Influence of dissipation on the extraction of quantum states via repeated measurements
aa r X i v : . [ qu a n t - ph ] J u l Influence of dissipation on the extraction of quantum states via repeatedmeasurements
B. Militello, ∗ K. Yuasa,
2, 3, † H. Nakazato, and A. Messina MIUR and Dipartimento di Scienze Fisiche ed Astronomichedell’Universit`a di Palermo, Via Archirafi 36, I-90123 Palermo, Italy Dipartimento di Fisica, Universit`a di Bari, I-70126 Bari, Italy Istituto Nazionale di Fisica Nucleare, Sezione di Bari, I-70126 Bari, Italy Department of Physics, Waseda University, Tokyo 169-8555, Japan (Dated: October 30, 2018)A quantum system put in interaction with another one that is repeatedly measured is subject toa non-unitary dynamics, through which it is possible to extract subspaces. This key idea has beenexploited to propose schemes aimed at the generation of pure quantum states (purification). Allsuch schemes have so far been considered in the ideal situations of isolated systems. In this paper, weanalyze the influence of non-negligible interactions with environment during the extraction process,with the scope of investigating the possibility of purifying the state of a system in spite of thesources of dissipation. A general framework is presented and a paradigmatic example consisting oftwo interacting spins immersed in a bosonic bath is studied. The effectiveness of the purificationscheme is discussed in terms of purity for different values of the relevant parameters and in connectionwith the bath temperature.
PACS numbers: 03.65.Ta, 03.65.Yz, 32.80.Pj, 32.80.Qk
I. INTRODUCTION
During the last decades, enormous progresses havebeen made in such physical contexts as cavity quantumelectrodynamics (CQED) [1], superconductor-based cir-cuits [2], and trapped ions [3]. In connection with the ap-plications in the field of quantum information [4], seminalexperimental results have been reached, such as the im-plementations of quantum logical gates [5, 6, 7] and therealizations of quantum teleportation [8]. Generation ofquantum states is a crucial issue in nano-technologies, be-cause it is the basis of initialization processes with whichmany experimental protocols begin. Incidentally, it alsoprovides the possibility of observing physical systems be-having according to the predictions of quantum mechan-ics once a nonclassical state has been created.Recently, a strategy for the generation of pure quan-tum states through extraction from arbitrary initialstates has been proposed [9, 10]. This procedure is basedon the idea of putting a quantum system in interactionwith another one that is repeatedly measured in order toinduce a non-unitary evolution which forces the formersystem onto a Hilbert subspace. If such a subspace isone-dimensional, the process reduces to the extraction ofa pure quantum state. For this reason, this procedure hasbeen addressed as a “purification” [4, 11]. On the basisof this general scheme, many applications have been pro-posed: it is possible to extract entanglement [10, 12], andthe initialization of multiple qubits would be useful forquantum computation [10, 13]; extensions of the scheme ∗ Electronic address: bdmilite@fisica.unipa.it † Electronic address: [email protected] enable us to establish entanglement between two spa-tially separated systems via repeated measurements onan entanglement mediator [14]; in single trapped ions, theextraction of angular-momentum Schr¨odinger-cat stateshas been proposed [15] and the possibility of steering theextraction of pure states through quantum Zeno effecthas been predicted [16]. In passing, we mention thatthe approach here recalled is related to quantum non-demolition measurements [17], introduced for gravita-tional wave detection [18] and exploited in different physi-cal systems for applications in quantum computation andinformation [19] and for quantum state generation in gen-eral. For instance, in the context of trapped ions, puttingthe vibrational degrees of freedom in interaction with theelectronic degrees of freedom, and repeatedly measuringthe atomic state, it is possible to generate Fock statesboth in one-dimensional [20] and two-dimensional con-texts [21].Until now, the extraction of pure states through re-peated measurements has been considered in ideal situa-tions, that is, the evolution is assumed to be unitary ex-cept for the change of state by the measurements. In thispaper, we discuss how the predictions change when thesystem is put in interaction with an environment and, asa consequence, is subject to a non-unitary evolution (evenbetween two successive measurements), which is assumedto be described by a Lindblad-type equation [22]. Firstof all in this paper, we discuss the behavior of a quan-tum system, whose dynamics is governed by the repeatedmeasurements represented by projection operators on theother system in interaction with the former and by thedissipative environmental interaction with the two quan-tum systems. A criterion for the extraction of pure statesin the presence of dissipation is derived and a quantumsystem composed of two two-level systems (qubits) im-mersed in a common bosonic bath is analyzed as a verysimple example. The ability or disability of extractingpure states for a one qubit system by this scheme is esti-mated numerically in terms of the purity of the densityoperator.The present paper is organized as follows. In Sec. II,the general aspects of the purification protocol in thepresence of interaction with an environment are dis-cussed. In Sec. III, a simple model of two mutually in-teracting two-level systems immersed in a bosonic bathis analyzed, and the main differences between the idealsituation (in the absence of dissipation) and the morerealistic situation here studied (in the presence of dissi-pation) are singled out. Finally, some conclusive remarksare given in Sec. IV.
