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Classification of nonproduct states with maximum stabilizer dimension David W. Lyons,∗ Scott N. Walck,† and Stephanie A. Blanda‡ Lebanon Valley College, Annville, PA 17003 (Dated: 11 September2007) Nonproduct n-qubit pure states with maximum dimensional stabilizer subgroups of the group of local unitary transformations are precisely the generalized n-qubit Greenberger-Horne-Zeilinger states and their local unitary equivalents, for n≥3,n6=4. We characterize the Lie algebra of the stabilizersubgroupforthesestates. Forn=4,thereisanadditionalmaximalstabilizersubalgebra, not local unitary equivalent to the former. We give a canonical form for states with this stabilizer as well. PACSnumbers: 03.67.Mn 8 0 0 I. INTRODUCTION and their local unitary equivalents for 3 or more qubits, 2 together with an additional class of states for the spe- n The desire to measure and classify entanglement for cial case of 4 qubits. In order to prove these results, we a J multiparty states of n-qubit systems has been moti- extend the stabilizer analysis of our previous work [11] vated by potentialapplications inquantum computation from stabilizers of pure states to stabilizers of arbitrary 9 2 and communication that utilize entanglement as a re- density matrices. source [1, 2]. Entanglement classification is also an in- ] terestingquestioninthefundamentaltheoryofquantum h II. THE MAIN RESULT information. p It is natural to consider two states to have the same - t entanglement type if one is transformable to the other The n-qubit local unitary group is n a byalocalunitarytransformation[3,4]. Thusanyentan- u glement measure, as a function on state space, must be G=G0×G1×·×Gn q invariantundertheactionofthelocalunitarygroup. One [ such invariant is the isomorphism class of the stabilizer whereG0 =U(1)isthegroupofphasesandGj =SU(2) 3 subgroup(thesetoflocalunitarytransformationsthatdo is the 3-dimensional group of single qubit rotations in qubit j for 1≤j ≤n. Its Lie algebra v notalteragivenstate)anditsLiealgebraofinfinitesimal 5 transformations. Inspiredby ideas originally laid out for g=g ⊕g ⊕···⊕g 0 3-qubit systems [3, 5], the authors of the present paper 0 1 n 1 have achieved a number of results for systems of arbi- is the set of infinitesimal local unitary transformations 1 . trarynumbers ofqubits onthe structureofstabilizerLie acting on the tangent level, where g0 is 1-dimensional 9 subalgebrasandinparticular,thosestabilizersthathave and each g =su(2), the set of skew-Hermitian matrices 0 j maximum possible dimension. Evidence that maximum with trace zero. Given an n-qubit state vector |ψi, let 7 stabilizer dimensionis aninteresting property is the fact 0 : that states that have such stabilizers also turn out to Stabψ ={g ∈G: g|ψi=|ψi} v maximize other known entanglement measures and play Xi key roles in quantum computational algorithms [6, 7]. denote the subgroup of elements in G that stabilize ψ, In [8] we describe a connection between stabilizer struc- and let r a tureandthe questionofwhenapurestate isdetermined by its reduced density matrices. Kψ ={X ∈g: X|ψi=0} States with maximum stabilizer dimension are prod- be the corresponding Lie