Protecting qutrit-qutrit entanglement by weak measurement and reversal
aa r X i v : . [ qu a n t - ph ] J un Protecting qutrit-qutrit entanglement by weak measurement and reversal
Xing Xiao , ∗ and Yan-Ling Li College of Physics and Electronic Information,Gannan Normal University, Ganzhou 341000, China Institute of Optoelectronic Materials and Technology,Gannan Normal University, Ganzhou 341000, China School of Information Engineering, Jiangxi University of Science and Technology, Ganzhou 341000, China
Entangled states in high dimensional systems are of great interest due to the extended possibil-ities they provide in quantum information processing. Recently, Sun [Phys. Rev. A 82, 052323(2010)] and Kim [Nat. Phys. 8, 117 (2012)] pointed out that weak measurement and quantum weakmeasurement reversal can actively combat decoherence. We generalize their studies from qubits toqutrits under amplitude damping decoherence. We find that the qutrit-qutrit entanglement can bepartially retrieved for certain initial states when only weak measurement reversals are performed.However, we can completely defeat amplitude damping decoherence for any initial states by thecombination of prior weak measurements and post optimal weak measurement reversals. The ex-perimental feasibility of our schemes is also discussed.
PACS numbers: 03.67.Pp, 03.65.Yz, 03.67.Bg
I. INTRODUCTION
Quantum entanglement is not only a remarkable char-acteristic which distinguishes the quantum realm fromthe classical one, but also a key resource for quantuminformation and quantum computation [1]. However, inrealistic quantum information processing, entanglementis inevitably affected by the interaction between the sys-tem and its environment, which leads to degradation and,in certain cases, entanglement sudden death (ESD) [2–4].Thus, it is very important to protect entanglement fromenvironmental noise.Weak measurements [5] are generalizations of von Neu-mann measurements and are associated with a positive-operator valued measure (POVM). For weak measure-ments [7, 8], the information extracted from the quan-tum system is deliberately limited, thereby keeping themeasured system’s state from randomly collapsing to-wards an eigenstate. Thus, it would be possible to re-verse the initial state with some operations. Recently, itwas pointed out that weak measurements and quantumweak measurement reversals can effectively protect thequantum states of a single qubit system from decoher-ence [9–11]; this idea has also been extended to protectthe entanglement of two-qubit systems [12–15] from am-plitude damping decoherence. Until now, probabilisticreversal with a weak measurement has already been ex-perimentally demonstrated on a superconducting phasequbit [16], as well as on a photonic qubit [13, 17].Most studies of weak measurements concerning theprotection of entanglement are restricted to two dimen-sional (2D) systems. However, quantum informationtasks require high dimensional bipartite entanglement.It is well known that high dimensional entangled sys- ∗ Corresponding author: [email protected] tems such as qutrits [18–20] can offer significant ad-vantages for the manipulation of information carriers.For instance, biphotonic qutrit-qutrit entanglement [21]enables more efficient use of communication channels[22]. Moreover, high dimensional entangled systems offerhigher information-density coding and greater resilienceto errors than 2D entangled systems in quantum cryp-tography [23]. However, practical applications of suchprotocols are only conceivable when the prepared highdimensional entangled states have sufficiently long co-herence times for manipulation.In this paper, we propose using weak measurementsto preserve the entanglement of two initially entan-gled qutrits which suffer independent amplitude damp-ing noise. Our schemes for protecting entanglement arebased on the fact that weak quantum measurement canbe reversed probabilisticlly. We specifically consider