Two Local Observables are Sufficient to Characterize Maximally Entangled States of N Qubits
aa r X i v : . [ qu a n t - ph ] N ov Two Local Observables are Sufficient to Characterize Maximally Entangled States of N Qubits
Fengli Yan , ∗ Ting Gao , † and Eric Chitambar ‡ College of Physics Science and Information Engineering and Hebei Advanced Thin Films Laboratory,Hebei Normal University, Shijiazhuang 050016, China College of Mathematics and Information Science,Hebei Normal University, Shijiazhuang 050016, China Center for Quantum Information and Quantum Control (CQIQC),Dept. of Electrical and Computer Engineering and Dept. of Physics,University of Toronto, Toronto, Ontario, M5S 3G4, Canada (Dated: December 2, 2018)Maximally entangled states (MES) represent a valuable resource in quantum information pro-cessing. In N -qubit systems the MES are N -GHZ states, i.e. the collection of | GHZ N i = √ ( | · · · i + | · · · i ) and its local unitary (LU) equivalences. While it is well-known that suchstates are uniquely stabilized by N commuting observables, in this Letter we consider the mini-mum number of non-commuting observables needed to characterize an N -qubit MES as the uniquecommon eigenstate. Here, we prove that in this general case, any N -GHZ state can be uniquelystabilized by only two observables. Thus, for the task of MES certification, only two correlatedmeasurements are required with each party observing the spin of his/her system along one of twodirections. PACS numbers: 03.65.Ud,03.65.Ta, 03.67.Dd
From both a theoretical and practical perspective,maximally entangled states (MES) play an importantrole in quantum information science. While there maybe different ways to consider some state more entangledthan another, one can work from an axiomatic perspec-tive to define “maximally” entangled states in the mul-tipartite setting. This is the approach taken by Gisinand Bechmann-Pasquinucci who identify N -GHZ statesas maximally entangled in N -qubit systems [1]. Theirjustification primarily comes from observing these statesto maximally violate the Bell-Klyshko inequalities, a gen-eralization of the Bell inequalities to more than two par-ties. Chen advanced the work of [1] by proving N -GHZstates to be the unique family of states which demon-strate such a maximal violation [2]. Hence, it becomesappropriate to regard N -GHZ states as the maximallyentangled multiqubit states.At the same time, MES have been recognized as key in-gredients in quantum information processing (QIP). Thepioneering bipartite tasks of quantum key distribution(QKD) [3, 4], teleportation [5], superdense coding [6], andquantum direct communication [7] all utilize the EPRstate | Ψ + i = √ ( | i + | i ) to achieve their powerfulnon-classical effects. Multipartite generalizations of theseprocedures have been developed [8–11], as well as novelschemes such as quantum secret sharing [8, 12], whichlike their bipartite ancestors, involve manipulations andmeasurements on MES. The general attraction of MESfor information processing is dual since they not only al-low for complete correlation between measurements onsubsystems, but their purity also ensures these correla-tions to exist exclusively within the system, i.e. no ex-ternal eavesdropper can be correlated with any of the subsystems.Since the use of MES is critical to the success of theaforementioned QIP schemes, it is important for the par-ties to verify that they indeed are encoding their informa-tion in MES and not other types of states. One methodof doing this is to prepare sufficiently more MES thanneeded for the given QIP task. From this larger popu-lation, a random subset of states is checked to be MES,and if this inspection passes, the remaining states arecertified