aa r X i v : . [ qu a n t - ph ] N ov The squeezed generalized amplitude damping channel
R. Srikanth
1, 2, ∗ and Subhashish Banerjee † Poornaprajna Institute of Scientific Research, Devanahalli, Bangalore- 562 110, India Raman Research Institute, Sadashiva Nagar, Bangalore - 560 080, India Raman Research Institute, Sadashiva Nagar, Karnataka, Bangalore - 560 080, India
Squeezing of a thermal bath introduces new features absent in an open quantum system inter-acting with an uncorrelated (zero squeezing) thermal bath. The resulting dynamics, governed by aLindblad-type evolution, extends the concept of a generalized amplitude damping channel, whichcorresponds to a dissipative interaction with a purely thermal bath. Here we present the Kraus rep-resentation of this map, which we call the squeezed generalized amplitude damping channel. As anapplication of this channel to quantum information, we study the classical capacity of this channel.
PACS numbers: 03.65.Yz, 03.67.Hk, 03.67.-a
I. INTRODUCTION
The concept of open quantum systems is a ubiquitous one in that any real system of interest is surrounded by itsenvironment (reservoir or bath), which influences its dynamics. They provide a natural route for discussing dampingand dephasing. One of the first testing grounds for open system ideas was in quantum optics [1]. Its application toother areas gained momentum from the works of Caldeira and Leggett [2], and Zurek [3] among others. Dependingupon the system-reservoir ( S − R ) interaction, open systems can broadly be classified into two categories, viz., quantumnon-demolition (QND) or dissipative. A particular type of quantum nondemolition (QND) S − R interaction is givenby a class of energy-preserving measurements in which dephasing occurs without damping the system. This maybe achieved when the Hamiltonian H S of the system commutes with the Hamiltonian H SR describing the system-reservoir interaction, i.e., H SR is a constant of the motion generated by H S [4, 5, 6]. A dissipative open system wouldbe when H S and H SR do not commute resulting in dephasing along with damping [7]. A prototype of dissipative openquantum systems, having many applications, is the quantum Brownian motion of harmonic oscillators. This modelwas studied by Caldeira and Leggett [2] for the case where the system and its environment were initially separable.The above treatment of the quantum Brownian motion was generalized to the physically reasonable initial conditionof a mixed state of the system and its environment by Hakim and Ambegaokar [8], Smith and Caldeira [9], Grabert,Schramm and Ingold [10], and for the case of a system in a Stern-Gerlach potential [11], and also for the quantumBrownian motion with nonlinear system-environment couplings [12], among others. The interest in the relevance ofopen system ideas to quantum information and quantum computation has burgeoned in recent times because of theimpressive progress made on the experimental front in the manipulation of quantum states of matter towards quantuminformation processing and quantum communication.A number of open system effects can be given an operator-sum or Kraus representation [13]. In this representation,a superoperator E due to interaction with the environment, acting on the state of the system is given by ρ −→ E ( ρ ) = X k h e k | U ( ρ ⊗ | f ih f | ) U † | e k i = X j E j ρE † j , (1)where U is the unitary operator representing the free evolution of the system, reservoir, as well as the interactionbetween the two, {| f i} is the environment’s initial state, and {| e k i} is a basis for the environment. The environmentand the system are assumed to start in a separable state. The E j ≡ h e k | U | f i are the Kraus operators, which satisfythe completeness condition P j E † j E j = I . It can be shown that any transformation that can be cast in the form (1)is a completely positive (CP) map [14].To connect the predicted effects to actual experiments, a detailed model of the interaction between the principalsystem and the environment is required. However, from the viewpoint of a number of applications to quantumcomputation and information processing, these details may not be of immediate relevance. In such a case, theKraus representation is useful because it provides an intrinsic description of the principal system, without explicitly ∗ Electronic address: [email protected] † Electronic address: [email protected] considering the detailed properties of the environment [14]. The essential features of the problem are contained inthe operators E k . This not only simplifies calculations, but often provides theoretical insight. An example we willencounter below is the interplay between environmental squeezing and thermal effects for the case of dissipativesystem-reservoir