Introduction

Bell’s celebrated theorem1 from 1964 proposed an experimental test to address the question of locality and determinism in quantum mechanics. More concretely, Bell’s theorem established that, for systems that admit a description in terms of a local hidden variable model, the correlations exhibited by measurements on separate subsystems must obey certain constraints, usually stated in the form of an inequality, typically referred to as a Bell inequality. Correlations that violate such an inequality defy an explanation in terms of local hidden variables and are commonly referred to as “Bell nonlocal”, or just “nonlocal”. Quantum theory allows for the existence of nonlocal correlations, implying that quantum phenomena cannot be reproduced by local hidden variables. Such Bell inequality violations have been demonstrated convincingly in several experiments2,3,4.

Bell nonlocality has a long tradition in the study of the foundations of quantum theory, and has received even more attention in the last three decades, as it has proven to be a crucial ingredient in several information processing tasks. The key insight is that the violation of a Bell inequality witnesses entanglement shared among the devices involved, without making any assumption about the inner workings of the devices themselves. Such an approach to certifying properties of quantum systems is referred to as device-independent. Its appeal is that it removes trust in the behavior of the quantum devices involved. The approach has been adopted in various cryptographic protocols, such as quantum key distribution5,6,7,8, randomness generation9,10,11, and verifiable delegated computation protocols12,13,14.

Remarkably, certain nonlocal correlations go one step beyond merely witnessing entanglement: they suffice to single out a unique state that is compatible with them, up to inherent degrees of freedom. Such correlations are said to self-test the quantum state. In fact, the concept of self-testing, formally introduced by Mayers and Yao15,16, applies not only to states, but also to quantum measurements and channels17,18. As an example, it has long been known that the maximal quantum violation of the Clauser–Horne–Shimony–Holt (CHSH) Bell inequality self-tests a maximally entangled pair of qubits, and the measurement of anticommuting observables by each party19,20,21,22. We remark that certain transformations, namely local isometries or complex conjugation of the states and measurement, do not affect the observed measurement statistics. Thus, such transformations represent an inherent degree of freedom, and the characterization provided by a self-testing correlations can at most be modulo all of these transformations.

The field of self-testing has progressed substantially since its inception (see ref. 23 for a detailed review). In the bipartite case, we now know that all pure entangled states can be self-tested24. However, despite the efforts, much less is known in the multipartite case, where self-testing is known only for limited classes of states. Notable examples include, in the qubit case, self-testing of Dicke states25,26,27,28, GHZ-states29 and graph states30,31, and, for arbitrary local dimension, all states that admit a Schmidt decomposition26. Recently, Hardy-type paradoxes have been used to show that some genuine multipartite nonlocal states can be self-tested32,33. Finally, it has been shown that all pure (multipartite) entangled states can be self-tested when incorporated into a quantum network34. Although this protocol can, in principle, self-test any pure multipartite entangled state, it does so only in a network setting that introduces additional resources: the construction requires auxiliary maximally entangled states and extra parties. Concretely, to self-test an n-partite target state, the protocol employs 2n parties and n singlets, each of them shared between one of the initial and one of the auxiliary parties (each main party is paired with an auxiliary partner); if one makes the stronger modeling assumption that the network sources, for the state to be self-tested and the n additional singlets, are mutually independent, an alternative implementation in ref. 34 reduces the number of parties to n + 1. These structural additions (auxiliary entanglement, additional parties and independence assumptions) are not part of the single-source Bell scenario studied in the present work. Thus, the question of whether all pure multipartite entangled states can be self-tested in the standard Bell scenario remains open. There are two crucial challenges:

  • While in the bipartite case any state \(\left|\Psi \right\rangle\) admits a Schmidt decomposition, i.e. local orthonormal bases \(\{\left|{u}_{i}\right\rangle \}\) and \(\{\left|{v}_{i}\right\rangle \}\) such that \(\left|\Psi \right\rangle={\sum }_{i}\sqrt{{\lambda }_{i}}\left|{u}_{i}\right\rangle \left|{v}_{i}\right\rangle\), for some λi, this is not the case for multipartite states. Already in the tripartite case, almost any state \(\left|\Psi \right\rangle\) does not admit a Schmidt decomposition, i.e. there do not exist orthonormal bases \(\{\left|{u}_{i}\right\rangle \}\), \(\{\left|{v}_{i}\right\rangle \}\), \(\{\left|{w}_{i}\right\rangle \}\), and numbers λi such that \(\left|\Psi \right\rangle={\sum }_{i}\sqrt{{\lambda }_{i}}\left|{u}_{i}\right\rangle \left|{v}_{i}\right\rangle \left|{w}_{i}\right\rangle\). The existence of a Schmidt decomposition is very helpful in self-testing results for the bipartite case24, and indeed the only self-testing result in the multipartite case that covers a large class of states is precisely for those that admit a Schmidt decomposition26.

