Self-testing multipartite entangled states through projections onto two systems

Finding ways to test the behaviour of quantum devices is a timely enterprise, especially in light of the rapid development of quantum technologies. Device-independent self-testing is one desirable approach, as it makes minimal assumptions on the devices being tested. In this work, we address the question of which states can be self-tested. This has been answered recently in the bipartite case (Coladangelo et al 2017 Nat. Commun. 8 15485), 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, stabiliser 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 one 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 stabiliser methods). Finally, we give the first self-test of a class of multipartite qudit states: we generalise the self-testing of partially entangled GHZ states by adapting techniques from (Coladangelo et al 2017 Nat. Commun. 8 15485), 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][3][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.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][14][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 |ψ .On its share of this state, party i can perform one of several projective measurements {M a i x i ,i } a i , labelled by x i ∈ X i , with possible outcomes a i ∈ A i .Here X i and A i stand for finite alphabets of possible questions and answers for party i.The experiment is characterised by a collection of conditional probabilities {p(a 1 , . . ., a N |x 1 , . . ., x N ) : is the probability of obtaining outputs a 1 , . . ., a N upon performing the measurements x 1 , . . ., x N1 .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 M a i x i 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(a 1 , . . ., a N |x 1 , . . ., x N ) : a i ∈ A i } x i ∈X i selftests the state |Ψ and measurements { Ma i x i ,i } a i , i = 1, . . ., N, if for any state and measurements |ψ and {M a i x i ,i } a i , i = 1, . . ., N, reproducing the correlation, there exists a local isometry where |junk 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 where α ≥ 0 and A i and B i 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 σ z and σ x are the standard Pauli matrices.
A typical construction of the isometry Φ is the one encoding the SWAP gate, as illustrated in Fig. 1.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 |ψ , the idea is that N − 2 chosen parties perform local measurements on their shares of |ψ 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 |ψ .Our approach is inspired by Ref. [10], which shows that any state in the class (|100 + |101 + α|001 )/ √ 2 + α 2 , 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, 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 a to denote the biggest integer n such that n ≤ a, while a 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 where 0 < c j < 1 for all i and ∑ d−1 j=0 c 2 j = 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 where θ ∈ (0, π/4] and |GHZ N (π/4) = |GHZ N is the standard N-qubit GHZ state.The form of this state is such that if any subset of N − 2 parties performs a σ X measurement, the collapsed state shared by the remaining two parties is cos θ|00 ± sin θ|11 , 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) .
Theorem 1.Let |ψ be an N-partite state, and let A 0,i , A 1,i be a pair of binary observables for the i-th party, for i = 1, . . ., N. Suppose the following correlations are satisfied: where h(a) denotes the parity of the number of "−" in a, and α = 2 cos 2θ/ 1 + sin 2 2θ.Let µ be such that tan µ = sin 2θ.
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) .

Corollary 1.
Let |ψ be an N-partite state, and let A 0,i , A 1,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 Proof: This follows as a special case (d = 2) of Lemma 2 stated below, upon defining P As one can expect, the ideal measurements achieving these correlations are: We 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) 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 |Ψ be a state of the form (6). Suppose there exist sets of unitaries {X , where the subscript l ∈ {1, . . ., N} indicates that the operator acts on the system of the l-th party, and sets of projections {P k=0 , that are complete and orthogonal for l = 1, . . ., N − 1 and need not be such for l = N, and they satisfy: for all k = 1, . . ., N.Then, there exists a local isometry Φ such that Φ(|ψ ) = |junk ⊗ |Ψ .
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 |Ψ = ∑ d−1 j=0 c j |j ⊗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 d , |(2m + 1) mod d }.
For example, [σ Z ] m = |2m 2m| − |2m + 1 2m + 1|.Similarly, we denote by [A] m the observable defined with respect to the basis {|(2m + 1) mod d , |(2m + 2) mod d }.We use the notation A i to denote the direct sum of observables A i .Let X i denote the question set of the i-th party, and let X i = {0, 1, 2} for i = 1, . . ., N − 1, and X N = {0, 1, 2, 3}.Let x i ∈ X i denote a question to the i-th party.The answer sets are Definition 2 (Ideal measurements for multipartite entangled Schmidt states).The N parties make the following measurements on the joint state |Ψ = ∑ d−1 j=0 c j |j ⊗n .
For i = 1, . . ., N − 1: • For question x i = 0, the i-th party measures in the computational basis {|0 , |1 , • • • , |d − 1 } of its system, • For x i = 1 and x i = 2: for d even, in the eigenbases of observables • For x N = 2 and x N = 3: for d even, the N-th party measures in the eigenbases of 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 X i be the question sets and A i the answer sets.A correlation specifies, for each possible question For example, we denote the correlation tables for the ideal correlations for multipartite entangled GHZ states from Theorem 1 as T ghz N (θ m ) x , where x ∈ {0, 1} N denotes the question.
The self-testing properties of the ideal correlations are: • For questions x ∈ {0, 1} N , we require T x to be block-diagonal with 2 ×N blocks C x,m := (c x corresponding to outcomes in {2m, 2m + 1} N , where the multiplication by the weight is intended entry-wise, and θ m := arctan c 2m+1 /c 2m .
• For questions with x i ∈ {0, 2}, for i = 1, . . ., N − 1 and x N ∈ {2, 3} we require T x to be blockdiagonal with the 2 ×N blocks "shifted down" by one measurement outcome.These should be D x,m := ) corresponding to measurement outcomes in {2m + 1, 2m + 2} N , where θ m := arctan c 2m+2 /c 2m+1 and f (0 We are now ready to state the main theorem of this section.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.

