Generalized Geometric Measure of Entanglement for Multiparty Mixed States
Tamoghna Das, Sudipto Singha Roy, Shrobona Bagchi, Avijit Misra, Aditi Sen De, Ujjwal Sen
aa r X i v : . [ qu a n t - ph ] N ov Generalized Geometric Measure of Entanglement for Multiparty Mixed States
Tamoghna Das, Sudipto Singha Roy, Shrobona Bagchi, Avijit Misra, Aditi Sen(De) and Ujjwal Sen
Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211 019, India
Computing entanglement of an arbitrary bipartite or multipartite mixed state is in general not an easy taskas it usually involves complex optimization. Here we show that exploiting symmetries of certain multiquditmixed states, we can compute a genuine multiparty entanglement measure, the generalized geometric measure,for these classes of mixed states. The chosen states have different ranks and consist of an arbitrary number ofparties.
I. INTRODUCTION
Characterization and quantification of quantum entangle-ment [1] lies at the heart of quantum information theory, sinceits early recognition as “spooky action at a distance” [2] in theEinstein-Podolsky-Rosen article [3]. Moreover, it has beensuccessfully identified as a key resource in several quantumcommunication protocols including superdense coding [4],teleportation [5], and quantum cryptography [6]. Entangle-ment has been shown to be a necessary ingredient in studyingquantum state tomography [7], quantum metrology [8], coop-erative quantum phenomena in many body systems like quan-tum phase transitions [9], etc. Quantification of entanglementis also essential for characterization of successful preparationsof quantum states, both in two party and multiparty domains,in the laboratories [10].The notion of entanglement is rather well-understood in thebipartite regime, especially for pure states [11–15]. Whileseveral entanglement measure can be computed for bipartitepure states, the situation for mixed states is difficult, and thereare only few entanglement measures which can be computedefficiently. The logarithmic negativity [14] can be obtainedfor arbitrary bipartite states, while the entanglement of for-mation [12, 13] can be computed for all two-qubit states.The situation becomes complicated even for the pure stateswhen the number of parties increase. However, there havebeen significant advances in recent times to quantify multi-partite entanglement of pure quantum states in arbitrary di-mensions [1]. They are broadly classified in two catagories − distance-based measures [16–19] and monogamy-based ones[6, 11, 20, 21]. On the other hand, quantifying entanglementfor arbitrary multiparty mixed states is still an arduous task.Recently, experiments by using photon polarization [22] andions [23] have been reported in which multiparty states of theorder of ten parties have been created successfully. Such phys-ical implementations demand a general tool to compute mul-tiparty entanglement measures for arbitrary mixed states. Re-cently there have been notable advancements in this direction[24]. Moreover, when an entanglement measure can only beevaluated for pure states, the entanglement-assisted study ofcooperative phenomena becomes restricted to only a systemwhich is at zero temperature.We address here the question of computing the generalizedgeometric measure (GGM) [19], a genuine multiparty entan-glement quantifier, for mixed states. The GGM of pure stateshas already been computed efficiently in several systems forarbitrary number of parties [25]. In this paper, we define the GGM for mixed states via the convex roof. To deal with theobstacle of evaluating the convex roof extension, we use sym-metry properties of certain multiparty quantum states and sim-plify the evaluation of GGM for these classes of mixed states,as prescribed in Refs. [26–28] (cf. [29]). Exploiting suchsymmetries, we are able to compute the GGM of differentparadigmatic classes of mixed states having different ranks. Inparticular, we first present the exact value of GGM for certainclasses of rank 2 and rank 3 mixed states with arbitrary num-ber of qubits. We then compute the GGM for a specific classof states which is a mixture of Greenberger-Horne-Zeilinger(GHZ) [30] and all the Dicke states [31], having a variety ofranks. The common property that all these