Particle A has the kinetic energy of the form, vA PA , which is the characteristic of, for instance, an ultrarelativistic particle or a one-dimensional Tomonaga–Luttinger particle [23,24]; HS of Eq. (3) represents a total energy of S. Detector D is composed of an array of N harmonic oscillators whose quanta are annihilated by al ’s and created by al† ’s with energy hω ¯ l . These constitute the unperturbed Hamiltonian H0 of Eq. (2). The Hamiltonian H1 of (5) represents the interaction between particle A and each constituent oscillator of D, where XA is the position operator of particle A and the interaction potential around the site position xl (xl+1 − xl = d) is given, for instance, by Ωl · d −(x−xl )2 /2δ 2 Ωl (x − xl ) = √ e , 2πδ 2
(6)
A (= 1 + I z ) imposes where Ωl on the right-hand side shows the interaction strength. The projection operator P+ A 2 a condition on the detector that the interaction occurs only when the incident particle spin is in an up state. That is, the interaction becomes effective only when the particle A with up spin component enters into the potential range (width of order δ) around xl . That is, IA does not flip by the interaction and no bit flip error is allowed. Usually, interactions are written in the relative form of l V (X − Ql ) where the lattice coordinate Ql is expanded in the phonon coordinate δQl (Ql = xl + δQl ) and thus the interaction term is given by V (X − xl )δQl . In this derivation, the interaction term is proportional to δQl ∝ al + al† . This usual form of interaction between S and D should be compared with the present Hamiltonian (5) where the exponential operators are inserted to take into account energy (momentum) conservation in a simple but intuitive manner. This is recognized by the properties of the displacement operators eiωl XA /vA and e−iωl XA /vA as follows: √ eiωl XA /vA al |pA ⊗ |nl = nl |pA + h¯ ωl /vA ⊗ |nl − 1, (7) −iωl XA /vA † al |pA ⊗ |nl = nl + 1 |pA − h¯ ωl /vA ⊗ |nl + 1, e (8)
S. Kitajima / Physics Letters A 334 (2005) 345–351
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where PA |pA = pA |pA and al† al |nl = nl |nl . That is, annihilation and creation of quanta give rise to increase and decrease in momenta of particle A. Particle A starts from x0 and passes through D toward particle B outside of D. We regard the extended BD model as a new dynamical system of QDC identifying the spin of particle A with Alice’s qubit. In order to implement the QDC protocol in the BD model, we impose the following initial condition on a density matrix W (t): W (0) = ρI (0) ⊗ ρψ (0) ⊗ ρD (0)
(9)
with ρI (0) = |I I |,
(10)
ρψ (0) = |ψψ|
(11)
where |I is one of Bell’s entangled spin states, AB 1 A B A B Φ 0 ⊗ 0 ± 1 ⊗ 1 , (12) ± = √ 2 AB 1 Ψ = √ 0 A ⊗ 1 B ± 1 A ⊗ 0 B , (13) ± 2 |0 and |1 being the spin-up state and the spin-down state, respectively. The orbital state of the particles is given by |ψ = |ψA ⊗ |ψB = dxA ψA (xA )|xA ⊗ dxB ψB (xB )|xB .
(14) (15)
The initial density matrix for the detector is of the form ρD (0) =
N
ρD,l (0),
(16)
l=1
where ρD,l (0) is a density matrix for the lth harmonic oscillator in thermal equilibrium. This is represented by †
ρD,l (0) =
¯ l al al e−β hω †
Tr e−β h¯ ωl al al 2 0 d zl (A) 0 0∗ ρD,l zl , zl , t = 0 zl0 zl0 , = π
(17) (18)
where |zl0 is a coherent state [25–27] for the lth harmonic oscillator and the antinormal function (P -function) is given by [25,28,29] (A) 0 0∗ ρD,l (19) zl , zl , t = 0 = eβ h¯ ωl − 1 exp − eβ h¯ ωl − 1 zl0∗ zl0 with β = 1/kB T , T the temperature. We encode two-bit classical information (0, 0), (0, 1), (1, 0) and (1, 1) by suitable choice of unitary operators U A ’s to obtain AB AB U A (0, 0)Φ+ (20) = Φ+ , AB AB A U (0, 1)Φ+ = Φ− , (21) AB U A (1, 0)Φ+ (22) = Ψ+AB
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and
AB AB = Ψ − . U A (1, 1)Φ+
(23)
When Alice sends her particle A to Bob who has particle B, particle A moves into D interacting with constituent harmonic oscillators. Bob receives particle A perturbed by the environment. We are thus able to incorporate the QDC protocol into the extended BD model. In the following we solve the dynamical problem of QDC exactly by taking environmental fluctuations at finite temperature into account, irrespective of the interaction strength between S and D. This is contrasted with existing treatments where the system–reservoir interaction is assumed to be weak.
