Fidelity of Fock-state-encoded qubits subjected to continuous variable Gaussian processes
aa r X i v : . [ qu a n t - ph ] D ec Fidelity of Fock-state-encoded qubits subjected to continuous variable Gaussianprocesses
Brian Julsgaard ∗ and Klaus Mølmer Department of Physics and Astronomy, Aarhus University,Ny Munkegade 120, DK-8000 Aarhus C, Denmark. (Dated: March 7, 2018)When a harmonic oscillator is under the influence of a Gaussian process such as linear damping,parametric gain, and linear coupling to a thermal environment, its coherent states are transformedinto states with Gaussian Wigner function. Qubit states can be encoded in the | i and | i Fockstates of a quantum harmonic oscillator, and it is relevant to know the fidelity of the output qubitstate after a Gaussian process on the oscillator. In this paper we present a general expression for theaverage qubit fidelity in terms of the first and second moments of the output from input coherentstates subjected to Gaussian processes.
PACS numbers: 03.67.-a, 03.67.Hk
I. INTRODUCTION
In analogy with the classical bit in computer science,the qubit forms the most basic building block within thefield of quantum information [1]. In order to performquantum computation one must, among other tasks, beable to initialize, manipulate, and read out the informa-tion encoded in qubits, and in a scalable implementationit is necessary to store quantum information and trans-port it from one place or medium to another. Such oper-ations are applied in quantum memories for few-photonlight pulses in single atoms [2, 3] and in quantum tele-portation between similar qubits [4, 5]. In parallel toqubit-based quantum information science there has alsobeen attention to continuous-variable versions of quan-tum computation [6], quantum teleportation [7–9], andquantum memories [10–14]. These protocols can be im-plemented in, e.g., quadrature variables of electromag-netic fields [7], atomic or solid state ensembles of spins[15, 16], or vibration modes of nano-mechanical oscilla-tors [17–21], which are all exact or excellent approximaterealizations of the quantum harmonic oscillator.While the discrete and continuous variable versions ofquantum information originally seemed as detached sci-entific domains, there have been demonstrations of sin-gle light quanta, discrete in nature, transferred into thecollective spin degrees of freedom of macroscopic atomicensembles, which are continuous in nature [22–24]. Morerecently, quantum memories for photonic qubits havebeen implemented benefiting from the increased collec-tive interaction strength of atomic ensembles comparedto single atoms [25–27]. In connection with the use of hy-brid physical systems for quantum information process-ing, multiple proposals exist, making use of the intercon-nection of mesoscopic qubit degrees of freedom and thecontinuous variables of ensembles of microscopic systems,nano-mechanical resonators and quantized field modes ∗ [email protected] [25, 28–35].The present manuscript addresses an important ques-tion in this context: If the transformation properties ofcontinuous variables are known for a particular processin a given physical system, then what can be said about aqubit encoded into the same system and subjected to thesame transformation? Specifically, if a harmonic oscilla-tor is subjected to a Gaussian process, characterized byits effect on the first and second moments of the conju-gate variables ˆ X and ˆ P , we present a general formula forthe qubit fidelity, i.e. the probability that the input stateof a qubit encoded into the | i and | i Fock states of theharmonic oscillator coincide with the output state afterthe system has been exposed to the process. A Gaussianprocess can be characterized completely by its action ona small set of coherent states [36], and as pointed outin Ref. [8] this is easier than preparing qubit states forexperimental determination of its fidelity. Also, from atheoretical perspective, as exemplified by Ref. [37], thequantum memory fidelity for qubit states can be calcu-lated more easily in a multi-mode set-up by using coher-ent input states and accounting solely for the first andsecond moments of the physical variables involved.The paper is arranged as follows: In Sec. II we showhow the observed first and second moments of outputstates, following from application of coherent input statesto the process, yield a convenient parametrization of theGaussian process. In Sec. III we derive the average fi-delity over all qubit states encoded in the | i and | i Fock states and subjected to the Gaussian process. InSec. IV, we present some specific examples, and in Sec. Vwe conclude the manuscript.