  • While in the bipartite case all pure entangled states are equivalent to their complex conjugates up to a local unitary transformation, which is a consequence of the existence of the Schmidt decomposition, already in the tripartite case there are examples of states for which this equivalence does not hold35,36. In fact, the set of pure states that are equivalent under local unitary transformations to their complex conjugate is of measure zero already for three-qubit systems35. In light of this, the best that one can achieve is to self-test a target pure state \(\left|\Psi \right\rangle\) in an arbitrary coherent superposition with its complex conjugate \(\left|{\Psi }^{*}\right\rangle\) through a “flag” qubit, i.e. self-test the state

    $$\sqrt{p}\left|\Psi \right\rangle \left|{{\bf{0}}}\right\rangle+\sqrt{1-p}\left|{\Psi }^{*}\right\rangle \left|{{\bf{1}}}\right\rangle \,,$$
    (1)

    where \(\left|{{\bf{0}}}\right\rangle\) and \(\left|{{\bf{1}}}\right\rangle\) are orthogonal flag states (see below for a precise definition), and p ∈ [0, 1] is an arbitrary real parameter determined by the physical realization; it specifies the relative weight between \(\left|\Psi \right\rangle\) and its complex conjugate \(\left|{\Psi }^{*}\right\rangle\) in the certified superposition.

In this work, we solve the problem of self-testing multipartite entangled states of an arbitrary number of qubits in the standard Bell scenario. We show the following:

Theorem 1

(Informal) Any pure entangled state of n≥2 qubits can be self-tested up to a local isometry, and up to complex conjugation (in the sense of Equation (1)).

More precisely, for each n-qubit state \(\left|\Psi \right\rangle\), we describe corresponding self-testing correlations on n parties, where each party receives O(n)-bit questions and returns 1-bit answers. One can view our result as saying that each n-qubit state \(\left|\Psi \right\rangle\) possesses a classical fingerprint. Our self-testing correlations are a combination of various “sub-tests” that together serve to pin down \(\left|\Psi \right\rangle\) uniquely.

Results

This article is organized as follows. In “Preliminaries", we introduce notation for the standard Bell scenario, along with the well-known “tilted CHSH inequality”37, which will serve as a building block in our self-test. In “The measurement lemma", we introduce a key workhorse of our result, the lemma that allows one to self-test the measurement of an arbitrary binary observable from the violation of CHSH and the tilted CHSH inequalities. Subsection “Self-testing tripartite states" describes our self-test in the case of three-qubit states. In “Recipe for self-testing multipartite states", we describe how the results from the tripartite case can be generalized to an arbitrary number of qubits, proving our main result.

Preliminaries

We consider a Bell scenario with n parties, each of which can perform m binary measurements. We denote the measurement input (i.e. the question) for party i by xi ∈ {0, …, m − 1}, and their outcome (i.e. the answer) by ai ∈ {0, 1}. In general, one could consider a larger answer set, but for this work it will suffice to consider binary answers. We introduce the shorthand notation x = (x1, …, xn) and a = (a1, …, an) to collectively denote the questions and answers of all n parties. Let \({\rho }^{{{\rm{E}}}}\equiv {\rho }^{{{{\rm{E}}}}_{1}\cdots {{{\rm{E}}}}_{N}}\), a mixed state acting on some Hilbert space \({\bigotimes }_{i=1}^{n}{{{\mathcal{H}}}}^{{{{\rm{E}}}}_{i}}\), denote the joint state of the n parties, where Ei denote the i-th party’s register. Let \({M}_{{a}_{i}| {x}_{i}}\) denote the positive semi-definite operator corresponding to party i obtaining outcome ai on question xi. So, for each i and xi, we have \({\sum }_{{a}_{i}}{M}_{{a}_{i}| {x}_{i}}={{\mathbb{1}}}^{{{{\rm{E}}}}_{i}}\). Then, the probability of observing outcome a on question x is

$$p\left({{\bf{a}}}| {{\bf{x}}}\right)={{\rm{tr}}}\left({\rho }^{{{\rm{E}}}}{M}_{{a}_{1}| {x}_{1}}\otimes \cdots \otimes {M}_{{a}_{n}| {x}_{n}}\right)\,.$$
(2)

We refer to the collection of conditional probabilities p(a∣x) as a “correlations”. Given that we restrict our attention to binary measurements, we can equivalently think of party i as measuring the observable \({A}_{{x}_{i}}={\sum }_{{a}_{i}}{(-1)}^{{a}_{i}}{M}_{{a}_{i}| {x}_{i}}\) on question xi. Then, the correlations from Eq. (2) are equivalently described by the “correlators”

$$\langle {A}_{{x}_{1}}{A}_{{x}_{2}}\cdots {A}_{{x}_{n}}\rangle={{\rm{tr}}}\left({A}_{{x}_{1}}\otimes {A}_{{x}_{2}}\otimes \cdots \otimes {A}_{{x}_{n}}{\rho }^{{{\rm{E}}}}\right).$$
(3)

These correlators may involve an observable for a subset of the parties, in which case, the operator acting on the remaining parties is the identity, e.g., \(\langle {A}_{{x}_{1}}{A}_{{x}_{2}}\rangle={{\rm{tr}}}({A}_{{x}_{1}}\otimes {A}_{{x}_{2}}\otimes {{\mathbb{1}}}^{{{{\rm{E}}}}_{3}}\otimes \cdots \otimes {{\mathbb{1}}}^{{{{\rm{E}}}}_{n}}{\rho }^{{{\rm{E}}}})\).

A self-testing protocol aims to establish equivalence between two experiments, which are usually called the “physical” and the “reference” experiment. An experiment is specified by a joint state of the n parties, and measurements for each party. The reference experiment involving the reference state \({\left|\Psi \right\rangle }^{{{\rm{E}}}^{\prime} }\equiv {\left|\Psi \right\rangle }^{{{{\rm{E}}}}_{1}^{{\prime} }\cdots {{\rm{E}}}{{\prime} }_{n}}\) and reference measurements {Ma∣x} is the “blueprint” with which we want to compare the physical experiment {ρE, Ma∣x} producing the observed correlations. In this work Hilbert spaces \({{{\mathcal{H}}}}^{{{\rm{E}}}^{\prime}_{i} }\) are for every i equal to \({{\mathbb{C}}}^{2}\). Self-testing is then defined as follows.