to also be MES with arbitrarily high probabil-ity. The task of verifying channel security then reducesto whether N parties can determine if some collectionof mutually shared states are all MES. In the bipartitecase, Ekert first proposed using Bell inequalities to as-certain whether two parties hold EPR states [4]. WhileBell inequalities involve the expectation values of fourdifferent observables, Bennett et al. later observed thatonly two local observables were necessary to detect thepossession of EPR pairs [13]. Specifically, the state | Ψ + i is the unique +1 eigenstate of the local spin measure-ments σ X ⊗ σ X and σ Z ⊗ σ Z , where σ X and σ Z are Paulimatrices. Consequently, repeating these measurementson some sample of states can detect the presence of apotential eavesdropper and ensure the protocol’s overallsafety.Using stabilizer formalism, this idea can be generalizedto check the safety of multipartite MES. The set of com-muting product Pauli operators having | GHZ N i as the unique common +1 eigenstate forms an Abelian group.Letting { P i } ki =1 denote a minimal set of generators forthis group and I the identity, the projector onto theircommon +1 eigenspace is given by k Π ki =1 ( I + P i ). Thedimension of this space is given by tr (cid:0) k Π ki =1 ( I + P i ) (cid:1) = N k , which means that at least N commuting local spinmeasurements are needed to determine whether the par-ties share | GHZ N i . In fact, the observables σ ⊗ NX , σ Z ⊗ σ Z ⊗ I ⊗ ( N − , σ Z ⊗ I ⊗ σ Z ⊗ I ⊗ ( N − , · · · , σ Z ⊗ I ⊗ ( N − ⊗ σ Z suffice. Nevertheless, a natural question is whetherfewer than N measurements are sufficient to certify thepossession of | GHZ N i if we do not require the measure-ments to commute. In this Letter, we find that remark-ably for any N , only two different observables are needed.More precisely, let unit vectors ~v l and ~w l describetwo arbitrary directions in which party l measures the“spin” of his/her system via observables A l := ~v l · ~σ and B l := ~w l · ~σ respectively. We consider the common +1eigenspace of operators A := N Nl =1 A l and B =: N Nl =1 B l .It is found that for any N -GHZ state | ψ i , there exists vec-tors ~v l and ~w l such that | ψ i is the unique +1 eigenstateof the two operators just given. We also investigate theconverse: for any two observables of the form N Nl =1 A l and N Nl =1 B l , under what conditions do they posses aone-dimensional eigenspace. Note that since each B l haseigenvalues of ±
1, local unitary operators can be applied,and without loss of generality we can assume B = σ ⊗ NZ .With a perhaps slight abuse of language, we say that astate is stabilized by some operator if it is a +1 eigenstate.Our results are summarized as follows: Theorem 1.
For observables A = N Nl =1 A l , B = σ ⊗ NZ with A l = (sin θ l cos φ l , sin θ l sin φ l , cos θ l ) · ~σ ,(i) if there exists no bit string ~m with m l ∈ { , } suchthat sin[( N Σ l =1 ( − m l θ l ) /
2] = 0 , then A and B haveno common eigenstates,(ii) if there exists exactly one bit string ~m such that m = 0 and sin[( N Σ l =1 ( − m l θ l ) /
2] = 0 , then some N -GHZ state is the unique common +1 eigenstateof A and B ; moreover, to every N -GHZ state | ψ i there exists θ l and φ l such that | ψ i is uniquely sta-bilized by A and B , and(iii) if there exists more than one one bit string ~m suchthat m = 0 and sin[( N Σ l =1 ( − m l θ l ) /
2] = 0 , then thecommon +1 eigenstates of A and B are given bythe solution to Eq. (4) . In statements (ii) and (iii), the condition m = 0 isadded just to avoid trivial redundancies. Since − sin x =sin( − x ), a string ~m will solve sin[( N Σ l =1 ( − m l θ l ) /
2] = 0 iffits complement string having components 1 − m l is alsoa solution. Proof.