II. PARAMETRIZING THE GAUSSIANPROCESS
We consider a process, which maps an input quantumstate of a single harmonic oscillator to an output stateon the same or a different oscillator. For instance thiscould represent the storage of a radiation-field state into
Linear process proces (a) (b) in out process FIG. 1. (a) A single mode of a harmonic oscillator is subjectedto a Gaussian process, which maps the input observables ˆ X in and ˆ P in into the output variables ˆ X out and ˆ P out , of the same ora different quantum system. The wavy arrows represent, e.g.,absorption losses or addition of thermal noise associated withthe possible coupling to environment degrees of freedom. (b)Schematic view of the transformation of a coherent input statewith Var( ˆ X in ) = Var( ˆ P in ) = . The solid arrows indicate themean values, and the circle and the ellipse show the standarddeviation of the continuous quadrature variables in the xp -plane. For the output state, θ denotes the angle betweenthe x -axis and the major axis of the uncertainty ellipse with σ ≥ σ . polarization modes of a spin ensemble [10], or it couldrepresent the teleportation of the quadrature amplitudesfrom one laser beam to another [7]. Figure 1(a) showsschematically how this process transforms the input har-monic oscillator mode ( ˆ X in , ˆ P in ) into the output mode( ˆ X out , ˆ P out ) under the possible influence of the environ-ment. For any input state density matrix ˆ ρ in this processis mathematically described by a map, ˆ ρ out = E (ˆ ρ in ), andour task is to (i) establish a suitable parametrization ofthis map, and to (ii) calculate the fidelity when a qubitstate is subjected to the process.The restriction to Gaussian processes relies on two as-sumptions about the map E . The first one is that itis linear in the sense that our input and output har-monic oscillator modes couple linearly to each otherand to all auxiliary reservoir modes. Thus we assumethat ( ˆ X in , ˆ P in ), ( ˆ X out , ˆ P out ), and the reservoir variables(ˆ x j , ˆ p j ) with j = 1 , . . . , n , obey the equation: ˆ X out ˆ P out ˆ x out1 ˆ p out1 ...ˆ x out n ˆ p out n = A ˆ X in ˆ P in ˆ x in1 ˆ p in1 ...ˆ x in n ˆ p in n , (1)where A is a 2( n + 1) × n + 1) matrix. To preserve canonical commutator relations, A must be a symplecticmatrix [38], which we shall of course assume to hold inthe following. The first two rows of this set of equationscan be rewritten as: (cid:20) ˆ X out ˆ P out (cid:21) = (cid:20) A A A A (cid:21) (cid:20) ˆ X in ˆ P in (cid:21) + (cid:20) ˆ F x ˆ F p (cid:21) , (2)where, ˆ F = [ ˆ F x ˆ F p ] T , are noise operators and representthe combined influence of the reservoir modes.Our second assumption about E is that all the reser-voir modes are described by Gaussian states and areuncorrelated to the input state, h ˆ X in ˆ F x i = h ˆ X in ih ˆ F x i ,etc. This means in particular that the operators ˆ F show Gaussian fluctuations, and in order to preservethe commutation relation of the output mode it is re-quired that [ ˆ F x , ˆ F p ] = i (1 − det( ˜ A )), where ˜ A is theupper 2 × A used in Eq. (2). The secondmoments of the input operators, the output operators,and the noise operators are all given by covariance