II. GENERAL FRAMEWORKA. Framework
The scheme of extraction we study in this paper isbased on the idea presented in Ref. [9]. Assume that weare interested in preparing a target system, S, into a purestate. To this end, put it in interaction with an ancillasystem, X, which is repeatedly measured and found inthe same state, say | Φ i X . As a consequence of both theinteraction with and the measurements on X, system Sis subject to a non-unitary dynamics which forces it in asubspace. If the subspace is one-dimensional, the relevantstate is extracted. We underline that if a negative resultis obtained measuring system X (i.e., if the ancilla systemis found in a state different from the expected one), thenthe trial can be considered as “failed” and the systemhas to be reset in order to restart the experiment, orwe can just continue the process as a new trial with theinitial condition given by the result of the unsuccessfulprojection.In the following, we consider this scheme but in thepresence of dissipation, in order to examine whether apure state can still be extracted. To this end, assume nowthat both S and X are immersed in a bath, so that thedynamics of the whole system S+X between the repeatedmeasurements is governed by the master equation dρ tot dt = L ρ tot , L = H + D , (1)where ρ tot is the density operator of S+X (the trace overthe bath degrees of freedom has already been performed). H is the Hamiltonian superoperator defined by H ρ tot = − i [ ˆ H, ρ tot ], while D is the Lindblad-type dissipator [22]which takes into account the interaction with the environ-ment, in the Markovian limit. Let M ( τ ) = e L τ denotethe superoperator describing the dissipative dynamics ofS+X and P the projection superoperator associated withthe measurement of the state | Φ i X , i.e. P ρ tot = | Φ i X h Φ | ρ tot | Φ i X h Φ | . (2) Assuming that the time elapsed between two successivemeasurements is always τ and that every of the N + 1measurements has confirmed X to be in the state | Φ i X ,the initial state ρ tot (0) is mapped into the final one, ρ tot ( τ, N ) ∝ PM ( τ ) · · · PM ( τ ) P ρ tot (0) = V N tot ( τ ) ρ tot (0) , (3)where V tot ( τ ) = PM ( τ ) P , (4)which, although rigorously speaking acts on Liouvillespace [23] of the whole system S+X, is substantially asuperoperator acting on the Liouville space of subsys-tem S, except for the trivial action on X (i.e. projectionon the state | Φ i X ). Therefore, it can be rewritten as V tot ( τ ) = V ( τ ) ⊗ P , where V ( τ ) is a superoperator actingonly on the Liouville space of S.If the master equation is given in the Lindbladform [22], the evolution of S+X in interaction with thebath is written down in the Kraus representation [24] as M ( τ ) ρ tot = G X k =0 ˆ T k ( τ ) ρ tot ˆ T † k ( τ ) , (5)where G + 1 is the number of Kraus operators ˆ T k ( τ ) in-volved in the decay process. Therefore, the superoperator V ( τ ) that maps the state of S just after a measurementto another after the next measurement reads V ( τ ) ρ = G X k =0 ˆ V k ( τ ) ρ ˆ V † k ( τ ) , (6)where ρ is the state of S andˆ V k ( τ ) = X h Φ | ˆ T k ( τ ) | Φ i X (7)is an operator acting on the Hilbert space of S. Note thatthe state of the whole system S+X after a measurementis factorized like ρ tot = ρ ⊗ | Φ i X h Φ | .The properties of V ( τ ) determine the fate of the stateof S, after many repetitions of measurements on X. Infact, assume that V ( τ ) is diagonalizable and considerits spectral decomposition in terms of its eigenprojec-tions [25]: V ( τ ) = X n λ n Π n , (8)where Π n is the eigenprojection belonging to the eigen-value λ n and satisfies the ortho-normality and complete-ness conditionsΠ m Π n = δ mn Π n , X n Π n = 1 . (9)The eigenvalues λ n are complex-valued in general andordered in such a way that | λ | = | λ | = · · · = | λ g | > | λ g +1 | ≥ | λ g +2 | ≥ · · · . If V ( τ ) is not diagonalizable,the Jordan decomposition applies instead of (8) [25].Generalization of the following argument to such casesis straightforward. See, for instance, an Appendix ofRef. [10].The evaluation of the N th power of V ( τ ) shows that,the larger the number N of the repeated measurements is,the more dominant the blocks belonging to the maximum(in modulus) eigenvalues are over the other blocks. Thatis, for a large enough N , the action of V N ( τ ) diminishesthe components of the system density operator which donot belong to the generalized eigenspaces correspondingto the maximum (in modulus) eigenvalues. Indeed, V N ( τ ) = X n λ Nn Π n N →∞ −−−−→ g X n =0 λ Nn Π n , (10)which, in the case wherein only one eigenvalue existswhose modulus is maximum (i.e. when g = 0), reducesto V N ( τ ) N →∞ −−−−→ λ N Π . (11)Therefore, in such a case, for a large enough (dependingon the structure of the spectrum) N , the action of V N ( τ )essentially reduces to the action of Π , hence projectingthe system in the relevant subspace.As for the outcome state, it is worth mentioning that, if λ is degenerated, the state of S extracted by Π dependson the initial state of S, ρ (0), since such a final state issubstantially proportional to Π ρ (0), which depends on ρ (0) if Π refers to a multi-dimensional space. On thecontrary, if λ is not degenerated and Π refers to a one-dimensional subspace, i.e.