classes possessesis that they remain invariant under the action of same sym-metric local unitary operators on each qubit. Moreover, wefind the GGM of a class of tripartite states of rank 4 which re-mains unaltered under different local unitaries on each party.Finally, we show that such symmetry properties can lead toan exact expression of GGM for a class of multiqudit stateshaving varied ranks.The paper is organized in the following manner. In Sec. II,we review the definition and the various properties of the gen-eralized geometric measure for pure states. In section III, wedefine GGM for mixed states via the convex roof construction.Here, we also discuss the Terhal-Vollbrecht-Werner techniqueof exploiting the symmetry of a quantum state for simplify-ing the evaluation of a convex roof extension. The same sec-tion also contains the computation of the GGM for differentclasses of mixed states. We present a summary in Sec. IV. II. GENERALIZED GEOMETRIC MEASURE
A pure state is said to be genuinely multiparty entangled ifit is not product in any bipartition. The generalized geometricmeasure (GGM) [19] (cf. [16]) of an N -party pure quantumstate, | ψ N i , is a computable entanglement measure that canquantify genuine multiparty entanglement. It is defined as anoptimized distance of the given state from the set of all statesthat are not genuinely multiparty entangled. Mathematically,it is given by E ( | ψ N i ) = 1 − Λ max ( | ψ N i ) , (1)where Λ max ( | ψ N i ) = max |h χ | ψ N i| , with the maximizationbeing over all | χ i that are not genuinely multiparty entangled.An equivalent form of the above equation is [19] E ( | ψ n i ) = 1 − max { λ I : L | I ∪ L = { A , . . . , A N } , I ∩ L = ∅} , (2)where λ I : L is the maximal Schmidt coefficient in the bipartitesplit I : L of | ψ N i .Let us enumerate some properties of the GGM which estab-lish it as a bona fide measure of genuine multiparty entangle-ment [19]. E ( | ψ N i ) ≥ , for all | ψ N i , E ( | ψ N i ) = 0 iff | ψ N i is not genuinely multiparty entangled, and E ( | ψ N i ) is nonin-creasing under local quantum operations at the N parties andclassical communication between them. III. GGM FOR MIXED STATES
We can now define the GGM of a general mixed quantumstate, in terms of the convex roof construction. For an arbitrary N -party mixed state, ρ N , the GGM can be defined as G ( ρ N ) = min { p i , | ψ iN i} X i p i E ( | ψ iN i ) , (3)where the minimization is over all pure state decompositionsof ρ N i.e., ρ N = P i p i | ψ iN ih ψ iN | . It is difficult to find theoptimal decomposition and the computation of GGM is ingeneral impossible even for moderate-sized systems. How-ever, the situation is different if the mixed quantum state un-der consideration possesses some symmetry [17, 27–29]. InRef. [27], Vollbrecht and Werner have provided a generalmethod to compute an entanglement measure, defined via theconvex roof extension, of a class of mixed states which areinvariant, on average, under a group of local unitaries. Be-low we briefly outline the same. Suppose ρ ′ N = ( U ⊗ U ⊗ . . . ⊗ U N ) ρ N ( U † ⊗ U † ⊗ . . . ⊗ U † N ) , where U i are the localunitary operators, acting on Hilbert spaces H i . The GGM of ρ N and ρ ′ N are the same. If it happens that ρ N = ρ ′ N , then ( U ⊗ U ⊗ . . . ⊗ U N ) is called a local symmetry of ρ N . Let G be a group of unitary operators U = ( U ⊗ U ⊗ . . . ⊗ U N ) and P be a twirl operator, such that, A P −→ R dU U AU † ≡ P ( A ) ,where the integral is carried out Haar uniformly. In case ofa mixed state ρ N , if there exist a twirl operator P such that P ( ρ N ) = ρ N , then the entanglement, G ( ρ N ) , can be obtainedfrom a pure | ψ i which satisfies P ( | ψ ih ψ | ) = ρ. (4)In principle, one can have a set of pure states, {| ψ i} = M ρ N ,which satisfies Eq. (4), and it is sufficient to perform the op-timization over this set. A further step is needed where weconvexify the optimized quantity over the parameters in ρ N ,if it is not already convex.We now show that this method can be utilized to evaluatethe GGM for several classes of multiparty states with arbi-trary number of parties having certain symmetries. We presentthese classes according to their ranks. G x FIG. 1. (Color online.) GGM of ρ N ( x ) = x | ψ N ih ψ N | + (1 − x ) | ψ ⊥ N ih ψ ⊥ N | against x. All the quantities are dimensionless. A. Classes of rank 2 multiqubit states