interactions. Moreover, the reduced dynamics of a number of, seemingly different, physical systemscould be modelled in the quantum operations formalism [14] by the same noisy channel. This would help in thedevelopment of insight into the common features of the reduced dynamics of the above systems. For example, for thecase of a two-level system interacting, via a quantum non-demolition (QND) interaction, either with a bath of two-level systems (in the weak coupling limit) or harmonic oscillators (at zero temperature T and zero bath squeezing),the reduced dynamics in the quatum operations formalism can be shown to be governed by the phase dampingchannel [15, 16]. Another example would be the reduced dynamics of a simplified Jaynes-Cummings model consistingof a two-level atom coupled to a single cavity mode which in turn is interacting with a vacuum bath of harmonicoscillators. This can, considering only a single excitation in the atom-cavity system, be shown to be generated byan amplitude damping channel. Since as shown below, an amplitude damping channel results generally from theinteractions governed by the Lindblad type of evolution (at zero T and zero bath squeezing), this enables us to getan understanding of the reduced dynamics of the above system without having to concern ourselves with particulardetails.In this paper we study an open system, taken to be two-level system or qubit, where the bath is taken to beinitially in a squeezed thermal state. The resulting dynamics, governed by a Lindblad-type evolution, generates acompletely positive map that extends the concept of a generalized amplitude damping channel, which correspondsto a dissipative interaction with a purely thermal bath. We present the Kraus representation of this map, which wecall the squeezed generalized amplitude damping channel. An advantage of using a squeezed thermal bath is that thedecay rate of quantum coherence can be suppressed leading to preservation of nonclassical effects [15, 17, 18]. It hasalso been shown to modify the evolution of the geometric phase of two-level atomic systems [16]. The preservation ofentanglement in the presence of a squeezed bath has been investigated in Ref. [19].The paper is organized as follows. In Section II, we obtain the evolution of a qubit in a dissipative (non-QND)interaction with its bath. In Section II A, we consider, in specific, a system interacting with a squeezed thermal bathin the weak Born-Markov rotating wave approximation. In Section II B, we consider a single-mode Jaynes-Cummingsmodel in a vacuum bath. The amplitude damping and generalized amplitude damping channels are introduced inSection III, where it is pointed out that the simplified Jaynes-Cummings model realizes an amplitude damping channel,while the weak Born-Markov interaction without bath squeezing realizes a generalized amplitude damping channel. Weintroduce the squeezed generalized amplitude damping channel, which extends the concept of generalized amplitudedamping noise, to the case where environmental squeezing is included, in Section IV. Of particular interest is the factthat unlike the case of a purely dephasing channel, where the action of squeezing and temperature are concurrentlydecohering, in the case of squeezed generalized amplitude damping channel, they can exhibit counteractive behavior[16, 20]. In specific, in Section V, where we study the classical capacity of a squeezed generalized amplitude dampingchannel, we show that squeezing can improve the channel capacity, whereas temperature necessarily degrades it. Wemake our conclusions in Section VI. II. TWO-LEVEL SYSTEM IN NON-QND INTERACTION WITH BATH
In this section we study the dynamics of a two-level system in a dissipative interaction with its bath, which istaken as one composed of harmonic oscillators. We first consider the case of the system interacting with a bath whichis initially in a squeezed thermal state, in the weak coupling Born-Markov, rotating wave approximation. Next weconsider a simple single mode Jaynes-Cummings model in a vacuum bath.
A. System interacting with bath in the weak Born-Markov, rotating-wave approximation