Π ρ = f ( ρ ) ρ (12)with ρ the relevant eigenstate (which is an element ofthe Liouville space of S) and f a suitable form on theLiouville space of S, the eigenstate ρ is extracted irre-spectively of the initial condition of the system, providedΠ ρ (0) = 0. As a first step, in this paper we shall con-centrate on such situations wherein there is a single ex-tracted eigenspace which in addition is one-dimensional.In an ideal case wherein any sources of dissipation areabsent, Eq. (6) reduces to V ( τ ) ρ = ˆ V ( τ ) ρ ˆ V † ( τ ) withˆ V ( τ ) = X h Φ | e − i ˆ Ht | Φ i X [9], so that, if ˆ V ( τ ) possesses anondegenerate and unique maximum (in modulus) eigen-value, then the final state to be extracted is the purestate | u i S h u | , since in this case Eq. (12) is supplementedby ρ = | u i S h u | and f ( ρ ) = S h v | ρ | v i S where | u i and h v | are the right- and left-eigenvectors of ˆ V ( τ ) be-longing to its largest (in modulus) eigenvalue, respec-tively. This is the basic idea of the purification schemebased on the repeated measurements, which was first pro-posed in Ref. [9] and has been analyzed and developed inRefs. [10, 12, 13, 14, 15, 16]. On the other hand, it is im-portant to stress that in the non-ideal case, even if thereis a single and nondegenerate eigenvalue that is maxi-mum in modulus, this does not guarantee that the state ρ is a pure state and hence there is no warranty that the final extracted state is pure, either. In this sense, ourstate-extraction scheme may not necessarily be an effec-tive purification scheme. However, we still try to seeka possibility of extracting a pure state even in the pres-ence of dissipation. The examination of such situationswherein we can extract a pure state is the main topic ofthis paper. Efficiency — This scheme for extracting quantumstates is a conditional one, in the sense that each timesystem X is measured it has to be found in the samestate, denoted by | Φ i X . In Refs. [9, 10, 12, 13, 14, 15, 16]it is proved that the probability of success of the ex-traction, that is the probability of finding system Xin the state | Φ i X successively N times, is given bythe normalization factor of the state extracted by (3),tr S {V N ( τ ) ρ (0) } = P n λ Nn tr S { Π n ρ (0) } , which behavesasymptotically as → λ N tr S { Π ρ (0) } as N → ∞ [or → P n ≤ g λ Nn tr S { Π n ρ (0) } in the more general situation,which from now on we shall not mention anymore for thesake of simplicity]. These expressions for the probabilityof success (still valid in the non-ideal case, provided theprojectors Π n are the appropriate ones) show that thestructure of the spectrum of V ( τ ) plays a crucial role forthe efficiency and fastness of the extraction. In partic-ular, on the one hand, the fact that λ has a modulusquite larger than those of the other eigenvalues makes V N ( τ ) quickly approach λ N Π , so that a smaller numberof measurements is required to well approximate the fi-nal result Π ρ (0). On the other hand, the closer to unitythe modulus of λ is, the greater the probability of suc-cess is, approaching just tr S { Π ρ (0) } (without decayingout completely) for λ ≃
1. On the contrary, for smallvalues of λ , the probability quickly approaches zero like λ N , which means that if a large number of measurementsis required to approach the state (tr S { Π ρ (0) } ) − Π ρ (0)the scheme becomes very inefficient. Therefore, the num-ber of measurements necessary to extract the target statewould be an important measure of efficiency. It can beroughly estimated by the following argument. The idea isto see how much the relevant part approaching the targetstate, λ N Π ρ (0), dominates over the rest: p = λ N k Π ρ (0) k P n | λ n | N k Π n ρ (0) k , (13)where k · k is a certain norm, for instance k A k := p tr S { A † A } . This measure approaches unity in the limitof an infinite number of measurements, p → N → ∞ .Then, we ask how many measurements are necessary forthis quantity to exceed a desired value 0 < p <
1. Afterrewriting (13) as λ N k Π ρ (0) k / ( P n =0 | λ n | N k Π n ρ (0) k ) = p/ (1 − p ), a sufficient condition for p ≥ p is given by λ N k Π ρ (0) k ( M − | λ | N R ( ρ (0)) ≥ p − p , (14)where M is the number of eigenvalues or equivalentlythe dimension of the Liouville space and R ( ρ (0)) =max n =0 k Π n ρ (0) k . The number of measurements neces-sary to get a better quality than p is therefore estimatedby N ≥ ln[ p / (1 − p )] + ln( M −
1) + ln R ( ρ (0)) k Π ρ (0) k ln | λ /λ | . (15)It is important to note that this threshold depends on ρ (0), according to the expectation that the larger is thenorm of the relevant part in the initial state, k Π ρ (0) k ,the smaller is the number of necessary measurements. B. Searching for Pure Eigenvectors: the Criterion
