Effectiveness of Depolarizing noise in causing sudden death of entanglement
aa r X i v : . [ qu a n t - ph ] O c t Effectiveness of Depolarizing noise in causing sudden death of entanglement
K.O. Yashodamma, P.J. Geetha, and Sudha
1, 2, ∗ Department of Physics, Kuvempu University, Shankaraghatta, Shimoga-577 451, India Inspire Institute Inc., Alexandria, Virginia, 22303, USA. (Dated: March 4, 2018)Continuing on the recent observation that sudden death of entanglement can occur even when asingle qubit of a 2-qubit state is exposed to noisy environment(Results in Physics,3,41–45 (2013)),we examine the local effects of several noises on bipartite qubit-qutrit and qutrit-qutrit systems. Inorder to rule out any initial interactions with environment, we consider maximally entangled purestates of qubit-qutrit and qutrit-qutrit systems for our analysis. We show that depolarizing andgeneralized amplitude damping noise can cause sudden death of entanglement in these states evenwhen they act only on one part of the system. We also show that sudden death of entanglementoccurs much faster under the action of depolarizing noise when compared to that due to generalizedamplitude damping. This result strengthens the observation (Results in Physics,3,41–45 (2013)) thatdepolarizing noise is more effective than other noise models in causing sudden death of entanglement.
PACS numbers: 03.67.Mn, 03.67.Ac, 42.50.Lc
I. INTRODUCTION
The pure multipartite maximally entangled quantumstates are of wide scope in various fields of quantum com-putation, quantum information and quantum communi-cation [1] as these states produce protocols with highprecision. But due to an inevitable coupling of quantumsystems with their surrounding environment, the stateslose their coherence as well as entanglement. This irre-versible interaction of quantum states with surroundingnoisy environments is called decoherence [2] and in addi-tion to making them more mixed, it causes degradationof the initial entanglement [2]. The study of dynamicsof the quantum states, pure as well as mixed, exposed tonoisy environment is an important area of study in or-der to devise ways such that the system can be protectedfrom the detrimental effects of the environment.Depending upon the environment surrounding thequantum state there may either be an asymptotic de-cay of entanglement or the entanglement may vanish infinite time. The phenomenon of finite time disentangle-ment or Entanglement Sudden Death (ESD) has beengiven enough importance [3–25] and considerable amountof work is also going on to avoid ESD [26–32].It has been shown in Ref. [4] that the entanglementloss occurs in finite time under the action of pure vac-cum noise in bipartite states of qubits. Such a finite timedisentanglement is shown to be due to non additivity ofdecay rates [6] of the two parts individually exposed toeither same or different noisy environments. In a recentwork [25] it is shown that, the single noise acting on a sin-gle qubit is also sufficient to cause ESD in 2-qubit states.It was also shown that when acted on one of the qubitsof a pure 2-qubit state, both amplitude and phase noisecan only cause asymptotic decay of entanglement while ∗ Electronic address: arss@rediffmail.com depolarizing noise causes sudden death of entanglementthus establishing the effectiveness of depolarizing noise incausing ESD than amplitude noise and dephasing noise.In this article, we continue on the work in Ref. [25] andexamine the effect of several noisy environments on a partof bipartite qubit-qutrit and qutrit-qutrit systems. Thenoise models that we consider include amplitude damp-ing, dephasing, depolarizing and generalized amplitudedamping (GAD). We show that the local action of depo-larizing noise and generalized amplitude damping noisecan cause sudden death of entanglement in pure 2 × × II. ACTION OF NOISE ON PUREQUBIT-QUTRIT STATES
