SSymmetric 3 Qubit State Invariants
Alexander Meill and David A. Meyer (Dated: November 9, 2018)For pure symmetric 3-qubit states there are only three algebraically independent entanglementmeasures; one choice is the pairwise concurrence C , the 3-tangle τ , and the Kempe invariant κ .Using a canonical form for symmetric n qubit states derived from their Majorana representation,we derive the explicit achievable region of triples ( C , τ, κ ). PACS numbers:
INTRODUCTION
Entanglement is a critical resource for quantum com-putational tasks such as teleportation [1] and cryptogra-phy [2] among many others, but analytic calculations ofentanglement quickly become challenging as the dimen-sion of the Hilbert space increases. Restricting those cal-culations to states which are symmetric under subgroupsof permutations of party labels can greatly reduce thenumber of parameters and simplify the calculations con-siderably. Symmetric states are particularly useful forthis reason; in addition, symmetric states are relevantin many experimental settings such as in Measurement-Based Quantum Computing [3], as an initial state forGrover’s Algorithm [4], and as ground states of varioustranslation invariant Hamiltonians [5]. In this paper, therestriction to completely symmetric states is used to sim-plify the calculation of 3 qubit local unitary (LU) invari-ants.The entanglement of an n qubit state, as defined byany measure, remains invariant under local unitary oper-ators of the form U = U ⊗ U ⊗ . . . ⊗ U n , where U i ∈ U (2).Polynomial invariants of a multi-qubit state are not lim-ited to entanglement monotones, though these are a par-ticularly relevant choice. In this paper, the invariants ofan arbitrary three qubit symmetric state are calculatedexplicitly and their full achievable region is calculated. Any multi-particle state has a set of polynomials inthe coefficients of the state which are invariant under theaction of various local operators [6]. In particular, a 3qubit state, under the action of unitaries which act onlylocally on one qubit, is known to have 5 algebraically in-dependent invariants (as well as the trace norm and Z invariant) [7]. There is some freedom in choosing 5 gen-erators of the algebra of invariant polynomials, as anypolynomial in invariants is additionally an invariant ofthe state. One set of generators that is particularly con-venient for 3 qubit states under local unitary operatorsis {C , , C , , C , , τ, κ } , (1) where C i,j is the pairwise concurrence between parties i and j [8], τ is the 3-tangle [9], and κ is the Kempeinvariant [10]. These quantities are defined for a 3 qubitstate, | ψ (cid:105) = (cid:80) i,j,k =0 c ijk | ijk (cid:105) , as follows: C i,j = max { , λ − λ − λ − λ } , (2) τ = 2 | (cid:15) i i (cid:15) i i (cid:15) j j (cid:15) j j (cid:15) k k (cid:15) k k × c i j k c i j k c i j k c i j k | , (3) κ = c i j k c i j k c i j k c ∗ i j k c ∗ i j k c ∗ i j k , (4)where the λ α in (2) are the square roots of the eigen-values, in decreasing order, of ρ i,j ( σ y ⊗ σ y ) ρ ∗ i,j ( σ y ⊗ σ y ).Here, ρ i,j is the reduced density operator of the statehaving traced out all parties other than i and j . Note,also, that in (3) and (4) we have adopted the conventionof summing over repeated indices. This choice of invari-ants is particularly useful as it uses some of the mostprevalent entanglement measures in the concurrence and3-tangle.It would be interesting to completely map the space ofthese 5 invariants, but the calculations are difficult forarbitrary states, and the 5 dimensional picture would beunwieldy to describe or visualize. Instead, by examininga particular subset of states, the number of parametersand invariants can be reduced to be more manageable. Inthe next section this is done for the subset of states whichare symmetric under permutation of the party labels. SYMMETRIC 3 QUBIT STATES