It is easy to show that a necessary and sufficient con-dition for a pure state being an eigenvector of the map V ( τ ) is that it is a simultaneous eigenstate of all the op-erators ˆ V k ( τ ) involved in the relevant map (see Eq. (6)).The proof of this statement proceeds as follows.= ⇒ Obviously, if the state | φ i S is a common eigen-state of all ˆ V k ’s, i.e., ˆ V k | φ i S = α k | φ i S and consequently S h φ | ˆ V † k = S h φ | α ∗ k , then one has G X k =0 ˆ V k ( τ ) | φ i S h φ | ˆ V † k ( τ ) = λ φ | φ i S h φ | , (16)where λ φ = X k α ∗ k α k ≥ . (17) ⇐ = Let the pure state | φ i S h φ | be an eigenvector of V ( τ ), then P Gk =0 ˆ V k ( τ ) | φ i S h φ | ˆ V † k ( τ ) = λ φ | φ i S h φ | . Con-sider now the overlap with a quantum state | φ ⊥ i S h φ ⊥ | orthogonal to | φ i S h φ | :0 = λ φ S h φ ⊥ | φ i S h φ | φ ⊥ i S = G X k =0 S h φ ⊥ | ˆ V k ( τ ) | φ i S h φ | ˆ V † k ( τ ) | φ ⊥ i S = G X k =0 | S h φ ⊥ | ˆ V k ( τ ) | φ i S | , (18)from which it follows that S h φ ⊥ | ˆ V k ( τ ) | φ i S = 0 for all k and whatever the state | φ ⊥ i S is, provided it is orthogonalto | φ i S . In other words, it means thatˆ V k ( τ ) | φ i S = α k | φ i S , ∀ k, (19)where α k is a suitable complex number. This completesthe proof. C. Searching for Pure Eigenvectors: Purity
In those cases in which we are not able to extract anexactly pure state, there is a possibility of extracting “al-most pure” states, that is, mixed states very close (in the sense of purity) to the pure states. To look for almostpure states which can be extracted, let us recall a measureof purity of a given state. We show later how the purityof the eigenstate of the linear map V ( τ ) correspondingto the maximum eigenvalue behaves as a function of theparameters of the scheme, i.e. the interval of time τ andthe repeatedly measured state of X, | Φ i X .The purity of a state is defined as the trace of thesquare of the relevant (normalized) density operator [26]: P ( ρ ) = tr S ρ . (20)This quantity is upper and lower bounded in accordancewith 1 /L ≤ P ( ρ ) ≤ L being the number of levels ofthe system under scrutiny. Observe that the maximumvalue [ P ( ρ ) = 1] corresponds to pure states, while theminimum value [ P ( ρ ) = 1 /L ] corresponds to maximallymixed states with maximal von Neumann’s entropy. D. Weak-Damping Case
It is possible to derive a formula for the purity of theextracted state for general systems in the weak-dampingregime. Such a formula would be useful for understand-ing which parameters spoil the purity and convenient foran optimization of the purification.Let us decompose the relevant map V ( τ ) into twoparts, V ( τ ) = V (0) ( τ ) + δ V ( τ ) , (21)where V (0) ( τ ) is the map in the absence of the environ-mental perturbation and the rest is treated as a pertur-bation to it, which is given in the weak-damping regimeby δ V ( τ ) ⊗ P ≃ Z τ dt P e H ( τ − t ) D e H t P . (22)Assuming that ˆ V ( τ ) is diagonalizable, let | u n i S and S h v n | denote its right- and left-eigenvectors, respectively,which form a complete ortho-normal set, P n | u n i S h v n | = S [27]. (We also normalize the right-eigenvectors as S h u n | u n i S = 1.) Then, the right-eigenvectors of the idealmap read V (0) ( τ ) σ (0) mn = λ (0) mn σ (0) mn , σ (0) mn = | u m i S h u n | . (23)These are orthogonal to the left-eigenvectors˜ σ (0) mn = | v m i S h v n | (24)in the sense(˜ σ (0) mn , σ (0) m ′ n ′ ) = δ mm ′ δ nn ′ , ( A, B ) = tr S { A † B } . (25)We are interested in a situation where we can purify S inthe absence of the environmental perturbation. That is, λ (0)00 is not degenerated and is the only eigenvalue that isthe largest in modulus.Now, the standard perturbative treatment yields thefirst-order correction to the right-eigenvector, δσ (1) mn = − X m ′ n ′ = mn σ (0) m ′ n ′ (˜ σ (0) m ′ n ′ , δ V ( τ ) σ (0) mn ) λ (0) m ′ n ′ − λ (0) mn + c σ (0) mn , (26)where the constant c is set equal to zero by the normal-ization condition (˜ σ (0) mn + δ ˜ σ (1) mn , σ (0) mn + δσ (1) mn ) = 1.This formula is valid when λ (0) mn is not degenerated.We are interested in the state to be extracted, i.e. ρ ≃ ( σ (0)00 + δσ (1)00 ) / (1+tr S δσ (1)00 ). Since λ (0)00 has been assumedto be nondegenerated, the formula (26) is valid for δσ (1)00 .The purity of ρ up to this order is therefore given by P ( ρ ) ≃ − (cid:0) tr S δσ (1)00 − S h u | δσ (1)00 | u i S (cid:1) = 1 − X mn =00 S h u n | ˆ Q | u m i S (˜ σ (0) mn , δ V ( τ ) σ (0)00 ) λ (0)00 − λ (0) mn , (27)where ˆ Q = S − | u i S h u | is a projection operator.This is the formula for the purity of the extracted stateup to the first order in the decay constants in the weak-damping regime. This shows that, if the state σ (0)00 to beextracted in the ideal case is an eigenstate of the pertur-bation, i.e. δ V ( τ ) σ (0)00 ∝ σ (0)00 , the first-order correction tothe purity vanishes and the purification is robust againstthe environmental perturbation, at least up to this or-der. This is a weaker version of the criterion discussed inSec. II B and is convenient since the dissipator of a masterequation, D , suffices to this criterion without knowing theKraus operators ˆ T k ( τ ) of the decay process, which mayrequire solving the master equation. Furthermore, thisformula would be useful for finding a parameter set thatoptimizes the purity (minimizes the first-order correctionto the purity).When S is a two-level system, the formula (27) is re-duced to P ( ρ ) ≃ − S h v | v i S (˜ σ (0)11 , δ V ( τ ) σ (0)00 ) λ (0)00 − λ (0)11 . (28) III. A SIMPLE MODEL
In this section, we apply the ideas presented above tothe case of a simple model. Such a system consists of twomutually-interacting spins immersed in a bosonic bath,one of which is repeatedly measured to purify the other.