ma-trices. For instance, for the output mode the covariancematrix reads γ out = h { δ ˆ y out · δ ˆ y Tout }i , where the vec-tor ˆ y out = [ ˆ X out ˆ P out ] T and where ˆ y out = h ˆ y out i + δ ˆ y out defines the fluctuations of operators around their meanvalues. The output mode covariance matrix thus reads: γ out = 2 (cid:20) Var( ˆ X out ) Cov( ˆ X out , ˆ P out )Cov( ˆ X out , ˆ P out ) Var( ˆ P out ) (cid:21) , (3)where Cov( ˆ X out , ˆ P out ) = h δ ˆ X out δ ˆ P out + δ ˆ P out δ ˆ X out i .Similar covariance matrices γ in and γ F are defined forthe input and the noise parts, respectively. The secondmoments of the operators, i.e., the covariance matrices,fulfill: γ out = ˜ A γ in ˜ A T + γ F . (4)It was shown recently that coherent states suffice as inputstates to fully characterize a process on harmonic oscilla-tor modes [39], and in the case of a Gaussian process withthe linear transformation (1) of the canonical variables, asmall discrete set of coherent states is enough to yield thecomplete information about the process [36]. Gaussianstates are described completely by their first and secondmoments, and the matrix ˜ A , the two mean values h ˆ F x i and h ˆ F p i , and the three real parameters of γ F are suffi-cient to describe the entire Gaussian process. We shallnow show how the process may equivalently be character-ized by the quantities indicated in Fig. 1(a), which areexperimentally available when applying coherent inputstates to the process.For the vacuum input state h ˆ X out i = h ˆ F x i and h ˆ P out i = h ˆ F p i map out the mean values of the noise operatorsof the environment, and then two other coherent inputstates with non-zero mean values suffice to map out theentries of the matrix ˜ A , since h y out i = ˜ A h y in i + h ˆ F i .In turn, since γ in is the identity matrix for any coher-ent state, the second moments of the output mode op-erators establish the relations, Var( ˆ F x ) = Var( ˆ X out ) − ( A + A ), Var( ˆ F p ) = Var( ˆ P out ) − ( A + A ), andCov( ˆ F x , ˆ F p ) = Cov( ˆ X out , ˆ P out ) − ( A A + A A ).In the following, we assume without loss of generalitythat h ˆ F x i = h ˆ F p i = 0, since any known non-zero meanvalue added to the output mode can be readily identifiedby experiment and subtracted by a simple displacement,which will not degrade our knowledge of the quantumstate. It is convenient to use the parametrization for thesecond moments of the output mode shown in Fig. 1(b),i.e., the variances, σ and σ , along the main axes of the“noise ellipse” and the angle θ between the x -axis andthe major axis of the noise ellipse. These variables relateto the parameters σ x = Var( ˆ X out ), σ p = Var( ˆ P out ), and C x,p = Cov( ˆ X out , ˆ P out ) of γ out by: σ = ¯ σ + δσ , σ = ¯ σ − δσ , tan(2 θ ) = 2 C x,p σ x − σ p , with ¯ σ = σ x + σ p , δσ = r
