A simple approach to self-testing multipartite entangled states ˇ Ivan Supi´ c,1 Andrea Coladangelo,2 Remigiusz Augusiak,3 and Antonio Ac´ın1, 4
arXiv:1707.06534v1 [quant-ph] 20 Jul 2017
1 ICFO-Institut
de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain∗ 2 Department of Computing and Mathematical Sciences, California Institute of Technology, 1200 E California Blvd, Pasadena, CA 91125, United States 3 Center for Theoretical Physics, Polish Academy of Sciences, Aleja Lotnik´ow 32/46, 02-668 Warsaw, Poland† 4 ICREA - Instituci´ o Catalana de Recerca i Estudis Avancats, 08011 Barcelona, Spain (Dated: July 21, 2017) A timely enterprise nowadays is understanding which states can be deviceindependently self-tested and how. This question has been answered recently in the bipartite case [25], while it is largely unexplored in the multipartite case, with only a few scattered results, using a variety of different methods: maximal violation of a Bell inequality, numerical SWAP method, stabilizer self-testing etc. In this work, we investigate a simple, and potentially unifying, approach: combining projections onto two-qubit spaces (projecting parties or degrees of freedom) and then using maximal violation of the tilted CHSH inequalities. This allows to obtain self-testing of Dicke states and partially entangled GHZ states with two measurements per party, and also to recover self-testing of graph states (previously known only through stabilizer methods). Finally, we give the first self-test of a class multipartite qudit states: we generalize the self-testing of partially entangled GHZ states by adapting techniques from [25] , and show that all multipartite states which admit a Schmidt decomposition can be self-tested with few measurements.
I.
INTRODUCTION
The rapid development of quantum technologies in recent years creates an urgent need for certification tools. Quantum computing and quantum simulation are state of the art tasks which require verifiable realizations. One way to certify the correct functioning of a quantum computer would be to ask it to solve a problem that is thought to be hard for a classical computer, like factoring large numbers and simply checking the correctness of the solution. However, it is conjectured that the class of problems that can be solved efficiently on a quantum computer (BQP) has elements outside the class of problems whose solution can be checked classically (NP) [1], which makes this type of verification incomplete. Thus, efforts have been made towards building reliable certification protocols for quantum systems performing universal quantum computing or quantum simulations [2–4]. A canonical way to approach this problem is to exploit tomographic protocols [5]. Unfortunately, quantum devices performing tasks such as quantum computation typically involve multipartite quantum states and the complexity of tomographic techniques scales exponentially with the number of particles involved. Moreover, they demand a set of trusted measurements, which in certain scenarios is not an available resource. An alternative technique able to positively address these problems is self-testing [6]. Contrary to quantum state and process tomography, self-testing is a completely device-independent task. ∗
†
[email protected] [email protected]
2 It aims to verify that a given quantum device operates on a certain quantum state, and performs certain measurements on it, solely from the correlations it generates. The building block for this, as well as for all other device-independent protocols is Bell’s theorem [7], which says that correlations violating Bell inequalities do not admit local hidden-variable models. Thus, correlations useful for self-testing must be non-local. Self-testing was formally introduced by Mayers and Yao [6]. Since then, there has been growing interest in designing self-testing methods [9, 13, 16, 18, 19], and studying their robustness [13–15]. An important recent development shows that all pure entangled bipartite states can be self-tested [25]. It is in fact the case that most of the currently known self-testing protocols are tailored to bipartite states, leaving the multipartite scenario rather unexplored. The known examples cover only the tripartite W state, a class of partially entangled tripartite states [10, 11] and graph states [12]. The aim of this paper is to extend the class of multipartite states that can be self-tested, by investigating a simple approach that exploits the well-understood self-testing of two-qubit states. At a high level, this is done by combining projections to two-qubit spaces and then exploiting maximal violation of tilted CHSH inequalities. Using this potentially unifying approach, we show self-testing of all Dicke states and partially entangled GHZ states with only two measurements per party. We also show that our method efficiently applies also to self-testing of graph states, previously known only through stabilizer state methods, with a slight improvement in the number of measurement settings per party. Finally, using techniques from [25] as a building block, we provide the first self-testing result for a class of multipartite qudit states, by showing that all multipartite qudit states which possess a Schmidt decomposition can be self-tested, with at most four measurements per party. II.
PRELIMINARIES
Self-testing is a device-independent task [17] whose aim is to characterize the form of the quantum state and measurements solely from the correlations that they generate. To introduce it formally, consider N non-communicating parties sharing some N-partite state |ψi. On its share of this state, party i can perform one of several projective measurements { Mxaii ,i } ai , labelled by xi ∈ Xi , with possible outcomes ai ∈ Ai . Here Xi and Ai stand for finite alphabets of possible questions and answers for party i. The experiment is characterised by a collection of conditional probabilities { p( a1 , . . . , a N | x1 , . . . , x N ) : ai ∈ Ai } xi ∈Xi , where p( a1 , . . . , a N | x1 , . . . , x N ) = hψ| Mxa11 ,1 ⊗ . . . ⊗ Mxa NN ,N |ψi
(1)
is the probability of obtaining outputs a1 , . . . , a N upon performing the measurements x1 , . . . , x N 1 . We refer to this as a correlation. It is sometimes convenient to describe correlations with the aid of standard correlators, where instead of measurement operators Mxaii one uses Hermitian observables with eigenvalues ±1. Now, we can formally define self-testing in the following way. Definition 1 (Self-testing). We say that a correlation { p( a1 , . . . , a N | x1 , . . . , x N ) : ai ∈ Ai } xi ∈Xi self˜ ai } a , i = 1, . . . , N, if for any state and measurements |ψi and tests the state |Ψi and measurements { M xi ,i i { Mxaii ,i } ai , i = 1, . . . , N, reproducing the correlation, there exists a local isometry Φ = Φ1 ⊗ . . . ⊗ Φ N such that ˜ a1 ⊗ . . . ⊗ M ˜ a N |ψ0 i). Φ( Mxa11 ,1 ⊗ . . . ⊗ Mxa NN ,N |ψi) = |junki ⊗ ( M x N ,N x1 ,1 1
(2)
We take the parties’ measurements to be projective, invoking Naimark’s dilation theorem. We take the joint state to be pure for ease of exposition, but we emphasize that all of our proofs hold analogously starting from a joint mixed state.
3 where |junki is some auxiliary state representing unimportant degrees of freedom. In some cases the existence of an isometry obeying (2) can be proven solely from the maximal violation of some Bell inequality. For instance, all two-qubit pure entangled states can be selftested with a one-parameter class of tilted CHSH Bell inequalities [9] given by αh A0 i + h A0 B0 i + h A0 B1 i + h A1 B0 i − h A1 B1 i ≤ 2 + α,
(3)
where α ≥ 0 and Ai and Bi are observables with outcomes ±1 measured by the parties. Note that for α = 0, Eq. (3) reproduces the well-known CHSH Bell inequality [8]. For further purposes let us briefly recall this result. Here σz and σx are the standard Pauli matrices. Lemma 1 ([9]). Suppose a bipartite state |ψi and dichotomic observables Ai and Bi achieve the maximal √ 2 quantum violation p of the tilted CHSH inequality (3) 8 + 2α , for some α. Let θ, µ ∈ (0, π/2) be such that sin 2θ = (4 − α2 )/(4 + α2 ) and µ = arctan sin 2θ. Let Z A = A0 , X A = A1 . Let ZB∗ and XB∗ be respectively ( B0 + B1 )/2 cos µ and ( B0 − B1 )/2 sin µ, but with all zero eigenvalues replaced by one, and define ZB = ZB∗ | ZB∗ |−1 and XB = XB∗ | XB∗ |−1 . Then, we have Z A | ψ i = ZB | ψ i,
(4)
cos θX A (1 − Z A )|ψi = sin θXB (1 + Z A )|ψi.
(5)
Moreover, there exists a local isometry Φ such that Φ( Ai ⊗ Bj |ψi) = |junki ⊗ ( A˜ i ⊗ B˜ j )|ψθ i, where |ψθ i = cos θ |00i + sin θ |11i, and A˜ 0 = σz , A˜ 1 = σx , and B˜ 0/1 = cos µσz ± sin µσx . A typical construction of the isometry Φ is the one encoding the SWAP gate, as illustrated in Fig. 1. |0i
H
|ψi
|0i
H
H ZA
XA
ZB
XB
|junki |ψθ i
H
Figure 1. Example of a circuit that takes as input a state |ψi satisfying (4)-(5), adds two ancillas, each in |0i, and outputs the state |ψθ i in tensor product with an auxiliary state |junki. Here H is the usual Hadamard gate.
Our aim in this paper is to exploit the above result to develop methods for self-testing multipartite entangled quantum states. Given an N-partite entangled state |ψi, the idea is that N − 2 chosen parties perform local measurements on their shares of |ψi and the remaining two parties check whether the projected state they share violates maximally (3) for the appropriate α (we can think of this as a sub-test). This procedure is repeated for various subsets of N − 2 parties until the correlations imposed are sufficient to characterize the state |ψi. Our approach is inspired by √ Ref. [10], which shows that any state in the class (|100i + |101i + α|001i)/ 2 + α2 , containing the three-qubit W state, can be self-tested in this way. We will show that this approach can be generalized in order to self-test new (and old) classes of multipartite states. The main challenge is to show that all the sub-tests of different pairs of parties are compatible. To be more precise,
4 for a generic state there will always be a party which will be involved in several different subtests and, in principle, will be required to use different measurements to pass the different tests. Consequently, isometries (Fig. 1) corresponding to different sub-tests are in principle constructed from different observables. However, a single isometry is required in order to self-test the global state. Overcoming the problem of building a single isometry from several different ones is the key step to achieve a valid self-test for multipartite states. For states that exhibit certain symmetries, this can be done efficiently with few measurements. We leave for future work the exploration for states that do not have any particular symmetry. In the N-partite scenario, parties will be denoted by numbers from 1 to N and measurement observables by capital letters with a superscript denoting the party. For a two-outcome observable W, we denote by W (±) = (I ± W )/2 the projectors onto the ±1 eigenspaces. We use the notation b ac to denote the biggest integer n such that n ≤ a, while d ae is the smallest n such that n ≥ a.