Here we take up the case of a two-level system interacting with a squeezed thermal bath in the weak Born-Markov, rotating wave approximation. The system Hamiltonian is given by H S = ( ~ ω/ σ z . The system interactswith the bath of harmonic oscillators via the atomic dipole operator which in the interaction picture is given as ~D ( t ) = ~dσ − e − iωt + ~d ∗ σ + e iωt where ~d is the transition matrix elements of the dipole operator. The evolution of thereduced density matrix operator of the system S in the interaction picture has the following form [7, 21] ddt ρ s ( t ) = γ ( N + 1) (cid:18) σ − ρ s ( t ) σ + − σ + σ − ρ s ( t ) − ρ s ( t ) σ + σ − (cid:19) + γ N (cid:18) σ + ρ s ( t ) σ − − σ − σ + ρ s ( t ) − ρ s ( t ) σ − σ + (cid:19) − γ M σ + ρ s ( t ) σ + − γ M ∗ σ − ρ s ( t ) σ − . (2)Here γ is the spontaneous emission rate given by γ = (4 ω | ~d | ) / (3 ~ c ), and σ + , σ − are the standard raising andlowering operators, respectively given by σ + = | ih | = ( σ x + iσ y ) and σ − = | ih | = ( σ x − iσ y ). Eq. (2) may beexpressed in a manifestly Lindblad form as [16] ddt ρ s ( t ) = X j =1 (cid:16) R j ρ s R † j − R † j R j ρ s − ρ s R † j R j (cid:17) , (3)where R = ( γ ( N th + 1) / / R , R = ( γ N th / / R † and R = σ − cosh( r ) + e i Φ σ + sinh( r ). This observationguarantees that the evolution of the density operator can be given a Kraus or operator-sum representation [14], apoint we return to in Section IV. If T = 0, then R vanishes, and a single Lindblad operator suffices to describe Eq.(2).A number of methods of generating bath squeezing have been proposed in the literature. A squeezed reservoir maybe constructed on the basis of establishment of squeezed light field [22]. A single-mode squeezed state created in adegenerate parametric amplifier operated in an appropriate cavity, when coupled to an infinite number of external“output” modes, transfers the squeezing into correlations between side-bands of the multimode light field. A sub-threshold optical parametric oscillator (OPO) can be used as the basis for the implementation of a stable, reliablesource of continuous-wave squeezed vacuum [23]. Experiments probing the squeezed-light-atom system have beencarried out in Refs. [24, 25]. In particular, the latter reference details a method in which an OPO operated belowthreshold downconverts a high energy photon into two correlated low energy photons generating a close-to-minimum-uncertainty squeezed vacuum state. At the output of the OPO, the squeezed vacuum is mixed on a 99-1 beam-splitterwith a phase-coherent reference oscillator with controlled relative phase to the squeezing, resulting in a combinedelectromagnetic field that is equivalent to a displaced squeezed state.An infinite array of beam-splitters can be used to model a squeezed reservoir [26]. The signal is injected from the leftof the array, with independent squeezed fields (all with the same properties) injected into the other ports. The outputof a given beam-splitter serves as input to the subsequent one. In the limit of an infinite number of beam-splitters, thedynamics generated by Eq. (2) is simulated. In Ref. [27], quantum reservoir engineering with laser-cooled trappedions is used to mimic the dynamics of an atom interacting with a squeezed vacuum bath (Eq. (2) with temperature T set to zero). Another way to engineer a quantum reservoir to mimic coupling with a squeezed bath has been proposedin Ref. [28]. They consider a four-level system with two (degenerate) ground and excited states, driven by weaklaser fields and coupled to a vacuum reservoir of radiation modes. Interference between the spontaneous emissionchannels in optical pumping leads to a squeezed bath type coupling for the two-level system constituted by the twoground levels. The properties of the squeezed bath are shown to be controllable by means of the laser parameters. Anexperimental study of the decoherence and decay of quantum states of a trapped atomic ion’s harmonic motion withseveral types of engineered reservoirs, including amplitude dampling and generalized amplitude damping channels,have been made in Refs. [29, 30].In Eqs. (2) and (3), we use the nomenclature | i for the upper state and | i for the lower state and σ x , σ y , σ z arethe standard Pauli matrices. In Eq. (2), N = N th (cosh ( r ) + sinh ( r )) + sinh ( r ) , (4)and M = − sinh(2 r ) e i Φ (2 N th + 1), and N th = 1 / ( e ~ ω/k B T − N th is the Planck distribution giving thenumber of thermal photons at the frequency ω and r , Φ are bath squeezing parameters [31]. The analogous case ofa thermal bath without squeezing can be obtained from the above expressions by setting these squeezing parametersto zero. We solve the Eq. (2) using the Bloch vector formalism as ρ s ( t ) = 12 ( I + h ~σ ( t ) i · ~σ ) = (cid:18) (1 + h σ z ( t ) i ) h σ − ( t ) ih σ + ( t ) i (1 − h σ z ( t ) i ) (cid:19) . (5)In Eq. (5) by the vector ~σ ( t ) we mean ( σ x ( t ) , σ y ( t ) , σ z ( t )) and h ~σ ( t ) i denotes the Bloch vectors which are solved usingEq. (2) to yield [16] h σ x ( t ) i = (cid:20) (cid:0) e γ at − (cid:1) (1 + cos(Φ)) (cid:21) e − γ (2 N +1+ a ) t h σ x (0) i − sin(Φ) sinh( γ at e − γ (2 N +1) t h σ y (0) i , h σ y ( t ) i = (cid:20) (cid:0) e γ at − (cid:1) (1 − cos(Φ)) (cid:21) e − γ (2 N +1+ a ) t h σ y (0) i − sin(Φ) sinh( γ at e − γ (2 N +1) t h σ x (0) i , h σ z ( t ) i = e − γ (2 N +1) t h σ z (0) i − N + 1) (cid:16) − e − γ (2 N +1) t (cid:17) . (6)In these equations a = sinh(2 r )(2 N th + 1). Using the Eqs. (6) in Eq. (5) and then reverting back to the Schr¨odingerpicture, the reduced density matrix of the system can be written as ρ s ( t ) = (cid:18) (1 + A ) Be − iωt B ∗ e iωt (1 − A ) (cid:19) , (7)where, in view of Eq. (5), A ≡ h σ z ( t ) i = e − γ (2 N +1) t h σ z (0) i − N + 1) (cid:16) − e − γ (2 N +1) t (cid:17) , (8) B = (cid:20) (cid:0) e γ at − (cid:1)(cid:21) e − γ (2 N +1+ a ) t h σ − (0) i + sinh( γ at e i Φ − γ (2 N +1) t h σ + (0) i . (9)From Eq. (6), it is seen that the system evolves towards a fixed asymptotic point in the Bloch sphere [15], whichin general is not a pure state, but the mixture ρ asymp = (cid:18) − q q (cid:19) , (10)where q = N +12 N +1 . If T = 0 and r = 0, then q = 1, and the pure state | i is reached asymptotically, an observationthat serves as the basis for the quantum deleter [32]. B. Simplified Jaynes-Cummings Model
Here we consider a simplified Jaynes-Cummings model taking into account the effect of the environment, which ismodelled as a zero temperature bath. In this model we consider the case of only a single excitation in the atom-cavitysystem with the bath modelling the effect of imperfect cavity mirrors. Also the cavity frequency is assumed to be inresonance with the atomic frequency [7, 33]. The total Hamiltonian is H = H S + H R + H SR = ω σ + σ − + X k ω k b † k b k + σ + X k g k b k + σ − X k g ∗ k b † k . (11)Here H S , H R and H SR stand for the Hamiltonians of the system, reservoir and system reservoir interaction, respec-tively. In the case of a single excitation in the atom-cavity system, the cavity mode can be eliminated in favour ofthe following effective spectral density I ( ω ) = 12 π γ κ ( ω − ω ) + κ . (12)Here ω is the atomic transition frequency and κ is the spectral width of the system-environment coupling. Tracingover the vacuum bath and assuming that initially there are no photons, the reduced density matrix of the atom(two-level system) can be obtained in the Schr¨odinger representation as ρ s ( t ) = (cid:18) a be − iω t b ∗ e iω t (1 − a ) (cid:19) , (13)where a = ρ , (0) e − κt (cid:20) cosh (cid:18) lt (cid:19) + κl sinh (cid:18) lt (cid:19)(cid:21) , (14a) b = ρ , − (0) e − κt (cid:20) cosh (cid:18) lt (cid:19) + κl sinh (cid:18) lt (cid:19)(cid:21) . (14b)Here l = p κ − γ κ , where γ , κ are as in Eq. (12). Initially the system is chosen to be in the state | ψ (0) i = cos( θ | i + e iφ sin( θ | i . (15)From the above equation, it can be easily seen that ρ , (0) = cos ( θ ) and ρ , − (0) = e − iφ sin( θ ). III. AMPLITUDE DAMPING AND GENERALIZED AMPLITUDE DAMPING CHANNELS
The generalized amplitude channel is generated by the evolution given by the master equation (2), with the bathsqueezing parameters r and Φ set to zero. Generalized amplitude damping channels capture the idea of energydissipation from a system, for example, in the spontaneous emission of a photon, or when a spin system at hightemperature approaches equilibrium with its environment (also cf. Refs. [29, 30]). A simple model of an amplitudedamping channel is the scattering of a photon via a beam-splitter. One of the output modes is the environment,which is traced out. The unitary transformation at the beam-splitter is given by B = exp (cid:2) θ ( a † b − ab † ) (cid:3) , where a, b and a † , b † are the annihilation and creation operators for photons in the two modes. A. Amplitude damping channel
This channel is generated by the evolution given by the master equation (2), with temperature T and bath squeezingparameters r and Φ set to zero. The corresponding Kraus operators are: E ≡ (cid:20) p − λ ( t ) 00 1 (cid:21) ; E ≡ (cid:20) p λ ( t ) 0 (cid:21) . (16)The effect of these operators is to produce the completely positive map X j E j (cid:18) A B ∗ B − A (cid:19) E † j = (cid:18) A (1 − λ ) √ − λB ∗ √ − λB − A + λA (cid:19) , (17)where, on comparison with Eq. (15), we see that A = cos ( θ / B = (1 / e iφ sin( θ ). The simplified Jaynes-Cummings model of the previous subsection is easily seen to realize an amplitude damping channel. It is straightfor-ward to verify that with the identification1 − λ ( t ) ≡ e − κt (cid:20) cosh (cid:18) lt (cid:19) + κl sinh (cid:18) lt (cid:19)(cid:21) . (18)the operators (16), acting on the state (15), reproduce the evolution (13) (in the interaction picture). B. Generalized amplitude damping channel