14 ( σ x − σ p ) + C x,p . (5)If C x,p is positive (negative), 0 < θ < π ( − π <θ < σ x = σ p and C x,p = 0 we assume θ = π sign( C x,p ) . For a coherent input state, ˆ ρ in = | α ih α | , the out-put state ˆ ρ out can always be described by a displaced,squeezed, thermal state, offering enough variables toparametrize the Gaussian state, illustrated in Fig. 1(b).We shall now provide a convenient expression of this out-put state as a function of the coherent state amplitude, α , and the parameters, A , A , A , A , σ , σ , and θ ,discussed above. To this end we define first the thermalstate: ˆ ρ = 1 π ¯ n Z d γe −| γ | / ¯ n | γ ih γ | , (6)where the integral is carried out over all coherent states | γ i . Applying the squeezing operator ˆ S ( r ) = e r (ˆ a − ˆ a † ) ,where r is a real parameter, to the thermal state we ob-tain the squeezed thermal state: ˆ ρ STS = ˆ S ( r )ˆ ρ ˆ S † ( r )with well-known properties [40]. With the standard def-initions ˆ X = ˆ a +ˆ a † √ and ˆ P = − i (ˆ a − ˆ a † ) √ this state hasVar( ˆ X ) = (¯ n + ) e − r and Var( ˆ P ) = (¯ n + ) e r . Bychoosing appropriately the values of ¯ n and r , σ = (¯ n + 12 ) e − r , σ = (¯ n + 12 ) e r . (7)and applying, finally, the rotation operator ˆ R ( θ ) = e iθ ˆ a † ˆ a we obtain a rotated squeezed thermal state,ˆ ρ r = ˆ R ( θ )ˆ ρ STS ˆ R † ( θ ) , (8)with precisely the noise properties indicated by the out-put ellipse shown in Fig. 1(b).The correct dependence of the output state mean val-ues on the amplitude of the input coherent state is re-produced by applying the displacement operator ˆ D (¯ α ) = e ¯ α ˆ a † − ¯ α ∗ ˆ a to ˆ ρ r such that a coherent input state is mappedto the output state, E ( | α ih α | ) = ˆ ρ α , (9)with ˆ ρ α = ˆ D (¯ α ) ˆ R ( θ ) ˆ S ( r )ˆ ρ ˆ S † ( r ) ˆ R † ( θ ) ˆ D † (¯ α ) , (cid:20) ¯ α R ¯ α I (cid:21) = (cid:20) A A A A (cid:21) (cid:20) α R α I (cid:21) , (10)where “R” and “I” refer to the real and imaginary parts,respectively, of input mean amplitude α and output meanamplitude ¯ α . It is convenient to introduce the equivalentrelations between α and ¯ α in complex notation:¯ α = Cα + Dα ∗ ,C = 12 ( A − iA + iA + A ) ,D = 12 ( A + iA + iA − A ) . (11)We note that rather than presenting a map on the in-put coherent state, Eq. (10) formally provides the out-put state as an α -dependent transformation of a definiteinput state: | α ih α | → E ( | α ih α | ) ≡ ˆ D (¯ α )ˆ ρ r ˆ D † (¯ α ). Thisform is, however, perfectly useful to characterize the pro-cess and it is a good starting point for our analysis of thequbit fidelity in the next section. III. DERIVATION OF THE QUBIT FIDELITYFORMULA
From the coherent-state expansion on the Fock-statebasis, | α i = e − | α | ∞ X n =0 α n √ n ! | n i , (12)we see that the Fock basis states can be formally ob-tained from expressions involving coherent states by | n i = √ n ! ∂ n ∂α n [ e | α | | α i ] | α =0 . In turn, due to the linearityof the map E , its action on a general Fock state outerproduct can be retrieved as: E ( | n ih m | ) = 1 √ n ! m ! ∂ n ∂α n ∂ m ∂α ∗ m [ e | α | E ( | α ih α | )] | α =0 . (13)Any qubit state expanded on the Fock states | n = 0 i and | n = 1 i can thus be mapped if we know the quantities E ( | n = 0 ih n = 0 | ) = E ( | α = 0 ih α = 0 | ), E ( | ih | ) = ∂∂α E ( | α ih α | ) | α =0 , E ( | ih | ) = ∂∂α ∗ E ( | α ih α | ) | α =0 , and E ( | ih | ) = (1 + ∂ ∂α∂α ∗ ) E ( | α ih α | ) | α =0 .The derivatives can be expressed in terms of ¯ α usingEq. (11): ∂∂α = C ∂∂ ¯ α + D ∗ ∂∂ ¯ α ∗ ,∂∂α ∗ = D ∂∂ ¯ α + C ∗ ∂∂ ¯ α ∗ , (14) ∂ ∂α∂α ∗ = CD ∂ ∂ ¯ α + ( | C | + | D | ) ∂ ∂ ¯ α∂ ¯ α ∗ + ( CD ) ∗ ∂ ∂ ¯ α ∗ . Only the displacement operators in Eq. (10) dependon the coherent state amplitudes, and their deriva-tives are given by ∂ ˆ D (¯ α ) ∂ ¯ α = (cid:16) ˆ a † − ¯ α ∗ (cid:17) ˆ D (¯ α ), ∂ ˆ D (¯ α ) ∂ ¯ α ∗ = − (cid:0) ˆ a − ¯ α (cid:1) ˆ D (¯ α ), and their hermitian conjugates. Thefirst and second derivatives of E ( | α ih α | ) with respect to¯ α and ¯ α ∗ are thus given by ∂E∂ ¯ α = ˆ a † ˆ ρ r − ˆ ρ r ˆ a † ,∂E∂ ¯ α ∗ = − ˆ a ˆ ρ r + ˆ ρ r ˆ a,∂ E∂ ¯ α = ˆ a † ˆ ρ r − a † ˆ ρ r ˆ a † + ˆ ρ r ˆ a † , (15) ∂ E∂ ¯ α ∗ = ˆ a ˆ ρ r − a ˆ ρ r ˆ a + ˆ ρ r ˆ a ,∂ E∂ ¯ α∂ ¯ α ∗ = − ˆ a † ˆ a ˆ ρ r + ˆ a † ˆ ρ r ˆ a + ˆ a ˆ ρ r ˆ a † − ˆ ρ r ˆ a ˆ a † , where ˆ ρ r is given in Eq. (8), and the right hand sides areformally independent of α (the derivatives are evaluatedat α = 0).The fidelity is defined as the overlap of the state sub-ject to the transformation E with the original qubit stateand thus requires matrix elements of the left-hand sideof Eq. (13) between the Fock states | i and | i . In turn,using Eqs. (13)-(15) this is equivalent to calculating ma-trix element of the right-hand side of Eq. (15) betweenthe Fock states | i and | i . Now, due to the raisingand lowering operators in this equation (up to quadraticorder) we end up with matrix elements on the form h n ′ | ˆ ρ r | m ′ i = e iθ ( n ′ − m ′ ) h n ′ | ˆ ρ STS | m ′ i , where the integers n ′ and m ′ may take values from 0 to 3. For instance, wehave h | E ( | ih | ) | i = C h | ˆ a † ˆ ρ r − ˆ ρ r ˆ a † | i + D ∗ h | − ˆ a ˆ ρ r +ˆ ρ r ˆ a | i = C [ h | ˆ ρ STS | i−h | ˆ ρ STS | i ] −√ D ∗ e iθ h | ˆ ρ STS | i ,and the first term in this expression can be calculated di-rectly as h | ˆ ρ STS | i = 1 π ¯ n Z d γe −| γ | / ¯ n |h | ˆ S ( r ) | γ i| = 1 π ¯ n cosh( r ) Z d γe − n n | γ | + γ γ ∗ tanh( r ) = 1 q(cid:2) ( + ¯ n ) e − r + (cid:3) (cid:2) ( + ¯ n ) e r + (cid:3) = 1[( σ + )( σ + )] / . (16) The first equality, in which h n = 0 | refers to the Fockbasis and | γ i to the coherent-state basis, follows from theexpansion (6) of the thermal state on coherent states, thesecond line exploits the Fock-state expansion of squeezedcoherent states [41]: h n | ˆ S ( r ) | γ i = e − | γ | + γ tanh( r ) p n ! cosh( r ) (cid:18)
12 tanh( r ) (cid:19) n × H n (cid:16) γ/ p sinh(2 r ) (cid:17) , (17)where H n is a Hermite polynomial, the third line car-ries out the γ -integration, and the last step applies therelations in Eq. (7). Similar calculations are readily per-formed for the remaining relevant matrix elements andyield: h | ˆ ρ STS | i = σ σ − [( σ + )( σ + )] / , h | ˆ ρ STS | i = σ − σ √ σ + )( σ + )] / , h | ˆ ρ STS | i = (cid:0) σ σ − (cid:1) + ( σ − σ ) [( σ + )( σ + )] / h | ˆ ρ STS | i = √ σ σ − )( σ − σ )4[( σ + )( σ + )] / . (18)By integrating the fidelity for any input qubit state, | ψ (Ω) i = cos θ | i + e iφ sin θ | i with 0 ≤ θ ≤ π and0 ≤ φ ≤ π , we determine the average qubit fidelity F q : F q = 14 π Z d Ω h ψ (Ω) | E ( | ψ (Ω) ih ψ (Ω) | ) | ψ (Ω) i = 13 [ h | E ( | ih | ) | i + h | E ( | ih | ) | i ]+ 16 [ h | E ( | ih | ) | i + h | E ( | ih | ) | i ]+ 16 [ h | E ( | ih | ) | i + h | E ( | ih | ) | i ] , (19)where | i and | i refer to Fock states. With the expres-sion derived above, we thus reach