III.
OUR RESULTS
In this work, we expand the class of self-testable multipartite states. More precisely, in subsection III A we show that all multipartite partially entangled GHZ (qubit) states can be self-tested with two measurements per party. Then, we make use of this result as a building block to extend self-testing to all multipartite entangled Schmidt-decomposable qudit states, of any local dimension d, with only three measurements per party (except one party has four). To the best of our knowledge, this is the first self-test for multipartite states of qudits, for d > 2. Finally, in subsections III B and III C we apply the approach used for multipartite partially entangled GHZ (qubit) states to the self-test the classes of Dicke states and graph states (previously known to be self-testable through stabilizer methods [12]).
A.
All multipartite entangled qudit Schmidt states
While in the bipartite setting all states admit a Schmidt decomposition, in the general multipartite setting this is not the case. We refer to those multipartite states that admit a Schmidt decomposition as Schmidt states. These, up to a local unitary, can be written in the form
|Ψi =
d −1
∑ c j | ji⊗ N
(6)
j =0
where 0 < c j < 1 for all i and ∑dj=−01 c2j = 1. Our proof that all multipartite entangled Schmidt states can be self-tested follows closely the ideas from [25], while leveraging as a building block our novel self-testing result for partially entangled GHZ states. Thus, we proceed by first proving a self-testing theorem for multipartite partially entangled qubit GHZ states. Multipartite partially entangled GHZ states. Multipartite qubit Schmidt states, also known as partially entangled GHZ states, are of the form
|GHZ N (θ )i = cos θ |0i⊗ N + sin θ |1i⊗ N
(7)
5 where θ ∈ (0, π/4] and |GHZ N (π/4)i = |GHZ N i is the standard N-qubit GHZ state. The form of this state is such that if any subset of N − 2 parties performs a σX measurement, the collapsed state shared by the remaining two parties is cos θ |00i ± sin θ |11i, depending on the parity of the measurement outcomes. As already mentioned, these states can be self-tested with the aid of inequality (3), which is the main ingredient of our self-test of |GHZ N (θ )i. Theorem 1. Let |ψi be an N-partite state, and let A0,i , A1,i be a pair of binary observables for the i-th party, for i = 1, . . . , N. Suppose the following correlations are satisfied: (+)
(+)
(+)
hψ| A0,i |ψi = hψ| A0,i A0,j |ψi = cos2 θ, hψ| hψ|
N −2
( ai )
i =1 N −2
( ai )
1 , 2 N −2
∀i, j ∈ {1, . . . , N − 1}
∏ A1,i
|ψi =
∏ A1,i
(αA0,N −1 + A0,N −1 A0,N + A0,N −1 A1,N + (−1)h(a) A1,N −1 A0,N
i =1
− (−1)
h( a)
(8)
∀ a ∈ {+, −} N −2
A1,N −2 A1,N −1 )|ψi =
√
(9)
8 + 2α2 , 2 N −2
∀ a ∈ {+, −} N −2
(10) (11)
p where h( a) denotes the parity of the number of “−” in a, and α = 2 cos 2θ/ 1 + sin2 2θ. Let µ be 0 = such that tan µ = sin 2θ. Define Zi = A0,i and Xi = A1,i , for i = 1, . . . , N − 1. Then, let ZN ∗ 0 ∗ ∗ ( A0,N + A1,N )/2 cos µ, and let ZN be ZN with zero eigenvalues replaced by 1. Define ZN = ZN | ZN |−1 . 0 = (A Define X N similarly starting from X N 0,N − A1,N ) /2 sin µ. Then, Z1 |ψi = · · · = ZN |ψi,
X1 · · · XD ( I − Z1 )|ψi = tan θ ( I + Z1 )|ψi.
(12) (13)
Proof: We refer the reader to Appendix A for the formal proof of this Theorem, while providing here an intuitive understanding of the correlations given above. The first equation (8) defines the existence of one measurement observable, whose marginal carries the information of angle θ. The straightforward consequence of it is Eq. (12), which is analogue to Eq. (4). On the other hand, eq. (9) involves a different measurement observable with zero marginal, while eq. (10) shows that when the first N − 2 parties perform this zero marginal measurement the remaining two parties maximally violate the corresponding tilted CHSH inequality, i.e. the reduced state is self-tested to be the partially entangled pair of qubits. Eq. (13) is analogue to Eq. (5). As a corollary, these correlations self-test the state |GHZ N (θ )i. Corollary 1. Let |ψi be an N-partite state, and let A0,i , A1,i be a pair of binary observables for the ith party, for i = 1, . . . , N. Suppose they satisfy the correlations of Theorem 1. Then, there exists a local isometry Φ such that Φ(|ψi) = |junki|GHZ N (θ )i
(14) (k)
Proof: This follows as a special case (d = 2) of Lemma 2 stated below, upon defining Pi = [ I + (−1)k Zi ]/2, for k ∈ {0, 1}. As one can expect, the ideal measurements achieving these correlations are: A0,i = σz , A1,i = σx , for i = 1, . . . , N − 1, and A0,N = cos θσz + sin θσx , A1,N = cos θσz − sin θσx . We
6 refer to the correlations achieved by these ideal measurements as the ideal correlations for multipartite entangled GHZ states. All multipartite entangled qudit Schmidt states. The generalisation of Theorem 1 to all multipartite qudit Schmidt states is then an adaptation of the proof in [25] for the bipartite case, with the difference that it uses as a building block the |GHZ N (θ )i self-test that we just developed, instead of the tilted CHSH inequality. We begin by stating a straightforward generalisation to the multipartite setting of the criterion from [20] which gives sufficient conditions for self-testing a Schmidt state. Then, our proof that all multipartite entangled qudit Schmidt states can be self-tested goes through showing the existence of operators satisfying the conditions of such criterion. Lemma 2 (Generalisation of criterion from [20]). Let |Ψi be a state of the form (6). Suppose there ( k ) −1 exist sets of unitaries { Xl }dk= 0 , where the subscript l ∈ {1, . . . , N } indicates that the operator acts (k)
−1 on the system of the l-th party, and sets of projections { Pl }dk= 0 , that are complete and orthogonal for l = 1, . . . , N − 1 and need not be such for l = N, and they satisfy: (k)
(k)
P1 |ψi = . . . = PN |ψi, c (0) (k) (k) (k) X1 . . . X N P1 |ψi = k P1 |ψi c0
(15) (16)
for all k = 1, . . . , N. Then, there exists a local isometry Φ such that Φ(|ψi) = |junki ⊗ |Ψi. Proof. The proof of Lemma 2 is a straightforward generalisation of the proof of the criterion from [20], and is included in the Appendix for completeness. We now describe the self-testing correlations for |Ψi = ∑dj=−01 c j | ji⊗n . Their structure is inspired by the self-testing correlations from [25] for the bipartite case, and they consist of three d-outcome measurements for all but the last party, which has four. We desribe them by first presenting the ideal measurements that achieve them, as we believe this aids understading. Subsequently, we extract their essential properties that guarantee self-testing. For a single-qubit observable A, denote by [ A]m the observable defined with respect to the basis {|2m mod di, |(2m + 1) mod di}. For example, [σZ ]m = |2mih2m| − |2m + 1ih2m + 1|. Similarly, we denote by [ A]0m the observable defined with respect to the basis {|(2m + 1) mod di, |(2m + 2) mod di}. We use the notation L Ai to denote the direct sum of observables Ai . Let Xi denote the question set of the i-th party, and let Xi = {0, 1, 2} for i = 1, . . . , N − 1, and X N = {0, 1, 2, 3}. Let xi ∈ Xi denote a question to the i-th party. The answer sets are Ai = {0, 1, . . . , d − 1}, for i = 1, . . . , N. Definition 2 (Ideal measurements for multipartite entangled Schmidt states). The N parties make the following measurements on the joint state |Ψi = ∑dj=−01 c j | ji⊗n . For i = 1, . . . , N − 1: • For question xi = 0, the i-th party measures in the computational basis {|0i, |1i, · · · , |d − 1i} of its system, L d −1
L d −1
• For xi = 1 and xi = 2: for d even, in the eigenbases of observables m2 =0 [σX ]m and m2 =0 [σX ]0m respectively, with the natural assignments of d measurement outcomes; for d odd, in the eigenbases of observables
L d−2 1 −1 m =0
[σX ]m ⊕ |d − 1ihd − 1| and |0ih0| ⊕
L d−2 1 −1 m =0
[σX ]0m respectively.
7 For i = N: • For x N = 0 and x N = 1, the party N measures in the eigenbases of
L d2 −1
L 2d −1
m=0 [cos ( µm ) σZ
+
sin (µm )σX ]m and m=0 [cos (µm )σZ − sin (µm )σX ]m respectively, with the natural assignments of d measurement outcomes, where µm = arctan[sin(2θm )] and θm = arctan(c2m+1 /c2m ); for d odd, he measures in the eigenbases of L d−2 1 −1 m =0
L d−2 1 −1 m =0
[cos (µm )σZ + sin (µm )σX ]m ⊕ |d − 1ihd − 1| and
[cos (µm )σZ − sin (µm )σX ]m ⊕ |d − 1ihd − 1| respectively.