A. Model
Two-spin system — Consider a system of two in-teracting spins or pseudo-spins, for instance a couple of identical two-level atoms subjected to a dipolar coupling.Assuming that the matrix elements of the dipole opera-tors are real, and neglecting the counter-rotating terms,one reaches the following Hamiltonian (for details, seeRefs. [10, 28]):ˆ H tot = X i =S , X Ω2 (1 + ˆ σ ( i ) z ) + ǫ (ˆ σ (S)+ ˆ σ (X) − + ˆ σ (S) − ˆ σ (X)+ ) (29)where ˆ σ ( i ) z = |↑i i h↑| − |↓i i h↓| , ˆ σ ( i )+ = |↑i i h↓| = (ˆ σ ( i ) − ) † , Ωis the Bohr frequency of the two-level system and ǫ thecoupling constant. We have set ~ = 1.The eigenstates of the Hamiltonian are the triplet andsinglet two-spin states: | i tot = |↑i S |↑i X , (30a) | i tot = 1 √ (cid:2) |↑i S |↓i X + |↓i S |↑i X (cid:3) , (30b) | i tot = |↓i S |↓i X , (30c) | s i tot = 1 √ (cid:2) |↑i S |↓i X − |↓i S |↑i X (cid:3) , (30d)which are common eigenstates of ˆ Σ and ˆΣ z with ˆ Σ = ( ˆ σ (X) + ˆ σ (S) ), whose eigenvalues are given by Σ(Σ + 1)and m Σ , respectively. The corresponding eigenenergiesare 2Ω, Ω + ǫ , 0, and Ω − ǫ , respectively. If we considerthe case Ω > ǫ , then | i is the ground state. Interaction with a bosonic bath — The interactionwith a bosonic bath, whose free Hamiltonian is given byˆ H B = R dk ω k ˆ a † k ˆ a k , is modelled through the system-bathinteraction Hamiltonianˆ H I = X i =S , X ( ˆ B i + ˆ B † i )(ˆ σ ( i )+ + ˆ σ ( i ) − ) , (31)where ˆ B i = R dk g k ( r i )ˆ a k , with r i the position of spin i and g k ( r i ) the coupling constant between the atom atposition r i and bath mode k . Following the standardderivation [29] and assuming the spins very close eachother in order to have g k ( r ) ≃ g k ( r ), we reach thefollowing master equation in the Schr¨odinger picture forthe density operator ρ tot of S+X: dρ tot dt = − i [ ˜ H tot , ρ tot ]+ γ (1 + n − ) D ρ tot + γ (1 + n + ) D ρ tot + γ n − D ρ tot + γ n + D ρ tot (32)with D ij ρ tot = | j i tot h i | ρ tot | i i tot h j | − {| i i tot h i | , ρ tot } , n ± the mean numbers of bosons in the bath modes of fre-quencies Ω ± ǫ , which are the Bohr frequencies betweenthe states involved in the transitions | i tot → | i tot (Ω − ǫ ) and | i tot → | i tot (Ω + ǫ ). γ and γ arethe decay rates related to such modes evaluated as thespectral correlation functions of ˆ B i + ˆ B † i , and are re-lated to g k ’s by γ = 2 π R dk | g k | δ ( ω k − Ω + ǫ ) and γ = 2 π R dk | g k | δ ( ω k − Ω − ǫ ). Finally, ˜ H tot is theLamb-shifted Hamiltonian of S+X. B. Extraction of Pure States under the Influenceof a Zero-Temperature Bosonic Bath
Consider now the special case wherein the bath is atzero temperature. The spin labelled with X is repeat-edly measured and found in the state | Φ i X = cos θ |↑i X + e iχ sin θ |↓i X , while the other spin, labelled with S, isdriven toward a quantum state through its interactionwith X. The same situation is discussed in Ref. [10], inthe absence of the environmental coupling, where it isfound that the extracted state can be made pure veryefficiently, in particular measuring the states |↑i X and |↓i X . The evolution of the damped system between twosuccessive measurements is easily evaluated, for instance,following the approach developed in Ref. [30], and is givenby (see Eq. 5) ρ tot ( t ) = X k =0 ˆ T k ( t ) ρ tot (0) ˆ T † k ( t ) (33a)with 4 Kraus operators,ˆ T ( t ) = | i tot h | + e − γ t e − i (Ω+ ǫ ) t | i tot h | + e − γ t e − i t | i tot h | + e − i (Ω − ǫ ) t | s i tot h s | , (33b)ˆ T ( t ) = √ − e − γ t | i tot h | , (33c)ˆ T ( t ) = r γ γ − γ ( e − γ t − e − γ t ) | i tot h | , (33d)ˆ T ( t ) = s γ e − γ t − γ e − γ t γ − γ | i tot h | . (33e)ˆ T ( t ) reduces to the unitary evolution operator in thecase γ = γ = 0, whereas the others, i.e. ˆ T k ( t ) for k ≥ τ dur-ing the dissipative dynamics (33). According to (19), inorder for the S state | φ i S h φ | , with | φ i S = cos η |↑i S + e iξ sin η |↓i S , be a pure eigenstate of the contracted map V ( τ ), it should satisfy s k ( τ ) ≡ S h φ ⊥ | X h Φ | ˆ T k ( t ) | Φ i X | φ i S = 0 , k = 0 , , , | φ ⊥ i S = sin η |↑i S − e iξ cos η |↓i S . Indeed, it is equiv-alent to look for the eigenstates of the contracted oper-ators ˆ V k ( τ ) defined in (7). It is straightforward to findthat s ( τ ) ∝ sin θ cos η (cid:0) e iξ cos θ sin η + e iχ sin θ cos η (cid:1) ,s ( τ ) ∝ cos θ cos η (cid:0) e − iχ sin θ sin η − e − iξ cos θ cos η (cid:1) ,s ( τ ) ∝ sin θ cos θ sin η cos η , (35) where the proportionality factors are the nonvanishingcoefficients in (33c)–(33e). From these expressions, it fol-lows that s ( τ ) = s ( τ ) = s ( τ ) = 0 is accomplished onlyfor cos η = 0. This condition is necessary and sufficientto make s k ’s vanish for k = 1 , ,