The rank 2 mixed state, which we are now going to consideris a mixture of two orthogonal N -party pure states, given by ρ N ( x ) = x | ψ N ih ψ N | + (1 − x ) | ψ ⊥ N ih ψ ⊥ N | , (5)where the subscript and superscript of ρ represent the numberof qubits and rank respectively. Here, | ψ N i and | ψ ⊥ N i lie intwo orthogonal mutually complementary subspaces of the N-party Hilbert space H ⊗ N . | ψ N i = P ⌊ N ⌋ i =0 a i | D ig i , with | D kg i = ( Nk ) X j =1 b kj | ... | {z } N − k .. | {z } k i , (6)where | D kg i ’s are the generalized Dicke states [31] with k number of excitations i.e. they are the general superpositionsof pure states with all permutations of ( N − k ) | i ’s and k | i ’s. And | ψ ⊥ N i = ⌊ N ⌋− X i =0 a ′ i | D i +1 g i . (7)We have chosen the coefficients in all pure and mixed statessuch that there are properly normalized.For ρ N ( x ) , we can find a group of local unitary op-erators consisting of two unitaries, U = I , and U = σ z , which, on average, keep ρ N ( x ) invariant. Here, I is the identity operator on the qubit Hilbert space and σ x , σ y , and σ z are the Pauli operators. One can checkthat ρ N ( x ) = P k =1 U ⊗ Nk | ψ N ( x ) ih ψ N ( x ) | U †⊗ Nk , where | ψ N ( x ) i = √ x | ψ N i + e iφ √ − x | ψ ⊥ N i is the only class ofpure states that is twirled to ρ N ( x ) by applying the twirl oper-ator corresponding to those unitaries. Hence, by following therecipe in [27], we can calculate the GGM of ρ N ( x ) . Since itinvolves several parameters, for illustration, we choose fullysymmetric states, i.e, when all the coefficients of | ψ N i and | ψ ⊥ N i are equal. The GGM of ρ N ( x, sym ) is the convex hull
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 G x x G FIG. 2. (Color online.) A plot of the GGM of ρ ( x , x ) = x | GHZ +3 ih GHZ +3 | + x | D ih D | + (1 − x − x ) | D ih D | with the state parameters x and x . All the axes are dimensionless. of the GGM of the pure states | ψ N ( x, sym ) i = √ x | ψ N i + √ − xe iφ min | ψ ⊥ N i . Here the phase, φ min , gives the mini-mum GGM among all the GGM with different φ values. Wethen find that GGM reaches its minimum for φ min = 0 .Therefore, the GGM of ρ N ( x, sym ) is given by G ( ρ N ( x, sym )) = 12 (1 − √ x √ − x ) , (8)since the right hand side is already convex as depicted inFig. 1. An important point to note here that the GGM of ρ N ( x, sym ) , given in Eq. (8), is independent of number ofparties, N . B. Classes of rank 3 multiqubit states
We now calculate the GGM for different classes of mixedstates, of rank 3.
1. Case 1
Let us now consider a three-qubit rank 3 mixed state, ρ ( x , x ) [17], which is a mixture of known | GHZ +3 i , | D i ,and | D i . Here, | GHZ +3 i = √ ( | i + | i ) [30], and | D i and | D i are given by | D g i and | D g i of Eq. (6) respec-tively, with b kj = √ for all j . It reads as ρ ( x , x ) = x | GHZ +3 ih GHZ +3 | + x | D ih D | + (1 − x − x ) | D ih D | . (9)Note that | D i is the well-known W-state [32]. The mix-ture is invariant under local unitaries given by U = I , U = (cid:18) e πi (cid:19) , and U = (cid:18) e − πi (cid:19) , when they act oneach qubit [17]. The corresponding pure state which after lo-cal unitary transformations, leads to ρ ( x , x ) , can be writtenas | ψ ( x , x ) i = √ x | GHZ + i + √ x e iφ | D i + √ − x − x e iφ | D i . (10)
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 G x x G FIG. 3. (Color online.) Plot corresponds to GGM of | ψ ,g i vs. themixing parameters x and x . Here, α = 0 . for the | gGHZ i state. Both convex and nonconvex regions are seen. The convex partcorresponds to the GGM of ρ ,g ( x , x ) . All quantities are dimen-sionless. The minimum of GGM among { φ , φ } is again obtainedwhen φ = φ = 0 . By computing the Hessian matrix,we find both analytically and numerically that the GGM of | ψ ( x , x ) i is convex with respect to x and x . Therefore,the GGM of ρ ( x , x ) is given by G (cid:0) ρ ( x , x ) (cid:1) = 16 (cid:18) − (cid:26) − x − x ( x −
1) +8 √ x x (cid:16) p x (1 − x − x ) − x − x (cid:17) +4 x (cid:16) p x (1 − x − x ) − x (cid:17) (cid:27) (cid:19) , (11)and is depicted in Fig. 2.