Let Π A ∩ B be the projector onto the common +1eigenspace of A and B , and choose | Ψ i SE to be any pu-rification of it in some larger Hilbert space. Here, S refers to the N -qubit system, and E refers to the envi-ronment or perhaps an eavesdropper. Thus, we assume( A ⊗ I E ) | Ψ i SE = ( B ⊗ I E ) | Ψ i SE = | Ψ i SE . We seekthe conditions for Π A ∩ B being a one-dimensional projec-tor, which is equivalent to | Ψ i SE being a product state: | ψ i S | e i E . In this case, | Ψ i SE is perfectly secure fromleaking any information to an eavesdropper.We begin by defining two sets S = { ~j ∈ Z N : ⊕ Nl =1 j l =0 } and S = { ~j ∈ Z N : ⊕ Nl =1 j l = 1 } . The bitwise innerproduct between two N -bit strings will be denoted by ~j · ~k = P Nl =1 j l k l .Any state stabilized by σ ⊗ NZ ⊗ I E is of the form | Ψ i = X ~i ∈S | ~i i| e ~i i , (1)where | e ~i i are states of the environment and | ~i i = N Nl =1 | i l i l with i l ∈ { , } . Since A l = (cid:18) cos θ l e − i φ l sin θ l e i φ l sin θ l − cos θ l (cid:19) , the action A l | i l i canbe conveniently expressed as (cos θ l ) i l ( e − i φ l sin θ l ) i l | i l +( e i φ l sin θ l ) i l ( − cos θ l ) i l | i l where i l = 1 − i l . Then theequality ( N Nl =1 A l ⊗ I E ) | Ψ i = | Ψ i becomes | Ψ i = X ~i ∈S N O l =1 [(cos θ l ) i l ( e − i φ l sin θ l ) i l | i l + ( e i φ l sin θ l ) i l ( − cos θ l ) i l | i l ] | e ~i i = X ~i ∈S | ~i i| e ~i i . (2)Contracting by h ~j | gives | e ~j i = X ~i ∈S N Y l =1 [(cos θ l ) i l ( e − i φ l sin θ l ) i l ] j l · [( e i φ l sin θ l ) i l ( − cos θ l ) i l ] j l | e ~i i . (3)Here we allow for ~j to be any string with obviously | e ~j i =0 for ~j ∈ S . The system’s state will be unentangledfrom the environment iff there exists complex scalars c ~i and some state | e i such that | e ~i i = c ~i | e i for all ~i ∈S . Substituting this into the previous equation givesthe system of 2 N − linear equations c ~j = X ~i ∈S N Y l =1 [(cos θ l ) i l ( e − i φ l sin θ l ) i l ] j l · [( e i φ l sin θ l ) i l ( − cos θ l ) i l ] j l c ~i , ∀ ~j ∈ S . (4)Thus, there exists a unique solution to this iff the state | ψ i = P ~i ∈S c ~i | ~i i is uniquely stabilized by both A and B .On the other hand, if there are multiple solutions, thendim(Π A ∩ B ) >
1, and if there is no solution, then Π A ∩ B = ∅ . At this point, we have essentially answered the ques-tion of whether two given observables uniquely stabilizea state since (4) can be efficiently solved. However, byfurther analysis, we can better understand its solutionset and obtain the converse result of part (ii) in Theorem1. Taking | e ′ ~i i = Q Nl =1 ( − i l / e − i φ l i l | e ~i i , e i β ( ~j ) = Q Nl =1 ( − j l / e i φ l j l and using the identity i l ⊕ j l = i l + j l − i l j l , Eq. (3) simplifies to | e ~j i = e i β ( ~j ) X ~i ∈S N Y l =1 cos θ l [( − − / tan θ l ] i l ⊕ j l | e ′ ~i i (5)where we take the convention 0 = 1. Now for ~j ∈ S ,the LHS becomes zero and we are left with the system of2 N − vector equations X ~i ∈S N Y l =1 cos θ l [( − − / tan θ l ] i l ⊕ j l | e ′ ~i i = 0 . (6)We can encode all this information in the followingway. For any ~m ∈ Z N , define the function f ~m : S →{− , +1 } by f ~m ( ~v ) := ( − ~m · ~v . Observe that if ~m = ~n ,then f ~m = f ~n . Indeed, there must exist some compo-nent k such that one and only one m k or n k is zero, andhence ( − ~m · ~e k = ( − ~n · ~e k with ~e k being the k th unitvector. At the same time, for every ~m its complement ~n is the only vector such that f ~m = − f ~n ( n l = 1 − m l ).Thus while there are 2 N different bit vectors ~m , there are2 N − linearly independent functions ( − ~m · ~v generatedby the ~m ∈ Z N . Consequently, all possible 2 N − linearlyindependent combinations formed by adding or subtract-ing the equations in (6) are contained in the equations X ~i ∈S ~j ∈S N Y l =1 ( − m l j l cos θ l [( − − / tan θ l ] i l ⊕ j l | e ′ ~i i = 0 (7)for any choice of m l ∈ { , } .We next use the facts that ( − m l j l =( − m l i l ( − m l ( i l ⊕ j l ) , and that for a fixed ~i , { i l ⊕ j l : j l ∈ S } = S since L Nl =1 i l = 0. Hence, X ~j ∈S N Y l =1 cos (( − m l θ l ) [( − − / tan (( − m l θ l )] j l · X ~i ∈S ( − ~m · ~i | e ′ ~i i = 0 . (8)By mathematical induction, it is not difficult to prove that X ~j ∈S N Y l =1 cos θ l [( − − / tan θ l ] j l = ( − − / sin N Σ l =1 θ l , X ~j ∈S N Y l =1 cos θ l [( − − / tan θ l ] j l = cos N Σ l =1 θ l . (9)Then from Eq. (8) we havesin( N Σ l =1 ( − m l θ l ) X ~i ∈S ( − ~m · ~i | e ′ ~i i = 0 (10)for every ~m ∈ Z N . Let M denote the set of all binaryvectors ~m ∈ M such that sin( N Σ l =1 ( − m l θ l ) = 0. Con-sequently, P ~i ∈S ( − ~m · ~i | e ′ ~i i = 0 for every ~m ∈ M c := Z N \ M . Multiplying both sides by ( − ~m · ~h with ~h ∈ S and summing over M c gives0 = X ~m ∈M c X ~i ∈S ( − ~m · ( ~h + ~i ) | e ′ ~i i = X ~i ∈S [ X ~m ∈ Z N ( − ~m · ( ~h + ~i ) | e ′ ~i i − X ~m ∈M ( − ~m · ( ~h + ~i ) | e ′ ~i i ]= 2 N ( | e ′ ~h i − X ~m ∈M ( − ~m · ~h | E ( ~m ) i ) (11)where | E ( ~m ) i = 2 − ( N − P ~i ∈S ( − ~m · ~i | e ′ ~i i . Here, inpassing from the second to the third equation, we haveused the general fact that P ~m ∈ Z N ( − ~m · ~v = 0 unless v l is even for all l , in which case it equals 2 N . From (11)we immediately see that M 6 = ∅ or else | e ′ ~h i = 0 for all ~h ∈ S . This proves statement (i) of the Theorem.If M contains only one string ~m and its complement,then | e ′ ~h i = ( − ~m · ~h | E ( ~m ) i for all ~h ∈ S . Substitutingthis back into (5) yields, | e ~j i = e i β ( ~j ) | E ( ~m ) i X ~i ∈S N Y l =1 ( − m l i l cos θ l [( − − / tan θ l ] i l ⊕ j l (12)with ~j ∈ S . We simplify this expression analogous toEq. (8) to obtain | e ~j i = e i β ( ~j ) ( − ~m · ~j | E ( ~m ) i X ~i ∈S N Y l =1 cos (( − m l θ l ) [( − − / tan (( − m l θ l )] i l . (13)The second identity in (9) then gives | e ~j i = e i β ( ~j ) ( − ~m · ~j cos( N Σ l =1 ( − m l θ l ) | E ( ~m ) i . (14)At the same time, since ~j ∈ S , we must have | e ′ ~j i = e − i β ( ~j ) | e ~j i = ( − ~m · ~j | E ( ~m ) i which gives the additionalcondition that cos( N Σ l =1 ( − m l θ l ) = 1. Then using (14),we return to | Ψ i SE by | Ψ i SE = X ~j ∈S e i β ( ~j ) ( − ~m · ~j | ~j i| E ( ~m ) i = X ~j ∈S N O l =1 | ˜ j l i l | E ( ~m ) i (15)where | ˜0 i = | i , | ˜1 i = ( − (1 / m l ) e i φ l | i . From parityconsiderations, it is straightforward to see that under thelocal rotation of | i → ( | i + | i ) and | i → ( | i − | i ) byeach party, the state | GHZ N i transforms as | · · · i + | · · · i → ( | i + | i ) ⊗ N + ( | i − | i ) ⊗ N = X ~j ∈S N O l =1 | j l i l . (16)This proves | Ψ i SE to be of the form | ψ i S | e i E where | ψ i S is an N -GHZ state; the two necessary and sufficient con-ditions are sin( N Σ l =1 ( − m l θ l ) = 0 and cos( N Σ l =1 ( − m l θ l ) =1 which we can combine into the single equationsin[( N Σ l =1 ( − m l θ l ) /
2] = 0.Furthermore, starting from (15), we can reverse theconstruction. For instance, if N is odd, then choosing θ l = 2 π/N for all l generates a unique solution (up to itscomplement) of ~m = ~ N Σ l =1 ( − m l θ l ) /