This channel is generated by the evolution governed by the master equation (2), with bath squeezing parameters r and Φ set to zero, but T not necessarily zero. The corresponding Kraus operators are: E ≡ √ p (cid:20) p − λ ( t ) 00 1 (cid:21) ; E ≡ √ p (cid:20) p λ ( t ) 0 (cid:21) ; E ≡ √ − p (cid:20) p − λ ( t ) (cid:21) ; E ≡ √ − p (cid:20) p λ ( t )0 0 (cid:21) , (19)where 0 ≤ p ≤ X j E j (cid:18) A B ∗ B − A (cid:19) E † j = p (cid:18) A (1 − λ ) √ − λB ∗ √ − λB − A + λA (cid:19) +(1 − p ) (cid:18) A + λ (1 − A ) √ − λB ∗ √ − λB − A + (1 − λ )(1 − A ) (cid:19) . (20)It is straightforward to verify that with the identification λ ( t ) ≡ − e − γ (2 N th +1) t ; p ≡ N th + 12 N th + 1 , (21)the operators (19) acting on the state (15) reproduce the evolution (6), with squeezing set to zero but temperaturenon-vanishing, by means of the map given by Eq. (1). If T = 0, then p = 1, reducing Eq. (19) to the amplitudedamping channel, given by Eq. (16). IV. THE SQUEEZED GENERALIZED AMPLITUDE DAMPING CHANNEL
This channel is generated by the evolution given by the master equation (2), with neither the bath squeezingparameters r and Φ nor the temperature T necessarily zero. Thus this is a very general (completely positive) mapgenerated by Eq. (2). To generalize (19) to include the effects of squeezing, we construct the following set of Krausoperators: E ≡ √ p (cid:20) p − α ( t ) 00 p − β ( t ) (cid:21) ; E ≡ √ p (cid:20) p β ( t ) p α ( t ) e − iφ ( t ) (cid:21) ; E ≡ √ p (cid:20) p − µ ( t ) 00 p − ν ( t ) (cid:21) ; E ≡ √ p (cid:20) p ν ( t ) p µ ( t ) e − iθ ( t ) (cid:21) . (22)It is readily checked that Eq. (22) satisfies the completeness condition X j =0 E † j E j = I , (23)provided p + p = 1 . (24)Substituting the Kraus operator elements given by Eq. (22) in Eq. (1), and using Eq. (5), yields the followingBloch vector evolution equations: h σ x ( t ) i = h ( p p (1 − α ( t ))(1 − β ( t ))) + p p (1 − µ ( t ))(1 − ν ( t )) + ( p p α ( t ) β ( t ) cos φ + p p µ ( t ) ν ( t ) cos θ ) i h σ x (0) i− h ( p p α ( t ) β ( t ) sin φ + p p µ ( t ) ν ( t ) sin θ ) i h σ y (0) i , (25a) h σ y ( t ) i = h ( p p (1 − α ( t ))(1 − β ( t )) + p p (1 − µ ( t ))(1 − ν ( t ))) − ( p p α ( t ) β ( t ) cos φ + p p µ ( t ) ν ( t ) cos θ ) i h σ y (0) i− h ( p p α ( t ) β ( t ) sin φ + p p µ ( t ) ν ( t ) sin θ ) i h σ x (0) i , (25b) h σ z ( t ) i = (1 − p ( µ ( t ) + ν ( t )) − p ( α ( t ) + β ( t ))) h σ z (0) i − p ( µ ( t ) − ν ( t )) − p ( α ( t ) − β ( t )) . (25c)Comparing Eqs. (25) with Eqs. (6), we can read off the corresponding terms. In fact, the system is underdeterminedas there are more variables than constraints. An inspection of Eqs. (6) shows that they yield a total of 5 constraintson the channel variables, p , p , µ , ν , α , β , θ and φ , with a further constraint coming from Eq. (24). The tworedundant variables may be conveniently chosen to be β and φ . Setting β = φ = 0, a comparison of Eqs. (25) and(6) produces the following relations: p p (1 − α ( t )) + p p (1 − µ ( t ))(1 − ν ( t )) = cosh (cid:18) γ at (cid:19) exp (cid:16) − γ N + 1) t (cid:17) , (26a) p p µ ( t ) ν ( t ) cos θ = cos(Φ) sinh (cid:18) γ at (cid:19) exp (cid:16) − γ N + 1) t (cid:17) , (26b) p p µ ( t ) ν ( t ) sin θ = sin(Φ) sinh (cid:18) γ at (cid:19) exp (cid:16) − γ N + 1) t (cid:17) , (26c) p α ( t ) + p ( µ ( t ) − ν ( t )) = 1(2 N + 1) (cid:16) − e − γ (2 N +1) t (cid:17) , (26d)1 − p α ( t ) − p ( µ ( t ) + ν ( t )) = e − γ (2 N +1) t , (26e)and the required squeezed generalized amplitude damping channel is given, in place of Eq. (22), by the Kraus operators E ≡ √ p (cid:20) p − α ( t ) 00 1 (cid:21) ; E ≡ √ p (cid:20) p α ( t ) 0 (cid:21) ; E ≡ √ p (cid:20) p − µ ( t ) 00 p − ν ( t ) (cid:21) ; E ≡ √ p (cid:20) p ν ( t ) p µ ( t ) e − iθ ( t ) (cid:21) . (27)It is seen from Eqs. (26) that at time t = 0, µ (0) = ν (0) = α (0) = 0. We now determine the remaining channelparameters. From Eqs. (26b) and (26c), we find tan θ = tan Φ , (28)allowing us to set θ = Φ, and to identify the channel parameter θ with the bath squeezing angle. The remainingchannel parameters may be identified as follows.From Eqs. (26d) and (26e), we find ν ( t ) = Np (2 N + 1) (1 − e − γ (2 N +1) t ) . (29)Substituting Eq. (29) into Eq. (26c), we find µ ( t ) = 2 N + 12 p N sinh ( γ at/ γ (2 N + 1) t/
2) exp (cid:16) − γ N + 1) t (cid:17) . (30)Using Eqs. (29) and (30) in (26e), we obtain α ( t ) = 1 p (cid:16) − p [ µ ( t ) + ν ( t )] − e − γ (2 N +1) t (cid:17) , (31)where ν ( t ) and µ ( t ) are given by Eqs. (29) and (30), respectively.Substituting Eqs. (29), (30), (31) and (24) into Eq. (26a), we obtain after some manipulations p ( t ) = 1( A + B − C − − D × (cid:2) A B + C + A ( B − C − B (1 + C ) − D ) − (1 + B ) D − C ( B + D − ± p D ( B − AB + ( A − C + D )( A − AB + ( B − C + D ) i , (32)where A = 2 N + 12 N sinh ( γ at/ γ (2 N + 1) t/
2) exp ( − γ (2 N + 1) t/ ,B = N N + 1 (1 − exp( − γ (2 N + 1) t )) ,C = A + B + exp( − γ (2 N + 1) t ) ,D = cosh ( γ at/
2) exp( − γ (2 N + 1) t ) . (33)
20 40 60 80 100 t0.20.40.60.81 Ν FIG. 1: ν ( t ) [Eq. (29)] with respect to t , with γ = 0 .