the final, explicit ex-pression for the average qubit fidelity in terms of themapping parameters of the Gaussian process: F q = 16 q ( σ + )( σ + ) ( σ σ − )( σ + )( σ + ) + Re { C + ˜ D ∗ } σ + + Re { C − ˜ D ∗ } σ + − | C + ˜ D ∗ | ( σ − σ + ) − | C − ˜ D ∗ | ( σ − σ + ) − | C + ˜ D ∗ | ( σ − ) + | C − ˜ D ∗ | ( σ − )2( σ + )( σ + ) ) , (20)where ˜ D = De − iθ . This is the main results of the arti-cle, and in the next section we shall consider the fidelityformula in various specific cases, corresponding to theexperimental storage and transfer schemes mentioned inthe Introduction.Let us briefly discuss the different effects contribut-ing to a reduction of the fidelity. First, we observe thatEq. (20) decreases when σ , become large. This is natu-ral, as the qubit occupies only the lowest two Fock states,while the output state is distributed toward higher num-ber states n ∝ σ , σ , and hence a corresponding smallerfraction of the population remains in the qubit space.Even with σ , close to the minimum allowed by theHeisenberg uncertainty relation, the values of C, D and θ can lead to large variations in the qubit fidelity. Thisis associated with the possibility for the map to yieldan (undesired) unitary operation on the qubit, e.g., inthe form of a rotation of the Bloch vector around the z -axis, caused by a rotation of the continuous quadraturevariables in the ( ˆ X, ˆ P ) phase space. Thus, the unitarymapping ˆ X → − ˆ X and ˆ P → − ˆ P , represented by A = A = − A = A = 0, and σ = σ = ,yields, according to Eq. (20), an average qubit fidelity F q = . The mapping, however, is perfect, if we onlyredefine the basis states by a simple phase change of − π after the process, and it makes sense to allow incorpo-ration of such a trivial transformation in the definitionof the average qubit fidelity. The effect on Eq. (20) of aphase rotation by θ ′ corresponds to setting C → Ce iθ ′ and ˜ D ∗ → ˜ D ∗ e iθ ′ , which affects only the two terms lin-ear in C and ˜ D ∗ in Eq. (20). The angles θ ′ yielding theextremal values of F q are thus given by: e iθ ′ = ( C ∗ + ˜ D )( σ + ) + ( C ∗ − ˜ D )( σ + )( C + ˜ D ∗ )( σ + ) + ( C − ˜ D ∗ )( σ + ) . (21)For the simple map with the π rotation, qubit fidelityextrema are found at θ ′ = nπ , where n is an integer, andfor odd n the rotation is counter-acted and a unit fidelityis recovered. IV. EXAMPLESA. Symmetric gain and variance
Consider the specific case where both ˆ X and ˆ P aremultiplied by the same gain coefficient g in the trans-formation process such that A = A = g and A = − − − − − − − g σ − FIG. 2. (Color online) A contour plot of the qubit fidelityfor symmetric gain and variances according to Eq. (22) asa function of the gain imperfection, 1 − g , and the excessvariance relative to the vacuum noise limit, 2 σ −
1. A fewvalues of F q are marked on the graph with the dashed curveenclosing the non-classical limit F q > . A = 0. Assume also the added noise to be symmetric, σ = σ = σ . Then the fidelity becomes F q = 6 σ + 3 σ + g (2 σ + 1) − g (3 σ − )6( σ + ) , (22)which is identical to the result found in [8]. The value of F q as a function of g and σ is shown in Fig. 2.By a projective qubit measurement, one obtains anoutcome that may be stored by classical means, and thecorresponding eigenstate may be reinstalled in the physi-cal output system at any later time. This classical proce-dure provides a qubit state with an average overlap withthe unknown initial state of 2 / F q = 2 / B. An oscillator coupled to a heat bath