Definition 1

(Self-testing quantum states) The correlations p(a∣x) self-test the state \(\left|\Psi \right\rangle\) if, for all states ρE compatible with p(a∣x) via Eq. (2), there exists a local unitary \(U={\otimes }_{i=1}^{n}{U}^{{{{\rm{E}}}}_{i},{{\rm{E}}}{{\prime} }_{i}}\) such that for any purification \({\left|\psi \right\rangle }^{{{\rm{E}}}{{\rm{P}}}}\) of ρE

$$\left(U\otimes {{\mathbb{1}}}^{{{\rm{P}}}}\right)\left({\left|\psi \right\rangle }^{{{\rm{E}}}{{\rm{P}}}}{\otimes }^{n}_{i=1}{\left|0\right\rangle }^{{{\rm{E}}}^{\prime} }\right)=\sqrt{p}{\left|\Psi \right\rangle }^{{{\rm{E}}}^{\prime} }\otimes {\left|{\xi }_{0}\right\rangle }^{{{\rm{E}}}{{\rm{P}}}}+\sqrt{1-p}{\left|{\Psi }^{*}\right\rangle }^{{{\rm{E}}}^{\prime} }\otimes {\left|{\xi }_{1}\right\rangle }^{{{\rm{E}}}{{\rm{P}}}},$$
(4)

for some orthogonal states \(\left|{\xi }_{0}\right\rangle\) and \(\left|{\xi }_{1}\right\rangle\) and a number 0 ≤ p ≤ 1.

This is the same definition used in ref. 34 and resembles that of self-testing of subspaces put forward in ref. 38, however, here the state \(\left|\Psi \right\rangle\) and its complex conjugation \(\left|{\Psi }^{*}\right\rangle\) are not orthogonal in general. In fact, the method we introduce here allows one to certify a subspace spanned by the target state \(\left|\Psi \right\rangle\) and its complex conjugate \(\left|{\Psi }^{*}\right\rangle\).

To set the stage, we revisit the one-parameter class of Bell inequalities often referred to in the literature as “tilted CHSH” Bell inequalities37 (recall that the superscript denotes the party),

$${I}_{\alpha } : = \alpha \langle {A}_{0}\rangle+\langle {A}_{0}{B}_{0}\rangle+\langle {A}_{0}{B}_{1}\rangle \,+\langle {A}_{1}{B}_{0}\rangle -\langle {A}_{1}{B}_{1}\rangle \le 2+\alpha,$$
(5)

where 0≤α≤2, and Ai and Bi are the binary observables applied by the first and second party, respectively, on question i. The maximal quantum violation of this inequality is \(2\sqrt{2}\sqrt{1+{\alpha }^{2}/4}\), which is achieved by the observables

$$\begin{array}{rcl}{A}_{0} &=& {\sigma }_{{{\rm{z}}}}\,\,\,\,{B}_{0}=\cos \mu \,{\sigma }_{{{\rm{z}}}}+\sin \mu \,{\sigma }_{{{\rm{x}}}},\\ {A}_{1} &=& {\sigma }_{{{\rm{x}}}}\,\,\,\,{B}_{1}=\cos \mu \,{\sigma }_{{{\rm{z}}}}-\sin \mu \,{\sigma }_{{{\rm{x}}}},\end{array}$$
(6)

on the two-qubit partially entangled state

$$\left|{\psi }_{\theta }\right\rangle=\cos \theta \left|00\right\rangle+\sin \theta \left|11\right\rangle,$$
(7)

where θ ∈ (0, π/4] is such that \(\alpha=2\cos 2\theta /\sqrt{1+{\sin }^{2}2\theta }\), and μ is such that \(\tan \mu=\sin 2\theta\). Here σx, σz are the standard Pauli X and Z matrices. It is, by now, well known39 that the maximal violation of the tilted CHSH inequality above self-tests the strategy described by (6) and (7). Notice that the tilted CHSH inequality also self-tests Pauli σx and σz measurements by party 1 on her half of the state \(\left|{\psi }_{\theta }\right\rangle\).

For our result, we will require a procedure that not only self-tests σx and σz measurements, but also σy, i.e. a “tomographically complete” set of measurements by one party for any two-qubit entangled state. Fortunately, a family of Bell inequalities with this property is known to exist40. It can be viewed as a combination of two tilted CHSH inequalities (with the same tilting parameter α) that involve overlapping measurements and one CHSH inequality. Party 1 has a total of three possible measurements A0, A1, A2 and party 2 has six. Each inequality is used to certify the anticommutation relation between a single pair of observables of the first party. Using the corresponding optimal partially entangled pair of qubits, the two tilted CHSH inequalities can be simultaneously maximally violated. On the other hand, the CHSH inequality cannot be maximally violated, but its optimal value conditioned on using the partially entangled pair of qubits is still achieved by anti-commuting measurements. Let us now be more precise and formally state the three above mentioned Bell expressions that we use to self-test the Pauli matrices. The first expression is exactly Iα introduced in Eq. (5) which involves the first two observables of the first party A0 and A1. It is then completed by the following two other expressions:

$${J}_{\alpha }: = \alpha \langle {A}_{0}\rangle+\langle {A}_{0}{B}_{2}\rangle+\langle {A}_{0}{B}_{3}\rangle+\langle {A}_{2}{B}_{2}\rangle -\langle {A}_{2}{B}_{3}\rangle,$$
(8)
$$L : = \langle {A}_{1}{B}_{4}\rangle+\langle {A}_{1}{B}_{5}\rangle+\langle {A}_{2}{B}_{4}\rangle -\langle {A}_{2}{B}_{5}\rangle \,.$$
(9)

where Jα is the second tilted CHSH expression, which has the same form as Iα but involves another pair of the first party’s measurements, A0 and A2. The third expression is the standard CHSH Bell expression which involves the last pair of measurements A1 and A2. Importantly, for each of these three expressions Bob performs a different pair of measurements. The numbering of the observables that we choose to use in Eqs. (8) and (9) is consistent with the “combined” scenario described above: this is a Bell scenario consisting of three measurements for party 1, A0, A1, A2, and six for party 2, B0, …, B5. Notice, for example, that observable A0 appears in both the Bell operators Iα and Jα. Then, a simultaneous maximal violation of Iα and Jα, combined with the highest possible value achievable on L by the partially entangled state \(\left|{\psi }_{\theta }\right\rangle\) implies that all three possible pairs of operators of party 1 anti-commute, and are thus equivalent up to an isometry to σx, σy, and σz. The following lemma states the self-testing property of the combination of these three Bell operators.

Lemma 1

((informal) Self-testing a partially entangled pair of qubits, and σx, σy, σz40)Let 0 ≤ α ≤ 2. A simultaneous maximal violation of Iα and Jα, combined with \(L=2\sqrt{2}\sin \theta\), self-tests:

  1. 1.

    The partially entangled state \(\left|{\psi }_{\theta }\right\rangle\), where θ and α are related as in the tilted CHSH inequality;

  2. 2.

    Observables σx, σy, and σz measured by party 1.

A formal statement of this lemma can be found in Supplementary note 2.

The measurement lemma

We will now describe one of our key technical contributions, namely a method to self-test the measurement of arbitrary binary observables by party 2 by extending the Bell scenario of Lemma 1. Suppose we already know that the joint state \(\left|\psi \right\rangle\), along with observables A0, A1, A2 and B0, …, B5 reproduce the correlations of Lemma 1. Recall that the significance of the latter is that it yields a self-test for a tomographically complete set of measurements by party 1 (on an arbitrary partially entangled state, which can be chosen by varying the tilting parameter α). Now, suppose that we consider an additional sixth observable B6 for party 2. Can we impose additional constraints on the correlations, which should involve B6, to enforce that B6 must be an arbitrary measurement of our choice (up to a local isometry)? It turns out that this can be done by leveraging the tomographically complete set of measurements that we have already self-tested on party 1’s side: namely, A0, A1, A2 are equivalent up to a local isometry to σz, σx, and σy respectively. Then, the constraints to impose are very natural. Suppose we wish to self-test the binary observable W. Then, we simply impose the constraints

$$\begin{array}{rc} \left\langle \psi \right|{A}_{0}\otimes {B}_{6}\left|\psi \right\rangle=\left\langle {\psi }_{\theta }\right|{\sigma }_{{{\rm{z}}}}\otimes W\left|{\psi }_{\theta }\right\rangle \,\,\left\langle \psi \right|{A}_{1}\otimes {B}_{6}\left|\psi \right\rangle=\left\langle {\psi }_{\theta }\right|{\sigma }_{{{\rm{x}}}}\otimes W\left|{\psi }_{\theta }\right\rangle \,,\\ \left\langle \psi \right|{A}_{2}\otimes {B}_{6}\left|\psi \right\rangle=\left\langle {\psi }_{\theta }\right|{\sigma }_{{{\rm{y}}}}\otimes W\left|{\psi }_{\theta }\right\rangle \,\,\,\,\,\left\langle \psi \right|I\otimes {B}_{6}\left|\psi \right\rangle=\left\langle {\psi }_{\theta }\right|I\otimes W\left|{\psi }_{\theta }\right\rangle \,.\end{array}$$

We show that these additional constraints are enough to enforce that B6 is equal to W, up to a local isometry. A bit more concretely, we show the following:

Lemma 2

((informal) Measurement lemma) Suppose \(\left|\psi \right\rangle\), A0, A1, A2 and B0, …, B5 reproduce the self-testing correlations of Lemma 1. Let W = αxσx + αyσy + αzσz for some real numbers αx, αy, αz such that ∣αx∣2 + ∣αy∣2 + ∣αz∣2 = 1. Suppose additionally that B6 satisfies

$$\left\langle \psi \right|{A}_{0}\otimes {B}_{6}\left|\psi \right\rangle={\alpha }_{z},\,\left\langle \psi \right|{A}_{1}\otimes {B}_{6}\left|\psi \right\rangle={\alpha }_{x}\sin 2\theta,$$
(10)
$$\left\langle \psi \right|{A}_{2}\otimes {B}_{6}\left|\psi \right\rangle={\alpha }_{y}\sin 2\theta,\left\langle \psi \right|{\mathbb{1}}\otimes {B}_{6}\left|\psi \right\rangle={\alpha }_{z}\cos 2\theta .$$
(11)

Then, A is equivalent to W or W * up to a local isometry.