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 J 2 of N qubits and its projection onto the z-axis J z .In a concise way they can be stated as 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: 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 k N = σ N x |D k N with σ N x denoting σ x applied to party N. Our self-test exploits the fact that every Dicke state can be written as which, after recursive application, allows one to express it in terms of the (k + 1)-partite W state, that is, where the first ket is shared by the parties 1, . . ., N − k − 1 and . Moreover, due to the fact that the Dicke states are symmetric, the above 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 |xW N , 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 |ψ and measurements Z i , X i for parties i = 1, . . ., N − 1 and D N and E N for the last party, satisfy the following conditions: with i = 1, . . ., N − 1, where, as before, B is the Bell operator between the parties i and N.Moreover, we assume that with i = 1, . . ., N − 1.Then, for the isometry We defer the detailed proof to Appendix D, presenting here only a sketch.The proof makes use of the fact that |xW N can be written as √ N, where (|00 + |11 ) i,N is the maximally entangled state between the parties i and N, and the state |rest 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 Z i on |ψ , the state held by the parties i and N violates maximally the CHSH Bell inequality.Conditions in (22) are needed to characterize |rest 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 ≤ N/2 is unitarily equivalent to a Dicke state with m ≥ N/2 , i.e., |D k for k = 0, . . ., N/2 .Thus, it is enough to consider the Dicke states with k ≥ N/2 .Second, due to the fact that Theorem 3 is formulated for |xW N , while in the decomposition (20) we have σ |xW N , one has to modify the conditions in Eqs. ( 21) and ( 22) as Then, to self-test the Dicke states one proceeds in the following way: and check whether the state corresponding to the remaining parties satisfies the conditions for |xD k k+1 = σ 2. For every sequence (i 1 , . . ., i N ) consisting of k + 1 ones on the first N − 1 positions, check that the state |ψ obeys the following correlations where Z 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 |ψ G = ∏ (a,b)∈E U a,b |+ ⊗N , where U a,b is the controlled-Z interaction between qubits a and b, the product goes over all edges of G, and |+ = (|0 + |1 )/ √ 2. Notice that |ψ G can also be written as where the sum is over all sequences i = (i 1 , . . ., i N ) with each i j ∈ {0, 1}, and µ(i) is the number of edges connecting qubits being in the state |1 for a given ket |i .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 σ z -basis, the two qubits i and j are left in one of the Bell states (cf.Ref. [26]): where m i is the number of parties from ν i,j \ {j} whose result of the measurement in the σ z -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 |ψ G , let ν i denote the set of of all neighbours of the qubit i (all qubits connected to i by an edge).Likewise, we denote by ν i,j the set of neighbours of qubits i and j.Let then |ν i | and |ν i,j | be the numbers of elements of ν i and ν i,j , respectively.Also, for simplicity, we label the qubits of |ψ G in such a way that the qubits N − 1 and N are connected and the qubit N is the one with the smallest number of neighbours.Denoting Z (τ) l , where τ is an |ν i,j |-element sequence with each τ l ∈ {0, 1} (the operator Z (τ) ν i,j acts only on the parties belonging to ν i,j ), we can state our result.
Theorem 4. Let |ψ and measurements Z i , X i with i = 1, . . ., N − 1 and D for every choice of the |ν i,j |-element sequence τ.The Bell operators B for all connected pairs of indices i = j except for 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 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.If we introduce notation X A = A 1 , X B = B 1 and X C = C 1 , then

d 2 − 1 m=0 [σ X ] m and d 2 − 1 m=0
[σ X ] m respectively, with the natural assignments of d measurement outcomes; for d odd, in the eigenbases of observables d−1