A study of Quantum Correlation for Three Qubit States under the effect of Quantum Noisy Channels
aa r X i v : . [ qu a n t - ph ] N ov A study of Quantum Correlation for Three Qubit States under the effect of QuantumNoisy Channels
Pratik K. Sarangi , , & Indranil Chakrabarty Center for Quantum Information & Quantum Computation,Dept of Physics, Indian Institute of Science, Bangalore, India. Center for Security, Theory & Algorithmic Research,International Institute for Information Technology, Hyderabad, India. The University of Arizona, Tucson, AZ 85721, United States
We study the dynamics of quantum dissension for three qubit states in various dissipative channelssuch as amplitude damping, dephasing and depolarizing. Our study is solely based on Markovianenvironments where quantum channels are without memory and each qubit is coupled to its ownenvironment. We start with mixed GHZ, mixed W, mixture of separable states, a mixed biseparablestate, as the initial states and mostly observe that the decay of quantum dissension is asymptoticin contrast to sudden death of quantum entanglement in similar environments. This is a clearindication of the fact that quantum correlation in general is more robust against the effect of noise.However, for a given class of initial mixed states we find a temporary leap in quantum dissension fora certain interval of time. More precisely, we observe the revival of quantum correlation to happenfor certain time period. This signifies that the measure of quantum correlation such as quantumdiscord, quantum dissension, defined from the information theoretic perspective is different fromthe correlation defined from the entanglement-separability paradigm and can increase under theeffect of the local noise. We also study the effects of these channels on the monogamy score of eachof these initial states. Interestingly, we find that for certain class of states and channels, there ischange from negative values to positive values of the monogamy score with classical randomness aswell as with time. This gives us an important insight in obtaining states which are freely sharable(polygamous state) from the states which are not freely sharable (monogamous). This is indeeda remarkable feature, as we can create monogamous states from polygamous states Monogamousstates are considered to have more signatures of quantum ness and can be used for security purpose.
1. INTRODUCTION
For a long time quantum entanglement was only of philo-sophical interest and researchers were mainly focusing onaddressing the questions that were related with the quan-tum mechanical understanding of various fundamentalnotions like reality and locality [1]. However, for thelast two decades world had seen that quantum entangle-ment is not only a philosophical riddle but also a realityas far as the laboratory preparation of entangled qubitsare concerned [2]. Researches that were conducted dur-ing these decades were not all concerned about its ex-istence but mostly about its usefulness as a resource tocarry out information processing protocols like quantumteleportation [3], cryptography [4], superdense coding [5],and in many other tasks [6] . It was subsequently evidentfrom various followed up investigations that quantum en-tanglement plays a pivotal role in all these informationprocessing protocols. Therefore, understanding the pre-cise nature of entanglement in bipartite and multipartyquantum systems has become the holy-grail of quantuminformation processing.However, the precise role of entanglement as a resource inquantum information processing is not fully understoodand it was suggested that entanglement is not the onlytype of correlation present in quantum states. This is be-cause lately some computational tasks were carried outeven in the absence of entanglement [7]. This providedthe foundation to the belief that there may be correla- tion present in the system even in the absence of entan-glement. Hence, researchers redefined quantum corre-lation from the information theoretic perspective. Thisgave rise to various measures [8–11] of quantum corre-lation, the predominant of them being quantum discord[8]. Though there are issues that need to be addressed,in much deeper level quantum discord temporarily sat-isfies certain relevant questions. Subsequently, quantumdiscord has been given an operational interpretation indifferent contexts like quantum state merging [12] andremote state preparation [13]. In addition, extension ofthe notion of quantum discord to multi qubit cases hasbeen proposed [10, 11].Many works were done in the recent past to investigatethe dynamics of quantum correlation in open systemsby comparing the evolution of different types of initialstates in specific models. These states are typically twoqubits coupled with two local baths or one common bath.In principle, there are several factors that can affect theevolution, namely, the initial state for the system andenvironment, the type of system-environment interactionand the structure of the reservoir. A more relevant ques-tion will be how robust are these measures when they aresubjected to the noise in quantum channels.It is mainly inspired by the studies of sudden death of en-tanglement for two qubits, having no direct interaction[14, 15]. Entanglement Sudden Death (ESD) is said tooccur when the initial entanglement falls and remains atzero after a finite period of evolution for some choices ofthe initial state. ESD is a potential threat to quantumalgorithms and quantum information protocols and thusthe quantum systems should be well protected againstnoisy environments. Another possible way to circum-vent such resource vanishing is to make use of resourceswhich do not suffer from sudden death. At this point,one can ask a similar question:
Does quantum discordpresent similar behavior?