Symmetric states offer a significant simplification tothe picture of 3 qubit invariants. Clearly if a state issymmetric under relabeling of parties, each of the two-party reduced density operators, ρ i,j , will be identical.This then causes C , = C , = C , = C and effectivelyreduces the number of invariants to 3, which will be de-noted, {C , τ, κ } . (5)The goal now is to calculate these invariants for a gen-eral 3 qubit symmetric state and describe the region of a r X i v : . [ qu a n t - ph ] N ov allowed and acheivable values for these invariants. Beforedoing so, however, we can further simplify the problem byexamining how the symmetric subspace of 3 qubit statesreduces the parameters on which the invariants depend.The most natural representation of an n qubit sym-metric state is in the Dicke basis [11], | ψ (cid:105) = n (cid:88) i =0 a i (cid:12)(cid:12)(cid:12) S ( n ) i (cid:69) , (6)where (cid:12)(cid:12)(cid:12) S ( n ) i (cid:69) = (cid:18) ni (cid:19) − / (cid:88) π ∈ S n π | ... (cid:124) (cid:123)(cid:122) (cid:125) n − i ... (cid:124) (cid:123)(cid:122) (cid:125) i (cid:105) (7)are the Dicke basis states which represent an equal su-perposition of all possible states with i “0” entries and n − i “1” entries. They are obviously symmetric since π permutes the parties of the state and the sum is an equalsuperposition of all possible permutations. In this rep-resentation, the 3 qubit symmetric state has 4 complexcoefficients, a i , which reduce to 6 real parameters afternormalization and the factoring out of an overall phase.While the invariants can be calculated from these 6 realparameters, it is useful to apply a set of local unitaries tothe state to reduce the number of parameters. The calcu-lation of the invariants then becomes more concise whilestill containing the same information since the invariantsshould not change under local unitaries. Such a simpli-fication was investigated in [11]. Using almost the sameargument, we show in the following that most 3-qubitsymmetric states are equivalent under local unitaries tostates of the form, | ψ (cid:48) (cid:105) = A (cid:16) | (cid:105) + ye iφ | θ (cid:105) ⊗ (cid:17) , (8)where y ∈ [0 , θ ∈ [0 , π ], φ ∈ [0 , π ), | θ (cid:105) =cos( θ/ | (cid:105) +sin( θ/ | (cid:105) is a single qubit state with purelyreal coefficients, and A is a normalization constant. Whatfollows is a proof of (8).Consider an arbitrary 3 qubit symmetric state givenby, | ψ (cid:105) = (cid:88) i =0 a i (cid:12)(cid:12)(cid:12) S (3) i (cid:69) . (9)One can compute the Majorana Polynomial [13] of | ψ (cid:105) ,by projecting it onto the unnormalized state | α (cid:105) =( | (cid:105) + α ∗ | (cid:105) ) ⊗ , where α is an arbitrary complex number.The resulting inner product is the following polynomialin α , (cid:104) α | ψ (cid:105) = a + √ a α + √ a α + a α . (10)By the first fundamental theorem of algebra, the rootsof this polynomial, α i , known as the Majorana roots, uniquely determine the coefficients a i up to some overallscaling factor. So to show that two states are the same, itsuffices to show that they have the same Majorana roots.We will exploit this fact to show that most states, | ψ (cid:105) ,can be represented as | ψ (cid:105) = c | Φ (cid:105) + c | Φ (cid:105) , (11)where | Φ j (cid:105) = (cid:0) cos θ j | (cid:105) + e iφ j sin θ j | (cid:105) (cid:1) ⊗ and c j ∈ C ,by showing that (11) has the same Majorana roots as (9)for some choice of c j , θ j and φ j . Before computing theMajorana roots of (11), we can factor out c from thestate to express it as, | ψ (cid:105) = A ( | Φ (cid:105) + c | Φ (cid:105) ) , (12)where A is a normalization constant and c = c /c ∈ C .The Majorana polynomial of (12) can be expressed as (cid:104) α | ψ (cid:105) = (cid:0) cos θ + α sin θ e iφ (cid:1) + c (cid:0) cos θ + α sin θ e iφ (cid:1) . (13)Note we have dropped the normalization factor, A , sincewe need only specify the polynomial up to a scaling fac-tor. We can further simplify by dividing by cos θ , whichleaves, (cid:104) α | ψ (cid:105) = (1 + αβ ) + c (cid:48) (1 + αβ ) , (14)where β j = tan θ j e iφ j and c (cid:48) = c (cos θ ) / (cos θ ). Thegoal is to demonstrate that one can solve for a choice of β j and c (cid:48) that will satisfy (14) having the same roots as(10). This creates the following constraints on β j and c (cid:48) ,0 = (1 + α β ) + c (cid:48) (1 + α β ) (15)0 = (1 + α β ) + c (cid:48) (1 + α β ) . (16)Additionally, we can require that the projection of (12)onto | α (cid:105) be the same as (10) when evaluated at α = 0,which provides the third constraint, a = c (cos θ ) + c (cos θ ) . (17)Equations (15-17) provide sufficient constraints on β j and c (cid:48) to identify a representation (11) which is the samestate as (9) so long as no Majorana root, α i is degener-ate with degree 2 [12]. One can then construct a localunitary operator of the form U = U ⊗ U ⊗ U , where U (cid:0) cos θ | (cid:105) + sin θ e iφ | (cid:105) (cid:1) = | (cid:105) . Applying this to (12)results in U | ψ (cid:105) = A ( | (cid:105) + c | χ (cid:105) ) , (18)where | χ (cid:105) = U | Φ (cid:105) = (cid:0) cos θ/ | (cid:105) + sin θ/ e iχ | (cid:105) (cid:1) ⊗ .Lastly, one can apply the local unitary operator, V = (cid:18) e − iχ (cid:19) ⊗ (19)to the state (18) to arrive finally at the desired result, | ψ (cid:48) (cid:105) = A (cid:16) | (cid:105) + ye iφ | θ (cid:105) ⊗ (cid:17) , (20)where | θ (cid:105) = cos θ/ | (cid:105) + sin θ/ | (cid:105) , and c = ye iφ .This canonical form is a useful tool in examining the lo-cal unitary invariants of 3-qubit symmetric states. Othercanonical forms and representations for 3-qubit states areknown, but this one provides considerable simplificationto symmetric states in particular and leaves them sym-metric after the rotation. If one were to transform (20)into one of the general 3-qubit canonical forms developedby Ac´ın et al. in [14] one would get the following state, | χ (cid:105) = A (cid:20) sin θ | (cid:105) + cos θ (cid:0) e − iθ + y cos θ (cid:1) | (cid:105) + y cos θ sin θ ( | (cid:105) + | (cid:105) ) + y sin θ | (cid:105) (cid:21) , (21)which is less obviously symmetric than (20). An effectivecanonical form utilizing the inner products of the vec-tors in the Majorana representation was presented in [15].This form was used to calculate a slightly different set ofinvariants, (cid:8) τ, κ, Tr( ρ ) (cid:9) , where ρ is the single-party re-duced density matrix of the symmetric state obtained bytracing out any two parties, and the states which maxi-mize and minimize the various invariants were examined.In what follows, we will calculate the {C , τ, κ } of (20) anddescribe the full achievable space of those variants.In terms of the parameters y , θ , and φ of (20), theinvariants are, τ = 2 y sin θ y + 2 y cos θ cos φ (22) C = y sin θ sin θ y + 2 y cos θ cos φ (23) κ = 18 (cid:0) y + 2 y cos θ cos φ (cid:1) × (cid:20) (cid:0) y (cid:1) (cid:0) y + 8 y + 9 y (4 cos θ + cos 2 θ ) (cid:1) +24 y cos θ (cid:0) y + 2 y + 3 y cos θ (cid:1) cos φ +48 y (cid:0) y (cid:1) cos θ φ + 16 y cos θ φ (cid:21) . (24)Figure 1 shows the invariants of 10 randomly gen-erated symmetric 3 qubit states, where the states weregenerated by sampling randomly over the allowed val-ues of y , θ , and φ . At a first glance, it is