The sense in which the latter equivalence holds is slightly subtle, and is the following: up to a unitary on party 2’s side, the observable B6 can be interpreted as making a sequential measurement where a first two-outcome measurement on an auxiliary system “decides” whether to measure a qubit using W or W*. More formally, there is a unitary UB on party 2, and an observable S such that

$${U}^{{{\rm{B}}}}\,{B}_{6}{{U}^{{{\rm{B}}}}}^{{\dagger} }= ({\alpha }_{x}\,{\sigma }_{{{\rm{x}}}}+{\alpha }_{y}\,{\sigma }_{{{\rm{y}}}}+{\alpha }_{z}\,{\sigma }_{{{\rm{z}}}})\otimes \left(\frac{{\mathbb{1}}+S}{2}\right) \\ +{({\alpha }_{x}{\sigma }_{{{\rm{x}}}}+{\alpha }_{y}{\sigma }_{{{\rm{y}}}}+{\alpha }_{z}{\sigma }_{{{\rm{z}}}})}^{*}\otimes \left(\frac{{\mathbb{1}}-S}{2}\right)\,.$$

Crucial for our construction is that, in Lemma 2, the unitary UB is consistent with the local unitary UA ⊗ UB that maps \(\left|\psi \right\rangle\), A0, A1, A2 and B0, …, B5 to their ideal counterparts (which is already guaranteed to exist from Lemma 1).

This lemma can be seen as an instance of the so-called “post-hoc” measurement self-testing, introduced in ref. 23 and formalized in ref. 18. Post-hoc self-testing refers to the situation where a state and a tomographically complete set of measurements on one side have already been self-tested, and one then identifies an additional, previously unknown, measurement by examining its correlations with the self-tested measurement set. Since the self-tested measurements form a complete operator basis, the joint statistics uniquely determine the form of the new measurement up to the usual isometries.

A formal statement of this lemma, and its proof, can be found in Supplementary note 3.

Self-testing tripartite states

We now address the core problem: self-testing an arbitrary genuinely multipartite entangled (GME) three-qubit state shared between Alice, Bob, and Charlie. A pure state is said to be GME if it does not have a tensor-product form across any bipartition. If a pure state is not GME, then it is separable across some bipartition, and its self-testing reduces to the known bipartite case39. We therefore focus on GME states. The main idea here is to employ a modular self-testing strategy. We impose constraints corresponding to the following: a party, say Alice, performs a local measurement, projecting Bob and Charlie into a bipartite pure state corresponding to each outcome. Each of these resulting states is then self-tested using established bipartite methods. We then switch the roles of the parties, and impose constraints corresponding to a different party performing the first measurement. We show that the combination of these constraints is sufficient to self-test the original tripartite state.

Theorem 2

Any pure GME three-qubit state can be self-tested.

In the rest of this section, we give an overview of the proof, which can be found in Supplementary note 4. The modular method of self-testing multipartite states that we employ was first used in ref. 25 to show that the three-qubit W-state can be self-tested, and later generalized to self-testing multipartite states (of more than three parties) that possess a Schmidt decomposition, such as Dicke states and graph states26.

Let the reference tripartite state be

$${\left|\Psi \right\rangle }^{{{\rm{A}}}^{\prime} {{\rm{B}}}^{\prime} {{\rm{C}}}^{\prime} }={\sum }_{i,j,k=0}^{1}{\lambda }_{ijk}{\left|i\right\rangle }^{{{\rm{A}}}^{\prime} }{\left|j\right\rangle }^{{{\rm{B}}}^{\prime} }{\left|k\right\rangle }^{{{\rm{C}}}^{\prime} }\,.$$
(12)

The first step is to consider the post-measurement state on registers \({{\rm{B}}}^{\prime}\) and \({{\rm{C}}}^{\prime}\) when Alice measures her qubit in the computational basis. Denote the post-measurement states as

$$\begin{array}{r}{\left|{\psi }_{a+}\right\rangle }^{{{\rm{B}}}^{\prime} {{\rm{C}}}^{\prime} }\propto {\sum }_{j,k}{\lambda }_{0jk}{\left|jk\right\rangle }^{{{\rm{B}}}^{\prime} {{\rm{C}}}^{\prime} },\,{\left|{\psi }_{a-}\right\rangle }^{{{\rm{B}}}^{\prime} {{\rm{C}}}^{\prime} }\propto {\sum }_{j,k}{\lambda }_{1jk}{\left|jk\right\rangle }^{{{\rm{B}}}^{\prime} {{\rm{C}}}^{\prime} },\end{array}$$
(13)

corresponding to Alice obtaining outcomes 0 and 1, respectively. We can always find local unitaries Va±, Wa± for Bob and Charlie that rotate their computational bases into the Schmidt bases of these states, such that:

$${V}_{{a}^{\pm }}^{{{\rm{B}}}^{\prime} }\otimes {W}_{{a}^{\pm }}^{{{\rm{C}}}^{\prime} }{\left|{\psi }_{a\pm }\right\rangle }^{{{\rm{B}}}^{\prime} {{\rm{C}}}^{\prime} }= \cos {\phi }_{{a}^{\pm }}{\left|{0}_{{a}^{\pm }}{0}_{{a}^{\pm }}\right\rangle }^{{{\rm{B}}}^{\prime} {{\rm{C}}}^{\prime} } \\ +\sin {\phi }_{{a}^{\pm }}{\left|{1}_{{a}^{\pm }}{1}_{{a}^{\pm }}\right\rangle }^{{{\rm{B}}}^{\prime} {{\rm{C}}}^{\prime} }\equiv {\left|{\Psi }_{a\pm }\right\rangle }^{{{\rm{B}}}^{\prime} {{\rm{C}}}^{\prime} }.$$
(14)

Here, the kets on the right-hand side are defined in the new, state-dependent Schmidt bases. An analogous state \(\left|{\psi }_{b+}\right\rangle\) for Alice and Charlie is defined by having Bob measure and obtain outcome 0.