algebra. uctsofsingletpairs(withanunentangledqubitwhenthe In [11] we show that the maximum possible dimen- number of qubits is odd) and their local unitary equiva- sion for K is n−1 when ψ is not a product state, for lents [9,10]. Inthe presentpaper,we showananalogous ψ n ≥ 3. Further, we show that for n 6= 4, we must have result for nonproduct states with maximum stabilizer di- dimP K =1 for all j, where P : g →g is the natural mension. Thesearethegeneralizedn-qubitGreenberger- j ψ j j projection. In fact this condition holds for the general- Horne-Zeilinger (GHZ) states ized n-qubit GHZ state α|00···0i+β|11···1i, α,β 6=0 α|00···0i+β|11···1i and its LU equivalents. For n=4,there is an additional possibleconditionforK toattainmaximumdimension, ∗Electronicaddress: [email protected] ψ †Electronicaddress: [email protected] andthatis dimPjKψ =3 forallj andKψ ∼=su(2). The ‡Electronicaddress: [email protected] main result of the present paper is the converse of these 2 previous results. We show that the generalized n-qubit Let GHZ states (and LU equivalents) are the only nonproduct states that have maximum stabilizer dimension, ex- Stab ={g ∈SU(2)n: gρg† =ρ} ρ cept for the 4-qubit case. For those 4-qubit states with stabilizer isomorphic to su(2), we give a canonical rep- denote the subgroupof elements that stabilizeρ, and let resentative for each LU equivalence class. Here is the formal statement. Kρ ={X ∈su(2)n: [X,ρ]=0} Theorem1. (Classificationofstateswithmaximal be the corresponding Lie algebra. We use multi-index stabilizer) Let |ψi be a nonproduct n-qubit state with notation I = (i ,i ,...,i ), where each i is a binary 1 2 n k stabilizer subalgebra Kψ of maximum possible dimension digit, to denote labels for the standard computational n−1, for some n≥3. One of two conditions must hold: basis for state space. We write Ic to denote the bitwise complement of I. (i) dimP K =1 for 1≤j ≤n, or j ψ OneexpectsanaturalcorrespondencebetweenK and ψ (ii) n = 4, dimPjKψ = 3 for 1 ≤ j ≤ n and Kψ ∼= Kρ in the case where ρ=|ψihψ|, and indeed there is. su(2). Proposition1. Letρ=|ψihψ|beapuren-qubitdensity If (i) holds, then |ψi is LU equivalent to a generalized n- matrix. Then K =PK , where P: g→g ⊕···⊕g is ρ ψ 1 n qubit GHZ state. If (ii) holds, then |ψi is LU equivalent the natural projection that drops the phase factor. to a state of the form Proof. We begin with a simple observation. Let X ∈ g |ψi = a(|0011i+|1100i) and let |φi=X|ψi. Skew-Hermicity of X gives us + b(|1001i+|0110i) + c(|1010i+|0101i) hψ|X =−hφ|. (1) for some complex coefficients a,b,c satisfying a > 0, To see that PK ⊂ K , let X ∈ K , say X = ψ ρ ψ abc6=0 and a+b+c=0. Furthermore, this is a unique (−it,X ,X ,...,X ). Let Y = PX, so Y |ψi = it|ψi. 1 2 n representative of the LU equivalence class for |ψi of this Then form. [Y,ρ]=Y |ψihψ|−|ψihψ|Y =itρ−itρ=0. The proof divides naturally into two sections, one for each of the two conditions on stabilizer in the statement To get the second equality, apply (1) to |φi = it|ψi = of the theorem. We consider conditions (i) and (ii) in Y |ψi. sections IV and V, respectively. It is convenient to work Conversely, we show that K ⊂ PK . Let Y ∈ K , ρ ψ ρ in terms of density matrices and their local unitary sta- and let |φi=Y |ψi. Using (1) again, we have bilizers; the following preliminary section establishes the necessary extension of our previous results to local uni- 0 = [Y,ρ]=Y |ψihψ|−|ψihψ|Y tary action on density matrices. = |φihψ|+|ψihφ|=|φihψ|+(|φihψ|)† so |φihψ| is skew-Hermitian. Thus there is some unitary III. PRELIMINARY PROPOSITIONS ON STABILIZER STRUCTURE U such that U|φihψ|U† is diagonal, with pure imaginary entries. In fact there is only one nonzero diagonal entry, say is for some real s, since |φihψ| has rank one. Forthesetofn-qubitpureandmixeddensitymatrices, ThereforeU|φi=isU|ψi, so |φi=is|ψi. Thus Y is the wemayomitthephasefactorinlocalunitaryoperations, projectionof (−is,Y ,Y ,...,Y ) in K . This concludes and take the local unitary group to be 1 2 n ψ the proof. SU(2)n =G ×···×G ⊂G 1 n LetususethesymbolP todenoteboththeprojection j and its Lie algebra to be P : g → g and also P : su(2)n → g . Applying P to j j j j j both sides of K =PK yields the following corollary. su(2)n =g ⊕···⊕g ⊂g. ρ ψ 1 n Corollary 1. Let ρ = |ψihψ| be a pure n-qubit density Given an n-qubit density matrix ρ, an element g ∈ matrix. Then P K =P K for 1≤j ≤n. SU(2)n acts on ρ by j ρ j ψ The next proposition establishes that we may work g·ρ=gρg†. with either K or K to calculate stabilizer dimension. ρ ψ An elementX ∈su(2)n actsonthe infinitesimallevelby Proposition2. Letρ=|ψihψ|beapuren-qubitdensity X ·ρ=[X,ρ]=Xρ−ρX. matrix. Then dimKρ =dimKψ. 3 Proof. LetX ,...,X ∈K belinearlyindependent. Let Wehave V =K if andonly if ψ is ageneralized n-qubit 1 r ψ ψ Y =PX for 1≤k ≤r and suppose that GHZ state. k k Proof. In Section V of [11] we show that if ψ is a gener- 0= a Y k k alized n-qubit GHZ state, then K = V. Now we prove ψ for some scalars a . SinceXY =X −it for some t , we the converse. k k k k k have Suppose that Kψ = V. Choose a multi-index I such that c 6= 0. If J is another multi-index different from I I akXk =i aktk. in indices K={k1,...,kp} and equal to I in some index r, let X X Since a X is in K , so is i a t , and so both of k k ψ k k these sums must be zero. Since the X are independent, k allthePak mustbezero,andhencePtheYk arealsolinearly X = (−1)ijAj − (−1)ijAr, independent. j∈K j∈K X X   We choose the basis sothatthesumofthecoefficientsoftheAj iszero,soX is inK . ViewingX asanelementofK andapplying(2), ψ ρ i 0 0 1 0 i A= , B = , C = . we have 0 −i −1 0 i 0 (cid:20) (cid:21) (cid:20) (cid:21) (cid:20) (cid:21) p forsu(2). WedenotebyAj theelementingthathasAin ζ(I,J)=2i (−1)iℓtℓ =2i (−1)2ikj =2ip6=0 thejthqubitslotandzeroselsewhere. Byslightabuseof ℓ:Xiℓ6=jℓ Xj=1 notation, we use the same symbol for the corresponding so ρ = c c∗ = 0 by Proposition 3, so c = 0. It element in su(2)n. Analogous notation applies for Bj followI,Js thatI|ψJi has the form |ψi = α|Ii +J β|Ici for and Cj, for 1≤j ≤n. some nonzero α,β. Finally, if I does not consist of all We will make repeated use of the following basic cal- zerosorallones,sayi =0,i =1,thenletX =A −A , k ℓ k ℓ culation. Let X = tkAk be an element of the local then ζ(I,Ic) = 4i 6= 0, so ρI,Ic would have to be zero, unitaryLiealgebrasu(2)n. Itisstraightforwardtocheck but this is impossible. Thus we conclude that |ψi = P that α|00···0i+β|11···1i. X ·ρ=[X,ρ]= ζ(I,J)ρ |IihJ|, (2) I,J