L 2d −1
0 m=0 [cos ( µm ) σZ + L 0 )] sin (µ0m )σX ]0m and m=0 [cos (µ0m )σZ − sin (µ0m )σX ]0m respectively, where µ0m = arctan[sin(2θm d − 1 L 2 −1 0 = arctan( c 0 and θm 2m+2 /c2m+1 ); for d odd, in the eigenbases of |0ih0| ⊕ m=0 [cos ( µm ) σZ + L d −1 − 1 sin (µ0m )σX ]0m and |0ih0| ⊕ m2=0 [cos (µ0m )σZ − sin (µ0m )σX ]0m , respectively.
• For x N = 2 and x N = 3: for d even, the N-th party measures in the eigenbases of d 2 −1
We refer to the correlation specified by the ideal measurements above as the ideal correlation for multipartite entangled Schmidt states. Next, we will highlight a set of properties of the ideal correlation that are enough to characterize it, in the sense that any quantum correlation that satisfies these properties has to be the ideal one. This also aids understanding of the self-testing proof (Proof of Theorem 2). In what follows, we will employ the language of correlation tables, which gives a convenient way to describe correlations. In general, let Xi be the question sets and Ai the answer sets. A correlation specifies, for each possible question x ∈ X1 × · · · × XN , a table Tx with entries Tx ( a) = p( a| x ) for a ∈ A1 × · · · × A N . For example, we denote the correlation tables for the ideal correlations for ghz (θm ) multipartite entangled GHZ states from Theorem 1 as Tx N , where x ∈ {0, 1} N denotes the question. Definition 3 (Self-testing properties of the ideal correlations for multipartite entangled Schmidt states). Recall that Xi = {0, 1, 2} for i = 1, . . . , N − 1, and X N = {0, 1, 2, 3}. Ai = {0, 1, . . . , d − 1}, for i = 1, . . . , N. The self-testing properties of the ideal correlations are: • For questions x ∈ {0, 1} N , we require Tx to be block-diagonal with 2× N blocks Cx,m := (c22m + ghz (θm ) c22m+1 ) · Tx N corresponding to outcomes in {2m, 2m +1} N , where the multiplication by the weight is intended entry-wise, and θm := arctan c2m+1 /c2m . • For questions with xi ∈ {0, 2}, for i = 1, . . . , N − 1 and x N ∈ {2, 3} we require Tx to be blockdiagonal with the 2× N blocks ”shifted down” by one measurement outcome. These should be Dx,m := ghz (θ 0 ) (c22m+1 + c22m+2 ) · T f (x1 )N,...,mf (x N−1 ),g(x N ) corresponding to measurement outcomes in {2m + 1, 2m + 0 : = arctan c 2} N , where θm 2m+2 /c2m+1 and f (0) = 0, f (2) = 1, g (2) = 0, g (3) = 1. We are now ready to state the main theorem of this section. Theorem 2. Let |Ψi = ∑dj=−01 c j | ji⊗ N , where 0 < c j < 1 for all i and ∑dj=−01 c2j = 1. Suppose N parties exhibit the ideal correlations for multipartite entangled Schmidt states from Definition 2 by making local measurements on a joint state |ψi. Then there exists a local isometry Φ such that Φ(|ψi) = |junki ⊗ |Ψi. As we mentioned, the proof of Theorem 2 follows closely the method of [25], and uses as a building block our self-testing of the n-partite partially entangled GHZ state. For the details, we refer the reader to Appendix C.
8 B.
Symmetric Dicke states
Let us now consider the symmetric Dicke states. These are simultaneous eigenstates of the square of the total angular momentum operator J2 of N qubits and its projection onto the z-axis Jz . In a concise way they can be stated as 1 | D kN i = q ∑ Pi (|1i⊗k |0i⊗( N−k) ), N (k) i
(17)
where the sum goes over all permutations of the parties and k is the number of excitations. For instance, for k = 1 they reproduce the N-qubit W state: 1 |WN i = √ (|0 . . . 01i + |0 . . . 10i + . . . + |10 . . . 0i). N
(18)
Interestingly, Dicke states have been generated experimentally [21] and have important role in metrology tasks [22] and quantum networking protocols [23]. We now show how to self-test Dicke states. For convenience, we consider the unitarily equivalent state | xD kN i = σxN | D kN i with σxN denoting σx applied to party N. Our self-test exploits the fact that every Dicke state can be written as √ 1 √ N − m |0i| xD kN −1 i + m |1i| xD kN−−11 i (19) | xD kN i = √ N which, after recursive application, allows one to express it in terms of the (k + 1)-partite W state, that is, q 1 (kk−+Ω1 ) k −Ω q | xD N i = |i1 , . . . , i N −k−1 i| xDkk+ (20) ∑ 1 i, N i1 ,...,i N −k−1 =0 (k) where the first ket is shared by the parties 1, . . . , N − k − 1 and Ω = i1 + . . . + i N −k−1 . Now, for i1 = . . . = i N −k−1 = 0, the corresponding state | xDkk+1 i is simply a rotated (k + 1)-partite W ⊗(k+1)
| xWk+1 i. Moreover, due to the fact that the Dicke states are symmetric, the above state σx decomposition holds for any choice of N − k − 1 parties among the first N − 1 parties. Thus, if we had a self-test for the N-partite W state | xWN i, we could use the above formula to generalize it to any Dicke state. Let us then show how to self-test any W state. Theorem 3. Let the state |ψi and measurements Zi , Xi for parties i = 1, . . . , N − 1 and D N and EN for the last party, satisfy the following conditions: * + * + √ N −1 N −1 O O 2 4 2 (+) (+) (+) Zl = , Zl ⊗ Bi,N = , (21) N N l =1,l 6=i l =1,l 6=i (+)
with i = 1, . . . , N − 1, where, as before, Bi,N = Zi ⊗ D N + Zi ⊗ EN + Xi ⊗ D N − Xi ⊗ EN is the Bell operator between the parties i and N. Moreover, we assume that * + N −1 O 1 1 (−) (+) (−) h Zi i = , Zi ⊗ Zi = (22) N N l =1,l 6=i with i = 1, . . . , N − 1. Then, for the isometry Φ N one has Φ N (|ψi) = |junki| xWN i.
9 We defer the detailed proof to Appendix D, presenting here only a sketch.√The proof makes use of the fact that | xWN i can be written as [|0i⊗ N −2 (|00i + |11i)i,N + |resti i]/ N, where (|00i + |11i)i,N is the maximally entangled state between the parties i and N, and the state |resti i collects all the remaining kets. We thus impose in Eq. (21) that if ( N − 2)-partite subset of the first N − 1 parties obtains +1 when measuring Zi on |ψi, the state held by the parties i and N violates maximally the CHSH Bell inequality. Conditions in (22) are needed to characterize |resti i, which completes the proof. Let us now demonstrate how the above result can be applied to self-test any Dicke state. First, let us simplify our considerations by noting that a Dicke state with k ≤ b N/2c is unitarily equivN −k alent to a Dicke state with m ≥ d N/2e, i.e., | D kN i = σz⊗ N | D N i for k = 0, . . . , b N/2c. Thus, it is enough to consider the Dicke states with k ≥ b N/2c. Second, due to the fact that Theorem 3 is ⊗(k+1) formulated for | xWN i, while in the decomposition (20) we have σx | xWN i, one has to modify (+) (−) the conditions in Eqs. (21) and (22) as Zi ↔ Zi for i = 1, . . . , N − 1, and D N → − EN and EN → − D N . Then, to self-test the Dicke states one proceeds in the following way: 1. Project any ( N − k − 1)-element subset Si of the first N − 1 parties of |ψi (there are ( NN−−1−1 k) such subsets) onto
N
j∈Si
(+)
Zj
and check whether the state corresponding to the remaining ⊗(k+1)
parties satisfies the conditions for | xDkk+1 i = σx
| xWk+1 i.
2. For every sequence (i1 , . . . , i N ) consisting of k + 1 ones on the first N − 1 positions, check that the state |ψi obeys the following correlations (i )
(i )
hψ| Z1 1 ⊗ . . . ⊗ ZNN |ψi = 0, (τi )
where Zi
(23)
= [1 + (−1)τi Zi ] /2.
The detailed proof that the above procedure allows to self-test the Dicke states is presented in Appendix E. Notice that our self-test exploits two observables per site and the total number of correlators one has to determine for every Dicke state in this procedure again scales linearly with N, in contrast with the exponential scaling of quantum state tomography.
C.