3. Dynamics of quantum dense coding process at finite temperature In the following, we will determine time-evolution of the total density matrix given by W (t) = e−iHt /h¯ W (0)eiHt /h¯ . This is done with the use of the relation, A A A A = e−i(H0 +H1 )t /h¯ P+ + P− + e−iH0 t /h¯ P− e−iHt /h¯ = e−iHt /h¯ P+ −iH t /h¯ † A B S VA (t)P+ DA {ωl X/vA } , = DA {ωl X/vA } e + P+
(24)
(25)
where N N † DA {φl } = Dl {φl } = e−iφl al al , l=1
(26)
l=1
N i VA (t) = exp − Θl (XA ; t) al† + al 2
(27)
l=1
and t Θl (x; t) =
dt Ωl (x + vA t − xl )
(28)
0 A + P A = 1. with P+ − After determining the time evolution of the total density matrix W (t), we eliminate other variables than spins of particles A and B. Then we find reduced density matrices for respective initial Bell states (20)–(23). When the AB , we obtain the reduced density matrices of the form initial encoded quantum state is |I = |Φ± 1 A A B B A A B B Φ 0 0 ⊗ 0 0 + 1 1 ⊗ 1 1 ρ± (t) = 2 2 −iω(+) t A A B B ± dxA ψA (xA ) e (29) C(N, T ; xA, t) 0 1 ⊗ 0 1 + h.c. ,
while for |I = |Ψ±AB , we obtain 1 A A B B A A B B Ψ 0 0 ⊗ 1 1 + 1 1 ⊗ 0 0 ρ± (t) = 2 2 −iω(−) t A A B B ± dxA ψA (xA ) e C(N, T ; xA , t) 0 1 ⊗ 1 0 + h.c. ,
(30)
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where C(N, T ; xA, t) =
N
exp − coth(β h¯ ωl /2)Θl (xA ; t)2 /8 ,
(31)
l=1
and t Θl (x; t) =
dt Ωl (x + vt − xl )
(32)
0
with ω(±) ≡ ωA ± ωB . Once the reduced density matrices (29) and (30) are obtained, we find immediately a quantum mechanical communication channel matrix P (t): 1 (+) , t) 1 − R(ω(+) , t) 0 0 2 + R(ω 2 1 − R(ω(+) , t) 1 + R(ω(+) , t) 0 0 2 P (t) = 2 (33) , 1 (−) , t) 1 − R(ω(−) , t) 0 0 + R(ω 2 2 0
0
1 2
− R(ω(−) , t)
1 2
+ R(ω(−) , t)
where
2 1 −iωt e dxA ψA (xA ) C(N, T ; xA, t). + eiωt 4 In Eq. (33) the matrix elements of P (t) are defined by Pf,i (t) = Tr ρf ρi (t) = f |ρi (t)|f R(ω, t) =
≡ P (f, t|i, 0),
(34)
(35) (36)
where ρi (t) is a reduced density matrix with an initial state |i numbered from 1 to 4 according to the order (12), Φ (t) and ρ (t) = ρ Φ (t). The density matrix ρ = |f f | specifies the final Bell measurement (13), e.g., ρ1 (t) = ρ+ 2 f − state |f : ρ1 = |Φ+ Φ+ |, ρ2 = |Φ− Φ− | and so on. From the channel matrix P (t), we find Shannon’s mutual information, IS (out : in) = H (in) + H (out) − H (out, in) P (f, t|i, 0)πi log P (f, t|i, 0) − log P (f, t) , =
(37) (38)
f,i
where H (in) = −
πi log πi ,
(39)
i
H (out) = −
P (f, t) log P (f, t),
(40)
f
and H (out, in) = −
P (f, t|i, 0)πi log P (f, t|i, 0)πi .
(41)
f,i
The probability for determining the final state |f at time t is given by P (f, t) = P (f, t|i, 0)πi . i
(42)
350
S. Kitajima / Physics Letters A 334 (2005) 345–351
Fig. 1. Time evolution of Shannon’s mutual information, IS (out : in) as a function of tˆ = vA t/d varying the temperature T¯ = kB T /hω ¯ l for N = 5 and Ωˆ l = Ωl d/vA = 0.2. The thick broken, thick solid, thin broken and thin solid lines correspond to T¯ = 0.1, 1, 10 and 100, respectively.
When the prior probabilities πi ’s are the same, πi = 1/4 (i = 1–4), we have Shannon’s mutual information for the BD model: 1 1 1 1 1 + R ω(+) , t log + R ω(+) , t + − R ω(+) , t log − R ω(+) , t IS (out : in) = 2 + 2 2 2 2 2
(−) (−) (−) 1 1 1 1 + R ω , t log + R ω , t + − R ω , t log − R ω(−) , t . (43) + 2 2 2 2 In Fig. 1, we show the time evolution of Shannon’s mutual information for N = 5, Ωˆ l = Ωl d/vA = 0.2, ωˆ A ≡ ωA d/vA = 1 and ωA = ωB . We used the interaction function of the form Ωl · d −(x−xl )2 /2δ 2 Ωl (x − xl ) = √ (44) e , 2πδ 2 where Ωl is the interaction strength and δ the potential range. In the figure we changed the temperature parameter T¯ ≡ kB T /h¯ ωl from 0.1 to 100. At the low temperature the oscillatory behavior is found in IS (out : in). This oscillation is due to the mechanical origin and the interference effect occurs in the constituent terms in Eq. (43). That is, in Fig. 1 we show the result with ωA = ωB giving ω(+) = 2ωA , ω(−) = 0. And thus the oscillation comes from R(ω(+) , t) which is written as a product of the mechanical factor cos(ω(+) t) and the damping factor. We note the small decay even for T¯ = 0.1 due to the quantum fluctuation of D. With increasing value of T¯ , the thermal fluctuation becomes dominant yielding the large decay. Especially at T¯ = 100, the information suffers sudden strong perturbation from the fluctuating detector. In addition, the particle spin suffers strong decoherence effect increasing interaction strength as is clearly seen from (31).
4. Concluding remarks We introduced a new dynamical model of QDC which is valid even for finite temperature. This theoretical model was exactly solved to give time evolution of Shannon’s mutual information as a function of time. We could discuss effects due to the thermal noise as well as the quantum noise. Our model can describe various aspects of decoherence in the quantum mechanical channel by changing the interaction strength, the number of constituent
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oscillators, and the temperature of D. We showed in this Letter only the temperature effect and in a future publication another aspects of the model will be examined further in details including the quantum mechanical information and Holevo capacity.
Acknowledgements The author is grateful to Professor F. Shibata for valuable discussions.
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