It is well known that an arbitrary pure qubit-qutritstate | ψ qq ′ i = X i,j a ij | i, j i (1)where i = 0 , j = 0 , , a ij are in general complex with P i,j | a ij | = 1. On Schmidtdecomposition (1) takes the form | ψ qq ′ i ≡ a | ′ i + p − a | ′ i (2)with α being a real number 0 ≤ a ≤ | i , | i are theSchmidt bases in the 2-dimensional space of qubits and | ′ i , | ′ i , | ′ i are the Schmidt bases in the 3-dimensionalqutrit space. From Schmidt decomposed form in Eq.(2) a maximally entangled pure qubit-qutrit state canbe readily seen to be | Ψ qq ′ i = 1 √ | ′ i + 1 √ | ′ i (3)We wish to examine the effects of different noisy environ-ments on the maximum possible entanglement containedin the state | Ψ qq ′ i . In order to do this, we employ nega-tivity of partial transpose N ( ρ ) as the suitable measure ofentanglement [33–35]. This measure being necessary andsufficient for 2 × | Ψ qq ′ i . A. Amplitude damping:
Amplitude damping noise gives the right descriptionfor energy dissipation from a quantum system. It charac-terizes processes such as spontaneous emission of a pho-ton from a quantum system, attainment of thermal equi-librium by a spin system at a high temperature and is oneof the well-studied noise models in the literature [1, 4, 25].The Kraus operators for a single qubit[4] amplitude noiseare given by E = (cid:18) η
00 1 (cid:19) ; E = (cid:18) p − η (cid:19) (4)where η = e − Γ t , with Γ being the decay factor of the am-plitude noise. The local action of the amplitude dampingchannel on the qubit (first subsystem) of the pure maxi-mally entangled state in Eq. (3) can be written as ρ Aqq ′ = X i =0 ( E i ⊗ I ) ρ qq ′ ( E i ⊗ I ) † (5)where I is the 3 × ρ qq ′ = | Ψ qq ′ ih Ψ qq ′ | is the density matrix of the state | Ψ qq ′ i . Ex-plicitly, the state ρ Aqq ′ is given by ρ Aqq ′ = 12 η η
00 0 0 0 0 00 0 0 0 0 00 0 0 1 − η η . (6) The partial transpose ¯ ρ Aqq ′ obtained by transposing thequbit indices is given by¯ ρ Aqq ′ = 12 η η η − η (7)and its non-zero eigenvalues are readily evaluated to be λ = λ = 12 ; λ = η λ = − η . (8)The only negative eigenvalue of ¯ ρ Aqq ′ being λ , the nega-tivity of partial transpose is given by N ( ρ Aqq ′ ) = η e − Γ t N ( ρ Aqq ′ ) with time is evidentthrough Eq. (9). It can be recalled here that similarsituation arose in the case of the local action of amplitudenoise on a 2-qubit pure state [25]. B. Phase damping:
The noise that describes the process of loss of quantuminformation without loss of energy is referred to as phasedamping noise [1] or dephasing noise. It is a uniquelyquantum mechanical noise [1] and the single qubit Krausoperators of phase damping channel are given by [9] P = (cid:18) γ (cid:19) ; P = (cid:18) p − γ (cid:19) (10)where γ = e − Γ t and Γ is the decay constant of the de-phasing noise. The time evolved density matrix of Ψ qq ′ with its first qubit being subjected to phase damping isgiven by ρ Pqq ′ = 12 γ
00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0 γ (11)The non-zero eigenvalues of the partially transposed den-sity matrix of ρ Pqq ′ , transposed with respect to the qubitindices, are identified to be 1 / , / , γ/ , − γ/ N ( ρ Pqq ′ ) is givenby N ( ρ Pqq ′ ) = γ − Γ t/
