Correlation evolution and monogamy of two geometric quantum discords in multipartite systems
aa r X i v : . [ qu a n t - ph ] O c t Correlation evolution and monogamy of two geometric quantum discords in multipartite systems
Yan-Kui Bai , , ∗ Ting-Ting Zhang , Li-Tao Wang , and Z. D. Wang † Department of Physics and Center of Theoretical and Computational Physics,The University of Hong Kong, Pokfulam Road, Hong Kong, China College of Physical Science and Information Engineering and Hebei Advance Thin Films Laboratory,Hebei Normal University, Shijiazhuang, Hebei 050024, China
We explore two different geometric quantum discords defined respectively via the trace norm (GQD-1) andHilbert-Schmidt norm (GQD-2) in multipartite systems. A rigorous hierarchy relation is revealed for the twoGQDs in a class of symmetric two-qubit X -shape states. For multiqubit pure states, it is found that both GQDsare related to the entanglement concurrence, with the hierarchy relation being saturated. Furthermore, we lookinto a four-partite dynamical system consisting of two cavities interacting with independent reservoirs. It isfound that the GQD-2 can exhibit various sudden change behaviours, while the GQD-1 only evolves asymptot-ically, with the two GQDs exhibiting different monogamous properties. PACS numbers: 03.65.Ud, 03.65.Yz
I. INTRODUCTION
Quantum correlation beyond entanglement is a typical char-acteristic of quantumness, which has attracted a lot of atten-tion for the last decade [1–4]. While quantum discord [5, 6]is a kind of basic measure for bipartite quantum correlations,it is difficult to evaluate even for two-qubit states. Daki´c et al introduced a geometric quantum discord via
Hilbert-Schmidtnorm (GQD-2), which is expressed as [7] D g ( ρ AB ) = 2 min χ ∈ Ω || ρ AB − χ AB || , (1)where the minimum runs over all the set Ω of zero-discordstates χ AB = P i | i ih i | A ⊗ ρ iB , and the geometric distance isquantified by the square norm || X − Y || = Tr ( X − Y ) . TheGQD-2 is effortlessly computable in many interesting cases[8–10], especially for two-qubit mixed states. Moreover, it isrecently shown that this measure is related to the fidelity ofremote state preparation [11–13].Unfortunately, the GQD-2 may increase under local oper-ations on the unmeasured party [14–16], which is differentfrom that of quantum discord and thus is argued to be a draw-back. To circumvent this problem, some other geometric dis-cords may need to be introduced, such as the geometric quan-tum discord based on the trace norm (GQD-1) [17–20] D g ( ρ AB ) = min χ ∈ Ω || ρ AB − χ AB || , (2)where the trace norm is || X || = Tr √ X † X . The GQD-1 hasan operational interpretation that it is in correspondence withthe one-shot state distinguishability between two states [21].However, the computability of GQD-1 is weaker than that ofGQD-2, and the analytical formula is available only for somespecific classes of two-qubit states [18, 22–24].For two-qubit Bell-diagonal states, it was identified boththeoretically and experimentally that the GQD-1 can exhibitthe freezing and sudden change behaviors under local noise ∗ Electronic address: [email protected] † Electronic address: [email protected] environments [19, 20, 25–27]. In particular, in the dynami-cal procedures preserving Bell-diagonal form, the two GQDsobey the following relation [18] D g [ ρ BD ( t )] ≥ D g [ ρ BD ( t )] . (3)For the dynamical evolution beyond Bell-diagonal form, therelation between two GQDs may need a further investigation.In particular, for multipartite quantum systems, a profoundunderstanding of the relation and difference between the twoGQDs is still awaited.In this paper, we first analyze a class of symmetric X -shapestates beyond the Bell-diagonal form and show that there is arigorous hierarchy relation between the two GQDs. For mul-tiqubit pure states, we show that the hierarchy relation is satu-rated and both GQDs are related to the entanglement concur-rence. Furthermore, we conduct a comparative investigationon the two GQDs for a four-partite dynamical system consist-ing of two cavities interacting with independent reservoirs. Itis found that the GQD-2 can exhibit different sudden changebehaviors, while, in contrast to the GQD-2, the GQD-1 onlyevolves asymptotically, and is not monogamous in both three-and four-qubit states. Finally, we discuss the monogamy prop-erty of the square of GQD-1 in multipartite systems. II. THE HIERARCHY AND MONOGAMY PROPERTIESOF TWO GQDS.