2. Case 2
Let us now move to a more general state while keeping therank fixed. Precisely, we consider a class of mixed states ofthe form ρ ,g ( x , x ) = x | gGHZ ih gGHZ | + x | D g ih D g | +(1 − x − x ) | D g ih D g | , (12)where | gGHZ i = α | i + √ − α | i is the generalizedGreenberger-Horne-Zeilinger state with ≤ α ≤ . The setof local unitaries that keep ρ ( x , x ) invariant, also keep thestate ρ ,g ( x , x ) invariant, and the class of pure state that areprojected to ρ ,g ( x , x ) is given by | ψ ,g ( x , x ) i = √ x | gGHZ i + e iφ √ x | D g i + e iφ √ − x − x | D g i . (13)In this case, we have ρ ,g ( x , x ) = P j =1 U ⊗ j | ψ ,g ( x , x ) ih ψ ,g ( x , x ) | U †⊗ j , where { U j , j = 1 , , } is the same as in Case 1. G x r = 0.98r = 0.96 FIG. 4. (Color.) Plot corresponds to GGM of | ψ ,g i vs. x , for twovalues of r = x − x . Here, α = 0 . for the | gGHZ i state. Theseare given by the dotted lines. The straight lines corresponds to theconvexified quantities. All quantities are dimensionless. Numerical simulation guarantees that the minimum of E ( | ψ ,g ( x , x ) i ) occurs for φ = φ = 0 . However, un-like the previous cases, we find that E ( | ψ ,g ( x , x ) i ) is notconvex for all values of x and x . In particular, we plot E ( | ψ ,g ( x , x ) i ) in Fig. 3, when α = 0 . and when thecoefficients in | D g i and | D g i are all equal. For certain re-gions of the parameter space, the figure is already convex,and hence the GGM of | ψ ,g ( x , x ) i ) in that region is theGGM of ρ ,g ( x , x ) . On the other hand, for the remain-ing regions, a convexification has to be carried out to obtainthe GGM of ρ g ( x , x ) . Specifically, E ( | ψ ,g ( x , x ) i ) = G ( ρ ,g ( x , x )) , when x is high while x is low. To obtainthe GGM in that region, the convexification is required. Toillustrate the process, we introduce a new variable, r = x − x ,and let us consider cases where r = 0 . and . . The con-vexification of the curves so generated are depicted in Fig. 4.
3. Case 3
Let us move to a class of states which is a multiqubit gen-eralization of ρ ( x , x ) . It is given by ρ N ( x , x ) = x | GHZ + N ih GHZ + N | + x | D ih D | +(1 − x − x ) | D N − ih D N − | , (14)where | GHZ + N i = √ ( | i ⊗ N + | i ⊗ N ) , and | D N − i is givenby | D N − g i of Eq. (6) with b kj = q ( Nk ) . Again, we have ρ N ( x , x ) = P j =1 U ⊗ Nj | ψ N ( x , x ) ih ψ N ( x , x ) | U †⊗ Nj ,where | ψ N ( x , x ) i is given in Eq. (10) with | D i being re-placed by | D N − i , for the same set of unitaries, given in Case1. Hence, we can compute the GGM of | ψ N ( x , x ) i andcheck its convexity. For φ = φ = 0 which gives the lowestGGM, Fig. 5 shows the GGM of | ψ ( x , x ) i with respect tothe parameters, x and x with N = 5 . From the figure, it
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0.2 0.25 0.3 0.35 0.4 0.45 0.5 G x x G FIG. 5. (Color online.) The plot of GGM for ρ ( x , x ) = x | GHZ +5 ih GHZ +5 | + x | D ih D | + (1 − x − x ) | D ih D | against x and x whenever it is convex. All axes are dimensionless. is clear that for example the GGM of | ψ ( x , x ) i is convexfor . ≤ x ≤ . and . ≤ x ≤ . and hence in thatregion, we have the GGM of ρ ( x , x ) . In the rest of theregion, to obtain the GGM of ρ ( x , x ) , we have to find theconvex hull of E ( | ψ ( x , x ) i ) . C. Higher rank multiqubit states
We now consider classes of mixed states with rank morethan three. First, we explore a class of multiparty states whichcan be dealt with symmetric unitaries. In other words, thisclass of states remain invariant, when the same unitary acts onall the parties, i.e. ρ NN = P j U ⊗ Nj ρ NN U †⊗ Nj . We will thenfind another class of states for which symmetric unitaries donot work.