In the first study [16] address-ing this question, researchers have compared the evolu-tion of concurrence and discord for two qubits, each sub-ject to independent Markovian decoherence (dephasing,depolarizing and amplitude damping). Looking at initialstates such as Werner states and partially-entangled purestates, the authors find no sudden death of discord evenwhen ESD does occur; quantum discord decays expo-nentially and vanishes asymptotically in all cases. How-ever, not much is known about the effects on multipartitecorrelation with time when they are transferred throughnoisy quantum channels.In this work, we study the dynamics of quantum dissen-sion of three qubit states which happens to be a measureof multi party quantum correlation, under the effect ofvarious quantum noisy channels. In addition, we alsostudy the dynamics of monogamy score of these threequbit states in presence of channel noise. In section 2,we provide a detail descriptions of quantum dissensionand monogamy score of quantum correlation. In section3, we study the effect of various noisy channels on quan-tum correlation and monogamy score when all the qubitsare transferred through them. Finally, we conclude inSection 4 by discussing future directions of explorations.
2. QUANTUM DISSENSION & MONOGAMYSCORE
In classical information theory [17], the total correlationbetween two random variables is defined by their mutualinformation. If X and Y are two random variables, themutual information is obtained by subtracting the jointentropy of the system from the sum of the individualentropies. Mathematically, this can be stated as: I ( X : Y ) = H ( X ) + H ( Y ) − H ( X : Y ) , (1)where H ( . ) defines Shannon entropy function.Another equivalent way of expressing mutual informationis by taking into account the reduction in uncertainty as-sociated with one random variable due to the introduc-tion of another random variable. Stated formally as, J ( X : Y ) = H ( X ) − H ( X | Y ) , (2)or K ( X : Y ) = H ( Y ) − H ( Y | X ) , (3)where H ( X | Y ) defines conditional entropy of X giventhat Y has already occurred and vice versa. All these above expressions are equivalent in classicalinformation theory. When we try to quantify corre-lation in quantum systems from an information theo-retic perspective, natural extension of these quantitieswill be obtained by replacing random variables with den-sity matrices, Shannon entropy with Von Neumann en-tropy and apposite definition of the conditional entropies.Stated mathematically, the quantum mutual informationis given by, I ( X : Y ) = S ( ρ X ) + S ( ρ Y ) − S ( ρ XY ) , (4)where ρ XY is the composite density matrix, ρ X and ρ Y are the local density matrices and S ( . ) defines Von Neu-mann entropy function.Similarly, by applying the argument of reduction of un-certainty associated with one quantum system with in-troduction of another quantum system, one can have thealternative definition of mutual information as, J ( X : Y ) = S ( ρ X ) − S ( ρ X | Y ) (5)and K ( X : Y ) = S ( ρ Y ) − S ( ρ Y | X ) . (6)Here S ( ρ X | Y ) is the average of conditional entropy andis obtained after carrying out a projective measurementon subsystem Y and vice versa. The projective mea-surement is done in the general basis {| u i = cos θ | i + e iφ sin θ | i , | u i = sin θ | i - e iφ cos θ | i} , where θ and φ have the range [0,2 π ]. Hence, the quantum conditionalentropy can be expressed as, S ( ρ X | Y ) = P j p j S ( ρ X | Π jY ) where p j = tr [( I X ⊗ Π jY ) ρ ( I X ⊗ Π jY )] , ( I X being iden-tity operator on the Hilbert space of the quantum system X ), gives the probability of obtaining the j the outcome.The corresponding post-measurement state of system X is ρ X | Π jY = p j tr Y [( I X ⊗ Π jY ) ρ ( I X ⊗ Π jY )] . It is impor-tant to note over here that S ( ρ X | Y ) is different from whatwill be the straightforward extension of classical condi-tional entropy. In quantum information, the meaning ofconditional entropy of the qubit X given that Y has oc-curred is the amount of uncertainty in the qubit X giventhat a measurement is carried out on the qubit Y .Consequently, the expressions I , J and K are not equiv-alent in the quantum domain. The differences between I − J and I − K are captured by quantum