3. In order to make s ( t )vanish too, it is necessary to have sin θ = 0 or cos θ = 0.In fact, the condition cos η = 0 means | φ i S = |↓i S , andevaluating s ( t ) in such a special situation provides s ( τ ) = S h↑| X h Φ | ˆ T ( τ ) | Φ i X |↓i S = − e − iχ e − i (Ω − ǫ ) τ √ θ θ − e − γ τ e − i ǫτ ) , (36)which, for τ = 0, vanishes only if sin θ = 0 or cos θ = 0.This analysis shows that in some special cases, that is,when the state of X is repeatedly measured and found in | Φ i X = |↑i X (sin θ = 0) or | Φ i X = |↓i X (cos θ = 0), thecontracted linear map V ( τ ) has the S state |↓i S h↓| as apure eigenstate. To reach the final conclusion about thepossibility of extracting such a pure state, one needs toknow whether the corresponding eigenvalue is the max-imum (in modulus) in the spectrum of the map. Weshall therefore diagonalize the contracted map in the twocases, sin θ = 0 and cos θ = 0.Representing the density operator of S as afour-dimensional vector, ρ = ( ρ ↑↑ , ρ ↓↓ , ρ ↑↓ , ρ ↓↑ ) =( S h↑| ρ |↑i S , S h↓| ρ |↓i S , S h↑| ρ |↓i S , S h↓| ρ |↑i S ), the contractedlinear map V ( τ ) is substantially represented by a 4 × Repeatedly measuring |↓i X ( θ = π ) — In the casewhere the X state |↓i X is repeatedly measured, the corre-sponding linear map V ( τ ) = V ↓ ( τ ) is represented by thefollowing matrix: V ↓ ( τ ) = | f ↓ ( τ ) | (1 − e − γ τ ) 1 0 00 0 f ↓ ( τ ) 00 0 0 f ∗↓ ( τ ) , (37)with f ↓ ( τ ) = e − i (Ω − ǫ ) τ (1 + e − γ τ e − i ǫτ ). The eigenval-ues of this matrix are λ = 1 , λ = λ ∗ = f ↓ ( τ ) , λ = | f ↓ ( τ ) | . (38)The right-eigenvector corresponding to the maximumeigenvalue λ = 1 is the pure state ρ = |↓i S h↓| . Thelarger the time τ is, the smaller the other three eigen-values of the map are and the faster the extraction of ρ = |↓i S h↓| is, in the sense that it requires a smallernumber of steps. Repeatedly measuring |↑i X ( θ = 0 ) — In the casewherein the X state |↑i X is repeatedly measured, the map V ( τ ) reduces to V ↑ ( τ ) = e − γ τ γ ( e − γ τ − e − γ τ )2( γ − γ ) | f ↑ ( τ ) | e − γ τ f ↑ ( τ ) 00 0 0 e − γ τ f ∗↑ ( τ ) (39)with f ↑ ( τ ) = e − i (Ω+ ǫ ) τ (1 + e − γ τ e i ǫτ ). This matrixis easily and exactly diagonalized as long as e − γ τ = | f ↑ ( τ ) | . There are two cases in the ordering of its eigen-values. Case I: if e − γ τ < | f ↑ ( τ ) | , λ = | f ↑ ( τ ) | , λ = λ ∗ = e − γ τ f ↑ ( τ ) , λ = e − γ τ . (40)Case II: if e − γ τ > | f ↑ ( τ ) | , λ = e − γ τ , λ = λ ∗ = e − γ τ f ↑ ( τ ) , λ = | f ↑ ( τ ) | . (41)In case I (which surely occurs in the strong-dampingregime γ τ → ∞ ), a pure state ρ (I) = |↓i S h↓| is extracted,while in case II, a mixed state is extracted, ρ (II) = p ↑↑ |↑i S h↑| + p ↓↓ |↓i S h↓| (42a)with p ↑↑ = 11 + α , p ↓↓ = 1 − p ↑↑ = α α , (42b) α = γ ( e − γ τ − e − γ τ )2( γ − γ )( e − γ τ − | f ↑ ( τ ) | ) . (42c)The latter is not in contradiction with the previous state-ment that one has a pure eigenstate for sin θ = 0. In-deed, the state |↓i S h↓| is still an eigenstate of the map,but it does not correspond to the maximum eigenvalueanymore, and then it is not the state to be extracted.The purity of the state ρ (II) in (42) is given by P ( ρ (II) ) = p ↑↑ + p ↓↓ = 1 + α (1 + α ) . (43)In the weak-damping case γ τ, γ τ ≪ ǫτ = 0 for simplicity), one has α ≃ γ τ ǫτ up to thefirst order in γ τ and γ τ , and hence P ( ρ (II) ) ≃ − α ≃ − γ τ sin ǫτ . (44)This formula, that alternatively can be directly derivedusing (28), shows that in the weak-damping regime (i)the purity is linearly affected by γ , while (ii) it is not in-fluenced by γ . Furthermore, (iii) the purity is optimizedby taking a nontrivial time interval τ ≃ . π/ǫ .It is worth noting that in the weak damping limit wecannot extract a pure state, while in the strong damp-ing limit a pure state can be obtained, which is the op-posite one would expect. To understand this fact, con-sider first of all that S+X has two stable states, | i tot h | and | s i tot h s | according to (33), and second that in thestrongly dissipative case the system has time to relaxonto the equilibrium state which is a mixture of the twostable states, whose statistical weights are determinedby the initial condition. Then, repeatedly measuring thestate |↑i X cuts the population of | i tot in the mixture andleaves only X h↑| s i tot h s |↑i X ∝ |↓i S h↓| .The case e − γ τ = | f ↑ ( τ ) | corresponds to a degeneratecase and hence, as clarified in Sec. II, is not in the scopeof this paper since it does not permit the extraction ofa precise state irrespectively of the initial state of thesystem.Notice that the general case corresponding to measur-ing a generic state | Φ i X can be discussed in the weakdamping limit through the perturbation analysis. C. At Finite Temperature