05. Asymptotically, ν ( t ) reaches the value 1. The solid and small-dashedcurves corresponds to the temperature (in units where ~ ≡ k B ≡ T = 1, with the bath squeezing parameter r = 0, 1,respectively, while the large-dashed curve corresponds to T = 3 and r = 1. If the squeezing parameter r is set to zero, then a = 0, and it follows from Eq. (32) and (33), that ˆ p = N th / (2 N th + 1), where the hat indicates that squeezing has been set to zero. It can be seen from Eq. (30), that forzero squeezing ( a = 0), ˆ µ ( t ) = 0. Substituting ˆ p = N th / (2 N th + 1) in Eq. (29), we find ˆ ν ( t ) = 1 − e − γ (2 N th +1) t .Now, a comparison of Eqs. (27) with Eqs. (19) shows that ˆ ν ( t ) = λ ( t ), given by Eq. (21).Further, in view of Eq. (24), ˆ p = ( N th + 1) / (2 N th + 1). Substituting Eq. (29) and the conditions a = 0 andˆ µ ( t ) = 0 into Eq. (31), it is easily verified thatˆ α ( t ) = 1 p (cid:16) − p [ˆ µ ( t ) + ˆ ν ( t )] − e − γ (2 N th +1) t (cid:17) = 2 N th + 1 N th + 1 (cid:18) ˆ ν ( t ) − N th N th + 1 ˆ ν ( t ) (cid:19) = 2 N th + 1 N th + 1 (cid:18) N th + 12 N th + 1 (cid:19) ˆ ν ( t )= ˆ ν ( t ) . (34)We thus have ˆ α ( t ) = ˆ ν ( t ) = λ ( t ), and hence the generalized amplitude damping channel (19) is recovered from Eq.(27) in the limit of vanishing squeezing.Figure 1 is a representative plot, showing that for large bath exposure time, ν ( t ) approaches 1. This Figure alsobrings out the concurrent behaviour of temperature and squeezing with respect to ν ( t ). At large t , α ( t ) also approachesunity. However, unlike the case of ν ( t ), temperature and squeezing can have a contrastive effect on α ( t ), as broughtout in Fig. 2. This contrastive effect of squeezing with respect to time has been observed in the case of mixedstate geometric phase [16] and quantum phase diffusion [20]. The dot-dashed curve in Fig. 2 represents a squeezedamplitude damping channel, i.e., a channel given by zero temperature but finite squeezing.The fact that as time progresses, squeezing effects tend to die out, leaving thermal effects alone to govern the systemevolution, is illustrated in Fig. 3. Squeezing of the bath modes introduces non-stationary effects due to correlationsbetween the modes. This Figure also shows the washing out of these effects with time being accentuated with increasein temperature.Unlike the case of the generalized amplitude damping channel, here the probabilities p ( t ) and p ( t ) are time-dependent on account of the presence of non-stationary effects introduced by the bath squeezing (Fig. 4), and p ( t )eventually reaches a stationary value of N/ (2 N + 1), as may be inferred from Eqs. (32) and (33). Substituting thisasymptotic value in Eqs. (29), (30) and (31), we find ν ( ∞ ) = 1, µ ( ∞ ) = 0 and α ( ∞ ) = 1, as was seen in Figs. 1, 3and 2, respectively.In the absence of squeezing, p ( ∞ ) becomes N th / (2 N th + 1), consistent with the expression for p in Eq. (21) forthe generalized amplitude damping channel. The solid line in Fig. 4 corresponds to the squeezed amplitude dampingchannel, which for the case of zero bath squeezing yields the action of a quantum deleter [32] via the amplitudedamping channel.If we have a = 0 and T = 0, then µ ( t | a = T = 0) = 0, as seen from Eq. (30), and p ( t ) = 0 because p ( t ) inEq. (32) reduces to N th / (2 N th + 1) when squeezing vanishes. Further, ν ( t | a = T = 0) = λ ( t | T = 0) by Eq. (34).Substituting these values in Eqs. (27), we obtain the amplitude damping channel. Eqs. (27) thus furnish a completerepresentation of a squeezed generalized amplitude damping channel.
20 40 60 80 100 t0.20.40.60.81 Α FIG. 2: α ( t ) [Eq. (31)] with respect to time t , with γ = 0 .
05, bringing out the counteractive effect of squeezing on temperature.Asymptotically, α ( t ) reaches 1. We find that increasing squeezing reduces α at any fixed temperature, and thus counteractsthe thermal effects. The solid and dot-dashed curves correspond to temperature (in units where ~ ≡ k B ≡ T = 0,with environment squeezing parameter r = 0 and 1, respectively. The small-dashed and large-dashed curves correspond totemperature T = 5, with r = 0 and 1, respectively.
20 40 60 80 100 t0.20.40.60.81 Μ FIG. 3: µ ( t ) [Eq. (30)] as a function of time t , with γ = 0 .