2] = 0. Asa result, the state | ψ i = P ~j ∈S N Nl =1 | ˜ j l i l is uniquely stabi-lized by A and B for any choice of φ l . Any other N -GHZstate can be written as N Ni =1 U i | ψ i for local unitaries U i ,and we compute this state to be uniquely stabilized by N Ni =1 U i A i U † i and N Ni =1 U i B i U † i . For even N , the proce-dure is identical with θ = πN +1 and θ l = πN +1 for l > M contains more than one ~m suchthat m = 0 and sin[( N Σ l =1 ( − m l θ l ) /
2] = 0. Then | e ′ ~h i = P ~m ∈M ( − ~m · ~h | E ( ~m ) i , and either (a) | e ′ ~h i 6∝| e ′ ~j i for some ~h,~j ∈ S , or (b) | e ′ ~h i = c ~h | e i for all ~h ∈ S and c ~h some complex scalar. In case (a),from Eq. (1) we see that the system and the envi-ronment are entangled which means dim(Π A ∩ B ) > A and B will have a unique +1 eigen-state iff Eq. (4) has a solution. In both cases,the global state is | Ψ i SE = P ~m ∈M m | GHZ N ( ~m ) i| E ( ~m ) i where each | GHZ N ( ~m ) i is an N -GHZ state of the form √ ( | k k . . . k N i + | ¯ k ¯ k . . . ¯ k N i ) k l ∈ { ˜0 , ˜1 } , with {| ˜0 i l , | ˜1 i l } a local basis of party l fixed for all ~m . Thiscompletes the proof of part (iii).In conclusion, we have considered the minimum num-ber of spin-direction measurements required to certify thepossession of maximally entangled states in N -qubit sys-tems. Our results are especially important to QKD wherea central task is verifying the purity of a quantum chan-nel and the absence of a possible eavesdropper. Specif-ically we have shown that for every N -qubit maximallyentangled state, only two different local measurementsare needed to accomplish this certification. Note thatin our analysis, we have mainly focused on the mutual+1 eigenspaces of A and B . However by considering allcombinations of ± A and ± B , we can learn whether thetwo observables share any unique eigenstate. A naturalquestion for future research is the minimum number ofmeasurements needed to test for MES in higher dimen-sional N -party systems.We graciously thank Hoi-Kwong Lo for providing sup-port and valuable discussions during the course of thisresearch. Additionally, this work was supported by theNational Natural Science Foundation of China underGrant No: 10971247, Hebei Natural Science Foundationof China under Grant Nos: F2009000311, A2010000344.E.C. was supported by the funding agencies CIFAR,CRC, NSERC, and QuantumWorks. ∗ Electronic address: fl[email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected][1] N. Gisin and H. Bechmann-Pasquinucci, Phys. Lett. A , 1 (1998).[2] Z. Q. Chen, Phys. Rev. Lett. , 110403 (2004).[3] C. H. Bennett and G. Brassard, in Proceedings of theIEEE International Conference on Computers, Systems,and Signal Processing, Bangalore (IEEE, New York,1984), pp.175-179.[4] A. K. Ekert, Phys. Rev. Lett. , 661 (1991).[5] C. H. Bennett, G. Brassard, C. Crpeau, R. Jozsa, A.Peres, and W. K. Wootters, Phys. Rev. Lett. , 1895(1993).[6] C. H. Bennett and S. J. Wiesner, Phys. Rev. Lett. ,2881 (1992).[7] K. Bostr¨om and T. Felbinger, Phys. Rev. Lett. ,187902 (2002).[8] K. Chen and H. K. Lo, in Information Theory, 2005. ISIT2005. Proceedings. International Symposium on (2005),pp. 1607-1611.[9] A. Karlsson and M. Bourennane, Phys. Rev. A , 4394(1998).[10] P. Agrawal and A. Pati, Phys. Rev. A , 062320 (2006).[11] T. Gao, F. L. Yan, and Z. X. Wang, J. Phys. A , 5761(2005).[12] M. Hillery, V. Bu˘zek, and A. Berthiaume, Phys. Rev. A , 1829 (1999). [13] C. H. Bennett, G. Brassard, and N. D. Mermin, Phys.Rev. Lett.68