XI,J IV. MAXIMAL STABILIZERS WITH where 1-DIMENSIONAL PROJECTIONS IN EACH QUBIT n ζ(I,J)=i tℓ[(−1)iℓ +(−1)jℓ+1]=2i (−1)iℓtℓ. In this section we consider the consequences of con- Xℓ=1 ℓ:Xiℓ6=jℓ dition (i) in the statement of Theorem 1. By virtue of As a consequence, we have the following. Propositions 1 and 2, we may work with K in place of ρ Proposition 3. Let ρ be an n-qubit density matrix, let Kψ. IfdimPjKρ =1forallj,wecanapplyanLUtrans- X = tkAk, and suppose that X ∈ Kρ. Then either formation so that PjX = t(X)A for all X ∈ Kρ, where t(X) is a scalar that depends on X. After this adjust- ζ(I,J)=0 or ρ =0 for all I,J. I,J P ment, the stabilizer K is a codimension 1 subspace of ρ We record here a proposition proved in [10] the subspaceofsu(2)n spannedby A ,A ,...,A . Thus 1 2 n (Lemma 3.8) with an alternative proof using Proposi- there is some nonzero real vector (m ,m ,...,m ) such 1 2 n tion 3. that Proposition 4. (Stabilizer criterion for an unen- n tangledqubit)Letρ=|ψihψ|beapuren-qubitdensity Kρ = tkAk: mktk =0 . matrix where |ψi = cI|Ii. If Aℓ ∈ Kρ, then the ℓth (Xk=1 X ) qubit is unentangled. In the following Proposition, we show that the only way P foranonproductstatetohavethisstabilizeristobe(LU Proof. Choose any nonzero state coefficient cI′. Apply Proposition3 to X =A , I =I′ andJ for whichj 6=i′. equivalent to) a generalized n-qubit GHZ state. ℓ ℓ ℓ Since ζ(I′,J) = ±2i 6= 0, we conclude that cJ must be Proposition 6. Let ρ = |ψihψ| be a pure nonproduct zero. n-qubit density matrix with stabilizer subalgebra Proposition 5. (Stabilizer criterion for general- n ized n-qubit GHZ states) Let |ψi = IcI|Ii be a Kρ =( tkAk: mktk =0) nonproduct n-qubit state vector for some n≥3, and let k=1 X X P where 06=(m ,m ,...,m )∈Rn. Then all the m have n 1 2 n j V = t A : t =0 . the same absolute value and |ψi is LU equivalent to a k k k ( ) generalized n-qubit GHZ state. k=1 X X 4 Proof. We may suppose, after a suitable LU trans- for some complex coefficients a,b,c satisfying a > 0, formation, that c 6= 0 (the LU transformation abc6=0 and a+b+c=0. Furthermore, this is a unique 00···0 exp(π/2C )=Id⊗···⊗Id⊗C ⊗Id⊗···⊗Idsends|Iito representative of the LU equivalence class for |ψi of this j j |I i, where Id is the 2×2 identity matrix and I denotes form. j j the multi-index obtained from I by complementing the Proof. Suppose that K =V. jth index, and changes the stabilizer element t A to ρ k k First we consider the consequences of the element Th(i−s1o)pδejkratktAiokn,mwahyercehδajnkgdeetnhoetessigtnheofKsroomneecPokferthdeelmta),. Ak in Kρ. By Proposition 3, for each I,J for which bPut does not change their absolute values. j ρPI,J =cIc∗J 6=0, we have If any mℓ = 0, then Aℓ ∈ Kρ. By Proposition 4, it ζ(I,J)= (−1)iℓ =0. follows that the ℓth qubit is unentangled. But |ψi is not aproduct, sowecanrule outthe possibility thatanym ℓ:Xiℓ6=jℓ ℓ is zero. From this it follows that if cI,cJ 6=0 and we have to flip Suppose now that there exist two coordinates of n0zerosandn1 onestotransformI intoJ,thenn0 =n1. (m ,...,m ) with different absolute values. Then If m is the number of indices which are zeros in both I 1 n (m1,...,mn) and J = (j1,...,jn) are linearly indepen- and J, then the number of zeros in I is n0+m, and