05 and r = 1. The asymptotic value of µ ( t ) is 0. The solid, large-dashed and small-dashed curves correspond to temperature (in units where ~ ≡ k B ≡ T equals 20, 5 and 1, respectively. V. CLASSICAL CAPACITY OF A SQUEEZED GENERALIZED AMPLITUDE DAMPING CHANNEL
A quantum communication channel can be used to perform a number of tasks, including transmitting classical orquantum information, as well as for the cryptographic purpose of creating shared information between a sender andreceiver, that is reliably secret from a malevolent eavesdropper [34]. A natural question is as to how informationcommunicated over a squeezed generalized amplitude damping channel (denoted E ), and given in the Kraus repre-sentation by Eq. (27), is degraded. In this Section, we briefly consider the communication of classical informationacross the channel [35]. The problem can be stated as the following game between sender Alice and receiver Bob:Alice has a classical information source producing symbols X = 0 , · · · , n with probabilities p , · · · , p n . She encodesthe symbols as quantum states ρ j (0 ≤ j ≤ n ) and communicates them to Bob, whose optimal measurement strategymaximizes his accessible information, which is bounded above by the Holevo quantity χ = S ( ρ ) − X j p j S ( ρ j ) , (35)where ρ = P j p j ρ j , and ρ j are various initial states [36]. In the present case, we assume Alice encodes her binarysymbols of 0 and 1 in terms of pure, orthogonal states of the form (15), and transmits them over the squeezedgeneralized amplitude damping channel.We further assume that Alice transmits her messages as product states, i.e., without entangling them across multiplechannel use. Then, the (product state) classical capacity C of the quantum channel is defined as the maximum of χ ( E ) over all ensembles { p j , ρ j } of possible input states ρ j [37, 38]. In Fig. 5, we plot χ ( E ) over pairs of orthogonalinput states ( θ , φ ) and ( θ + π, φ ) } , which correspond to the symbols 0 and 1, respectively, with probability of the0
50 100 150 200 t0.20.40.60.81p FIG. 4: Probability p [Eq. (32)] as a function of time t approaches the asymptotic value of N/ (2 N + 1). Here γ = 0 .
05. Thesolid curve corresponds to temperature (in units where ~ ≡ k B ≡ T = 0 and r = 0 .
05. The small-dashed and large-dashedcurves correspond to T = 2, with r equal to 0.1 and 0.5, respectively. We note that the solid line depicts the transformation ofthe squeezed amplitude damping channel to the amplitude damping channel. Θ Φ Χ Θ FIG. 5: Plotting the Holevo bound χ [Eq. (35)] for a squeezed amplitude damping channel with Φ = 0 and f = 0 .
5, overthe set { θ , φ } , which parametrizes the ensemble of input states { ( θ , φ ) , ( θ + π, φ ) } . Here temperature (in units where ~ ≡ k B ≡ T = 5, γ = 0 .
05, time t = 5 . r = 1. The channel capacity C is seen to correspondto the optimal value of θ = π/ √ ( | i ± | i ) for φ = 0]. input symbol 0 being f = 0 .
5. Here we take Φ = 0, and the optimum coding is seen to correspond to the choice( θ = π/ , φ = nπ ), where n ∈ I , i.e., the input states √ ( | i ± | i ) or √ ( | i ∓ | i ).Fig. 6 depicts χ ( E ) for various channel parameters, with the pair of orthogonal input states given by ( θ , φ = 0)and ( θ + π, φ = 0). As expected, longer exposure to the channel, or higher temperature, degrades information more,but the optimal choice of input states remains the same as before. Interestingly, squeezing improves the accessibleinformation for input states in a certain range of θ , but impairs it in other. This is consistent with the understandingthat the benefits of squeezing are quadrature-dependent. Fig. 7 demonstrates the contrastive effects of temperatureand squeezing on C . Comparing the solid and small-dashed curves, one notes that thermal effects tend to degrade C , whereas bath squeezing can improve it, as seen by comparing the small- and large-dashed curves. In fact, theimprovement due to squeezing is brought out dramatically by a comparison of the solid and large-dashed curves. Thishighlights the possible usefulness of squeezing to noisy quantum communication. VI. CONCLUSIONS
In this paper we have have obtained a Kraus representation of a noisy channel, which we call the squeezed generalizedamplitude damping channel, corresponding to the interaction of a two-level system (qubit) with a squeezed thermal1 Θ Χ FIG. 6: Optimal source coding for the squeezed amplitude damping channel, with χ plotted against θ corresponding to the“0” symbol. Here Φ = 0, γ = 0 .
05 and f = 0 .
5. It is seen that χ is maximized for states of the form (15) when the pair of inputstates are given by ( θ = π , φ = 0) and ( θ = π + π, φ = 0) [i.e., states √ ( | i ± | i )]. The solid and small-dashed curvesrepresent temperature (in units where ~ ≡ k B ≡ T = 0 and bath squeezing parameter r = 0, but t = 1 and 2, respectively.The large-dashed and dot-dashed curves represent T = 5 and t = 2, but with r = 0 and 2, respectively. A comparison of thesolid and small-dashed (small-dashed and large-dashed) curves demonstrates the expected degrading effect on the accessibleinformation, of increasing the bath exposure time t (increasing T ). A comparison of the large-dashed and dot-dashed curvesdemonstrates the dramatic effect of including squeezing. In particular, whereas squeezing improves the accessible informationfor the pair of input states √ ( | i ± | i ), it is detrimental for input states ( θ , φ ) given by (0 ,
0) (i.e., | i ) and ( π,
0) (i.e., | i ). FIG. 7: Interplay of squeezing and temperature on the classical capacity C of the squeezed amplitude damping channel (withinput states √ ( | i ± | i ), and f = 1 /
2, corresponding to the optimal coding). Here Φ = 0 and γ = 0 .