2] (12)The exponential decay of N ( ρ Pqq ′ ) due to the local ac-tion of dephasing noise on the first subsystem (qubit) ofthe pure maximally entangled qubit-qutrit state | Ψ qq ′ i isthus evident through Eq. (12). Here too we can recallthat the dephasing noise acting on the first (or second)qubit of the pure 2-qubit state resulted in an asymptoticdecay as detailed in Ref. [25]. C. Generalized Amplitude Damping :
Generalized Amplitude Dampling(GAD) is a quantumnoise that describes the effect of dissipation to environ-ment at finite temperature [1]. The relaxation processesdue to coupling of spins to their surrounding lattice whichis at a high temperature than the temperature of thespins are ably modelled by GAD [1]. Such processes of-ten occur in the NMR implementation of quantum com-puters and we believe that the study of the effect of GADon pure maximally entangled states is also important asregards any future protocols using entangled pure states.The Kraus operators corresponding to GeneralizedAmplitude Damping on a single qubit are representedby [30] G = p − p (cid:18) η (cid:19) G = p − p (cid:18) p − η (cid:19) G = √ p (cid:18) η
00 1 (cid:19) G = √ p (cid:18) p − η (cid:19) (13)where η = e − Γ t/ and Γ is the decay factor corre-sponding to generalized amplitude damping. The state ρ Gqq ′ = P i =0 ( G i ⊗ I ) ρ qq ′ ( G i ⊗ I ) † and its partial trans-pose ¯ ρ Gqq ′ (transposed either with respect to the qubit orwith the qutrit) can be readily evaluated. They are givenby ρ Gqq ′ = − p (1 − η )2 η (1 − p )(1 − η )2 p (1 − η )2 η p (1 − η )+ η
00 0 0 0 0 0 ¯ ρ Gqq ′ = − p (1 − η )2 (1 − p )(1 − η )2 η η p (1 − η )2 η p (1 − η )+ η
00 0 0 0 0 0
An explicit evaluation of N ( ρ Gqq ′ ) shows that an asymp-totic decay of entanglement happens at p = 0, p = 1and this is expected because the GAD reduces to ampli-tude damping when p = 0 or 1. Also, the variation of N ( ρ Gqq ′ ) with Γ t can be seen to be symmetric over thevalue p = 1 /
2. A plot of the negativity of partial trans-pose N ( ρ Gqq ′ ) versus Γ t for different values of p are asshown in Fig. 1. G t0.10.20.30.40.5N H Ρ L p = (cid:144) = (cid:144)
4, 3 (cid:144) = (cid:144)
8, 7 (cid:144) =
0, 1
FIG. 1: Variation of N ( ρ Gqq ′ ) with respect to Γ t for differentvalues of p . The sudden death of entanglement is seen tooccur when 0 < p < p = 1 / It is readily seen through Fig.1 that there is finitetime disentanglement due to the local action of GADon the qubit of maximally entangled pure qubit-qutritstate | Ψ qq ′ i when 0 < p <
1. It can also be seen thatthe decay time is shortest when p = 1 /
2. We thereforeconclude that GAD is capable of causing sudden death ofentanglement in pure maximally entangled qubit-qutritstates due to its action on one of the subsystems.
D. Depolarizing Noise:
A quantum noise that is capable of depolarizing a qubitwith probability α and keeping it undisturbed with aprobability 1 − α is the depolarizing noise [1]. Here ‘depo-larizing’ a qubit means converting the qubit into a com-pletely mixed state I / D = √ − α (cid:18) (cid:19) D = r α (cid:18) (cid:19) D = r α (cid:18) − ii (cid:19) D = r α (cid:18) − (cid:19) ; (14)Here α = 1 − e − Γ t/ and Γ is the decay factorof the depolarizing noise. On evaluating ρ Dqq ′ = P i =1 ( D i ⊗ I ) ρ qq ′ ( D i ⊗ I ) † and its partial transpose¯ ρ Dqq ′ , we readily obtain the negativity of partial transposeof ρ Dqq ′ as N ( ρ Dqq ′ ) = (1 − α )2 = 2 exp[ − Γ t/ − . (15)Notice that N ( ρ Dqq ′ ) becomes zero when α = or whenΓ t = 1 . | Ψ qq ′ i , we wish to exam-ine which among these is more effective. The followinggraph (Fig. 2) compares the entanglement decay time ofGAD (for p = 1 /