Graph states
We finally demonstrate that our method applies also to the graph states. These are N-qubit quantum states that have been widely exploited in quantum information processing, in particular in quantum computing, error correction, and secret sharing (see, e.g., Ref. [24]). It is thus an interesting question to design efficient methods of their certification, in particular self-testing. Such a method was proposed in Ref. [12] however, in general it needs three measurements for at least one party. Below we show that the approach based on violation of the CHSH Bell inequality provides a small improvement, as it requires only two measurements at each site. Before stating our result, we introduce some notation. Consider a graph G = (V, E) with V and E denoting respectively the N-element set of vertices of G and the set of edges connecting elements of V. A graph state corresponding to G is an N-qubit state given by |ψG i =
10 ∏(a,b)∈E Ua,b |+i⊗ N , where Ua,b is the controlled-Z interaction between qubits a and b, the product √ goes over all edges of G, and |+i = (|0i + |1i)/ 2. Notice that |ψG i can also be written as 1 | ψG i = √ 2N
∑
i∈{0,1}n
(−1)µ(i) |ii,
(24)
where the sum is over all sequences i = (i1 , . . . , i N ) with each i j ∈ {0, 1}, and µ(i) is the number of edges connecting qubits being in the state |1i for a given ket |ii. The main property of the graph states underlying our self-test is that by measuring all the neighbours of a pair of connected qubits i, j in the σz -basis, the two qubits i and j are left in one of the Bell states (cf. Ref. [26]): 1 m √ (σzmi ⊗ σz j )(|0+i + |1−i) 2
(25)
where mi is the number of parties from νi,j \ { j} whose result of the measurement in the σz -basis was −1. In (25) we neglect an unimportant −1 factor that might appear. Having all this, we can now state formally our result. Given a graph G and the corresponding graph state |ψG i, let νi denote the set of of all neighbours of the qubit i (all qubits connected to i by an edge). Likewise, we denote by νi,j the set of neighbours of qubits i and j. Let then |νi | and |νi,j | be the numbers of elements of νi and νi,j , respectively. Also, for simplicity, we label the qubits of |ψG i in such a way that the qubits N − 1 and N are connected and the qubit N is the one with the (τ ) (τ ) smallest number of neighbours. Denoting Zνi,j = ⊗l ∈νi,j Zl l , where τ is an |νi,j |-element sequence (τ )
with each τl ∈ {0, 1} (the operator Zνi,j acts only on the parties belonging to νi,j ), we can state our result. Theorem 4. Let |ψi and measurements Zi , Xi with i = 1, . . . , N − 1 and D N , EN , ZN ≡ ( D N − √ √ (τ ) EN )/ 2, N ≡ ( D N + EN )/ 2, satisfy h ZνN −1,N i = 1/2|νN −1,N | , and D
(τ ) ZνN −1,N
⊗
(m −1 ,m N ) BN −N1,N
E
=
√ 2 2
(26)
2|νN −1,N | (m
,m )
(m
,m )
−1 N −1 N for every choice of the |νi,j |-element sequence τ. The Bell operators BN −N1,N are defined as BN −N1,N = m m N N − 1 ZN −1 ⊗ ( D N − EN ). Additionally, we assume that (−1) X N −1 ⊗ ( D N + EN ) + (−1) D E D E 1 (−1)m j (τ ) (τ ) Zνi,j = |ν | , Zνi,j ⊗ Zi ⊗ X j = |ν | (27) 2 i,j 2 i,j
for all connected pairs of indices i 6= j except for ( N − 1, N ). Then Φ N (|ψi) = |ψG i. The proof of this statement may be found in Appendix F. It is worth noting that the above approach exploits violations of the CHSH Bell inequality between a single pair of parties [cf. Eqs. (26)], and not between every pair of neighbours. IV.
CONCLUSION AND DISCUSSION
We investigated a simple, but potentially general, approach to self-testing multipartite states, inspired by [10], which relies on the well understood method of self-testing bipartite qubit states based on the maximal violation of the tilted CHSH Bell inequality. This approach allows one to
11 self-test, with few measurements, all permutationally-invariant Dicke states, all partially entangled GHZ qubit states, and to recover self-testing of graph states (which was previously known through stabilizer-state methods). In our work, we also generalize self-testing of partially entangled GHZ qubit states to the qudit case, using techniques from [25]. We obtain the first self-testing result for a class of multipartite qudit states, by showing that all multipartite qudit states that admit a Schmidt decomposition can be self-tested. Importantly, our self-tests have a low complexity in terms of resources as they require up to four measurement choices per party, and the total number of correlators that one needs to determine scales linearly with the number of parties. As a direction for future work, we are particularly interested in extending this approach to self-test any generic multipartite entangled state of qubits (which is local-unitary equivalent to its complex conjugate in any basis). The main challenge here is to provide a general recipe to construct a single isometry that self-tests the global state from the different ones derived from various subtests (i.e. from projecting various subsets of parties and looking at the correlations of the remaining ones). This appears to be challenging for states that do not have any particular symmetry. Finally, notice that all presented self-tests which rely on the maximal violation of the CHSH Bell inequality can be restated and proved in terms of the other available self-tests. In particular, any self-test discussed in [18] would work in case of two measurements per site, and self-tests in [19] would work for higher number of inputs. Note added: After finishing this work, we learned about works [27] and [28], where the authors obtain self-testing of N-partite W-states and Dicke states, respectively. Acknowledgments. The authors thank Flavio Baccari, Marc Roda, Alexia Salavrakos and Thomas Vidick for useful discussions. This work was supported by Spanish MINECO (QIBEQI FIS2016-80773-P and Severo Ochoa SEV-2015-0522), the AXA Chair in Quantum Information Science, Generalitat de Catalunya (CERCA Programme), Fundacio´ Privada Cellex and ERC CoG QITBOX. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 705109. I. ˇ acknowledges the support of ”Obra Social La Caixa 2016”. A.C. is supported by AFOSR YIP S. award number FA9550-16-1-0495.
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Appendix A: Proof of Theorem 1
For ease of exposition, we prove the Theorem in the case N = 4, with the extension to general N being immediate. Let A0 , A1 , B0 , B1 , C0 , C1 , D0 , D1 , be the pairs of observables for the four parties. For an observ-
13 able D, let PDa = [1 + (−1) a D ]/2, and for brevity let cθ and sθ denote respectively cos θ and sin θ. For clarity, we recall the correlations from Theorem 1, for the case N = 4:
hψ| PA0 0 |ψi = hψ| PB00 |ψi = hψ| PC00 |ψi = hψ| PA0 0 PC00 |ψi = hψ| PB00 PC00 |ψi = c2θ , (A1a) 1 (A1b) hψ| PAa 1 PBb1 |ψi = , for a, b ∈ 0, 1 4 √ 8 + 2α2 hψ| PAa 1 PBb1 αC0 + C0 D0 + C0 D1 + (−1) a+b (C1 D0 − C1 D1 ) |ψi = , for a, b ∈ 0, 1 (A1c) 4 q where tan 2θ = α22 − 12 . Equations (A1a) imply, by Cauchy-Schwartz inequality, that PA0 0 |ψi = PB00 |ψi = PC00 |ψi
(A2)
PA1 0 |ψi = PB10 |ψi = PC10 |ψi.
(A3)
and consequently
Notice that equation (A1b) implies k PAa 1 PBb1 |ψik = 1/2, for a, b ∈ {0, 1}, and that the equations in (A1c) describe maximal violations of tilted CHSH inequalities by the normalized state 2PAa 1 PBb1 |ψi, for a, b ∈ {0, 1} (the ones for a ⊕ b = 1 are tilted CHSH inequalities upon relabelling D1 → − D1 ). 0 = Let µ be such that tan µ = s2θ . Define X A := A1 , XB := B1 and XC := C1 . Then, let ZD ∗ be Z 0 where we have replaced the zero eigenvalues with 1. Define ( D0 + D1 )/2 cos µ, and let ZD D ∗ ∗ − 1 0 = ( D − D ) /2 cos µ. Let P a = [1 + ZD = ZD | ZD | . Define XD similarly starting from XD 0 1 ZD (−1) a ZD ]/2. The maximal violations of tilted CHSH from (A1c) imply, thanks to Lemma 1 1, that PCa 0 = PZa D ,
for a ∈ {0, 1},
sθ PAa 1 PBb1 XC XD PC00 |ψi = (−1) a+b cθ PAa 1 PBb1 PC10 |ψi, for a, b ∈ {0, 1}.
(A4) (A5)
If we introduce notation X A = A1 , XB = B1 and XC = C1 , then X A XB XC XD PA1 0 |ψi = ( PA0 1 − PA1 1 )( PB01 − PB11 ) XC XD PC10 |ψi
= PA0 1 PB01 XC XD PC10 |ψi − PA0 1 PB11 XC XD PC10 |ψi − PA1 1 PB01 XC XD PC10 |ψi
+ PA1 1 PB11 XC XD PC10 |ψi (A6) sθ 0 1 0 sθ 0 1 0 sθ 1 1 0 sθ 0 0 0 = PA1 PB1 PA0 |ψi + PA1 PB1 PA0 |ψi + PA1 PB1 PA0 |ψi + PA1 PB1 PA0 |ψi cθ cθ cθ cθ sθ 0 = PA0 |ψi, (A7) cθ
where we used equation (A5) to obtain the third line, and ∑ a,b∈{0,1} PAa 1 PBb1 = 1 to obtain the last. Conditions (12) and (13) of Theorem 1 follow immediately from the above.
Appendix B: Proof of Lemma 2
In this section, we provide a proof of Lemma 2. We explicitly construct a local isometry Φ such that Φ(|ψi) = |junki ⊗ |Ψi for any Schmidt state |Ψi = ∑dj=−01 c j | ji⊗ N , where 0 < c j < 1 for all j and ∑dj=−01 c2j = 1, and | junki is some auxiliary state.
14 (k)
−1 Proof. Recall that { Pl }dk= 0 are complete sets of orthogonal projections for l = 1, . . . , N − 1 by (i ) ( j )
(i ) ( j )
hypothesis. Then, notice that for i 6= j we have, using condition (15), PN PN |ψi = PN P1 |ψi = ( j ) (i )
(k)
P1 P1 |ψi = 0, i.e., the PN are “orthogonal when acting on |ψi”. (k)
Let A be the unital algebra generated by { P1 }. Let H0 = A|ψi, where A|ψi = { Q|ψi : Q ∈ (k) (k) (k) (k) A}. Let P˜ = P |H0 be the restriction of P to H0 . Then, { P˜ }d−1 is a set of orthogonal N
N
N
N
k =0
projections. This is because, thanks to (15), one can always move the relevant operators to be in front of |ψi, as in the simple example (i ) ( j ) (k) ( k ) (i ) ( j ) P˜N P˜N ( P1 |ψi0 ) = P1 P˜N P˜N |ψi = 0.