discord i.e. D ( X : Y ) = I ( X : Y ) − J ( X : Y )= S ( ρ Y ) + min { Π jY } S ( ρ X | Y ) − S ( ρ XY ) , (7) D ( Y : X ) = I ( X : Y ) − K ( X : Y )= S ( ρ X ) + min { Π jX } S ( ρ Y | X ) − S ( ρ XY ) . (8)One variant of quantum discord is the geometric quantumdiscord which is defined as the distance between a quan-tum state and the nearest classical (or separable) state,[18]. Quantum discord has been established as a non-negative measure of correlation for any quantum states.Subsequent researches were carried out to obtain an an-alytical closed form of quantum discord and was foundfor certain class of states [19, 20]. An unified geometricview of quantum correlations which includes discord, en-tanglement along with the introduction of the conceptslike quantum dissonance was given in [21].One of the natural extension of quantum discord fromtwo qubit to three qubit systems is quantum dissension[10]. Introduction of three qubits naturally brings in oneand two-particle projective measurement into considera-tion. These measurements can be performed on differentsubsystems leading to multiple definitions of quantumdissension. In other words a single quantity is not suffi-cient enough to capture all aspects of correlation in mul-tiparty systems. Quantum dissension in this context canbe interpreted as a vector quantity with values of cor-relation rising because of multiple definitions as variouscomponents. However, in principle when we define corre-lation in multi qubit situations, measurement in one sub-system can enhance the correlation in other two subsys-tems and thereby making quantum dissension to assumenegative values [22]. We emphasize on all possible one-particle projective measurements and two-particle pro-jective measurements.The mutual information of three classical random vari-ables in terms of entropies and joint entropies, are givenby I ( X : Y : Z ) = H ( X ) + H ( Y ) + H ( Z ) − [ H ( X, Y ) + H ( X, Z ) + H ( Y, Z )] + H ( X, Y, Z ) . (9)It is also possible to obtain an expression for mutual in-formation I ( X : Y : Z ) that involves conditional entropywith respect to one random variable: J ( X : Y : Z ) = H ( X, Y ) − H ( X | Y ) − H ( Y | X ) − H ( X | Z ) − H ( Y | Z ) + H ( X, Y | Z ) . (10)One can define another equivalent expression for classi-cal mutual information that includes conditional entropywith respect to two random variables: K ( X : Y : Z ) = [ H ( X ) + H ( Y ) + H ( Z )] − [ H ( X, Y ) + H ( X, Z )] + H ( X | Y, Z ) . (11)These equivalent classical information-theoretic defini-tions forms our basis for defining quantum dissension inthe next subsections. Let us consider a three-qubit state ρ XY Z where
X, Y and Z refer to the first, second and the third qubit un-der consideration. The quantum version of I ( X : Y : Z ) obtained by replacing random variables with density ma-trices and Shannon entropy with Von Neumann entropyreads, I ( X : Y : Z ) = S ( ρ X ) + S ( ρ Y ) + S ( ρ Z ) − [ S ( ρ XY ) + S ( ρ Y Z ) + S ( ρ XZ )] + S ( ρ XY Z ) . (12)The quantum version of J ( X : Y : Z ) , obtained by ap-propriately defining conditional entropies, is given by J ( X : Y : Z ) = S ( ρ XY ) − S ( ρ Y | Π jX ) − S ( ρ X | Π jY ) − S ( ρ X | Π jZ ) − S ( ρ Y | Π jZ ) + S ( ρ X,Y | Π jZ ) , (13)where Π nj refer to a one particle projective measurementon the subsystem ′ n ′ performed on the basis {| u i =cos θ | i + e iφ sin θ | i , | u i = sin θ | i - e iφ cos θ | i} where θ and φ lies in the range [0,2 π ].Quantum dissension function for single particle projec-tive measurement is given by the difference of I ( X : Y : Z ) and J ( X : Y : Z ) , i.e. D ( X : Y : Z ) = J ( X : Y : Z ) − I ( X : Y : Z ) . (14)Quantum dissension is given by the quantity δ =min( D ( X : Y : Z ) ), where the minimization is takenover the entire range of basis parameters in order for D to reveal maximum possible quantum correlation. The natural extension of K ( X : Y : Z ) in the quantumdomain is given by, K ( X : Y : Z ) = [ S ( ρ X ) + S ( ρ Y ) + S ( ρ Z )] − [ S ( ρ XY ) + S ( ρ XZ )] + S ( ρ X | Π jY Z ) . (15)The two-particle projective measurement is carried outin the most general basis: | v i = cos θ | i + e iφ sin θ | i , | v i = sin θ | i - e iφ cos θ | i , | v i = cos θ | i + e iφ sin θ | i , | v