twosimple schemes as shown in Fig. 1. Similar schemeshave been discussed only in one or two-qubit systems[12, 13, 17], while we consider a qutrit-qutrit version inthis paper. The first scheme is “ amplitude damping + weak measurement reversal ”. In this case, unlike theentanglement decaying exponentially to zero in ampli-tude damping decoherence, we show that the weak mea-surement reversal procedure partially recovers the entan-glement under most conditions. The limitation of thisscheme is that ESD still occurs in some particular sit-uations. As an improvement on the former, the secondscheme is “ weak measurement + amplitude damping + weak measurement reversal ”. In this case, we find thecombination of prior weak measurement and post weakmeasurement reversal can actively combat decoherence.Moreover, it can effectively circumvent ESD. The physi-cal mechanism of the second scheme is that a prior weakmeasurement intentionally moves each qutrit close to itsground state. The amplitude damping decoherence isnaturally suppressed in this ‘lethargic’ state, and the en-tanglement is therefore preserved [24].This paper is organized as follows: In Section II, we in-troduce amplitude damping noise operators for the qutritcase, then we generalize the weak measurement and weakmeasurement reversal operators from qubit to qutrit. InSection III, we propose two different schemes to protectqutrit-qutrit entanglement. In Section IV, we give a briefdiscussion of the experimental feasibility of our schemes.Finally, we summarize our conclusions in Section V. II. BASIC THEORYA. amplitude damping for qutrits
The amplitude damping noise is a prototype model ofa dissipative interaction between a quantum system andits environment [1]. For example, the amplitude dampingnoise model can be applied to describe the spontaneousemission of a photon by a two-level system into an en-vironment of photon or phonon modes at zero (or verylow) temperature in (usually) the weak Born-Markov ap-proximation. weak meas.reversalamplitudedamping weak meas.reversalamplitudedamping weak meas.reversalamplitudedamping weak meas.reversalamplitudedampingweakmeas.weakmeas. (a) scheme one(b) scheme two d D d D d D d D FIG. 1: (color online) Schemes for protecting entanglementfrom decoherence using weak measurement and weak mea-surement reversal: (a), Two entangled qutrits go throughindependent amplitude damping channels and then a weakmeasurement reversal is performed on each qutrit. (b), Sim-ilar to (a), but a weak measurement is applied before eachqutrit undergoes decoherence.
For qutrits, the situations are more complicated asthere are three configurations of the 3-level system tobe taken into account [25]. Here, we will focus on theso called V-configuration. We denote the lower level as | i and the two upper levels as | i and | i , respectively.We assume that only dipole transitions between levels | i → | i and | i → | i are allowed. If the environ-ment is in a vacuum state, the amplitude damping noisewhich corresponds to the spontaneous emission from theV-configuration qutrit can be represented by the follow- ing map [26]: | i S | i E → | i S | i E , | i S | i E → √ − d | i S | i E + √ d | i S | i E , (1) | i S | i E → √ − D | i S | i E + √ D | i S | i E , where d, D ∈ [0 ,
1] represents the decay rates of the upperlevels | i and | i , respectively. B. weak measurement for qutrits