Consider a harmonic oscillator, e.g. a cavity field withresonance frequency ω , coupled by an energy-decay rate γ to an external heat bath at temperature T . Thecharacteristic number of excitations in the heat bath is¯ N = [exp( ¯ hω k B T ) − − , and the quantum Langevin equa-tions for the oscillator mode ˆ a can be written [42]: ∂ ˆ a∂t = − iω ˆ a − γ a − √ γ ˆ b in , (23) (a) γt F q (b) − − ¯ N γ T F q = / FIG. 3. (Color online) (a) The decay of qubit fidelity whenthe harmonic oscillator hosting the qubit is coupled to a heatbath by a decay rate γ . Each curve corresponds to a spe-cific bath temperature with ¯ N denoting the mean excitationlevel of the oscillator in equilibrium. From above: ¯ N = 0(red), ¯ N = 0 . N = 1 (blue), ¯ N = 3 (cyan), and¯ N = 10 (magenta). The horizontal dashed line denotes thenon-classical limit F q > and the vertical dashed lines markthe time T F q =2 / at which this limit is reached. This charac-teristic time is also shown in (b) on the vertical axis (in unitsof γ − ) as a function of the equilibrium excitation level ¯ N . where ˆ b in is the input thermal field, which in the broad-band approximation satisfies h ˆ b in ( t )ˆ b † in ( t ′ ) i = ( ¯ N +1) δ ( t − t ′ ) and h ˆ b † in ( t )ˆ b in ( t ′ ) i = ¯ N δ ( t − t ′ ). Eq. (2) yields thesolution of Eq. (23) with g ≡ A = A = e − γt and A = A = 0. From the properties of ˆ b in we de-duce that Var( ˆ F x ) = Var( ˆ F p ) = ( ¯ N + )(1 − e − γt ) andCov( ˆ F x , ˆ F p ) = 0, and hence for a coherent-state inputthe variances of the output state is σ ≡ σ = σ = + ¯ N (1 − e − γt ). The qubit fidelity now follows frominserting the parameters g and σ into Eq. (22), and theresulting fidelity is shown in Fig. 3.We observe that the decay of fidelity occurs faster whenthe heat bath temperature is increased. In Fig. 3(a) theinitial linear decrease in F q follows the approximate for-mula: F q ≈ − (2+5 ¯ N ) γt . In the asymptotic limit t → ∞ the fidelity converges, F q → ¯ N + ( ¯ N +1) , i.e. for ¯ N = 0 thequbit decays to the ground state | i which has a 50 %chance of reproducing the random input qubit, and forlarge ¯ N the oscillator is most likely excited away from thequbit space spanned by | i and | i leading to a vanishingfidelity.The horizontal dashed line with F q = 2 / ǫ F q FIG. 4. (Color online) The qubit fidelity from Eq. (25) asa function of ǫ , which is conveniently used to parametrizethe asymmetry in gain and variance by g x = g ǫ , g p = g /ǫ , σ x = σ ǫ , and σ p = σ /ǫ . From the top, g = 1 and σ = (black), g = 1 and σ = . (red), g = 0 . σ = (green), and g = 0 . σ = . (blue). which occurs at the ¯ N -dependent times marked by thevertical dashed lines. In Fig. 3(b) the these times areshown more generally as a function of ¯ N . The ¯ N → γT F q =2 / → − ln( √ − ≈ .
88, i.e. for anexponentially decreasing coherence, the process super-sedes the classical benchmark for times less than 88 % ofthe coherence time.