i = sin θ | i - e iφ cos θ | i , where θ , φ ǫ [0,2 π ]. In this case, the average quantum conditionalentropy is given as S ( ρ X | Y Z ) = P j p j S ( ρ X | Π jY Z ) with p j = tr [( I X ⊗ Π jY Z ) ρ ( I X ⊗ Π jY Z )] and ρ X | Π jY Z = p j tr Y Z [( I X ⊗ Π jY Z ) ρ ( I X ⊗ Π jY Z )] .To define quantum dissension for two-particle projectivemeasurement, we once again take the difference of theequivalent expressions of mutual information, i.e. D ( X : Y : Z ) = K ( X : Y : Z ) − I ( X : Y : Z )= S ( ρ X | Π jY Z ) + S ( ρ Y Z ) − S ( ρ XY Z ) . (16)The discord function D is also interpreted as quantumdiscord with a bipartite split of the system. One can min-imize D over all two-particle measurement projectors toobtain dissension as δ =min( D ( X : Y : Z ) ). This isthe most generic expression since it includes all possibletwo-particle projective measurements. Both δ and δ to-gether form the components of correlation vector definedin context of projective measurement done on differentsubsystems. Monogamy of quantum correlation is an unique phe-nomenon which addresses distributed correlation in amultiparty setting. It states that in a multipartite sit-uation, the total amount of individual correlations ofa single party with other parties is bounded by theamount of correlation shared by the same party withthe rest of the system when the rest are consideredas a single entity. Mathematically, given a multipar-tite quantum state ρ ...N shared between N parties,the monogamy condition for a bipartite correlation mea-sure Q should satisfy Q ( ρ ) + Q ( ρ ) + ... + Q ( ρ N ) ≤ Q ( ρ ...N ) where ρ j = tr ... ( j − j +1) ...N ρ ...j...N . It hadbeen shown that certain entanglement measures satisfythe monogamy inequality . However, there are certainmeasures of quantum correlation, including quantum dis-cord, which behave differently as far as the satisfying ofmonogamy inequality is concerned. By the term ’vio-lation of monogamy inequality for certain measure’, weactually refer to a situation where we can indeed findentangled states which violates the inequality for thatmeasure. In case of quantum discord, it had been seenthat W states violates the inequality and are polyga-mous in nature. More specifically, researchers consideredthe monogamy score δ m = D ( ρ AB ) + D ( ρ AC ) - D ( ρ A : BC ) ,(where ρ AB and ρ AC are the traced out density matri-ces from ρ ABC and D is quantum discord) and checkedwhether three-qubit states violate or satisfy the inequal-ity δ m ≤
3. EFFECT OF NOISY CHANNELS ONQUANTUM DISSENSION & MONOGAMYSCORE
In this section, we investigate the dynamics of quan-tum dissension when three-qubit states are transferredthrough noisy quantum channels. Moreover, we alsostudy the change of the monogamy score for various ini-tial states with time and purity of the state. We con-sider initial states to be mixed GHZ, mixed W, classi-cal mixture of two separable states, a mixed biseparablestates and the quantum channels to be amplitude damp-ing, phase damping and depolarizing.Given an initial state for three qubits ρ (0) , its evolutionin the presence of quantum noise can be compactly writ-ten as, ρ ( t ) = X l,m,n K l,m,n ρ (0) K † l,m,n , (17) where K l,m,n are the Kraus operators satisfying P l,m,n K † l,m,n K l,m,n =I for all t [23, 24]. For indepen-dent channels, K l,m,n = K l ⊗ K m ⊗ K n where K { l } describes one-qubit quantum channel effects. We ana-lytically present the dynamics of each initial state withrespect to the individual channels. In other words wepresent the dynamics of each of δ , δ and δ m . In eachcase, we apply the channel for sufficient time i.e. t=10seconds. In this subsection, we consider the effect of generalizedamplitude damping channel on various three-qubit quan-tum states. The amplitude damping channel describesthe process of energy dissipation in quantum processessuch as spontaneous emission, spin relaxation, photonscattering and attenuation etc. It is described by single-qubit Kraus operators K = √ q diag(1, √ − γ ), K = √ qγ ( σ + iσ )/2, K = √ − q diag( √ − γ ,1), K = p (1 − q ) γ ( σ − iσ )/2, where q defines the final probability dis-tribution when T → ∞ (q=1 corresponds to the usualamplitude damping channel). Here γ =1- e − Γ t , Γ repre-senting the decay rate.