2) and depolarizing noise. G t0.10.20.30.40.5N H Ρ L DepolarizingGAD
FIG. 2: The negativity of partial transpose of maximally en-tangled pure qubit-qutrit state | Ψ qq ′ i with its first qubit beingunder the action of generalized amplitude damping and depo-larizing noise are plotted as a function of time. The suddendeath of entanglement occurs faster in the case of depolarizingnoise than in the case of GAD. Fig. 2 clearly indicates that depolarizing causes anearly onset of entanglement sudden death than that dueto Generalized Amplitude Damping noise when acting onthe qubit of pure maximally entangled state qubit-qutritstate.While we have considered the action of different noisemodels on the first subsystem (qubit) of the qubit-qutritstate | Ψ qq ′ i , it is not difficult to see that the results willnot be any different when the second subsystem (qutrit)alone is exposed to noise. In the following section, weexamine whether the results obtained here in the caseof 2 × × III. LOCAL ACTION OF NOISE ON PUREQUTRIT-QUTRIT STATES
A maximally entangled pure qutrit-qutrit state is givenby, | Φ i q ′ q ′ = 1 √ | ′ ′ i + | ′ ′ i + | ′ ′ i ) . (16)with | i ′ r i , i = 0 ′ , ′ , ′ , r = 1 , E = η
00 0 η , E = p − η
00 0 00 0 0 E = p − η ; η = e − Γ t . (17)On explicit evaluation, the negativity of partial transposeof the state ρ Aq ′ q ′ = P i =0 ( E i ⊗ I ) ρ q ′ q ′ ( E i ⊗ I ) † , where ρ q ′ q ′ = | Φ q ′ q ′ ih Φ q ′ q ′ | , is obtained as N ( ρ Aq ′ q ′ ) = 56 η = 56 e − Γ t (18)This indicates asymptotic decay of entanglement for thepure qutrit-qutrit state ρ Aq ′ q ′ as in the case of qubit-qutritstate ρ Aqq ′ subjected to local action of amplitude noise.The action of phase noise on a single qutrit can berepresented by the Kraus operators P = γ
00 0 γ , P = p − γ
00 0 0 P = p − γ ; γ = e − Γ t (19)Here too, as in the case of amplitude noise, the negativityof partial transpose given by N ( ρ Pq ′ q ′ ) = γ γ + 2) = e − Γ t/ (cid:16) e − Γ t/ + 2 (cid:17) (20)shows asymptotic decay of entanglement.The single qutrit Kraus operators for Generalized Am-plitude Damping noise are given by, G = √ p η
00 0 η , G = √ p p − η
00 0 00 0 0 G = √ p p − η , G = p − p η η
00 0 1 G = p − p p − η , (21) G = p − p p − η with η = e − Γ t . A plot of the negativity of partial trans-pose N ( ρ Gq ′ q ′ ) versus Γ t for different values of p are asshown in Fig. 3. G t0.20.40.60.81.0N H Ρ L p = (cid:144) = (cid:144)
4, 3 (cid:144) = (cid:144)
8, 7 (cid:144) =
0, 1
FIG. 3: Variation of N ( ρ Gq ′ q ′ ) with respect to dimensiolesstime parameter Γ t for different values of p . It can be seen that under the action of GAD on a singlequtrit of the state | Φ q ′ q ′ i , the decay of entanglement isalmost smooth (See Fig. 3). The variation is symmetricabout p = 1 / p = 0, p = 1, an exponential decayoccurs for both these values.The single qutrit Kraus operators for depolarizingnoise [20, 36] are given by, D = √ − α I , D = r α Y,D = r α Z, D = r α Y ,D = r α Y Z, D = r α Y Z, (22) D = r α Y Z , D = r α Y Z D = r α Z where I is the 3 × Y = , Z = ω
00 0 ω ; (23) ω = 1; α = 1 − e − Γ t . The negativity of partial transpose of the state ρ Dq ′ q ′ = P i =0 ( D i ⊗ I ) ρ q ′ q ′ ( D i ⊗ I ) † is obtained as N ( ρ Dq ′ q ′ ) = 12 (2 − α ) = 12 (3 e − Γ t/ − N ( ρ Dq ′ q ′ ) = 0 when α = or whenΓ t = 2 .