(B1)
(k) Thus, the set { P˜B , I − PB0 }, where PB0 is the sum of all other projections, is a complete set of orthogonal projections. −1 k ( k ) d −1 k ˜ ( k ) d −1 ˜ ( k ) Now, define Zl := ∑dk= 0 ω Pl , for l = 1, . . . , N − 1, and Z N : = ∑k =0 ω PN + 1 − ∑k =0 PN . (k) In particular, the Zl are all unitary. Notice, moreover, that 1 − ∑k P˜ |ψi = 0, by using (15) and N
(k) { Pl }
are complete. the fact that the Define the local isometry
Φ :=
N O
Rll 0 F¯l 0 Sll 0 Fl 0 Appl ,
(B2)
l =1
where Appl : Hl → Hl ⊗ Hl 0 is the isometry that simply appends |0i0l , F is the quantum Fourier transform, F¯ is the inverse quantum Fourier transform, Rll 0 is defined so that |φil |k il 0 7→ (k) Xl |φil |k il 0 ∀|φi, and Sll 0 is defined so that |φil |kil 0 7→ Zlk |φil |k il 0 ∀|φi. We compute the action (k) of Φ on |ψi. For ease of notation with drop the tildes from the P˜ , while still referring to the new N
orthogonal projections. N
l
1
F0
|ψi ⊗ |0i⊗ N −→l
d N/2
∑
k1 ,...,k N
|ψi ⊗
N
1
l Sll 0
−→
d N/2
∑
k1 ,...,k N
O l
|k l il 0
(B3)
!ki (j ) ∑ ω ji P i
N −1
∏
∑ω
i
i =1
ji
jN
(j ) PN N
jN
+1−∑ k
(j ) PN N
!k N
|ψi ⊗
O l
|k l il 0 (B4)
= = = F¯ 0
1 d N/2 d N/2
1 dN
=
1 dN
l
∑
N
∑ ∏ ω j k P1
( ji )
i i
∑ ∑ ω j(∑ k ) P1
( j)
i i
k1 ,...,k N j
∑ ∑ ∑
k1 ,...,k N j m1 ,...,m N
|ψi ⊗
O
|ψi ⊗
O
|ψi ⊗
l
( j)
l
|k l il 0
(B5)
|k l il 0
(B6)
|k l il 0
(B7)
ω j(∑i ki ) ∏ ω −mr kr P1 |ψi ⊗ ( j)
r
i
k1 ,...,k N j m1 ,...,m N
l
O
∑ ∑ ∑ ∏ ωk ( j−m ) P1
= ∑ P1 |ψi ⊗ | ji⊗ N j
( ji )
i i
k1 ,...,k N j1 ,...,jN i =1
1 d N/2
N
∑ ∏ ω j k Pi
k1 ,...,k N j1 ,...,jN i =1
1
−→l
N
∑
i
i
( j)
|ψi ⊗
O l
O l
|ml il 0
|ml il 0
(B8) (B9) (B10)
15 N
l
Rll 0
−→
∑ ∏ j
=∑ j
=
i
( j) Xi
!
( j)
P1 |ψi ⊗ | ji⊗ N
(B11)
c j (0) P |ψi ⊗ | ji⊗ N c0 1
(B12)
1 (0) P |ψi ⊗ ∑ c j | ji⊗ N c0 1 j
(B13)
=|extrai ⊗ |ψtarget i,
(B14)
where to get (B12) we used condition (15). It is an easy check to see that the whole proof above can be repeated by starting from a mixed joint state, yielding a corresponding version of the Lemma that holds for a general mixed state. Appendix C: Proof of Theorem 2
As mentioned, we work in the tripartite case, as the general n-partite case follows analogously. The measurements of Alice, Bob and Charlie can be assumed to be projective, since we make no assumption on the dimension of the system. For ease of notation, the proof assumes that the joint state is pure, but one easily realizes that the proof goes through in the same way by rephrasing everything in terms of density matrices (see [25] for a slightly more detailed discussion). Let |ψi be the unknown joint state, and let PAa x be the projection on Alice side corresponding obtaining outcome a on question x. Define PBby and PCc z similarly on Bob and Charlie’s side. The proof structure follows closely that of [25], and goes through explicitly constructing projectors and unitary operators satisfying the sufficient conditions of Lemma 2. +1 +1 Define Aˆ x,m = PA2mx − PA2mx +1 , Bˆ y,m = PB2m − PB2m and Cˆ z,m = PC2m − PC2m , for x, y, z ∈ {0, 1}. y z y z 2m+1 2m m Let 1m and similarly define 1m By and 1Cz for x, y, z ∈ {0, 1}. Now, A x = PA x + PA x q 2m k PA0 k = hψ| PA2m0 |ψi v u d −1 d −1 u j = thψ| PA2m0 ∑ PBi 0 ∑ PC0 |ψi i =0
j =0
= c2m ,
(C1)
m 2 2 1/2 m and k PA2m0 +1 k = c2m+1 . Similarly, we derive k1m A x | ψ ik = k1 By | ψ ik = k1Cz | ψ ik = ( c2m + c2m+1 ) for any m and x, y, z ∈ {0, 1}. Notice then that m m m h ψ | 1m A x 1 By | ψ i = h ψ |1 A x 1 By
=
d −1
∑ PCi |ψi 0
i =0 m m h ψ | 1m 1 1 A x By C0 | ψ i
= c22m + c22m+1 ,
(C2)
where the second last equality is from the block-diagonal structure of the correlations. Since m 2 2 1/2 , then Cauchy-Schwartz inequality implies 1m | ψ i = k 1m A x | ψ ik = k1 By | ψ ik = ( c2m + c2m+1 ) Ax 1m | ψ i . So, we have By m m 1m A x | ψ i = 1 By | ψ i = 1Cz | ψ i
(C3)
16 for all x, y, z ∈ {0, 1}. The correlations are, by design, such that Aˆ 0,m , Aˆ 1,m , Bˆ 0,m , Bˆ 1,m , Cˆ 0,m , Cˆ 0,m , j j j the associated projections PAi , PBi , PCi , j ∈ {2m, 2m + 1} and |ψi reproduce the correlations (c22m + ghz 0 i := c22m+1 ) · Cx,y,z 3,2,θm . In order to apply Theorem 1, we need to define the normalised state |ψm 2 2 1/2 and the “unitarized” versions of the operators above, namely D ˆ (1m i,m : = A0 | ψ i) / ( c2m + c2m+1 ) Di ˆ i,m , for D ∈ { A, B, C }. It is easy to check that then Aˆ , Bˆ and Cˆ satisfy the 1 − 1m + D i,m
i,m
0 i. Thus, letting we have that, conditions of Theorem 1 (for N = 3) on state |ψm
X A,m XB,m XC,m (1 −
i,m
0 0 0 Z A,m |ψm i = ZB,m |ψm i = ZC,m |ψm i,
0 Z A,m )|ψm i
= tan(θm )(1 +
0 Z A,m )|ψm i.
(C4) (C5)
Define the subspace Cm = range(1Cm0 ) + range(1Cm1 ), and the projection 1Cm onto subspace Cm . Then, notice from the way ZC,m is defined, that it can be written as ZC,m = 1 − 1Cm + Z˜ C,m , where Z˜ C,m is some operator living entirely on subspace Cm . This implies that ZC,m |ψm i = Z˜ C,m |ψm i = Z˜ C,m |ψi, where we have used (C3) and the fact that 1Cm0 |ψi = 1Cm1 |ψi =⇒ 1Cm |ψi = 1Cmi |ψi .
(C6)
Hence, from (C4) it is not difficult to deduce that Aˆ 0,m |ψi = Bˆ 0,m |ψi = Z˜ C,m |ψi. (2m)
A0 : = (1m + Aˆ 0,m )/2 = PA2m0 , Constructing the projections of Lemma 2. Define projections PA (2m+1) (2m+1) (2m) B0 A0 B0 : = (1m − Bˆ 0,m )/2 = : = (1m − Aˆ 0,m )/2 = P2m+1 , P : = (1m + Bˆ 0,m )/2 = P2m , P P A
(2m)
+1 PB2m , PC 0
B0
B (2m+1)
A0
B
:= (1Cm + Z˜ C,m )/2 and PC (2m)
(2m+1)
:= (1Cm − Z˜ C,m )/2. are indeed projections, since Z˜ C,m has all ±1 eigenvalues correspond-
Note that PC , PC ing to subspace Cm , and is zero outside. We also have, for all m and k = 2m, 2m + 1, (k)
(k)
PB |ψi = PA |ψi =
1 A0 1 B0 [1 + (−1)k Aˆ 0,m ]|ψi = [1m + (−1)k Aˆ 0,m ]|ψi 2 m 2 1 (k) = [1Bm + (−1)k Z˜ B,m ]|ψi = PC |ψi. 2
(C7)
0 i = [1 A0 + (−1)k A 0 i = [1 A0 + (−1)k A ˆ 0,m ]|ψm ˆ 0,m ]|ψi = Further, notice that [1 + (−1)k Z A,m ]|ψm m m
(k)
PA |ψi. Substituting this into (C5), gives (2m+1)
X A,m XB,m XC,m PA
(2m)
|ψi = tan(θm ) PA
|ψi =
c2m+1 (2m) P | ψ i. c2m A
(C8)
0 0 Now, for the ”shifted” blocks, we can similarly define Aˆ 0x,m , Bˆ x,m and Cˆ x,m as Aˆ x,m = PA2mx +1 − PA2mx +2 and similar. Then, analogously, we deduce the existence of hermitian and unitary operators 0 , Y0 0 YA,m B,m and YC,m such that (2m+2)
YA,m YB,m YC,m PA
|ψi =
c2m+2 (2m+1) P | ψ i. c2m+1 A
(C9)
Constructing the unitary operators of Lemma 2. We will now directly construct unitary operators (k) satisfying conditions (15,16) of Lemma 2. Define X A/B/C as follows: if k = 0, 1, (k)
XA =
X A,0 YA,0 X A,1 YA,1 . . . X A,m−1 YA,m−1 X A,m , if k = 2m + 1, X Y X Y . . . X if k = 2m, A,0 A,0 A,1 A,1 A,m−1 YA,m−1 ,
(C10)
17 (k)
(k)
(k)
(k)
and analogously for XB and XC . Note that X A and XB are unitary since they are product of unitaries. Finally, we are left to check that (k)
(k)
(k) (k)
X A XB XC PA |ψi =
c k (0) P | ψ i. c0 A
(C11)
The case k = 0 holds trivially. For k = 2m + 1, For k = 2m + 1, (k)
(k)
(k) (k)
X A XB XC PA |ψi
= X A,0 YA,0 XB,0 YB,0 XC,0 YC,0 . . . X A,m−1 YA,m−1 XB,m−1 YB,m−1 XC,m−1 YC,m−1 (2m+1)
× X A,m XB,m XC,m PA |ψi (C8) c2m+1 (2m) X A,0 YA,0 XB,0 YB,0 XC,0 YC,0 . . . X A,m−1 YA,m−1 XB,m−1 YB,m−1 XC,m−1 YC,m−1 PA |ψi = c2m c2m (C9) c2m+1 · X A,0 YA,0 XB,0 YB,0 XC,0 YC,0 . . . X A,m−2 YA,m−2 XB,m−2 YB,m−2 = c2m c2m−1 (2m−1)
× XC,m−2 YC,m−2 PA
|ψi
= ... c c2m c c (0) 2 = 2m+1 · . . . · 1 PA |ψi c c c0 c 1 2m−1 2m c2m+1 (0) = PA |ψi c0
(C12)
which is indeed (C11) as 2m + 1 = k. The case k = 2m is similar. This concludes the proof of Theorem 2.