The null-result weak measurement that we consideris the POVM or partial-collapse measurement originallydiscussed in Refs. [7, 8]. It is different from amplitudedamping in the sense that we add an ideal detector tomonitor the environment function as follows: the detec-tor clicks with a probability p if there is an excitationin the environment and never clicks with a probability1 − p if no excitation is detected in the environment.For the qutrit case, we can construct the POVM ele-ments as: M = diag (0 , √ p, M = diag (0 , , √ q ) and M = diag (1 , √ − p, √ − q ), where p and q representthe weak measurement strengths of transitions | i → | i and | i → | i , respectively. The measurement operators M and M are identical to the normal projection mea-surements in which the state of the qutrit is irrevocablycollapsed to the ground state and an excitation is emittedfrom system to environment. They are not reversible andwe therefore discard the result from experiments whichproduced clicks, thereby removing the terms √ p | i S | i E and √ q | i S | i E from the state map. Fortunately, themeasurement operator M is a weak (or partial-collapse)measurement for the single qutrit that we are interestedin in this paper. We rewrite M as | i S | i E → | i S | i E , | i S | i E → p − p | i S | i E , (2) | i S | i E → p − q | i S | i E , C. weak measurement reversal for qutrits
Except for von Neumann projective measurements,any weak or partial-collapse measurement could be re-versed [27]. According to Ref. [27], it is easy to con-struct the reversed weak measurement operator of thenull-result weak measurement as shown in Eq. (2). Thesingle-qutrit reversing measurement ( M r ) is also a non-unitary operation that can be written as M r = p (1 − p r )(1 − q r ) 0 00 √ − q r
00 0 √ − p r , (3)where p r and q r are the strengths of the reversing mea-surements. As the matrix is non-unitary, the probabilityof successful reversal will always be less than unity. III. PROTECTION OF QUTRIT-QUTRITENTANGLEMENTA. scheme one
We first check the efficiency of the first scheme asshown in Fig. 1(a). For simplicity, we assume two iden-tical qutrits are initially prepared in the following state | Ψ i = α | i + β | i + γ | i , (4)where | α | + | β | + | γ | = 1. Such a qutrit-qutrit entan-gled state can be experimentally prepared by utilizingthe orbital angular momentum of photons [18, 20] . Weassume they suffer independent but identical amplitudedamping noise (i.e., d = d = D = D = D ). Then theinitial pure state inevitably evolves into a mixed state inthe presence of noise. ρ d = X i =1 ε i | Ψ ih Ψ | ε † i , (5)where ε i = E j ⊗ E k , ( j, k = 0 , ,
2) are the Kraus opera-tors. In the standard product basis {| j, k i = | j + k +1 i} ,the non-zero elements of ρ d are: ρ = | α | + D ( | β | + | γ | ) ,ρ = ρ ∗ = (1 − D ) αβ ∗ ,ρ = ρ ∗ = (1 − D ) αγ ∗ , (6) ρ = ρ = D (1 − D ) | β | ,ρ = ρ = D (1 − D ) | γ | ,ρ = (1 − D ) | β | ,ρ = ρ ∗ = (1 − D ) βγ ∗ ,ρ = (1 − D ) | γ | , After the amplitude damping decoherence, we performquantum measurement reversal operations on each qutritas shown in Eq. (3). The non-zero elements of the finalreduced density matrix ρ r are: ρ = [(1 − p r ) | α | + D (1 − p r ) ( | β | + | γ | )] /C ,ρ = ρ ∗ = (1 − D )(1 − p r ) αβ ∗ /C ,ρ = ρ ∗ = (1 − D )(1 − p r ) αγ ∗ /C , (7) ρ = ρ = D (1 − D )(1 − p r ) | β | /C ,ρ = ρ = D (1 − D )(1 − p r ) | γ | /C ,ρ = (1 − D ) | β | /C ,ρ = ρ ∗ = (1 − D ) βγ ∗ /C ,ρ = (1 − D ) | γ | /C , where C = (1 − p r ) | α | +[(1 − D ) +2 D (1 − D )(1 − p r )+ D (1 − p r ) ]( | β | + | γ | ) is the normalization parameter.To quantify the qutrit-qutrit entanglement change un-der amplitude damping noise and weak measurement re-versal, we need an effective measure of mixed qutrit stateentanglement since damping causes the pure states to evolve into mixed states. One usually takes the entan-glement of formation [28] as such a measure, but in prac-tice it is not known how to compute this measure formixed states of d ⊗ d dimensional systems in the casewhen d >