interestingto note that the 3-tangle and Kempe invariants achievetheir maximum values of 1 on the symmetric subspace,but the concurrence does not due its monogamy con-straints [9]. A straightforward maximization over the ! FIG. 1: Scatterplot from two points of view of invariants ofrandomly generated symmetric 3 qubit states. state parameters reveals a maximum concurrence of 2 / n = 3. The points of Figure 1 appear to lie al-most on a surface, but closer inspection reveals that theyin fact fill a narrow volume, the boundaries of which canbe calculated. We can invert the expressions (22-24) bya Gr¨obner basis calculation to find,cos θ C√C + τ (25)cos φ = 4 − τ − C − κ C (26) y = 6 τ + 9 C + 4 κ −
43 ( τ + C ) / − (cid:118)(cid:117)(cid:117)(cid:116)(cid:32) τ + 9 C + 4 κ −
43 ( τ + C ) / (cid:33) − . (27)The constraints on the state parameters then provideconstraints on these functions of the invariants. The ex-trema of these constraints are the surfaces which formthe boundaries of the invariant space. The boundariesare formed when equality is achieved in the following re-lations.0 ≤ − τ − C − κ + 3 C (28)0 ≥ − τ − C − κ − C (29)0 ≥ − τ − C − κ + 3 (cid:0) τ + C (cid:1) / . (30)These three surfaces, which are shown in Figure 2, formboundaries for the possible space of the invariants andserve as additional monogamy relations for symmetric3 qubit entanglement. Note that the state parameterconstraints lead to more constraints on the invariants,but (28-30) is the minimum set of constraints requiredto describe the region. Because there is a bijective mapbetween the invariants and the state parameters, eachinvariant triple which lies within the region satisfying (28-30) can be mapped to a 3 qubit symmetric state, andtherefore the entire region is achievable. FIG. 2: View of a slice of the boundaries of the volume ofsymmetric 3 qubit invariants superimposed over the points ofFigure 1. The contour achieving equality in equation (28) isshown in green, (29) in blue, and (30) in red.
We should at this point address the states which donot admit a representation of the form (8), which we de-note (cid:12)(cid:12) ¯ ψ (cid:11) . It is shown in [12] that 3 qubit states whichhave a degenerate Majorana root with degree 2 cannotbe expressed in this canonical form. Instead, we willparametrize states of that form and show that the in-variants of this subset of state likewise satisfy (28-30).An arbitrary 3 qubit state with a degenerate Majoranaroot of degree 2 can be expressed in the Majorana repre-sentation as (cid:12)(cid:12) ¯ ψ (cid:11) = 1 A (cid:88) π ∈ S π | φ (cid:105) ⊗ | φ (cid:105) ⊗ | φ (cid:105) , (31)where φ i are single qubit states. We can again use localunitaries to simplify states of this form to (cid:12)(cid:12) ¯ ψ (cid:48) (cid:11) = 1 A (cid:88) π ∈ S π | (cid:105) ⊗ | (cid:105) ⊗ | θ (cid:105) , (32)where | θ (cid:105) is the same as in (8) for θ ∈ (0 , π ]. The invari-ants of (32) are τ = 0 , (33) κ = 2 + 48 cos θ + 141 cos θ + 52 cos θ θ ) , (34) C = 2 − θ θ . (35)It is then easy to verify that (33-35) satisfy (28-30) for θ ∈ (0 , π ]. This is perhaps unsurprising given that thestates with a degenerate Majorana root of degree 2 area limiting case of states which admit the canonical form.Now all 3 qubit symmetric states have been consideredand it can be concluded that (28-30) do indeed describethe full achievable region for 3 qubit symmetric states. A similar approach could be used to analyze the invari-ants of n qubits for the symmetric subspace. The repre-sentation in [12] extends to symmetric n qubit states re-sulting in 2 n − n > n − n − n − n qubit case, though it is certainly worth examining. Acknowledgements
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