According to the generalized Bloch representation, a two-qubit quantum state can be written as [28] ρ = 14 ( I ⊗ I + X i =1 x i σ i ⊗ I + X i =1 y i I ⊗ σ i + X i,j =1 Γ ij σ i ⊗ σ j ) , (4)where x i = Tr ρ ( σ i ⊗ I ) , y i = Tr ρ ( I ⊗ σ i ) are componentsof local Bloch vectors with σ i being the Pauli matrix, and Γ is the correlation matrix with element Γ ij = Tr ρ ( σ i ⊗ σ j ) . Inthis case, the GQD-2 has an analytical formula [7, 9] D g ( ρ AB ) = 12 ( k + k + k − k max ) , (5)where k i s are the eigenvalues of matrix K = ~x~x T + ΓΓ T with k max being the maximum.So far, analytical results of GQD-1 have only been availablefor some specific classes of two-qubit states, where a typicalclass is the X -shape states in the form ρ XAB = ρ ρ ∗ ρ ρ ∗ ρ ρ ρ ρ . (6)Ciccarello et al proved the analytical formula of GQD-1 [24] D g ( ρ XAB ) = s γ γ max − γ γ min γ max − γ min + γ − γ , (7)where γ = 2( ρ + ρ ) , γ = 2( ρ − ρ ) , and γ =1 − ρ + ρ ) being the singular values of correlation matrix Γ , and γ min = min { γ , γ } and γ max = max { γ , γ + x } with x being the z -component of local Bloch vector.We consider a class of symmetric two-qubit X -shape state ρ XsAB , for which matrix elements in Eq. (6) satisfy the condi-tions ρ = 0 or ρ = 0 . For this class of quantum states, wehave the following hierarchy relation. Theorem 1 . For symmetric X -shape states, two GQDs obeythe following hierarchy relation D g ( ρ XsAB ) ≥ D g ( ρ XsAB ) , (8)where D g = | γ | and D g = ( γ + γ ′ min ) / with γ ′ min = min { γ , γ + x } , respectively. Proof . For X -shape states, the matrix K of GQD-2 in Eq.(5) is diagonal and the eigenvalues have the form k = γ =4( ρ + ρ ) , k = γ = 4( ρ − ρ ) , and k = γ + x =[1 − ρ + ρ )] + [2( ρ + ρ ) − . In the symmetriccase of ρ = 0 or ρ = 0 , we further have k = k = γ which results in D g ( ρ XsAB ) = ( γ + γ ′ min ) / . The GQD-1in the symmetric X -shape states has the property γ = γ ,and the measure in Eq. (7) can be simplified to D g = | γ | when the parameters γ max = γ min . For the two parameters γ max and γ min , there are three kinds of relations among thesingular values and local Bloch vector: i) γ ≥ γ and γ ≥ γ + x , ii) γ ≥ γ and γ ≤ γ + x , iii) γ ≤ γ and γ ≤ γ + x , respectively. When γ max = γ min , we canobtain | γ | = | γ | = | γ | for all the three kinds of relations,which corresponds to the homogeneous case and the GQD-1has the form D g = | γ | [24]. Therefore, for all symmetric X -shape states, the GQD-1 is formulated as a unified form D g ( ρ XsAB ) = | γ | . Noting that the fact γ ≥ γ ′ min , we havethe hierarchy relation in Eq. (8) and complete the proof.In the pure state case, quantum correlation is usually equiv-alent to quantum entanglement. Here, for N -qubit pure states,we have the following theorem. Theorem 2 . In an arbitrary N -qubit pure state | ψ A A ··· A n i ,the GQDs under partition A | A · · · A n are related to the en-tanglement