The analysis on the model has so far been focused onthe zero-temperature case and showed that the only purestate that can be extracted at the zero temperature is |↓i S when the state of X is repeatedly measured and found in |↑i X or |↓i X . The question of what happens in the caseof non-zero temperature naturally arises. To answer thisquestion, we resort to numerical calculations.The linear map V ( τ ) depends on τ , the measured state | Φ i X (which is individualized in the Bloch-sphere by thepolar and azimuthal angles, θ and χ , respectively), andin general the temperature of the environment, T . Giventhe map, the eigenvector associated with the maximumeigenvalue, and its purity are functions of all such quan-tities ( τ , θ , χ and T ).In Fig. 1, the purity of the state to be extracted isshown as a function of the parameters τ and θ with χ = 0being fixed, in four different situations concerning thebath. For the numerical calculations, we have set Ω /ǫ =10, γ /ǫ = 0 .
1, and γ /γ = 0 . τ , θ and χ are.In the other three figures, the behavior of purity in thepresence of interaction with the bath at different tem-peratures is shown. Figure 1(b) refers to the case of verylow temperature, effectively zero, and shows that in someregions of the parameter space, there is a possibility ofextracting pure or almost pure states. This is not in con- (a) P ǫτ θπ (b) P ǫτ θπ (c) P ǫτ θπ (d) P ǫτ θπ FIG. 1: The purity of the extracted state vs the parameters ǫτ and θ/π ( χ = 0 for all cases), in different situations: (a)in the ideal case, i.e. in the absence of interaction with the bath, and in the presence of interaction with the bath with (b) k B T / ~ Ω = 0 .
01, (c) k B T / ~ Ω = 1, and (d) k B T / ~ Ω = 10. In all cases, ratios between salient physical quantities are fixed asΩ /ǫ = 10, γ /ǫ = 0 . γ /γ = 0 . tradiction with the analysis in Sec. III B, where it hasbeen found that a pure state can be extracted only for θ = 0 , π . This result refers to an exactly pure eigenstate,while the numerical calculations here reported show thevalue of purity, which can be very close to unity althoughnot exactly 1.In Fig. 1(c), one can see that in practice there is no re-gion in the parameter space corresponding to pure states:the purity is visibly smaller than unity everywhere. Fi-nally, in Fig. 1(d), we see that at a higher temperature( k B T = 10 ~ Ω, with k B the Boltzmann constant), the pu-rity of the state to be extracted is equal to the minimumvalue for the two-level system, , almost everywhere, thatis, irrespectively of the values of parameters τ and θ .In Fig. 2, the purity is plotted as a function of the tem-perature T and of the time interval τ between successivemeasurements, when a fixed state of system X character-ized by θ = 3 π/ χ = 0 is repeatedly measured. Itis well visible that the more the temperature increases,the more the purity of the extracted state approaches theminimum value, that is, .All these results express in a clear way that the inter- action with an environment deteriorates the reliability ofthe purification scheme based on repeated measurements,although at the zero temperature pure states can still beextracted. D. Efficiency
In a realistic situation, the probability of extracting thetarget state as well as the number of measurements onehas to perform are important factors to consider. Accord-ing to the discussion at the end of Sec. II A, the probabil-ity of success is asymptotically given by λ N tr S { Π ρ (0) } .Therefore, except for those situations wherein the max-imum eigenvalue (in modulus) is unity, the most rele-vant condition to get a good efficiency is that the num-ber of required measurements is very low, which implies | λ /λ | ≪
1, or, better, that the denominator in thethreshold given in (15) is high, i.e. ln | λ /λ | ≫
1. There-fore, the peaks of ln | λ /λ | correspond to the maximaof the efficiency (i.e., the minima of the required num-ber of measurements). To better fix the idea, if we P T ǫτ
FIG. 2: The purity of the extracted state as a function ofthe temperature T (in units of ~ Ω /k B ) and ǫτ . The otherparameters are: θ = 3 π/ χ = 0, Ω /ǫ = 10, γ /ǫ = 0 . γ /γ = 0 . ask that the target state is obtained with a precision p = 0 .