Appendix D: Self-testing of the W states
In this section we provide a detailed proof of self-testing of the |WN i state 1 |WN i = √ (|0 . . . 01i + |0 . . . 010i + . . . + |10 . . . 0i). N
(D1)
For our convenience we show how to self-test the following unitarily equivalent state 1 | xWN i = √ (|0 . . . 0i + |0 . . . 011i + . . . + |10 . . . 01i). N
(D2)
which is obtained from |WN i by applying σx to the last qubit of |WN i. This is because | xWN i can be written as i 1 h ⊗ N −2 | xWN i = √ |0i (|00i + |11i)i,N + |resti i , N
(D3)
where (|00i + |11i)i,N stands for the two-qubit maximally entangled state distributed between the parties i and N with i = 1, . . . , N − 1, and the vectors |resti i contain the remaining kets. This decomposition explains the conditions we impose below. Let us now prove the following theorem.
18 Theorem 5. Assume that for a given state |ψi and measurements Zi , Xi for parties i = 1, . . . , N − 1 and D N and EN for the last party, the following conditions are satisfied * + * + √ N −1 N −1 O O 4 2 2 (+) (+) (+) Zl = , Zl ⊗ Bi,N = , (D4) N N l =1 l =1 l 6 =i
l 6 =i
(+)
with i = 1, . . . , N − 1, where, as before, Bi,N is the Bell operator between the parties i and N corresponding to the CHSH Bell inequality (+)
Bi,N = Zi ⊗ D N + Zi ⊗ EN + Xi ⊗ D N − Xi ⊗ EN .
(D5)
Moreover, we assume that (−) h Zi i
*
1 = , N
N −1 O
+ (+) Zi
l =1 l 6 =i
⊗
(−) Zi
=
1 N
(D6)
with i = 1, . . . , N − 1. Then, for the isometry Φ N one has Φ N |ψi|0i⊗ N = |junki| xWN i.
(D7)
√ √ Proof. Denoting ZN = ( D N + EN )/ 2 and X N = ( D N − EN )/ 2, the action of the isometry can be explicitly written as (τ ) τN (τ1 ) (D8) Z1 . . . ZN N |ψi|τ1 . . . τN i, Φ N |ψi|0i⊗ N = ∑ X1τ1 . . . X N τ ∈{0,1} N
where τ = (τ1 , . . . , τN ) with each τi ∈ {0, 1} and Zi i = [1 + (−1)τi Zi ]/2. It should be noticed that in general the operators ZN and D N might not be unitary, and one e N and Z e N , which by constructions are unitary. However, as already explained should consider X in Sec. ??, their action on |ψi is the same as the action of X N and ZN , thus, for simplicity, we use these operators. The first bunch of conditions (D4) implies that the norm of (τ )
(+)
|ψi i = Z1
(+)
(+)
(+)
. . . Zi−1 Zi+1 . . . ZN −1 |ψi
(D9)
√ is 2/N, which √ together with the second set of conditions in Eq. (D4) implies that the normalized ei i = N/2 |ψi i violate maximally the CHSH Bell inequality between the parties i and N states |ψ for i = 1, . . . , N − 1. This, by virtue of what was said in Sec. ??, yields the following identities ( Zi − ZN )|ψei i = 0
(D10)
[ Xi ( I + ZN ) − X N ( I − Zi )]|ψei i = 0 { Zi , Xi }|ψei i = 0.
(D11) (D12)
They immediately imply that all terms in Eq. (D8) for which one element of τ equals one and the rest equal zero vanish. To see it explicitly, let τi = 1 and τj = 0 for j 6= i. Then, for this τ, (−)
(+)
(−)
|ψτ i = Xi Zi ZN |ψi i. Applying (D10) to the latter and exploiting the fact that Zi one finally finds that |ψτ i = 0.
(+)
Zi
= 0,
19 Let us now consider those components of Eq. (D8) for which τ obeys τi = τN = 1 with i = 1, . . . , N − 1 and τj = 0 with j 6= i, N. Then, the following chain of equalities holds (+)
Z1
(+)
(−)
. . . Zi−1 Xi Zi
(+)
(+)
(−)
(−)
Zi+1 . . . ZN −1 X N ZN |ψi = Xi Zi
(−)
= Xi Zi
(−)
= Xi Zi
(+)
= Zi
(+)
= Z1
(−)
X N ZN |ψi i (−)
X N Zi
(+)
|ψi i
Xi ZN |ψi i
(+)
ZN |ψi i (+)
. . . Z N | ψ i,
(D13)
where the second equality stems from Eq. (D10), the third from Eq. (D11), and, finally, the fourth equality is a consequence of the anticommutation relation (D12) and the fact that Xi2 = 1. With all this in mind it is possible to group the terms in Eq. (D8) in the following way Φ N (|ψi|0i⊗ N ) = |junki| xWN i + |Ωi,
(D14)
√ (+) (+) where |junki = N Z1 . . . ZN |ψi and |Ωi contains all those terms for which τ contains more than two ones or exactly two ones but τN = 0. Now, our aim is to prove that |Ωi = 0. To this end we first notice that Eqs. (D6) imply the following correlations (−)
hψ| Zi
(+)
Zj
|ψi =
1 , N
(D15)
≤ 1, which in turn implies that each of correlators in (D15) is bounded from above by and from (+) (+) (+) (+) (−) (+) below by hψ| Z1 . . . Zi−1 Zi Zi+1 . . . ZN −2 ZN −1 |ψi and both are assumed to equal 1/N [cf. Eq. (D6)]. (−) (+) The first relation in Eq. (D15) together with Eq. (D6) and the fact that Zj + Zj = 1 yields (±)
where i 6= j and i, j = 1, . . . , N − 1. This is a direct consequence of the fact that Zi
(−) hψ| Zi |ψi
(−)
hψ| Zi
(−)
Zj
(−)
|ψi = 0. This, due to the fact that Zi (−)
Zi
(−)
Zj
(−)
Zj
is a projector, allows one to write
|ψi = 0
(D16)
for i, j = 1, . . . , N − 1. This is enough to conclude that |Ωi = 0, which when plugged into Eq. (D3), leads directly to Eq. (F6) because each component in |Ωi has either three τi which equal 1, or two τi that equal one but then τN = 0. Since the self-test relies on the maximal violation of the CHSH Bell inequality by a set of states |ψei i (i = 1, . . . , N − 1), it also inherits self-testing of the optimal CHSH measurements, meaning that (i ) Φ N Zi |ψi|0i⊗ N = |junki ⊗ σz | xWN i (i ) Φ N Xi |ψi|0i⊗ N = |junki ⊗ σx | xWN i (D17) for i = 1, . . . , N − 1, and
(i ) (i ) σz + σx √ Φ N D N |ψi|0i⊗ N = |junki ⊗ | xWN i, 2 (i ) (i ) σz − σx ⊗N √ Φ N EN |ψi|0i = |junki ⊗ | xWN i. 2
(D18)
20 This completes the proof. It should be noticed that our self-test of the W state exploits two observables per site and, as in the case of the partially entangled GHZ state, the number of correlators one needs to determine is 2N, and thus scales linearly with N.