2. In order to compare the entanglement decay inthe case of GAD and depolarizing noise, we have plottedbelow (Fig. 4) the negativity of partial transpose in bothcases with respect to time. G t0.20.40.60.81.0N H Ρ L DepolarizingGAD
FIG. 4: The variation of negativity of partial transpose withrespect to dimensionless time parameter Γ t under the localaction of depolarizing and generalized amplitude damping. The fact that depolarizing noise is more effective thanGAD in disentangling a pure maximally entangled qutrit-qutrit state, even when acting on a single qutrit of thestate, is clearly shown in Fig. 4.It is to be mentioned here that though negativity ofpartial transpose is only a sufficient criterion for entan-glement of 3 × IV. CONCLUSION
In this article, we have analyzed the effects of variousnoisy environments on a pure maximally entangled 2 × × Acknowledgment
K.O. Yashodamma and P.J. Geetha acknowledge the sup-port of Department of Science and Technology (DST),Govt. of India through the award of INSPIRE fellow-ship. [1] M. A. Nielsen, I. L. Chuang, Quantum Computation andQuantum Information (Cambridge Univ. Press, Cam- bridge, 2000). [2] W. H. Zurek. Rev. Mod. Phys.
715 (2003); M.Schlosshawer, Rev. Mod. Phys. , 1267 (2005);C. Si-mon, J. Kempe, Phys. Rev. A ,052327 (2002); W. D¨ur,H. J. Briegel, Phys. Rev. Lett. , , 080501 (2008)[3] L. Di´osi In: Benatti F, Floreanini R, editors. Irreversiblequantum dynamics. Berlin: Springer; 2003;[4] T. Yu, J. H. Eberly, Phys. Rev. Lett. , 140403(2006).[7] K. Roszak, P. Machnikowski Phys. Rev. A , 022313(2006)[8] M. Y¨onac, T. Yu, J. H. Eberly J. Phys. B , S621 (2006)[9] T. Yu and J.H. Eberly, Optics Comm., , 393 (2006).[10] T. Yu, J. H. Eberly, Science , 555 (2007)[11] T. Yu and J. H. Eberly, Quant. Inform. and Comput. ,459 (2007)[12] H. T. Cui, K. Li, X. X. Yi, Phys. Lett. A. , 44 (2007)[13] M. P. Almeida et.al. Science , 579 (2007)[14] K. Ann, G. Jaegar, Phys. Rev. A , 044101 (2007)[15] Mazhar Ali, A. R. P. Rau and Kedar Ranadequantph/arXiv:0710.2238.[16] K. Ann and G. Jaeger, Phys. Lett. A. , 579-583(2008).[17] J. Li, K. Chalapat, G. S Paraoanu, J. Low Temp. Phys. , 294 (2008).[18] B. Bellomo et.al., Phys. Rev. A , 062309 (2010).[19] B. Bellomo et.al., Phys. Scr. T147 , 014004 (2012)[20] S. Khan, quant-ph/arxiv:1012:1028.[21] R. Lo Franco et.al., Phys. Scr.
T147 , 014019 (2012) [22] R. Lo Franco, B. Bellomo, S. Maniscalco, G. Compagno,Int. J. Mod. Phys. B , 1245053 (2013)[23] Qian Xiao-Feng, J.H. Eberly, Phys. Lett. A. , 2931(2012)[24] R. Lo Franco, E. Andersson, B. Bellomo, G. Compagno,Phys. Rev. A , 032318 (2012)[25] K. O. Yashodamma and Sudha, Results in Physics 3, 41(2013).[26] A. R. P. Rau, Mazhar Ali and G. Alber, Eur. Phys. Lett.
459 (2009).[30] M. Siomau and Ali. A. Kamli, Phys.Rev. A. ,1 (1996)[35] G. Vidal, R. F. Werner, Phys. Rev. A. , 032314 (2007).[36] S. Salimi, M. M. Soltanzadeh, International Journal ofQuantum Infor- mation , 615 (2009)[37] J. Gregg, J. Mod. Opt.
1, 2007[38] M. Ramzan, M. K. Khan, Quant. Inf. Process11