1. Symmetric unitary case
Let us now consider a class of mixed states with arbitrarynumber of parties, which can be obtained by generalizing ρ ( x , x ) . The state, ρ NN ( x , x , . . . , x N − ) , is a mixture ofgeneralized GHZ and all the Dicke states. It reads as ρ NN ( x , x , . . . , x N − ) = (1 − X i x i ) | gGHZ N ih gGHZ N | + N − X i =1 x i | D ig ih D ig | , (15)with | gGHZ N i = α | i ⊗ N + √ − α | i ⊗ N . Rank of theabove state spans the integers in [1 , N ] . One can check that ρ NN ( x , . . . , x N − ) = N X j =1 U ⊗ Nj ρ NN ( x , . . . , x N − ) U †⊗ Nj , (16)
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 G x x G FIG. 6. (Color online.) GGM of ρ = x P [ GHZ +5 ] + x ( P [ D ] + P [ D ]) + − x − x ( P [ D ] + P [ D ]) . All axes are dimensionless. where the set of local unitaries, { U j } Nj =1 consists of I and (cid:18) e πijN (cid:19) with j = 1 , . . . , ( N − . We have to now showthat ρ NN ( x , x , . . . , x N − ) = X j U ⊗ Nj | ψ NN ( x , . . . , x N − ) ih ψ NN ( x , . . . , x N − | U †⊗ Nj , (17)where | ψ NN ( x , . . . , x N − ) i = p − P i x i | gGHZ N i + P N − i =1 √ x i | D ig i . To prove this, the we note the actions of lo-cal unitaries on each off-diagonal terms which e.g. are givenby U ⊗ Nj | D qg ih D rg | U †⊗ Nj = e πi ( q − r ) N | D qg ih D rg | . (18)We use the identity P j e πi ( q − r ) N = δ qr in the analysis. Simi-larly, X j U ⊗ Nj | D qg ih gGHZ N | U †⊗ Nj = e πiqN | D qg ih gGHZ N | = 0 . (19)All off-diagonal terms therefore vanish. We can now cal-culate the GGM of | ψ NN ( x , . . . , x N − ) i and check whether E ( | ψ NN ( x , . . . , x N − ) i ) is convex or not. If it is convex, then E ( | ψ NN ( x , . . . , x N − ) i ) = G ( ρ NN ( x , . . . , x N − )) . Other-wise, we have to perform convexification to obtain the exactvalue of G ( ρ NN ( x , . . . , x N − )) . To illustrate this example, weconsider a five-qubit state which is of the form ρ = x | GHZ +5 ih GHZ +5 | + x | D ih D | + | D ih D | )+ 1 − x − x | D ih D | + | D ih D | ) . (20)Following the aforementioned prescription, we compute FIG. 7. (Color online.) Plot of GGM of ρ with respect to the pa-rameters, x and y . The GGM of the corresponding unique pure state, | ψ ( x, y ) i = √ x | ζ i − i p y/ | ζ i − | ζ i ) + √ − x − y | ζ i hasa kink along the lines shown on the surface, in the plot. The GGM ofthe pure state is non-convex around these lines, and hence convexifi-cations are required thereat. E ( | ψ ( x , x ) i ) with | ψ ( x , x ) i = √ x | GHZ +5 i + r x X k =1 e iφ k | D k i + r − x − x X k =3 e iφ k | D k i . (21)For φ k = 0 , k = 1 , . . . , which gives the infimum of GGM, E ( | ψ ( x , x ) i ) is plotted with x and x in Fig. 6. By usingthe Hessian technique, we find that it is convex for the entirerange of x and x . Therefore, G ( ρ ) is obtained for all x and x and is given by G ( ρ ) = 12 − (cid:18) − (cid:26) x + 4 x + 310 7 − x − x − (cid:16)r x x
20 + r x (1 − x − x )20 + 2 x √ − x − x )5 √
2+ 310 p x (1 − x − x ) (cid:17) (cid:27)(cid:19) ! . (22)Comparing Figs. 5 and 6 with the situations obtained before,it seems that higher rank states, for a fixed total number ofqubits of the entire systems, have a greater affinity for beingconvex, when their GGMs are considered.