quantified by concurrence D g ( ψ A | A ··· A n ) = C A | A ··· A n ( ψ ) = 2 p λ λ ,D g ( ψ A | A ··· A n ) = C A | A ··· A n ( ψ ) = 4 λ λ , (9) where the hierarchy relation is saturated with λ i s being theeigenvalues of reduced density matrix ρ A . Proof . According to the Schmidt decomposition, an N -qubit pure state is equivalent to a logical two-qubit state un-der the partition A | A · · · A n . Up to local unitary trans-formations, the pure state has the form | ψ A A ··· A n i = √ λ | i A | i A ··· A n + √ λ | i A | i A ··· A n , which belongsto the class of symmetric X -shape states. Therefore, theGQD-1 for the N -qubit pure state is D g ( ψ A | A ··· A n ) = | γ | = 2 √ λ λ in terms of theorem . On the otherhand, the concurrence for the logical two-qubit state is C A | A ··· A n ( ψ ) = 2 a a = 2 √ λ λ [29], leading tothe first equation in the theorem. The GQD-2 for the purestate is D g ( ψ A | A ··· A n ) = 2(1 − P λ i ) [30]. Utilizingthe relation λ + λ ) = 2 λ λ + P λ i , we have D g ( ψ A | A ··· A n ) = 4 λ λ = C A | A ··· A n ( ψ ) which is justthe second equation in the theorem, and then the proof is com-pleted.Recently, Hu and Fan present a measure of measurement-induced nonlocality based on trace norm (MIN-1) N ( ρ AB ) = max Π A || ρ AB − Π A ( ρ AB ) || and the non-locality in a ⊗ N pure state is N ( ψ ) = 2 √ λ λ [31].According to theorem 2, the GQD-1 is also equivalent to theMIN-1 in this case.For mixed states, the GQDs are not equal to the entangle-ment in general, but the GQD-1 is still related to the concur-rence in symmetric X -shape states. Lemma 1.
In the class of symmetric two-qubit X -shapestates, the GQD-1 is not less than the concurrence, i.e., D g ( ρ XsAB ) ≥ C ( ρ XsAB ) . (10) Proof.
For the symmetric X -shape states, the concurrencecan be expressed as C ( ρ XsAB ) = max { , | γ | − η } in which η = √ ρ ρ when the matrix element ρ = 0 and η = √ ρ ρ when ρ = 0 . According to theorem 1, we have D g ( ρ XsAB ) = | γ | , from which we can obtain the lemma aftera direct comparison to the concurrence.In many-body quantum systems, one of the most importantproperties is that entanglement is monogamous [32], and, as isknown, this property can be used for constructing multipartiteentanglement measures [32–34]. For three-qubit pure states,the GQD-2 is monogamous [35], and this property is also sat-isfied by the square of quantum discord which results in agenuine three-qubit quantum correlation measure [36, 37]. Incomparison to the GQD-2, the GQD-1 has a merit of contrac-tility under local operations on the unmeasured party. There-fore, it is natural to ask whether the GQD-1 is monogamous inthe case of several qubits. Unfortunately, it will be seen in thenext section that the GQD-1 is not monogamous for the three-qubit pure states and mixed states as well as the four-qubitpure states. III. THE DYNAMICAL PROPERTY OF TWO GQDS INMULTIPARTITE CAVITY-RESERVOIR SYSTEMS.