2. A computable measure of distillable entan-glement of mixed states was proposed in Ref. [29]. It isbased on the trace norm of the partial transposition ρ T ofthe state ρ . From the Peres’ criterion of separability [30],it follows that if ρ T is not positive, then ρ is entangled.Hence one defines the negativity of the state ρ as N = || ρ T || − . (8) N is equal to the absolute value of the sum of nega-tive eigenvalues of ρ T and is an entanglement monotone[29], but it cannot detect bound entangled states [31]. D N e g a ti v it y N d N r D N e g a ti v it y N d N r (a)(b) Fig.2. (color online) Negativity as a function of D , where wehave chosen the optimal reversing measurement strength tobe p r = D . (a) | Ψ i = 1 / √ | i + | i + | i ), the reversednegativity is always higher than N d . When p r → N d goesto 0 while N r is finite. (b) | Ψ i = p / | i + p / | i +0 | i , N r is not always higher than N d , and both N d and N r go to 0 when ESD appears. According to Eq. (8), it is easy to calculate the dampednegativity ( N d ) and the reversed negativity ( N r ). How-ever, the general analytic expressions of negativity for ρ d and ρ r are too complicated to present as they de-pend on the relationships between the initial parame-ters α , β , γ and the decoherence parameter D . Hencewe will present numerical results in the discussions be-low. In Fig. 2, we show how N d and N r behave fortwo particular initial states under amplitude dampingdecoherence and corresponding optimal reversing mea-surements. We have chosen the optimal reversing mea-surement strength to be p r = D which gives the max-imum amount of entanglement of the two-qutrit state ρ r . For | Ψ i = 1 / √ | i + | i + | i ), we note thatthe damped negativity N d decays as the decoherencestrength D increases, while the reversed negativity N r approaches a finite value. The reversed negativity N r ishigher than the negativity N d regardless of the decoher-ence strength parameter, as shown in Fig. 2(a). How-ever, for | Ψ i = p / | i + p / | i + 0 | i , we findthe reversed negativity N r is not always higher than thenegativity N d . Moreover, the reversed negativity N r suf-fers sudden death as well as N d , as shown in Fig. 2(b).The reason is straightforward as all operations are local,and no entanglement can be created between two inde-pendent qutrits in a separable state. The above resultsfor qutrits are quite in accordance with those for qubitsdiscussed in Ref. [12].As the weak measurement reversals are non-unitaryoperations, this scheme naturally has less than unity suc-cess probability. Under the optimal reversing weak mea-surements (i.e., p r = D ), the corresponding success prob-ability is: P = (1 − D ) (cid:2) | β | + | γ | )(2 D + D ) (cid:3) . (9)It is clear that P → D → B. scheme two
As shown above, the first scheme has some limitationsregarding the protection of entanglement and circumven-tion of ESD. In this section, we show that an improvedscheme first proposed by Kim et al. [13] can completelycircumvent the decoherence and protect the qutrit-qutritentanglement even if ESD occurs. The key difference isthat a prior weak measurement is applied on each qutritbefore it suffers amplitude damping decoherence, as de-picted in Fig. 1(b).The whole procedure is as follows: for each qutrit, firsta prior weak measurement with strength p is performed,then it goes through the amplitude damping channel, andfinally a post weak measurement reversal with strength p r is carried out. After these operations, the non-zero elements of the reduced density matrix ρ wr are: ρ = [(1 − p r ) | α | + (1 − p ) D (1 − p r ) ( | β | + | γ | )] /C ,ρ = ρ ∗ = (1 − p )(1 − D )(1 − p r ) αβ ∗ /C ,ρ = ρ ∗ = (1 − p )(1 − D )(1 − p r ) αγ ∗ /C , (10) ρ = ρ = (1 − p ) D (1 − D )(1 − p r ) | β | /C ,ρ = ρ = (1 − p ) D (1 − D )(1 − p r ) | γ | /C ,ρ = (1 − p ) (1 − D ) | β | /C ,ρ = ρ ∗ = (1 − p ) (1 − D ) βγ ∗ /C ,ρ = (1 − p ) (1 − D ) | γ | /C , where C = (1 − p r ) | α | + (1 − p ) [(1 − D ) + 2 D (1 − D )(1 − p r ) + D (1 − p r ) ]( | β | + | γ | ). D N d α = β = γ =1 / √ α = p / , β = p / , γ =0 p N w r / N i α = β = γ =1 / √ α = p / , β = p / , γ =0 (a) (b) Fig.3. (color online) For two particular initially entangledqutrit-qutrit states | Ψ i = 1 / √ | i + | i + | i ) (solid line)and | Ψ i = p / | i + p / | i + 0 | i (dashed line): (a)Negativity N d as a function of decoherence strength D . (b)The ratio of N wr to N i as a function of weak measurementstrength p when D = 0 . Following