2. Asymmetric unitary case
Until now, we have considered the states which remain un-altered under local symmetric unitaries of the form U ⊗ N i .Let us now illustrate a class of three-qubit mixed stateswhich remains unchanged under the local unitaries of the form U i ⊗ U j ⊗ U k . The class of mixed state having rank 4, reads ρ = X i x i | ζ i ih ζ i | , (23)where | ζ i = 12 ( | i + | i − | i + | i ) , | ζ i = 12 ( − i | i − i | i + | i + | i ) , | ζ i = 12 ( i | i + i | i + | i + | i ) , and | ζ i = 12 ( | i + | i + | i − | i ) . It is invariant under { U i , i = 1 , . . . } , which are given by U = I ⊗ I ⊗ I,U = iσ y ⊗ H ′ ⊗ H ′ ,U = I ⊗ σ y ⊗ σ y , and U = − iσ y ⊗ H ′ T ⊗ H ′ T , with H ′ = √ (cid:18) − (cid:19) . Note that these unitaries form aclosed group. The only pure states that are twirled to theabove mixed states are of the form | ψ i = P i √ x i e iφ i | ζ i i .We compute the GGM of | ψ i and minimize it over φ i ’s. TheGGM of ρ is given by the minimum of the E ( | ψ i ) for dif-ferent values of φ i s provided the quantity is convex itself.To visualize its GGM, let us consider, x = x = y , i.e.the state is of the form ρ = x | ζ i h ζ | + y | ζ i h ζ | + | ζ i h ζ | )+(1 − x − y ) | ζ i h ζ | . (24)In this case, we find that the minimum GGM of | ψ ( x, y ) i fordifferent values of φ i ’s is obtained when φ = − φ = − π and φ = 0 . We find the GGM of ρ ( x, y ) by convexifyingthe GGM of | ψ ( x, y ) i ) . D. Cases of multiqudit states
In the previous sections, we have evaluated the GGM ofcertain multiqubit systems. We will now show that a similarmethod can be extended to obtained the analytical expressionof GGM of multiqudit mixed states. Specifically, we consideran N -qudit mixed state of rank d , in the Hilbert space H ⊗ Nd ,of the form ρ dN,d = d X k =1 p k | Ψ i k h Ψ | k , (25)where | Ψ i k = P { j } q j j ...j N | j j ...j N i ( k ) and ( P m j m )( mod d ) = k . Our aim is to evaluate theGGM of the state ρ dN,d . Therefore, like previous cases, weconstruct a twirling operator, consisting of unitary operators Z d which are d -dimensional, non-hermitian generalization ofthe σ z and given by Z d = d − X j =0 e πijd | j ih j | . (26) Here, each of the unitary operators act locally andsymmetrically on ρ dN,d as Z ⊗ Nd . Note that the set (cid:26) I d , Z ⊗ Nd , (cid:16) Z ⊗ Nd (cid:17) , .., (cid:16) Z ⊗ Nd (cid:17) d − (cid:27) forms a group and thecorresponding twirling operator keeps ρ dN,d invariant. Now,we have to find the set of all pure states | Ψ i dN,d thatare projected to ρ dN,d under the action of the aforemen-tioned twirling operator. It can be easily checked that | Ψ i dN,d = P dk =1 e iφ k | Ψ i k are the only class of pure statesthat are mapped to ρ dN,d under the twirling operator, i.e., P d − q =0 (cid:16) Z ⊗ Nd (cid:17) q | Ψ i dN,d h Ψ | dN,d (cid:16) Z †⊗ Nd (cid:17) q = ρ dN,d . In this casealso, the minimum of the GGM’s of | Ψ i dN,d over the phases { φ k } gives the GGM of ρ dN,d provided the minimum GGM isalready a convex function of the state parameters. Otherwiseone has to convexify the function to obtain the GGM of ρ dN,d .Until now, we have considered systems with the same di-mensions of the local Hilbert spaces. However, this formal-ism can be further extended where the local Hilbert spaces’dimensions are not equal, i.e., for quantum systems belongingin H d ⊗H d ⊗ . . . ⊗H d N , with d = d = ...d N . In that case,we have two different scenarios. Firstly, a d = a d = ... = d N , where { a i } N − i =1 ∈ I + . Without