Here we conduct a comparative study on GQD-1 and GQD-2 in a four-partite dynamical system consisting of two cavities ακ t ακ t κ t α (a) (b)(c)IIIIII ba D g2 (c c ) D g2 (r r ) FIG. 1: (Color online) Dynamical evolution of GQD-2 as func-tions of the time evolution κt and the initial state parameter α : (a) D g ( ρ c c ) , (b) D g ( ρ r r ) , (c) three types of correlation evolutionclassified by the sudden change lines of cavity photons (blue line)and reservoirs (red line). interacting with independent reservoirs, where the interactionHamiltonian of a single cavity-reservoir system is [38, 39] ˆ H = ~ ω ˆ a † ˆ a + ~ N X k =1 ω k ˆ b † k ˆ b k + ~ N X k =1 g k (ˆ a ˆ b † k + ˆ b k ˆ a † ) . (11)The initial state is | Φ i = ( α | i + β | i ) c c | i r r inwhich the dissipative reservoirs are in the vacuum state. Inthe limit of N → ∞ for a reservoir with a flat spectrum, theoutput state of the multipartite system will be [38] | Ψ t i = α | i c r c r + β | φ t i c r | φ t i c r , (12)where | φ t i = ξ ( t ) | i + χ ( t ) | i with the amplitudes being ξ ( t ) = exp ( − κt/ and χ ( t ) = [1 − exp ( − κt )] / .In the dynamical evolution, the density matrix of two cavityphotons does not preserve the Bell-diagonal form ( x = 0 )and has the form ρ c c = α + β χ αβξ β ξ χ β ξ χ αβξ β ξ , (13)which is just the symmetric X -shape state with element ρ =0 . We first consider the correlation evolution of GQD-2 in thesubsystem. According to theorem 1, we have D g ( ρ c c ) = [ γ ( ρ c c ) + γ ′ min ( ρ c c )] / , (14)where the parameters are γ = 4 α β ξ and γ ′ min = min { γ , γ + x } with γ = (1 − β ξ χ ) and x =(1 − β ξ ) , respectively. For the subsystem of two reser-voirs, its density matrix is similar to that in Eq. (13) and the κ t α κ t α κ t α κ t α (a) (b)(c) (d) FIG. 2: (Color online) Asymptotical evolution of GQD-1 in differentsubsystems as functions of the time evolution κt and the initial stateamplitude α : (a) D g ( ρ c c ) , (b) D g ( ρ r r ) , (c) D g ( ρ c r ) , (d) D g ( ρ c r ) . relation ρ r r = S ξ ↔ χ [ ρ c c ] is satisfied in which S ξ ↔ χ is atransformation exchanging the parameters ξ and χ . Therefore,the GQD-2 of two reservoirs can be expressed as D g ( ρ r r ) = S ξ ↔ χ [ D g ( ρ c c )] . (15)In Fig.1(a) and Fig.1(b), D g ( ρ c c ) and D g ( ρ r r ) are plot-ted as functions of the time evolution κt and the initial stateamplitude α , where we can see that the sudden change behav-ior exists. Due to the intrinsic relation in Eq. (15), whenthe GQD-2 of two cavity photons experiences the suddenchange, the same behavior must happen in the reservoir sys-tems, which is similar to the relation of entanglement suddendeath and sudden birth in the subsystems [38, 40, 41].For both cavity photons and reservoirs subsystems, the sud-den change condition is γ = γ + x , (16)and from which we plot the sudden change lines for D g ( ρ c c ) (blue line) and D g ( ρ r r ) (red line) in Fig1.(c).As seen from the figure, there are three types of correlationevolution. In type I with α ∈ (0 , / √ , two D g s experi-ence double sudden changes with a revival. The revival oc-curs at the time κt = ln and the blue dashed line ab indicatesthe minimum of D g s in the overlap area marked out by thesudden change lines. In type II with α ∈ (1 / √ , . ,although the revival behavior disappears, the double suddenchanges still exist except for the case α = 1 / √ exhibitingsingle sudden change behavior. In type III with α ≥ . ,two quantum correlations