99, since we have ln(4 −
1) + ln[ p / (1 − p )] ≈ . | λ /λ | ∼
4, the process requires one or two measure-ments when the system starts with an initial conditionsatisfying k Π ρ (0) k ∼ R ( ρ (0)), which for instance is usu-ally the case for the maximally mixed state.In Fig. 3(a), we consider the ideal case, while inFigs. 3(b)–3(d), we refer to non-ideal situations at zero,intermediate, and high temperature. The plots clearlyshow that the interaction with a nonzero temperature en-vironment negatively affects the efficiency, lowering thepeaks and extending the valleys. Nevertheless, at zerotemperature, various peaks are still present, and in fact,at zero temperature, the degradation with respect to theideal case is not so dramatic. IV. SUMMARY
Let us summarize the results reported in this paper.Putting a system in interaction with a repeatedly mea-sured one forces the former system onto a subspace, hencerealizing, under suitable conditions, the extraction ofpure states. In a more realistic situation, the two systemsare interacting with their environment too, and thereforeare subjected to dissipation. Such an interaction practi-cally reduces the chance to extract pure states.From the mathematical point of view, the main dif-ference between the two situations is represented by thefact that in the ideal case one extracts eigenvectors of amap onto a Hilbert space, whereas in the non-ideal caseone extracts eigenvectors of a map onto a Liouville space. We have explored the general framework and studied avery simple physical system (two spins interacting witha bosonic bath) in order to bring to light fundamentalfeatures of repeated-measurement based extraction pro-cesses in the presence of dissipation. In Sec. III B, wehave shown that a mixed state is extracted instead ofa pure state. Actually, this is what generally happens,especially at high temperatures. Nevertheless, with azero-temperature bath, it is still possible to extract pureand almost pure states (see Fig. 1(b)) with still fairlygood efficiency. Indeed, the efficiency, though negativelyaffected by the environment, is still good at zero temper-ature.Overall, we have considered the case wherein a verylarge number of measurements (evaluating the mathe-matical limit for an infinite number of measurements) isperformed on the ancilla system, as clearly expressed by(10) and (11). We conclude this paper expecting that insome cases a reduction of the number of measurementsperformed on the ancilla system entails an increase ofthe purity of the output state. See, for example, Fig. 4,wherein we have plotted the purity of the resulting quan-tum state as a function of the number of measurementsperformed on the ancilla system, starting from the maxi-mally mixed initial state, with a particular parameter set.It is well visible that the purity, starting from its min-imum value ( ), increases at the second measurement,and then decreases down to its asymptotic value. There-fore, in such a case, the highest value of purity is obtainedfor a smaller number of measurements ( N = 2). We willdiscuss this aspect of our scheme in the next future. ACKNOWLEDGMENTS
This work is partly supported by the bilateral Italian-Japanese Projects II04C1AF4E on “Quantum Informa-tion, Computation and Communication” of the ItalianMinistry of Education, University and Research, and15C1 on “Quantum Information and Computation” ofthe Italian Ministry for Foreign Affairs, by the Grantfor The 21st Century COE Program at Waseda Uni-versity and the Grant-in-Aid for Young Scientists (B)(No. 18740250) from the Ministry of Education, Cul-ture, Sports, Science and Technology, Japan, and by theGrants-in-Aid for JSPS Postdoctoral Fellowship for For-eign Researchers (Short-term) and for Scientific Research(C) (No. 18540292) from the Japan Society for the Pro-motion of Science. One of the authors (K.Y.) is sup-ported by the European Union through the IntegratedProject EuroSQIP. Moreover, the authors acknowledgepartial support from University of Palermo in the contextof the bilateral agreement between University of Palermoand Waseda University, dated May 10, 2004.0 ln ¯¯¯ λ λ ¯¯¯ ǫτ θπ (a) ln ¯¯¯ λ λ ¯¯¯ ǫτ θπ (b) ln ¯¯¯ λ λ ¯¯¯ ǫτ θπ (c) ln ¯¯¯ λ λ ¯¯¯ ǫτ θπ (d) FIG. 3: The quantity ln | λ /λ | vs the parameters ǫτ and θ/π ( χ = 0 for all cases), in different situations: (a) in the ideal case,i.e. in the absence of interaction with the bath, and in the presence of interaction with the bath with (b) k B T / ~ Ω = 0 .
01, (c) k B T / ~ Ω = 1, and (d) k B T / ~ Ω = 10. In all cases, ratios between salient physical quantities are fixed as Ω /ǫ = 10, γ /ǫ = 0 . γ /γ = 0 . P N FIG. 4: The purity of the state of system S as a function ofthe number of measurements on X. The initial state is ρ (0) =0 . |↑i S h↑| + 0 . |↓i S h↓| and its purity is 0 .
5. Observe that thepurity reaches the maximum value at N = 2 and then decaysto an asymptotic value. Here, the parameters are χ = 0, θ = 2 .
25, and ǫτ = 7 .
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