1. Mixed GHZ State-
We consider the three-qubit mixed GHZ state ρ GHZ =(1 − p ) I + p | GHZ ih GHZ | (we universally take p as theclassical randomness) as the initial state. The matrixelements of the density operator for a certain time t , orfor a certain value of parameter γ are given by, ρ = 18 (1 + γ )[(1 + γ ) + 3 p (1 − γ ) ] ,ρ = ρ = ρ = 18 (1 − γ )[(1 + γ ) − p (1 − γ )(3 γ + 1)] ,ρ = ρ = ρ = 18 (1 − γ ) [(1 + γ ) + p (3 γ − ,ρ = 18 (1 + 3 p )(1 − γ ) , ρ = p − γ ) . (18)It is evident from Fig.1, δ and δ attains values ( − . , . at t = 0 and p = 1 and decays asymp-totically till each of them approaches . The ampli-tude damping channel leaves the final population stateat (diag [1 , ⊗ which contains no quantum dissension.In other words, we have a steady decay of quantum dis-sension for the mixed GHZ state with time in an am-plitude damping channel. The reduced density matricesare separable states and contain zero quantum discordfor all values of time and purity. Thus, the monogamyscore δ m is negative of δ . Since δ m is always negativein this case, the state remains monogamous through outthe evolution period. δ δ FIG. 1: δ and δ dynamics of mixed GHZ state in GADChannel with q = 1
2. Mixed W State-
For the three-qubit mixed W state ρ W = (1 − p ) I + p | W ih W | , the dynamics of the state in terms of the matrixelements at time t is given by, ρ = 18 [(1 + γ ) + p (1 − γ )( γ + 4 γ − ,ρ = ρ = ρ =124 (1 − γ )[3(1 + γ ) − p (3 γ + 6 γ − ,ρ = ρ = ρ = 18 (1 − p )(1 − γ ) (1 + γ ) ,ρ = 18 (1 − p )(1 − γ ) ,ρ = ρ = ρ = 13 p (1 − γ ) . (19)The initial values of δ and δ for a pure W state are(-1.75,0.92) respectively. As shown in Fig.2, δ and δ starts asymptotic decay from (-1.75,0.92) at t = 0 and p = 1 till they approach after sufficient channel action.The final population distribution at the limit of γ → [1 , ⊗ resulting in zero quantum dissension. InFig.3(a), we study the evolution of monogamy score withtime and interestingly we find that for certain values ofthe parameter p , the monogamy score δ m changes fromnegative to positive. This is a clear indication of thefact that the states which are initially monogamous areentering into the polygamous regime with time.
3. Mixture of Separable States-
We take classical mixture of separable states | i and | + ++ i given by the density matrix ρ = p | ih | +(1 − p ) | + ++ ih + + + | . The dynamics of this mixtureunder the action of amplitude damping channel in terms δ δ FIG. 2: δ and δ dynamics of mixed W state in GAD Channelwith q = 1 δ m δ m FIG. 3: δ m dynamics in GAD Channel with q = 1 of (a) MixedW state, (b) Mixture of separable states: p | ih | + (1 − p ) | +++ ih +++ | of the matrix elements is as follows ρ = 18 [(1 + γ ) + p (1 − γ )( γ + 4 γ + 7)] ,ρ = ρ = ρ = 18 (1 − p ) p − γ (1 + γ ) ,ρ = ρ = ρ = ρ = ρ = ρ = 18 (1 − p )(1 − γ ) ,ρ = ρ = ρ = ρ = 18 (1 − p )(1 − γ ) ,ρ = ρ = ρ = 18 (1 − p )(1 − γ )(1 + γ ) ,ρ = ρ = ρ = ρ = ρ = ρ =18 (1 − p ) p − γ (1 − γ ) ,ρ = ρ = ρ = 18 (1 − p )(1 − γ ) (1 + γ ) ,ρ = ρ = ρ = ρ = ρ = ρ = 18 (1 − p )(1 − γ ) ,ρ = ρ = ρ = 18 (1 − p )(1 − γ ) ,ρ = 18 (1 − p )(1 − γ ) . (20)At t = 0 , the maximum values (-1.015,0.15) of quantumdissensions δ and δ are obtained for p = 1 / [Fig.4].In this particular dynamics, we observe an interestingphenomenon that there is no exact asymptotic decay ofquantum dissension δ . We observe the revival of quan-tum correlation for a certain period of time in the initialphase of the dynamics. This is something different fromthe standard intuition of asymptotic decay of quantumcorrelation when it undergoes dissipative dynamics. Thisremarkable feature can be interpreted as that the dissipa-tive dynamics is not necessarily going to decrease quan-tum correlation with passage of time. On the contrary,depending upon the initial state it can enhance the quan-tum correlation for a certain period of time. We refer tothis unique feature as revival of quantum correlation indissipative dynamics . However, the other dissension δ follows the standard process of asymptotic decay withtime.In Fig.3(b), we also compute the monogamy score andfind that states which are initially polygamous are be-coming monogamous with the passage of time. This iscontrary to what we observed in case of mixed W states.In this case, the states initially freely shareable (polyg-amous) are entering into not freely shareable (monoga-mous) regime due to channel action. This is a remarkablefeature as this helps us to obtain monogamous state frompolygamous state. This is indeed helpful as monogamyof quantum correlation is an useful tool for quantum se-curity. δ pt δ FIG. 4: δ and δ dynamics of mixture of separable states | i and | + ++ i in GAD Channel with q = 1