Appendix E: Complete self-test of all the symmetric Dicke state
Here we show that the above self-test of the N-partite W state can be used to construct a selftest of all the Dicke states. Let us recall that the latter are pure symmetric states from (C2 )⊗ N that can be written in the following way 1 | Dm Pi (|0i⊗ N −m |1i⊗m ), ∑ Ni = q N (m ) i
(E1)
where the sum is over all permutation of an N-element set and m = 0, . . . , N (there is N + 1 such states). Let us notice that a Dicke state with m ≤ b N/2c is unitarily equivalent to a Dicke state N −m ⊗N with m ≥ d N/2e, i.e., | D m i for m = 0, . . . , b N/2c. For this reason below we N i = σz | D N consider the Dicke states with m ≥ b N/2c. Interestingly, to self-test Dicke states we can exploit the previously demostrated self-test of the W state. This follows from the fact that any Dicke state can be written as √ 1 √ m m −1 √ | Dm N − m | 0 i| D i + m | 1 i| D i i = N N −1 N −1 N
(E2)
which, after recursive application, allows one to express | D m N i in terms of the Dicke states of smaller number of particles
| Dm Ni
1
∑
=
i1 ,...,i N −m−1 =0
q
+1 (mm− Σ) m−Σ q |i1 , . . . , i N −m−1 i| Dm +1 i, N (m)
(E3)
where Σ = i1 + . . . + i N −m−1 . Due to the fact that | D m N i is symmetric, the above decomposition holds true for any choice of N − m − 1 parties in the first ket in Eq. (E3). Having settled some basic information about the symmetric Dicke states, we can now move on to demonstrating how they can be self-tested. To facilitate our considerations we show how to self-test the following unitarlity equivalent state (N)
| xD m N i = σz
| xD m Ni 1
=
∑
i1 ,...,i N −m−1 =0
q
+1 (mm− Σ) m−∑ q |i1 , . . . , i N −m−1 i| xDm +1 i. N (m)
(E4)
m i = We then notice that the state corresponding to i1 = . . . = i N −m−1 = 0 is exactly | xDm +1
⊗(m+1)
| xWm+1 i with | xWm+1 i defined in Eq.(D1). Moreover, since | xD m N i is symmetric on the ⊗(m+1) first N − 1 parties, the state σx | xWm+1 i will appear in any decomposition of the form (E4) in which any choice of N − m − 1 parties from the first N − 1 ones are in state |0i. Importantly, we σx
21 ⊗(m+1)
already know how to self-test the W state σx | xWm+1 i. However, due to the transformation ⊗(m+1) σx we have to modify the conditions specified in Theorem 5 in the following way: (+)
Zi
(−)
↔ Zi
(i = 1, . . . , N − 1),
D N → − EN
EN → − D N .
and
(E5)
Now, to self-test a Dicke state | D m N i for any m ≥ b N/2c we can proceed in the following way: −1 1. Project any ( N − m − 1)-element subset Si of the first N − 1 parties of |ψi (there are ( NN −1− m )
such subsets) onto
N
j∈Si
(+)
Zj
and check whether the state corresponding to the remaining ⊗(m+1)
m i = σ parties satisfies the conditions for | xDm x +1
| xWm+1 i.
2. For every sequence τ = (τ1 , . . . , τN ) consisting of m + 1 ones on the first N − 1 positions, check that the state |ψi obeys the following correlations (τ1 )
(τ )
hψ| Z1 (τi )
where Zi
⊗ . . . ⊗ ZN N |ψi = 0,
(E6)
= [1 + (−1)τi Zi ] /2
Let us now see in more details how the above procedure allows one to self-test | D m N i. It is not difficult to see that the first condition leads us to the following decomposition # " i h O (+) ⊗( N −m−1) m (E7) | xDm Φ N (|ψi|0i⊗ N ) = Zl |φi i ⊗ |0iSi +1 i + | Φ i i l ∈Si
−1 for any i = 1, . . . , ( NN −1−m), where all Si stand for different ( N − m − 1)-element subsets of the ( N − 1)-element set {1, . . . , N − 1}, and |φi i is defined as
|φi i =
(−)
O
Xl Zl
l ∈{1,...,N }\Si
| ψ i.
(E8) (−)
In other words, to construct |φi i from |ψi one has to act on the latter with Xl Zl on all parties who do not belong to Si . Finally, |Φi i is some state from the global Hilbert space collecting the remaining terms. Let us now show that all the states
|φei i =
O l ∈Si
(+)
Zl
|φi i
(E9)
are the same. To this end, we will exploit the conditions (D10) and (D12), which are clearly preserved under the transformation (E5), and also the fact that [cf. Eq. (??)]:
( Xi − X N )|ψi = |ψi
(E10)
ei i and |φ ej i such that the corresponding sets Si for any i = 1, . . . , N − 1. Consider two vectors |φ and S j share N − m − 2 elements (remember that these sets are equinumerous). Let q and p be the two elements by which these sets differ, i.e., p ∈ Si (q ∈ S j ) and p ∈ / S j (q ∈ / Si ). Then, using (−)
(+)
the condition (D12) we turn the operator Xt Zt into Zt Xt at positions t = q and t = N for the ei i, and, analogously, at positions t = p and t = N for the state |φ ej i. We utilize the fact state |φ ei i = |φ ej i. that Xi X N |ψi = |ψi for all i = 1, . . . , N − 1 stemming from (E10), which shows that |φ
22 Finally, repeating this procedure for all pairs of states for which the corresponding sets Si differ by two elements, one finds that |φei i ≡ |φi for all i. As a result, the state (E7) simplifies to Φ N (|ψi|0i⊗ N ) = |φi| xD m N i + | Φ i,
|Φi is a vector from the global Hilbert space defined as (τ ) τN (τN ) |Φi = ∑ X1τ1 Z1 1 ⊗ . . . ⊗ X N Z N | ψ i ⊗ | τ i,
(E11)
(E12)
τ
where the summation is over all sequences τ = (τ1 , . . . , τN ) that contain less than N − m − 1 zeros (or, equivalently, more than m ones) on the first N − 1 positions. Now, to prove that |Φi = 0 it suffices to exploit the second step in the above procedure. That is, the condition (E6) is equivalent to (τ1 )
Z1
(τ )
⊗ . . . ⊗ ZN N |ψi = 0
(E13)
for every sequence (τ1 , . . . , τN ) consisting of m + 1 ones at the first N − 1 positions. Then, every component of the vector in Eq. (E12) contains a sequence of at least m + 1 Z (−) operators, which by virtue of (E13) implies that |Φi = 0. This completes the proof. For the self-testing of measurements the same argumentation as in the case of W-state self-test applies: (i )
Φ( Zi |ψi|0 . . . 0i) = |φi ⊗ σz | xD m Ni (i )
Φ( Xi |ψi|0 . . . 0i) = |φi ⊗ σx | xD m Ni Φ( ZN |ψi|0 . . . 0i) = |φi ⊗ Φ( X N |ψi|0 . . . 0i) = |φi ⊗
(i = 1, . . . , N − 1),
(i = 1, . . . , N − 1), ! (N) (N) σz + σx √ | xD m N i, 2 ! (N) (N) σz − σx √ | xD m N i. 2
Analogously to the N-partite W state the total amount of correlators necessary for self-testing of any Dicke state scales linearly with N. This is because one essentially needs to obtain the same correlators as for the W state. Appendix F: Self-testing the graph states
In this section we provide the detailed proof of the self-test of graph states, stated in Theorem 4. Before proving the theorem we need to make some introductory remarks. Let |ψG i be an Nqubit graph state that corresponds to a graph G = {V, E}, where V = {1, . . . , N } and E stand for the sets of vertices and edges, respectively. Recall that any graph state can be written in the following form 1 | ψG i = √ 2N
∑
i ∈{0,1} N
(−1)µ(i) |i i,
(F1)
23 where the summation is over all sequences i = (i1 , . . . , i N ) with i j = 0, 1, and µ(i ) is the number of edges connecting qubits being in the state |1i in ket |i i (without counting the same edge twice). Let then νi be the set of neighbours of the ith qubit, that is, all those qubits that are connected with i by an edge, while by |νi | we denote the number of elements in νi . Likewise, we denote by νi,j the set of neighbours of a pair of qubits i and j, i.e., all those qubits that are connected to either i or j (notice that νi,j = νj,i ), and |νi,j | the number of elements of νi,j . We also assume that the parties are labelled in such a way that qubits N − 1 and N are connected and the party N has the smallest number of neighbours, i.e., |νN | ≤ |νi | for all i. The main property of the graph states underlying our simple self-test is that by measuring all the neighbours of a pair of connected qubits i, j in the σz basis, the two qubits i and j are left in one of the Bell states [cf. Prop. 1 in Ref. [4]]: 1 m √ (σzmi ⊗ σz j )(|0+i + |1−i) 2
(F2)
where mi (m j ) is the number of parties from set νi,j \ j (νi,j \ i) whose result of a measurement in σz basis was −1, and where we have neglected an unimportant −1 factor that might appear. (τ ) (τ ) We are now ready to prove the main theorem. Let us denote Zνi,j = ⊗l ∈νi,j Zl l , where τ is an (τ )
|νi,j |-element sequence with each τl ∈ {+, −} (the operator Zνi,j acts only on the parties belonging to νi,j ).