loss of generality, d N istaken to be the largest dimension and the corresponding uni-taries are of the form Z d ⊗ Z d .. ⊗ Z d N with its subsequentpowers upto d N − , such that the composite unitary matri-ces form a group. Evidently, the case of equal dimensions isa special case of this. Thus, the pure states over which wehave to perform the minimization still have the same form,with a slightly different version of the condition given by P m j m ( mod d N ) = k . The second one is the situation whenall the dimensions are prime to each other, and in this case,we have to take unitaries upto the power of (cid:0) d d ...d N (cid:1) − ,where the form of pure states remain the same, with the mod-ified condition, P m j m (cid:0) mod d d ...d N (cid:1) = k . Therefore, ingeneral, we have to take the maximum power of the unitarieswhich is the lowest common multiple of d , d , ..., d N to ap-ply the similar prescription. In the next paragraph, we illus-trate this with an example.For simplicity, we consider the following three-qutritstate, ρ , = P k =0 x k | Ψ i k h Ψ | k , where | Ψ i k = P j q j j j | j j j i ( k ) and j + j + j ( mod
3) = k . Theexact form of the pure states {| Ψ k i} k =0 reads as | Ψ i = 13 ( X i =0 | iii i + X perm | i ) , | Ψ i = 13 ( X perm | i + X perm | i + X perm | i ) , and | Ψ i = 13 ( X perm | i X perm | i + X perm | i ) . (27)For this case, the unitaries which construct the twirling op-erators are given as { I , Z , Z } . Note that the unitaries ofthe form Z i ⊗ Z i ⊗ Z i form a group for i ranging from to and ρ , is evidently invariant under the corresponding
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 G x x G FIG. 8. (Color online.) Plot of GGM of ρ , against x and x . TheGGM of the corresponding unique pure state, | Ψ , i = √ x | Ψ i + e iφ √ x | Ψ i + e iφ √ − x − x | Ψ i is plotted with φ = φ =0 . The GGM of the pure state is convex everywhere, as evident fromthis plot and hence E ( | Ψ , i ) = G ( ρ , ) . twirling operator. The pure state that is mapped to ρ , un-der the action of the aforesaid twirling operator is of the form | Ψ , i = √ x | Ψ i + e iφ √ x | Ψ i + e iφ √ − x − x | Ψ i .It can be easily found that minimum GGM of | Ψ , i is ob-tained for φ = φ = 0 and it is a convex function of theparameters x and x . Hence, the GGM of ρ , is given by G ( ρ , ) = { − √ x x − p x { − x − x }− p x { − x − x }} . G ( ρ , ) is depicted in Fig. 8 and the convexity of the functioncan be visualized from the same. IV. CONCLUSION
Computing entanglement of an arbitrary mixed state is aformidable task. The entanglement of mixed states is gen-erally defined by constructing the convex roof over all pos-sible pure states which is practically impossible to computein most of the cases. Although there exists a few bipartitemeasures which can be obtained for arbitrary states, the eval-uation of entanglement for a mixed state in multiparty domainis still a challenging task. In this paper, we have computed agenuine multiparty entanglement measure known as general-ized geometric measure of some classes of mixed states witharbitrary number of parties and dimensions by using certainsymmetries. We evaluate the measure for several classes ofmultiqubit and multiqudit states having different ranks. Themethod, we exploited, uses a pure state that contains the sameamount of entanglement as the given mixed state, and leads tothe mixed state by action of a certain twirling operation.
Note added:
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