4. Mixed Biseparable State-
Now we provide another example where action ofquantum noisy channel can revive quantum dissen-sion for a short period of time in a much smoothmanner compared to our previous example. Here,we consider a mixed biseparable state: ρ = (1 − p ) I + p [ | i| ϕ + ih |h ϕ + | + | i| ψ − ih |h ψ − | ]. The dynamics of this density matrix at time t is given by, ρ = 18 (1 + γ ) [(1 + γ ) + p (1 − γ )] ,ρ = ρ = 18 (1 − γ )[(1 + γ ) + p (1 − γ )] ,ρ = 18 (1 − γ ) [(1 + γ ) + p (1 − γ )] ,ρ = 18 (1 − p )(1 − γ )(1 + γ ) ,ρ = ρ = 18 (1 − p )(1 − γ ) (1 + γ ) ,ρ = 18 (1 − p )(1 − γ ) ,ρ = − ρ = p − γ ) . (21)At t = 0 for this state, both δ and δ are having thevalue 0. However, quite surprisingly, we find that in theinitial phase both dissension δ and δ increase and attainmaximum values (0.00133,0.00183) and in the subsequentphases the values lower down and finally reach 0 [Fig.5].This reiterates the fact that for certain initial states thedissipative dynamics acts as a catalyst and helps in re-vival of quantum correlation. This dynamics is differentfrom our previous dynamics in the sense that here revivalof quantum correlation is much more than the quantumcorrelation present in the initial state. This is indeeda strong signature that in multi-qubit cases the channeldynamics can take a zero-correlated to a correlated state.Though the rise of correlation is not very high, however,in NMR systems [25] this rise is significant as one startingwith a zero-correlated state can use the state for compu-tation at subsequent phases of time instead of trashing itaway. The reduced density matrices are separable statesfor all values of time and purity, making making theirdiscord equal to zero. Here once again we have, δ m = - δ and hence channel action does not change the monogamyproperty of the mixed biseparable state. −3 tp δ −3 tp δ FIG. 5: δ and δ dynamics of mixed biseparable state inGAD channel with q = 1 .
1. Mixed GHZ State-
The density matrix elements of mixed GHZ at time t for q = 1 / are given as, ρ = ρ = 18 [1 + 3 p (1 − γ ) ] , ρ = p − γ ) ,ρ ii = 18 [1 − p (1 − γ ) ] , i = 2 , ..., . (22)(23)Here δ and δ starts decaying from (-3.00,1.00) at t = 0 , p = 1 and approaches after sufficient time[Fig.6]. The decay of δ is not exactly asymptoticin contrast to the action of GAD channel with q=1.The decay of δ is asymptotic as in the case of GADchannel with q = 1 . The initial state evolves to finalpopulation distribution (diag [1 / , / ⊗ , which con-tains no quantum dissension. Moreover, the reduceddensity matrices are separable states and do not con-tribute towards monogamy score. Therefore, dynamics of δ m is same as that of δ , only differing by a negative sign. δ δ FIG. 6: δ and δ dynamics of mixed GHZ state in GADChannel with q = 1 /
2. Mixed W State-
The density matrix evolution of the mixed W state attime t is given by, ρ = 18 [1 − p (1 − γ )( γ − γ + 1)] ,ρ = ρ = ρ = 124 [3 + p (1 − γ )(3 γ − γ + 5)] ,ρ = ρ = ρ = 124 [3 − p (1 − γ )(3 γ − γ + 3)] ,ρ = 18 [1 + p (1 − γ )( γ − γ − ,ρ = ρ = ρ = p − γ )(2 − γ ) ,ρ = ρ = ρ = p γ (1 − γ ) . (24) The quantum dissensions δ and δ attain values of (-1.75,0.92) at t = 0 and p = 1 [Fig.7]. As in thecase of mixed GHZ state, the curve for δ is not ex-actly asymptotic while the curve for δ is asymptotic.In the limit γ → , a final population distribution of(diag [1 / , / ⊗ is left resulting in zero quantum dissen-sion. For purity values closer to 1, the initial states arepolygamous and they enter into the monogamy regimedue to action of GAD channel [Fig.8(a)]. The states withpurity values closer to 0 are monogamous and do not ex-perience any such transition. Hence once again we haveone such example where there is a useful transition frompolygamous to monogamous regime. δ δ FIG. 7: δ and δ dynamics of mixed W state in GAD Channelwith q = 1 / δ m δ m FIG. 8: δ m dynamics in GAD Channel with q = 1 / of (a) Mixed W state (b) Mixture of separable states: p | ih | + (1 − p ) | +++ ih +++ |