Theorem 6. Let |ψi and measurements Zi , Xi with i = 1, . . . , N − 1 and ZN , D N , EN , ZN ≡ D N√+ EN 2
D N√− EN ,N 2
satisfy the following conditions D
(τ ) ZνN −1,N
E
=
1 2|νN −1,N |
,
D
(τ ) ZνN −1,N
⊗
(m −1 ,m N ) BN −N1,N (m
E
=
(m
,m N )
(F3)
2|νN −1,N |
,m N )
−1 for every choice of the |νi,j |-element sequence τ. The Bell operators BN −N1,N −1 BN −N1,N
√ 2 2
≡
are defined as
= (−1)m N X N −1 ⊗ ( D N + EN ) + (−1)m N−1 ZN −1 ⊗ ( D N − EN ),
(F4)
where m N −1 and m N are the numbers of neighbours of the qubits, respectively, N − 1 and N (excluding the Nth qubit and N − 1th qubit, respectively) which are projected onto the eigenvector of Zi− . We then assume that D E D E 1 (−1)m j (τ ) (τ ) Zνi,j = |ν | , Zνi,j ⊗ Zi ⊗ X j = |ν | (F5) 2 i,j 2 i,j (τ )
(τl )
for all connected pairs of indices i 6= j except for 6= ( N, N − 1) . As before, Zνi,j = ⊗l ∈νi,j Zl the isometry Φ N one has Φ N |ψi|0i⊗ N = |junki|ψG i. Proof. The conditions in Eq. (F3) imply that the normalized state p (τ ) (τ ) |ψeN −1,N i = 2|νN−1,N | ZνN−1,N |ψi
. Then, for
(F6)
(F7)
violates maximally the CHSH Bell inequality, which in turn implies that (τ )
(τ )
{ X N −1 , ZN −1 }|ψeN −1,N i = 0 and { X N , ZN }|ψeN −1,N i = 0,
(F8)
24
√ √ where X N = ( D N + EN )/ 2 and ZN = ( D N − EN )/ 2. These identities hold true for any of 2|νi,j | e(τ ) i, and therefore it must also hold for the initial state |ψi, i.e., projected states |ψ N −1,N { X N −1 , ZN −1 }|ψi = 0 and { X N , ZN }|ψi = 0.
(F9)
This is because one can always decompose |ψi in the eigenbasis of the operator ZνN −1,N which is a tensor product of Z operators acting on the neighbours of i, j. Then, let us focus on the second bunch p of conditions (F5). They imply that the length of the (τ )
(τ )
projected vectors |ψi,j i = Zνi,j |ψi is 1/
(τ )
(τ )
2|νi,j | , so is the norm of Zi |ψi,j i and X j |ψi,j i for any (τ )
(τ )
connected pair i 6= j. This together with (F5) mean that the vectors Zi |ψi,j i and X j |ψi,j i are parallel or antiparallel, or, more precisely, that (τ )
(τ )
(−1)m j Zi |ψi,j i = X j |ψi,j i
(F10)
for any connected pair of parties i 6= j. Let us now consider one of the parties connected to the party N − 1 (there must be at least one such party as otherwise the Nth one would not be the one with the smallest number of neighbours or the graph would be bipartite). We label this vertex by N − 2. It then follows from conditions (F10) that for the particular pair of vertices N − 2, N − 1, one has the following identities (τ )
(τ )
(τ )
(τ )
X N −2 |ψN −2,N −1 i = (−1)m N −2 ZN −1 |ψN −2,N −1 i
(F11)
and ZN −2 |ψN −2,N −1 i = (−1)m N −1 X N −1 |ψN −2,N −1 i
(F12)
hold true for all configurations of τ. With their aid the following sequence of equalities (τ )
(τ )
X N −2 ZN −2 |ψN −2,N −1 i = (−1)m N −1 X N −2 X N −1 |ψN −2,N −1 i (τ )
= (−1)m N−1 +m N−2 X N −1 ZN −1 |ψN −2,N −1 i (τ )
= −(−1)m N−1 +m N−2 ZN −1 X N −1 |ψN −2,N −1 i (τ )
= −(−1)m N−2 ZN −1 ZN −2 |ψN −2,N −1 i (τ )
= − ZN −2 X N −2 |ψN −2,N −1 i
(F13)
holds true for any choice of τ, where first and the second equality stems from Eqs. (F12) and (F11), respectively, the third one is a consequence of the anticommutativity of X N −1 and ZN −1 . Finally, the fourth and the fifth equality follows again from Eqs. (F12) and (F11), respectively. Since the identity (F13) is obeyed for any configuration of τ, it must also hold for the state |ψi, that is, { X N −2 , ZN −2 }|ψi = 0. Taking into account the conditions (F3), this procedure can be recursively applied to any pair of connected particles, yielding (together with (F9))
{ Xi , Zi }|ψi = 0
(F14)
for any i = 1, . . . , N. The action of the isometry is given by Φ N (|ψi|0i⊗ N ) =
1
∑
i1 ,...,i N =0
(i )
(i )
iN X1i1 . . . X N Z1 1 . . . ZNN |ψi|i1 . . . i N i
(F15)
25 Let us now consider a particular term in the above decomposition in which the sequence i1 , . . . , i N has k > 0 ones at positions j1 , . . . , jk , i.e., ij
ij
X j11 . . . X jkk
O
(+) O
Zl
(−)
Zl
l∈ I
l∈ /I
| ψ i,
(F16)
where I = { j1 , . . . , jk }. By using the previously derived relations, we want to turn this expression (+) (+) into one that is proportional to the junk state Z1 ⊗ . . . ⊗ ZN |ψi. To this end, let us first focus on the party jk and consider one of its neighbours which we denote by l. For this pair of parties, the conditions (F10) imply that X jk |ψjk ,l i = (−1)m jk Zl |ψjk ,l i,
(F17)
where, to recall, m jk is the number of neighbours of jk being in the state |1i except for l. The above identity together with the anticommutativity relation { X jk , Zjk }|ψi = 0 allow us to replace (−)
in Eq. (F16) the operator X jikk Zjk
(+)
by (−1)m jk Zjk Zl . Now, if the value of il in the corresponding (+)
ket |i1 , . . . , i N i is zero, the last operator Zl can be simply absorbed by Zl
uses that fact that
(−) = − Zl , meaning that m0 (−) (+) X jikk Zjk into (−1) jk Zjk ,
(−) Zl Zl
, while if il = 1, one
one has an additional minus sign. Altogether
this turns the operator where m0jk is the number of neighbours of jk (including il ) which are in the state |1i. Plugging this into Eq. (F16) we can rewrite the latter as
(−1)
m0j
k
ij
ij
O
X j11 . . . X jkk−−11
l∈ / I0
(+) O
Zl
l∈ I0
(−)
Zl
| ψ i,
(F18)
where now I 0 = I \ {il }, and so we have lowered the number of elements of I by one. It should be stressed that this affects the numbers of neighbours of those parties that are still in I 0 , which will be of importance for what follows. Now, we can apply recursively the same reasoning to the remaining elements of I 0 , keeping in mind that at each step one element is removed from I 0 . We thus arrive at 0
(+)
(−1)µ (i) Z1
(+)
⊗ . . . ⊗ Z N | ψ i,
(F19)
with µ0 defined as µ0 (i ) =
k
∑ m>j ,
l =1
(F20)
l
where m> jl is the number of neighbours of i jl being in the state |1i and having smaller indices than jl , or, in other words, those elements of I = { j1 , . . . , jk } smaller than jl that are neighbours of i jl . One immediately notices that µ0 (i ) equals µ(i ) for a given i, and therefore by applying the above reasoning to every term in Eq. (F15), one arrives at (+) (+) Φ N (|ψi|0i⊗ N ) = Z1 ⊗ . . . ⊗ ZN |ψi ⊗ ∑(−1)µ(i) |i i, (F21) i
which after normalizing both terms can be written as Φ N (|ψi|0i⊗ N ) = |junki|ψG i
(F22)
26
√ (+) (+) with |junki = (1/ 2 N )( Z1 ⊗ . . . ⊗ ZN |ψi). Once relation (F14) is satisfied for all is the proof for measurement self-testing goes along the same lines as the proof for the self-testing of the state. Let us check how isometry Φn acts on the state Xi˜|ψi. Eq. (F18) takes the form: Φ N ( Xi˜|ψi|0i⊗ N ) =
=
∑(−1)
m0j
I,l
∑ (−1)
k
ij
ij
O
X j11 . . . X jkk−−11
m0j +1 k
˜ I ⊕i,l
ij
ij
l∈ / I0
(+) O
Zl
l∈ I0
O
X j11 . . . X jkk−−11
l∈ / I0
(−)
Zl
(+) O
Zl
l∈ I0
Xi˜|ψi (−)
Zl
|ψi
˜ (+) (+) = Z1 ⊗ . . . ⊗ ZN |ψi ⊗ ∑(−1)µ(i⊕i) |i i i
(i˜)
∀i˜ ∈ {1, 2, . . . , N − 1}
= |junki ⊗ σx |ψG i,
where I ⊕ i˜ is equal to I/i˜ if i˜ ∈ I and to I ∪ i˜ otherwise, and µ(i ⊕ i˜) is the number of edges ˜ . . . 0)i (without counting the same connecting qubits being in the state |1i in ket |i ⊕ (0 . . . 0i0 edge twice). Similarly it can be shown that ˜
Φ N ( Zi˜|ψi|0i⊗ N ) = |junki ⊗ σz (i) |ψG i, (N)
Φ N ( D N |ψi|0i⊗ N ) = |junki ⊗
+σ √ z 2
σx
(N)
Φ N ( EN |ψi|0i
⊗N
) = |junki ⊗
∀i˜ ∈ {1, 2, . . . , N }, ! (N)
−σx
| ψG i , (N)
+σ √ z 2
!
| ψG i .
Finally, for self-testing graph-states one has to measure 3 + | E| correlators, where | E| is the total number of edges, which even for the fully connected graph is strictly better scaling than the complexity of quantum state tomography.
[1] C. Bamps and S. Pironio, Sum-of-squares decompositions for a family of Clauser-Horne-Shimony-Holt-like inequalities and their application to self-testing Phys. Rev. A 91 052111, (2015) [2] M. McKague and M. Mosca, Generalized Self-testing and the Security of the 6-State Protocol Theory of Quantum Computation, Communication, and Cryptography: 5th Conference (Lecture Notes inComputer Sciences vol 6519), p 113-130 (2011) [3] A. Ac´ın, S. Massar, S. Pironio, Randomness versus Nonlocality and Entanglement, Phys. Rev. Lett. 108, 100402 (2012). [4] M. Hein, J. Eisert, H. J. Briegel, Phys. Rev. A 69, 062311 (2004).