the dent, whereJ is any nonzerovectorwhoseentries areall number of zerosin J is n1+m=n0+m. It follows that 0’s and 1’s. Let J be a multi-index not equal to 00···0 the numberofzerosisconstantfor allmulti-indices I for with 1’s in positions k1,...,kp. We may choose a vec- which cI 6=0. tor(t1,...,tn)thatisperpendicularto(m1,...,mn)but Next we consider Ck in Kρ. Since exp(π/2C)=C, not perpendicular to J. It follows that X = t A lies we have p k k P in K , but t 6= 0, so we conclude that c = 0 ρ ℓ=1 kℓ P J ρ=exp(π/2 Ck)·ρ= ρIc,Jc|IihJ|, by Proposition 3. This means that |ψi is the completely unentangledPstate |00···0i. But this is a contradiction, X XI,J so since |ψi is not a product. This establishes that all the mj have the same absolute value. cIc∗J =cIcc∗Jc (3) Thus |ψi has (possibly after an LU transformation) the stabilizer of a generalized n-qubit GHZ state, and for all I,J. Applying (3) to the pair I,I, we obtain therefore |ψi must be a generalized n-qubit GHZ state |cI| = |cIc|. Applying (3) to the pair I,Ic, we see that by Proposition 5. cIc∗Ic is real, so it must be that cI = cIc for all I. Since the number of zeros is constant for all multi-indices for nonzero state coefficients, it follows that this number V. THE 4-QUBIT CASE must be two. Thus we have so far that |ψi must be of the form Inthissectionweconsidercondition(ii)ofTheorem1. |ψi = a(|0011i+|1100i) It follows from Lemma 1 and Proposition 1 of [11] that + b(|1001i+|0110i) afteranLUtransformation,ifnecessary,wemaytakeK ρ + c(|1010i+|0101i) to be for some a,b,c. If a = 0, then A −A would be in K , 4 4 4 1 2 ρ violating our assumption. Similarly, neither b nor c can K = A , B , C . ρ k k k * + be zero. k=1 k=1 k=1 X X X Toseethata+b+cmustequalzero,notethat( C )· k From this assumption, we derive a canonical form for ρ = 0 means that ( Ck)·|ψi = it|ψi for some t. A P unique representatives of each LU equivalence class. straightforwardcalculation shows that P Proposition 7. (Classification of 4-qubit states C ·|ψi=i c |Ii, with su(2) stabilizer) Let ρ=|ψihψ| be a pure 4-qubit k Ik ! density matrix, where |ψi= IcI|Ii, and let (cid:16)X (cid:17) XI Xk where I denotes the multi-index obtained from I by k 4 4P 4 complementing the kth index. The coefficient of |1000i V = Ak, Bk, Ck . ontherighthandsideis(a+b+c)|1000i,butzeroonthe *k=1 k=1 k=1 + lefthandside,soa+b+c=0. Finally,wemaytakeato X X X be positiveby applyingaphase adjustment, ifnecessary. Then K =V if and only if ρ Conversely,suppose that ψ has the form |ψi = a(|0011i+|1100i) |ψi = a(|0011i+|1100i) + b(|1001i+|0110i) + b(|1001i+|0110i) + c(|1010i+|0101i) + c(|1010i+|0101i) 5 for some a,b,c, abc 6= 0 and a+b+c = 0. It is easy determined. If b = 0, then b is not completely deter- 1 2 to check that K contains V. By our analysis of sta- mined; there remains the ambiguity of conjugation for b ρ bilizer structure in [11], we have dimK ≤ 6. We know and c. To settle this, let α(r,s,t) denote the state ρ dimK =6ifandonlyif|ψiis(LUequivalentto)aprod- ρ uct of two singlet pairs ([10]), but we can rule this out r(|0011i+|1100i) (it is easy to check that |ψi is not a product). Lemma 1 + s(|1001i+|0110i) in[11]rulesoutthepossibilitythatdimK couldbe4or ρ 5, so dimK must equal 3 and we must