the methods demonstrated in Refs. [10, 13],the optimal reversing measurement strength that givesthe maximum amount of entanglement for the two-qutritstate ρ wr is calculated to be p r = p + D ¯ p where ¯ p = 1 − p .We still consider the same initial states in scheme one andcompare the effectiveness of these two schemes for sup-pressing amplitude damping noise. In Fig. 3, we showthe protection of entanglement from decoherence by us-ing a weak measurement and weak measurement rever-sal. As we know, the qutrit-qutrit entanglement decaysmonotonously with increasing D and even experiencesESD when | Ψ i = p / | i + p / | i + 0 | i .However, it is clear that the qutrit-qutrit entangle-ment can be protected by the combined action of priorweak measurements and post weak measurement re-versals. In Fig. 3(a), we note that the negativity of | Ψ i = 1 / √ | i + | i + | i ) is 0.04 when D = 0 . p → N wr (negativity after the se-quence of weak measurement, decoherence and revers-ing measurement) to the initial negativity N i to high-light the entanglement recovery efficiency. To demon-strate the scheme’s ability to circumvent ESD, we choose D = 0 .
8, at which point ESD appears for the initial state | Ψ i = p / | i + p / | i + 0 | i in Fig. 3(a). Wefind the entanglement can be completely recovered witha certain probability by the sequence of weak measure-ment and weak measurement reversal, which is similarto that in Ref. [13] where a two-qubit entangled state isconsidered.Similarly to the first scheme, the success probabilityunder the optimal reversing weak measurements (i.e., p r = p + D ¯ p ) can be written as P = (1 − D ) ¯ p (cid:2) | β | + | γ | )(2 D ¯ p + D ¯ p ) (cid:3) . (11)We observe that for p →
1, the success probability P → p , the closer the initial qutrit is re-versed towards the | i state. Once the system is in | i ,then it will be immune to amplitude damping decoher-ence. In the first scheme, no prior weak measurementis carried out before the qutrits go through the ampli-tude damping channel, thus the amount of reversed en-tanglement highly depends on the initial states and thedecoherence strength D . In the second scheme, priorweak measurements are performed to move the state to-wards the | i state, which does not experience ampli-tude damping decoherence. Then optimal weak measure-ment reversals are performed to revert the qutrits backto the initial state. Therefore, the amount of reversedentanglement is not related to the decoherence strength D but depends on the weak measurement strength p . Ini-tial entanglement can entirely be recovered for any initial state by the combined prior weak measurements and postweak measurement reversals when p → C. Discussions
In the above analyses, we have assumed that the twoqutrits are identical and the decoherence parameters d and D are the same for states | i and | i . In fact, thesetwo schemes are universal for the most general case (i.e., d = d = D = D ). Following the same calculationprocedure as above, we plot in Fig. 4 the numerical re-sults for the two weak measurement schemes against theamplitude damping decoherence. For the first scheme,the optimal reversing measurement strength is calculatedto be p r k = d k and q r k = D k ( k = 1 , p r k = p k + d k ¯ p k and q r k = q k + D k ¯ q k for the secondscheme. D N e g a ti v it y N d N r p N w r / N i (a)(b) Fig.4. (color online) Entanglement protection of state | Ψ i =1 / √ | i + | i + | i ) by weak measurement and weakmeasurement reversal: (a) scheme one with the decoherenceparameters d = D , d = 0 . D , D = 0 . D and D = 0 . D ,(b) scheme two with the decoherence parameters d = 0 . d = 0 . D = 0 . D = 0 . IV. EXPERIMENTAL FEASIBILITY
It is necessary to give a brief discussion of some keyproblems which are related to the experimental imple-mentation of our procedure. Here, we restrict our dis-cussions to the cavity QED system which we think isthe best candidate for the experimental realization of ourscheme.