05. The solid andsmall-dashed curves correspond to zero squeezing r , and temperature (in units where ~ ≡ k B ≡ T = 0 and 5, respectively.The large-dashed curve corresponds to T = 5 and r = 2. A comparison between the solid and large-dashed curves shows thatsqueezing can improve C . bath via a dissipative interaction. The resulting dynamics, governed by a Lindblad-type evolution, generates acompletely positive map that extends the concept of a generalized amplitude damping channel, which corresponds toa dissipative interaction with a purely thermal bath. The physical motivation for studying this channel is that using asqueezed thermal bath the decay rate of quantum coherence can be suppressed, leading to preservation of nonclassicaleffects. This is in contrast to the case of a purely dephasing channel, where the action of squeezing, like temperature,tends to decohere the system. We studied the characteristics of the squeezed generalized amplitude damping channel,including its classical capacity C . We showed that as a result of bath squeezing, it is possible by a judicious choice ofthe input states, to improve C over the corresponding unsqueezed case. This could have interesting implications forquantum communication. [1] W. H. Louisell, Quantum Statistical Properties of Radiation (John Wiley and Sons, 1973).[2] A. O. Caldeira and A. J. Leggett, Physica A , 587 (1983). [3] W. H. Zurek, Phys. Today , 36 (1991); Prog. Theor. Phys. , 281 (1993).[4] J. Shao, M-L. Ge and H. Cheng, Phys. Rev. E , 1243 (1996).[5] D. Mozyrsky and V. Privman, Journal of Stat. Phys. , 787 (1998).[6] G. Gangopadhyay, M. S. Kumar and S. Dattagupta, J. Phys. A: Math. Gen. , 5485 (2001).[7] H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University Press, 2002).[8] V. Hakim and V. Ambegaokar, Phys. Rev. A , 423 (1985).[9] C. M. Smith and A. O. Caldeira, Phys. Rev. A , 3509 (1987); ibid , 3103 (1990).[10] H. Grabert, P. Schramm and G. L. Ingold, Phys. Rep. , 115 (1988).[11] S. Banerjee and R. Ghosh, Phys. Rev. A , 042105 (2000).[12] S. Banerjee and R. Ghosh, Phys. Rev. E , 056120 (2003).[13] K. Kraus, States, Effects and Operations (Springer-Verlag, Berlin, 1983).[14] M. Nielsen and I. Chuang,
Quantum Computation and Quantum Information (Cambridge University Press, Cambridge,2000).[15] S. Banerjee and R. Ghosh, to appear in J. Phys. A: Math. Theo.; eprint quant-ph/0703054.[16] S. Banerjee and R. Srikanth, to appear in Euro. Phys. J. D; eprint quant-ph/0611161.[17] T. A. B. Kennedy and D. F. Walls, Phys. Rev. A , 152 (1988).[18] M. S. Kim and V. Buˇzek , Phys. Rev. A , 610 (1993).[19] D. Wilson, J. Lee and M. S. Kim, Jl. Mod. Optics , 1809 (2003); eprint quant-ph/0206197.[20] S. Banerjee and R. Srikanth, eprint arXiv:0706.3633.[21] M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, Cambridge, 1997).[22] V. Buˇzek, P. L. Knight and I. K. Kudryavtsev, Phys. Rev. A , 1931 (1991).[23] G. Breitenbach, T. M¨uller, S. F. Ferreira, J.-Ph. Poizat, S. Schiller and J. Mlynek, J. Opt. Soc. Am. B , 2304 (1995).[24] N. Ph. Georgiades, E. S. Polzik, K. Edamatsu. H. J. Kimble and A. S. Parkins, Phys. Rev. Lett. , 3426 (1995).[25] Q. A. Turchette, N. Ph. Georgiades, C. J. Hood, H. J. Kimble and A. S. Parkins, Phys. Rev. A , 4056 (1998).[26] M. S. Kim and N. Imoto, Phys. Rev. A , 2401 (1995).[27] J. F. Poyatos, J. I. Cirac and P. Zoller, Phys. Rev. Lett. , 4728 (1996).[28] N. L¨utkenhaus, J. I. Cirac and P. Zoller, Phys. Rev. A , 548 (1998).[29] C. J. Myatt, B. E. King, Q. A. Turchette, C. A. Sackett, D. Kielpinski, W. M. Itano, C. Monroe and D. J. Wineland,Nature , 269 (2000).[30] Q. A. Turchette, C. J. Myatt, B. E. King, C. A. Sackett, D. Kielpinski, W. M. Itano, C. Monroe and D. J. Wineland,Phys. Rev. A , 3068 (1985); B. L. Schumacher and C. M. Caves, Phys. Rev. A ,3093 (1985).[32] R. Srikanth and S. Banerjee, to appear in Phys. Lett. A; eprint quant-ph/0611263.[33] B. M. Garraway, Phys. Rev. A , 2290 (1997).[34] B. Schumacher and M. D. Westmoreland, Phys. Rev. Lett. , 5695 (1998).[35] B. Schumacher and M. D. Westmoreland, Phys. Rev. A , 131 (1997).[36] A. S. Holevo, Probl. Peredachi Inf. , 3 (1993). [Probl. Inf. Transm. (USSR) , 197 (1973).[37] P. Hausladen, R. Josza, B. Schumacher, M. Westmoreland and W. K. Wootters, Phys. Rev. , 54 (1996).[38] B. Schumacher, M. Westmoreland and W. K. Wootters, Phys. Rev. Lett.76