We can choose the global computational basis (via local unitaries) such that all the projected states \((\left|{\psi }_{a\pm }\right\rangle,\left|{\psi }_{b+}\right\rangle )\) are entangled, and both pairs of coefficients λ000, λ001 and λ100, λ101 have different complex phases. This is always possible for GME states41 and is crucial for the final part of the proof.

It is important to distinguish between the different types of bases used throughout the proof. We fix a global computational basis for all parties, defined by the specific choice of local reference frames. In this basis, the state’s coefficients λijk satisfy the conditions mentioned above. When a party (say, Alice) is instructed to measure in this computational basis, an act that projects the other two parties’ state, her corresponding physical measurement is denoted by the observable A♢. The Pauli σx measurement in this same local context is denoted by A♦. The same notation, B♢/B♦ and C♢/C♦, is used for Bob and Charlie. In contrast, the Schmidt basis is not fixed but depends on which party projects the state and what outcome they get. For example, the Schmidt basis for Bob and Charlie after Alice measures A♢ and gets outcome 0 is defined by the state \(\left|{\psi }_{a+}\right\rangle\) and is generally different from the computational basis.

Our self-testing correlations are built from three distinct sub-tests.

  1. 1.

    Sub-test 1 (Alice measures A♢, gets 0): This test is designed to self-test the state \(\left|{\psi }_{a+}\right\rangle\) between Bob and Charlie. We require the observed correlations to satisfy the conditions from Lemma 1, self-testing this specific entangled state and the corresponding Pauli observables σz, σx, σy for Bob in his Schmidt basis. Hence, up to local isometries the projected state must be \(\left|{\Psi }_{a+}\right\rangle \otimes \left|{\xi }_{{a}^{+}}\right\rangle\). Furthermore, we use the Measurement Lemma Lemma 2) and its alternative form presented in Supplementary note 3 to self-test additional measurements for Bob and Charlie. Specifically, we self-test Bob’s measurement B♢ (which will later be identified with σz in the computational basis) and B♦ (to be identified with σx). Similarly, we self-test Charlie’s measurements C♢ and C♦. These measurements are self-tested in relation to Bob’s and Charlie’s Pauli observables in their Schmidt bases, and as explained in Sec. 2 they are self-tested up to complex conjugation. Thus, the state of Bob and Charlie written in the eigenbasis of B♢ and C♢ is, up to local isometries, proportional to \(\left|{\psi }_{a+}\right\rangle \otimes \left|{\xi }_{a+}^{+}\right\rangle+\left|{\psi }_{a+}^{*}\right\rangle \otimes \left|{\xi }_{a+}^{-}\right\rangle\), where \(\left|{\xi }_{a+}^{+}\right\rangle\) and \(\left|{\xi }_{a+}^{-}\right\rangle\) are mutually orthogonal “junk” states.

  2. 2.

    Sub-test 2 (Alice measures A♢, gets 1): This test is analogous to Sub-test 1 but is designed to self-test the state \(\left|{\psi }_{a-}\right\rangle\) between Bob and Charlie. It yields a similar conclusion: there exist isometries that extract the state \(\left|{\psi }_{a-}\right\rangle \otimes \left|{\xi }_{a-}^{+}\right\rangle+\left|{\psi }_{a-}^{*}\right\rangle \otimes \left|{\xi }_{a-}^{-}\right\rangle\) and certify the appropriate Pauli measurements in both computational and Schmidt bases.

  3. 3.

    Sub-test 3 (Bob measures B♢, gets 0): This test self-tests the state \(\left|{\psi }_{b+}\right\rangle\) shared by Alice and Charlie after Bob measures his system in the computational basis. The structure is identical to the first two sub-tests but with the roles of the parties cycled. It provides isometries that certify the state \(\left|{\psi }_{b+}\right\rangle \otimes \left|{\xi }_{b+}^{+}\right\rangle+\left|{\psi }_{b+}^{*}\right\rangle \otimes \left|{\xi }_{b+}^{-}\right\rangle\), and the measurements Pauli’s σz and σx in the computational basis for Alice and Charlie.