C. Asymmetric gain and variance along the samemajor axes
In most practical cases with asymmetric gain and vari-ance, the asymmetries materialize along the same axes in( ˆ X, ˆ P )-space. One example is the degenerate parametricamplifier [44], for which the transformations are h ˆ X out i = G h ˆ X in i , h ˆ P out i = G − h ˆ P in i , σ x = G , σ p = G , and C x,p = 0, i.e. the coordinate system is chosen, withoutloss of generality, such that the mean value transforma-tion ˜ A is diagonal and at the same time it turns outthat θ = 0, i.e. the ( ˆ X, ˆ P )-axes form also the major axesfor the covariance matrix γ out . Another example can befound in spin-ensemble based quantum memories, whichencode quantum information into the transverse compo-nents ˆ X ≡ ˆ S x / p | S z | and ˆ P ≡ − ˆ S y / p | S z | of a macro-scopic spin polarized along the negative z -direction. Foran inhomogeneous distribution of spin frequencies thestored information is “diffused” into the spin ensembleand recalled as a spin echo using a set of π pulses forinverting the ensemble population. These π pulses em-ploy a spin rotation around a certain axis and therebybreak the symmetry of the ( ˆ X, ˆ P )-space, and especiallyfor non-ideal π pulses the transformations (2) and (4)become asymmetric (in some cases even squeezed) [45].In this case also, ˜ A and γ turn out diagonal in a com-mon coordinate system, and the two above examples thusmotivate a closer look on the particular transformation:˜ A = (cid:20) g x g p (cid:21) , γ out = (cid:20) σ x
00 2 σ p (cid:21) . (24)As long as the Heisenberg uncertainty relation, σ x σ p ≥ ,is satisfied we allow σ x and σ p to take any value meet-ing the constraints 2 σ ≥ g x and 2 σ ≥ g p imposed byEq. (4) and the positivity of ( γ F ) and ( γ F ) . Wemay think of this transformations as a noisy paramet-ric amplifier (the version discussed above is a minimum-uncertainty case). When the properties of (24) are in-serted into the general formula (20) we find: F q = 16 q ( σ x + )( σ p + ) ( σ x σ p − )( σ x + )( σ p + )+ g x σ x + + g p σ p + − g x ( σ x − σ x + ) − g p ( σ p − σ p + ) − g x ( σ p − ) + g p ( σ x − )2( σ x + )( σ p + ) ) . (25)In order to illustrate how the asymmetry affects the qubitfidelity, we show in Fig. 4 a number of curves, where foreach curve the products g x g p ≡ g and σ x σ p ≡ σ remainconstant but the degree of asymmetry is changed alongthe horizontal axis, see the figure caption for explanation.We note that the upper curve corresponds to the specialcase of a noiseless parametric amplifier for which F q = √
23 cosh(2 r )+2 cosh( r )+3[1+cosh(2 r )] / , where we parametrized the gainas G = ǫ = e r . This expression for F q stays above theclassical benchmark for ǫ < ∼ . V. SUMMARY
In this paper we have presented calculations yieldingthe average fidelity for storage and transfer of qubit states which are encoded in the | i and | i Fock states of a har-monic oscillator, subjected to a Gaussian process. Sincecoherent states form a complete basis for the harmonicoscillator, the parameters characterizing a Gaussian pro-cess can be determined by its action on coherent states,and subsequently the action of the process on any class ofquantum states can be obtained. The main result of ourcalculation is the explicit expression, Eq. (20), for the av-erage qubit fidelity for a general Gaussian process. Thisexpression shows how imperfect gain and added noiseboth contribute to the infidelity of protocols handlingqubits in oscillator degrees of freedom. It also shows,however, that part of the infidelity may be recovered bymerely redefining the phases of the qubit basis states.There has already been considerable efforts to deter-mine the fidelity of Gaussian operations acting on oscil-lators prepared in coherent states, squeezed states andqubit states, and in connection with the beam-splitterlike coupling of light modes and atomic ensembles, theaverage qubit fidelity has been calculated in Ref. [8]. Ourtheory, indeed, reproduces that result when we restrict tosymmetric gain and noise. Currently, however, there isa growing experimental interest in hybrid quantum sys-tem architectures, where, e.g., effective two-level systemsare used for preparation and processing of qubit states,while oscillator systems are used for storage and trans-port. These systems apply different coupling schemesand frequently the couplings to the quadratures of theelectromagnetic, mechanical or collective spin oscillatorsdiffer, leading to asymmetries in the ( ˆ X, ˆ P ) phase-space.Another source of asymmetry may occur during process-ing of the individual oscillator modes, as exemplified by π pulses applied to spin ensembles in Ref. [45]. The generalexpression Eq. (20) and the examples given in Sec. IVproperly describe the fidelity of qubit manipulation insuch hybrid systems. ACKNOWLEDGMENTS