Initial state preparation.
The V-type qutrit-qutrit en-tanglement of Eq. (4) could be generated by sending apair of momentum and polarization-entangled photons totwo spatially separated cavities in which a V-type atomis trapped [32]. For the atomic level structure, we cantake Rb as our choice. The state | i corresponds to | F = 1 , m F = 0 i of 5 S / , while the states | i and | i correspond to the degenerate levels | F = 1 , m F = 1 i and | F = 1 , m F = − i of 5 P / , respectively. The transi-tions | i → | i and | i → | i emit right-circularly andleft-circularly polarized photons, so we can distinguishthe parameters p and q during the weak measurement. Amplitude damping decoherence.
In a cavity QED sys-tem, the amplitude damping decoherence is the naturalspontaneous emission of a photon from the excited stateof an atom to its ground state. The dynamic map ofEq. (1) describes a dissipative interaction between a V-type qutrit and its vacuum environment [26].
Weak measurement.
As shown in Sect. II, we notethat the only difference between the AD decoherencemap Eq. (1) and weak measurement map Eq. (2) is theinclusion of the √ p | i S | i E and √ q | i S | i E terms. Inthis sense, we can add an ideal single-photon detector tomonitor the cavity. Whenever there is a detector click,we discard the result. This postselection removes the √ p | i S | i E and √ q | i S | i E terms and hence a null-resultweak measurement is implemented. Weak measurement reversal.
To reverse the effect ofthe weak measurement ( M ), we only need to apply theinverse of M M − = √ − p
00 0 √ − q (12)since M − can be re-written as M − = 1 p (1 − p )(1 − q ) F M F M F = 1 p (1 − p )(1 − q ) M r , (13)where F is the trit-flip operation F = . (14) Thus, the weak measurement reversal procedure ofEq. (3) can be constructed by the following five sequen-tial operations on each system qutrit: trit-flip ( F ), weakmeasurement ( M ), trit-flip ( F ), another weak measure-ment ( M ), and trit-flip ( F ). The trit-flip operation F can be realized by a π pulse applied on the transition | i ↔ | i and followed by another π pulse to interchangethe populations between | i and | i . (i.e., by the seriesof two π pulses π | i↔| i π | i↔| i ) [33]. V. CONCLUSIONS
In conclusion, we have demonstrated that weak mea-surement reversal can indeed be useful for combating am-plitude damping decoherence and recovering the entan-glement of two qutrits. In particular, we have exam-ined two simple schemes: one is “ amplitude damping + weak measurement reversal ” and the other is “ weak mea-surement + amplitude damping + weak measurement re-versal ”. We have shown that the first scheme can par-tially recover qutrit-qutrit entanglement for certain ini-tial states, but it has some limitations with respect toentanglement protection efficiency and ESD circumven-tion. For the second scheme, in which prior weak mea-surements and post weak measurement reversals are car-ried out sequentially, the amplitude damping decoherencecan be completely suppressed for any initial states evenif ESD occurs. Even though the method is risky (i.e., astronger procedure is required for a longer preservation,which decreases the probability of success), this proce-dure for entanglement preservation is useful in entangle-ment distillation protocols and some quantum communi-cation tasks. Acknowledgments
We thank the copy editor for proofreading ourmanuscript. This work is supported by the Special Fundsof the National Natural Science Foundation of Chinaunder Grant Nos. 11247006 and 11247207, and by theNatural Science Foundation of Jiangxi under Grant Nos.20132BAB212008 and by the Scientific Research Foun-dation of the Jiangxi Provincial Education Departmentunder Grants No. GJJ13651. [1] M.A. Nielsen, I.L. Chuang,
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