The three sub-tests give only local or marginal guarantees. To elevate them to a certification of the full tripartite state, we apply a SWAP isometry that uses the self-tested ♢ and ♦ measurements to coherently extract the physical state onto three auxiliary qubits (Fig. 1). This produces two orthogonal branches corresponding to \(\left|\Psi \right\rangle\) and \(\left|{\Psi }^{*}\right\rangle\) on the auxiliaries; the third sub-test identifies the branches and yields the claimed form up to complex conjugation:

$$\Phi \left({\left|\psi \right\rangle }^{{{\rm{A}}}{{\rm{B}}}{{\rm{C}}}{{\rm{P}}}}\otimes {\left|+++\right\rangle }^{{{\rm{A}}}^{\prime} {{\rm{B}}}^{\prime} {{\rm{C}}}^{\prime} }\right)={\left|{\xi }_{0}\right\rangle }^{{{\rm{A}}}{{\rm{B}}}{{\rm{C}}}{{\rm{P}}}}\otimes {\left|\Psi \right\rangle }^{{{\rm{A}}}^{\prime} {{\rm{B}}}^{\prime} {{\rm{C}}}^{\prime} }+{\left|{\xi }_{1}\right\rangle }^{{{\rm{A}}}{{\rm{B}}}{{\rm{C}}}{{\rm{P}}}}\otimes {\left|{\Psi }^{*}\right\rangle }^{{{\rm{A}}}^{\prime} {{\rm{B}}}^{\prime} {{\rm{C}}}^{\prime} },$$
(15)
Fig. 1: The isometry for self-testing the tripartite states. Schematic of the local extraction isometry used to self-test a general three-qubit state.
Fig. 1: The isometry for self-testing the tripartite states. Schematic of the local extraction isometry used to self-test a general three-qubit state.The alternative text for this image may have been generated using AI.
Full size image

The circuit takes as input the physical (unknown) joint state \({\left|\psi \right\rangle }^{{{\rm{A}}}{{\rm{B}}}{{\rm{C}}}{{\rm{P}}}}\) (the register P holds any purification and is acted on trivially) together with three auxiliary single-qubit registers \({{\rm{A}}}{\prime},{{\rm{B}}}{\prime},{{\rm{C}}}{\prime}\) each initialized in \(\left|+\right\rangle=(\left|0\right\rangle+\left|1\right\rangle )/\sqrt{2}\). The boxes labelled A♢, B♢, C♢ and A♦, B♦, C♦ denote the two dichotomic measurements implemented by each party in the protocol: in the self-testing derivation the lozenge and black lozenge observables are shown to anticommute (on the same local space) and therefore are unitarily equivalent to Pauli σz and σx respectively. The Hadamard gates (H) on the auxiliary qubits and the controlled actions defined by the certified lozenge / black lozenge measurements implement the SWAP-type extraction that coherently transfers the certified quantum information onto the three auxilliary qubits. Under the ideal correlations the isometry produces two orthogonal branches on \({{\rm{A}}}{\prime},{{\rm{B}}}{\prime},{{\rm{C}}}{\prime}\) corresponding to the target state \(\left|\Psi \right\rangle\) and its complex conjugate \(\left|{\Psi }^{*}\right\rangle\); more precisely one obtains an output of the form \({\left|{\xi }_{0}\right\rangle }^{{{\rm{A}}}{{\rm{B}}}{{\rm{C}}}{{\rm{P}}}}\otimes {\left|\Psi \right\rangle }^{{{\rm{A}}}{\prime} {{\rm{B}}}{\prime} {{\rm{C}}}{\prime} }+{\left|{\xi }_{1}\right\rangle }^{{{\rm{A}}}{{\rm{B}}}{{\rm{C}}}{{\rm{P}}}}\otimes {\left|{\Psi }^{*}\right\rangle }^{{{\rm{A}}}{\prime} {{\rm{B}}}{\prime} {{\rm{C}}}{\prime} }\), which is Eq. (15) in the text.

Full expressions and the branch-merging argument are given in Supplementary note 4.

Recipe for self-testing multipartite states

We now show how the ideas discussed in the previous section can be extended to achieve self-testing of any n-partite qubit state. An arbitrary pure qubit state has the form

$$\left|\Psi \right\rangle={\sum }_{\vec{a}\in {\{0,1\}}^{n}}{\lambda }_{\vec{a}}\left|\vec{a}\right\rangle$$
(16)

in a product basis \(\{\left|\vec{a}\right\rangle \}\) and for normalized coefficients \(\{{\lambda }_{\vec{a}}\}\).

Our main self-testing result states the following.

Theorem 3

For any n-qubit pure state there exists a self-testing protocol requiring at most 9 ⋅ 2n−2 − 4 two-outcome measurements per party.

Any non-GME n-qubit state can always be decomposed into a tensor product of a number of GME states (including bipartite entangled states) and one-partite states. Putting the one-partite states apart, each of these GME states can be self-tested using our method for the corresponding number of parties, or earlier results in the case of bipartite states. Consequently, our analysis focuses on GME states.

The self-testing procedure generalizes the approach previously developed for tripartite states, extending it to the n-party scenario.

We assign a unique number to each party, ranging from 1 to n, and structure the self-testing procedure into n − 1 sub-tests. In the j-th sub-test, the parties {1, ⋯  , n}/{1, j + 1} perform measurements corresponding to the input ♢. These measurements prepare states for parties 1 and j + 1 that are then self-tested. In the j-th sub-test we name parties 1 and j + 1 tested parties, while all the remaining are called projecting parties. Party 1 serves as the tested party in every sub-test, while each of the other parties acts as a projecting party in all sub-tests except the one where it is itself being tested. In each sub-test, the n − 2 projecting parties collectively produce 2n−2 different combinations of measurement outputs, resulting in 2n−2 distinct post-measurement states for the two tested parties. However, not all these post-measurement states are self-tested within a given sub-test. The subset of states that undergo self-testing depends on the sub-test in question. Specifically, in sub-test k, the procedure self-tests the post-measurement states corresponding to all possible outputs of parties {k + 2, ⋯  , n}, but only those states corresponding to output 0 for parties {2, ⋯  , k}. Using the recipe provided in Lemma 1, in every sub-test the post-measurement states are self-tested, and using Lemma 2 also the observables corresponding to the inputs ♢ and ♦ of the tested parties. For the sake of illustration, Fig. 2 displays the construction in the case of five-partite states.