have K =V. + t(|1010i+|0101i) ρ ρ To show that |ψi is the unique representative of its LU classwith the form statedin the Proposition,we use for state coefficients r,s,t. The (not local unitary) op- local unitary invariants. First, we consider invariants of eration that permutes qubits 3 and 4 takes α(a,b,c) to the form tr((tr ρ)2) where A is a subsystem consisting α(a,c,b). If b is pure imaginary, then c is not, so our in- A of a subset of the 4 qubits. From these trace invariants, variantP3 tells us that α(a,c,b) is not LU equivalent σ,τ,φ astraightforwardderivationproducesinvariantsI ,I ,I to α(a,c∗,b∗). It follows that α(a,b,c) is not LU equiva- 1 2 3 in the table below. lenttoα(a,b∗,c∗),foriftheLUoperationg ⊗g ⊗g ⊗g 1 2 3 4 takes α(a,b,c) to α(a,b∗,c∗), then the LU operation Qubit pair in A Invariant g ⊗g ⊗g ⊗g takes α(a,c,b) to the LU inequivalent (1,2) I (ψ)=|a||b| 1 2 4 3 1 state α(a,c∗,b∗). (1,3) I (ψ)=|a||c| 2 (1,4) I (ψ)=|b||c| Having exhausted all cases, we see that |ψi is the 3 uniquestateinitsLUequivalenceclassoftheformstated It follows that the norms of the state coefficients for in the Proposition. |ψi are LU invariant (for example, we have |a| = I (ψ)I (ψ)/I (ψ)). Since we are taking a to be pos- 1 2 3 itive, and since a + b + c = 0, a simple geometric or p algebraic argument shows that b and c are determined VI. CONCLUSION up to conjugate. To determine the imaginary parts of b and c, we use a We have described the stabilizer subalgebra structure class of invariants given by andhaveacompletedescriptionofstateswithstabilizers of maximum dimension, both for product and nonprod- Pσm,τ,φ(ψ)= cI1···cImc∗J1···c∗Jm uctstates. Wehaveshownevidencethatwarrantsfurther I1,IX2,...,Im studyofstabilizerstructure. Thereareatleastthreenat- ural directions in which to further pursue this analysis: where I1,...,Im is an m-tuple of 4-qubit multi-indices seekcompletedescriptions,perhapsaclassification,ofall Ik =(ik,...,ik), and 1 4 possiblestabilizersubalgebras;seekcompletedescription Jk =(ik,ik ,ik ,ik ) andperhapscanonicalLU formsin the spiritofProposi- 1 σ(1) τ(1) φ(1) tion 7 for states that have those stabilizers; and extend these pursuits to mixed states. for1≤k ≤m [3,12]. We considerthe invariantfor m= 1 2 3 1 2 3 1 2 3 3,σ = ,τ = , and ϕ = . 3 2 1 2 1 3 2 3 1 (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) The imaginary part of this invariant acting on ψ is VII. ACKNOWLEDGMENTS −24a2b b (b2+b2+ab ) 1 2 1 2 1 The authors thank the National Science Foundation where b ,b are the real and imaginary parts of b, re- for their support of this work through NSF Award No. 1 2 spectively. From this we see that if b 6= 0, then b is PHY-0555506. 1 2 [1] M. Nielsen and I. Chuang, Quantum Computation and [6] A. Higuchi and A. Sudbery, Phys. Lett. A 273, 213 QuantumInformation, (2000). (2000), arXiv:quant-ph/0005013v2. [2] S.Gudder, Amer.Math. Monthly 110, 181 (2003). [7] S. Brierley and A. Higuchi, J. Phys. A 40, 8455 (2007), [3] N. Linden and S. Popescu, Fortschr. Phys. 46, 567 arXiv:0704.2961v1 [quant-ph]. (1998), arXiv:quant-ph/9711016. [8] S. Walck and D.Lyons, arXiv:0707.4428v1 [quant-ph]. [4] N.Linden,S.Popescu,andA.Sudbery,Phys.Rev.Lett. 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