3. Mixture of Separable States-
We consider initial density matrix ρ = p | i + (1 − p ) | + ++ i whose dynamics at time t is as follows, ρ = 18 [1 + p (1 − γ )( γ − γ + 7)] ,ρ = ρ = ρ = 18 [1 − p (1 − γ )( γ − γ + 1)] ,ρ = ρ = ρ = 18 [1 + p (1 − γ )( γ − γ − ,ρ = 18 [1 + p ( γ − ,ρ = ρ = ρ = ρ = ρ = ρ = ρ = ρ = ρ = ρ = ρ = ρ = 18 (1 − p ) p − γ,ρ = ρ = ρ = ρ = ρ = ρ = ρ = ρ = ρ = ρ = ρ = ρ = 18 (1 − p )(1 − γ ) ,ρ = ρ = ρ = ρ = 18 (1 − p )(1 − γ ) . (25)Once again it is evident from Fig[9], δ and δ achievemaximum values (-1.015,0.15) at t = 0 and p = 1 / .However, the decay profile of δ is much smoother thanthat of δ . The evolution of monogamy score [Fig. 8(b)]is quite different for q = 1 / than that of q = 1 . Herealso, all the initial polygamous density matrices enterinto the monogamy regime irrespective of the values ofparameter p. δ δ FIG. 9: δ and δ dynamics of mixture of separable states | i and | + ++ i in GAD Channel with q = 1 /
4. Mixed Biseparable State-
We also studied the dynamics of the mixed biseparablestate in presence of GAD channel for q = 1 / and wefound that both dissensions remain at zero starting fromthe initial state. In this subsection, we consider the dephasing channeland its action on various three-qubit states. A dephasingchannel causes loss of coherence without any energy ex-change. The one-qubit Kraus operators for such processare given by K =diag (1, √ − γ ) and K =diag(0, √ γ ).
1. Mixed GHZ State-
We once again consider the mixed GHZ state subjectedto dephasing noise. The density matrix elements of themixed GHZ at a time t are given by, ρ = ρ = 18 (1 + 3 p ) , ρ = p − γ ) ,ρ ii = 18 (1 − p ) , i = 2 , ..., . (26)(27)Here we observe that the diagonal elements are leftintact whereas the off-diagonal elements undergo changeas a consequence of dephasing noise. Interestingly, wefind that δ is not at all influenced by dephasing channelwhereas δ follows a regular asymptotic path [Fig.10].The degradation observed in δ is due to progressivelylower purity levels and is unaffected by dephasingnoise. The reduced density matrices do not contributetowards monogamy score, thus making the dynamics ofmonogamy score just negative of δ . δ δ FIG. 10: δ and δ dynamics of mixed GHZ state in dephasingchannel.
2. Mixed W State-
The dynamics of mixed W state subjected to dephasingnoise is as follows: ρ ii = 18 (1 − p ) , i = 1 , , , , ,ρ = ρ = ρ = 124 (3 + 5 p ) ,ρ = ρ = ρ = 13 p (1 − γ ) . (28)We noticed that for p = 1 , δ has a slower decay rate com-pared with other purity values and hence a finite amountof δ is present for all t ≤
10 at p = 1 [Fig.11]. The decayof δ is asymptotic. For certain values of purity, the ini-tial mixed W state is monogamous. However, they enterinto the polygamous regime as a consequence of phasedamping noise [Fig.12(a)]. After sufficient time, δ m de-cays down to zero for all purity values. δ δ FIG. 11: δ and δ dynamics of mixed W state in dephasingchannel. δ m δ m FIG. 12: δ m dynamics in dephasing channel of (a) MixedW state, (b) Mixture of separable states: p | ih | + (1 − p ) | +++ ih +++ |
3. Mixture of Separable States-
The dynamics of ρ = p | ih | + (1 − p ) | + ++ ih + + + | under the influence of phase damping channel is givenby: ρ = 18 (1 + 7 p ) , ρ ii = 18 (1 − p ) , i = 2 , ..., ,ρ = ρ = ρ = ρ = ρ = ρ = ρ = ρ = ρ = ρ = ρ = ρ = 18 (1 − p ) p − γ,ρ = ρ = ρ = ρ = ρ = ρ = ρ = ρ = ρ = ρ = ρ = ρ = 18 (1 − p )(1 − γ ) ,ρ = ρ = ρ = ρ = 18 (1 − p )(1 − γ ) . (29)Here, δ exhibits a strong revival all throughout thechannel. However, the decay profile of δ is perfectlyasymptotic [Fig.13]. Prior to channel action, i.e. at t = 0 ,all density matrices are polygamous. With the action ofthe dephasing channel, density matrices with mixed nesscloser to 1 enter into the monogamous regime [Fig.12(b)].