Fig. 2: A schematic representation of the self-testing scenario for five-partite states.
Fig. 2: A schematic representation of the self-testing scenario for five-partite states.The alternative text for this image may have been generated using AI.
Full size image

Every party in a given sub-test plays one of two roles: a projecting party (blue) is a party whose local dichotomic measurement denoted with lozenge is used to collapse the global state and thereby prepare different two-party reduced states for certification; a tested party (purple) is one of the two parties whose post-measurement two-party state is being certified in that sub-test. In each sub-test for a five-partite state, three projecting parties receive the lozenge-input,and the resulting projected states of the two tested parties are then self-tested. The correlations corresponding to the output 1 of a projecting party contribute to the self-testing of the tested parties' states only in sub-tests that occur before the projecting party itself assumes the role of a tested party. In this case, the first sub-test self-tests 23 distinct states, corresponding to 23 different global outputs from the projecting parties. In the second sub-test, this number reduces to 22, then to 2 in the third sub-test, and finally, the last sub-test self-tests a single state corresponding to one specific global output.

The total number of measurement settings required for this self-testing procedure depends on the party. Let us analyze this, focusing on the number of different inputs needed by each party beyond the ♢ and ♦ measurements common to each party:

  • Party n is tested only in the final sub-test. Its projected sub-state is self-tested only when all projecting parties obtain the outcome 0 for their ♢-measurements. To reproduce the required correlations (as specified in Lemma 1), party n needs 6 additional measurements, while party 1 requires 3 measurements.

  • Party n − 1 is a tested party only in a sub-test where its shared state with party 1 is self-tested for both outcomes of party n’s ♢-measurement (and only outputs 0 of the other projecting parties). This means that two different states are self-tested, requiring 6 measurements for one state and 3 for the other, totaling 9 measurements for both tested parties.

  • Continuing this analysis, party n − j requires 9 ⋅ 2(j−1) measurements because its shared state with party 1 is tested for 2j global outcomes of the remaining parties. Each tested state alternates between requiring 6 and 3 measurements.

  • The total number of measurements required by party 1 is the sum of its measurements across all sub-tests discussed above, which equals 3 + 9(2n−2 − 1).

The full proof of Theorem 3, including constructive details of the procedure, is provided in Supplementary note 5. Observables corresponding to dichotomic ♢ and ♦ measurements are used to construct a SWAP isometry. After taking into account all the results from different sub-tests, the output of the SWAP isometry has the form

$$\left|\xi \right\rangle \otimes \left|\Psi \right\rangle+\left|\xi ^{\prime} \right\rangle \otimes \left|{\Psi }^{*}\right\rangle,$$

which completes the proof of the theorem, c.f. Eq. (4) with p = 〈ξ∣ξ〉 and \(1-p=\langle \xi ^{\prime} | \xi ^{\prime} \rangle\).

Discussion

We have shown that all pure entangled states of n qubits (for any n ≥ 2) can be self-tested. We did so by explicitly describing a “classical fingerprint”, or self-testing correlations, on n parties for each such state. Our result opens the door to the possibility of certifying large entangled states experimentally using single-qubit measurements in a completely device-independent way, i.e. placing no trust in the experimental setup. An important direction for future work is to study “robustness”. Our self-test works in the exact setting, namely it establishes that there is a unique state that exactly produces the corresponding self-testing correlations. However, can one come up with a test that distinguishes states that are “ϵ-close” to ideal from states that are “2ϵ-far” from ideal? We expect that, given the universality of our test, one might have to settle for a less desirable parameter regime, where the robustness incurs a dimension dependent loss. Either way, understanding the landscape of robustness is a key question for the use of such tests in practice.

Another direction for future work is to reduce the number of possible measurements that each party could perform. Our self-tests require O(n)-bit questions for each party (thus exponentially many possible measurements). For example, alternative methods for self-testing partially entangled qubit pairs, such as those introduced in refs. 42 and 43, could be employed to reduce the number of measurements. Furthermore, for highly structured families of states, the number of measurements per party can be reduced to a constant, as demonstrated for W-states and graph states in ref. 26. It is an interesting question to understand if there are inherent limitations to reducing the number of measurements in “universal” and constructive self-tests like ours, or an exponentially large number of measurements is necessary.

Finally, an interesting open problem is whether the method developed here can be adapted to obtain a self-testing procedure for all multipartite qudit states. On the one hand, we believe such a generalization is plausible: the qubit-based method relies on a modular construction that reduces the multipartite self-test to appropriately chosen self-tests of partially entangled two-qubit pairs, together with a tomographically complete set of binary measurements on each side. On the other hand, extending this to qudits would require analogous building blocks, namely self-tests for partially entangled qudit pairs, with sufficiently rich measurement sets. In the current literature, self-testing of non-maximally entangled qudit pairs remains highly limited in this sense: the only existing result24 restricts to a very narrow class of block-diagonal measurements, which are far from being tomographically complete. Hence, bridging this gap and constructing suitable qudit-pair self-tests with general measurements appears as an important step towards the fully general setting of qudit multipartite self-testing.