evolve asymptotically, and the sud-den change behavior disappears. Similarly, for the subsystems c r and c r , we can also use the sudden change conditionin Eq. (16) to determine the type of correlation evolution.We now analyze two-qubit GQD-1 in multipartite cavity-reservoir systems, in which all the two-qubit states are in the −0.8−0.6−0.4−0.200.20.40.60.8 κ t M onoga m y F un c t i on s M ( ρ )M ( ψ ) M ( ψ )5 × M’ ( ψ )5 × M’ ( ψ )5 × M’ ( ρ ) FIG. 3: (Color online) Monogamy analysis of GDD-1 and GQD-2 inmultipartite cavity-reservoir systems, where M (Ψ) (blue solid line), M ( ρ ) (purple solid line), M (Ψ) (red solid line), and M ′ ( ρ ) (pur-ple dash-dotted line) are not monogamous but M ′ (Ψ) (blue dash-dotted line) and M ′ (Ψ) (red dash-dotted line) are monogamous. symmetric X -shape form. Based on theorem 1, the correlationequates to the first singular value | γ | and we have D g ( ρ c c ) = 2 αβξ , D g ( ρ r r ) = 2 αβχ ,D g ( ρ c r ) = 2 β ξχ, D g ( ρ c r ) = 2 αβξχ, (17)respectively. In Fig.2, the four D g s are plotted as func-tions of parameters κt and α , where all the quantum corre-lations evolve asymptotically. Via the hierarchy relation intheorem 1, the GQD-1 is related to the GQD-2. For exam-ple, D g ( ρ c c ) = D g ( ρ c c ) when the sudden change of D g ( ρ c c ) does not occur, and when the sudden change ap-pears the relation is strictly D g ( ρ c c ) > D g ( ρ c c ) in thesudden change area. The case for other subsystems is similar.The GQD-2 was indicated to be monogamous in three-qubitpure states [35]. Whether the GQD-1 is still monogamous inthese states? To answer this question, we turn to analyze themonogamy property of the two GQDs in multipartite cavity-reservoir systems.In the output state of Eq. (12), subsystem c r can be re-garded as a logic qubit, and then the reduced state ρ c c r is equivalent to a two-qubit state which is in the symmet-ric X -shape form and its GQD-1 is D g ( ρ c ( c r ) ) = 2 αβξ .In multipartite cavity-reservoir systems, the monogamy prop-erty of three-qubit pure states can be checked via thequantity M ( | Ψ t i ) = D g ( | Ψ i c | r ( c r ) ) − D g ( ρ c r ) − D g ( ρ c | ( c r ) ) in which the multiqubit GQD-1 can be cal-culated by theorem 2. Similarly, for the cases of three-qubit mixed states and four-qubit pure states, we can utilize M ( ρ t ) = D g ( ρ c ( c r ) ) − D g ( ρ c c ) − D g ( ρ c r ) and M ( | Ψ t i ) = D g ( | Ψ i c | r c r ) − D g ( ρ c r ) − D g ( ρ c c ) − D g ( ρ c r ) , respectively. In the same way, we can definethe quantities M ′ ( | Ψ t i ) , M ′ ( ρ t ) , and M ′ ( | Ψ t i ) with the cor-responding D g s. In Fig.3, these monogamy quantities areplotted as functions of the time evolution κt , where the ini-tial state amplitude is chosen to be α = 1 / √ . The func- tion M ′ ( | Ψ t i ) (blue dash-dotted line) is always nonnegative,which coincides with the fact that the GQD-2 is monogamousin three-qubit pure states [35]. In three-qubit mixed states, M ′ ( ρ t ) can be positive or negative (purple dash-dotted line),and then it is not monogamous. In four-qubit pure states, al-though M ′ ( | Ψ t i ) (red dash-dotted line) is monogamous, butthe property in general case cannot be guaranteed. For theGQD-1 in multipartite cavity-reservoir systems, we find allthe functions M ( | Ψ t i ) (blue solid line), M ( ρ t ) (purple solidline), and M ( | Ψ t i ) (red solid line) are negative, which indi-cates the GQD-1 is not monogamous. IV. DISCUSSION AND CONCLUSION.