4. Biseparable State-
For the initial state, ρ = (1 − p ) I + p [ | i| ϕ + ih |h ϕ + | + δ δ FIG. 13: δ and δ dynamics of mixture of separable states | i and | + ++ i in dephasing Channel | i| ψ − ih |h ψ − | ], the dynamics is given as: ρ ii = 18 (1 + p ) , i = 1 , , , ,ρ ii = 18 (1 − p ) , i = 5 , , , ,ρ = − ρ = p − γ ) . (30)Both δ and δ are zero throughout the channel operationtime and do not show any revival. In the final subsection of this section, we considerthe effect of the depolarizing channel on three-qubitstates. Under the action of a depolarizing channel,the initial single qubit density matrix dynamicallyevolves into a completely mixed state I /2. The Krausoperators representing depolarizing channel actionare K = p − γ/ I , K = p γ/ σ x , K = p γ/ σ y , K = p γ/ σ z . (where σ x , σ y , σ z are Pauli matrices)
1. Mixed GHZ State-
The dynamics of a mixed GHZ state when subjected todepolarizing channel is spelled out as, ρ = ρ = 18 [1 + 3 p (1 − γ ) ] , ρ = p − γ ) ,ρ ii = 18 [1 − p (1 − γ ) ] i = 2 , ..., . (31)Both δ and δ start decaying from the initial valuesof (-3.00,1.00) [Fig.14]. Quite interestingly, δ exhibitssmooth asymptotic decay in contrary to the anomaliesobserved in case of q = 1 / GAD channel and dephas-ing channel. This instance underlines the fact that acertain noisy environment can largely influence the dy-namics of multipartite quantum correlation. The depo-larizing channel transfers the initial mixed GHZ stateinto I /8 which contains zero quantum dissension. Here0 δ δ FIG. 14: δ and δ dynamics of mixed GHZ state in depolar-izing channel. the monogamy score δ m of mixed GHZ state is just thenegative of δ .
2. Mixed W State-
The dynamics of the mixed W state under the action ofdepolarizing channel is given by, ρ = 18 [1 − p (1 − γ )( γ − γ + 1)] ,ρ = ρ = ρ = 124 [3 + p (1 − γ )(3 γ − γ + 5)] ,ρ = ρ = ρ = p − γ )(1 − γ ) ,ρ = ρ = ρ = 124 [3 − p (1 − γ )(3 γ − γ + 3)] ,ρ = ρ = ρ = p γ (1 − γ ) ,ρ = 18 [1 + p (1 − γ )( γ − γ − . (32)Here, δ and δ attain maximum values of (-1.75,0.92)at t = 0 and p = 1 [Fig.15]. The initial mixed W stateevolves to I /8 in the limit of γ → resulting in zeroquantum dissension. δ follows a perfect asymptotic pathin contrast to the dynamics observed in case of q = 1 / GAD channel and dephasing channel. The monogamyscore δ m evolves as shown in Fig.16. For high purityvalues closer to 1, the initially polygamous states enterinto monogamous regime owing to depolarizing channelaction. On the other hand, states with low purity valueswhich are initially monogamous do not experience anysuch transition.
4. CONCLUSION
In this work, we have extensively studied the dynamicsof quantum correlation (quantum dissension) of variousthree qubit states like, mixed GHZ, mixed W, mixtureof separable states and a mixed biseparable state whenthese states are transferred through quantum noisychannels such as amplitude damping, dephasing and depolarizing. In most cases, we find that there is an δ δ FIG. 15: δ and δ dynamics of mixed W state in depolarizingchannel. δ m FIG. 16: δ m dynamics in depolarizing channel of Mixed Wstate, (b) asymptotic decay of quantum dissension with time.However, in certain cases, we have observed the revivalof quantum correlation depending upon the nature ofinitial state as well as channel. This is quite interestingas we can explicitly see enhancement of multiqubitcorrelation in presence of local noise; similar in the lineof quantum discord.In addition, we have studied dynamics of monogamyscore of three qubit states under different quantumnoisy channels. Remarkably, we have seen that there arecertain states which on undergoing effects of quantumchannels change itself from monogamous to polygamousstates. It is believed that monogamy property of thestate is a strong signature of quantumness of the stateand can be more useful security purpose compared topolygamous state. This study is useful from a futuristicperspective where we are required to create monogamousstate from polygamous state for various cryptographicprotocols. Acknowledgment
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