The 3d4d Bell scenario

We work in a bipartite Bell scenario, and we refer to Alice and Bob as operating the uncharacterized devices (or rather as the devices themselves). They receive inputs x and y, respectively, from the classical verifier, corresponding to their choice of measurement settings, and they return outcomes a and b respectively. In the particular scenario that we will consider, Alice has three possible measurement settings and Bob has four, while they have d possible outcomes each. So the inputs are x∈{0, 1, 2} and y∈{0, 1, 2, 3} and the outputs are a, b∈{0, 1, 2, ⋯, d−1}. We refer to this as a [{3, d},{4, d}] Bell scenario (Fig. 1a). The result of this Bell experiment can be fully described by the probabilities P(a, b|x, y) of obtaining a pair of outcomes a, b on measurement settings x, y. In the device-independent approach, the dimensionality of the measured system is not bounded a priori. Hence, the measurements made on the system can be assumed to be projective, with the projection corresponding to Alice obtaining outcome a on measurement setting x, and likewise for on Bob’s side. No further characterization of either the state or the measurements is required, and estimating the P(a, b|x, y) is all that has to be done in the lab.

Figure 1: The scheme of self-testing. (a) First, measurement inputs and outputs from a Bell experiment in a laboratory are recorded. (b) Using the recorded experimental data, one can estimate the correlations of the Bell experiment. (c) A local isometry Φ is constructed mathematically, as in the circuit diagram. Gates F and in this diagram denote the quantum Fourier transform and inverse quantum Fourier transform respectively. Gates R and S, which act jointly on and the ancillary system, are controlled unitaries defined precisely in the Supplementary Methods. (d) If one can show, using the correlations, that the local isometry is such that , then we conclude that the correlations self-test . Full size image

Self-testing of all pure bipartite entangled states

We state our main theorem.

Theorem 1: for every bipartite entangled state of qudits , there exist [{3, d},{4, d}] quantum correlations that, when reproduced by Alice and Bob through local measurements on a joint state ρ, imply the existence of a local isometry Φ such that , where is some auxiliary state. Moreover, under the isometry Φ, the local measurements on ρ are equivalent to measurements that act trivially on and as the ideal measurements on (described exactly in the Supplementary Methods).

The proof of Theorem 1 now proceeds at the mathematical level (Fig. 1c), and we provide an overview of the main ideas. The full details are contained in the Supplementary Methods. For ease of exposition, we take Alice and Bob’s shared state to be a pure state , but our proof goes through in the same way for a general ρ. Initially, the verifier has no knowledge about the state shared by the two devices, and he wishes to certify that it is a specific state of two qudits. We can think of providing Alice and Bob with a qudit each (A′ and B′), initialized in an arbitrary state ; then trying to swap information from the two black-boxes into these qudits. If at the end of the swap one finds , one concludes that the boxes contained the state ⊗ before the swap, where the precise state is not important, and is just ancillary. The physical and mathematical parts of self-testing are connected by the existence of a swap operation, which acts as desired thanks to the constraints given by the P(a, b|x, y). In mathematical terms, what we have just explained amounts to constructing a local isometry Φ such that . If such an isometry exists, one says that these correlations self-test . Invoking the Schmidt decomposition, self-testing all bipartite entangled states reduces to self-testing all states of the form.

where 0<c i <1 for all i and .

One may wonder whether mixed states could also be self-tested, that is, if some P(a, b|x, y) is uniquely compatible with a mixed state (or with its purified version, but with measurements acting trivially on the purifying system). The answer is negative: any P(a, b|x, y) produced by a bipartite mixed state can be reproduced by a bipartite pure state of the same dimension25. Hence, in the bipartite scenario, the best one can hope for is to self-test every pure state. To illustrate how we construct self-testing correlations for such a target state as in equation (1), we look at the case d=4, so that . We already know that with correlations having two inputs per party, one can self-test any two-qubit state (that is, d=2)17,18. So, the idea is that for x, y∈{0, 1}, we choose P(a, b|x, y) so that the probabilities for a, b∈{0, 1} certify , while those for a, b∈{2, 3} certify . All the other P(a, b|x, y), that is, those where (a, b) {0, 1}2∪{2, 3}2, are set to zero. Then, one similarly uses measurement settings x∈{0, 2} and y∈{2, 3}, but with a block structure certifying and .

In other words, our correlations rely on detecting a pattern of two-qubit correlations compatible exclusively with , across a suitable direct-sum decomposition of the Hilbert space in which the joint state lies. The recipe is clearly not restricted to d=4: with the same number of measurement settings, and naturally generalized block-diagonal correlations, one can self-test any bipartite entangled pure state of any dimension (see Fig. 2 for an illustration for d even; the argument carries on to d odd as well).

Figure 2: Block-diagonal correlations as two-qubit fingerprints. (a) In blue, the block-diagonal correlations for measurement settings x, y∈{0, 1} ‘certify’ the ‘even-odd’ pairs, while, in red, the block-diagonal correlations for measurement settings x∈{0, 2}, y∈{2, 3} certify the odd–even pairs. (b) The correlation table describes the structure of the block-diagonal correlations required for self-testing. The blocks in blue correspond to the correlations for measurement settings x, y∈{0, 1}, and the red blocks correspond to measurement settings x∈{0, 2}, y∈{2, 3}. Please refer to Supplementary Tables 1, 2, 6 and 7, for the full correlation tables. Full size image

Proof outline of Theorem 1

While the recipe is intuitive, the formal proof must follow the scheme illustrated in Fig. 1, and thus construct the local isometry. All the technical details are given in the Supplementary Methods, and here we outline how the proof proceeds.

First, we need to formalize the intuition that the two-qubit blocks are certified by the block-diagonal correlations described earlier. Consider the ‘tilted CHSH’ Bell-type inequality26

where x, y, a, b∈{0, 1}, α∈[0, 2), and . It is known, thanks to Yang and Navascués17, and Bamps and Pironio18, that maximal violation of this inequality, corresponding to , self-tests the state , with . However, when we try to apply this certification to each consecutive pair of two outcomes, we find that the value of the left hand side (LHS) of inequality (2) in each block, computed from the P(a, b|x, y) we described earlier, is the maximal violation multiplied by the probabilistic weight of that block: in other words, it is not the maximal violation itself. To recognize the covert maximal violation that indeed resides in each block, and the certification that follows from it, one has to realize that the state which achieves the maximal violation is not the joint state , but rather its projection onto each 2 × 2 block. From each such maximal violation, one can construct the four operators , , with support on the (2m, 2m+1) block (or , with support on the (2m+1, 2m+2) block), that are used in the self-testing isometry from Yang and Navascués17, and Bamps and Pironio18.

Second, one has to tie together the certifications in the different blocks, and explicitly construct the overall local isometry Φ such that . A sufficient condition for the existence of such an isometry has been formulated by Yang and Navascués17: one needs complete sets of orthogonal projections and and unitary operators satisfying the following conditions for all k=0, 1, ..., d−1:

where ω=e2πi/d. In our construction, the projections are chosen from Alice and Bob’s projection measurements, and each operator is the product of all the and (formally extended to the whole space, and denoted and respectively in the Supplementary Methods) covering all 2 × 2 blocks up to k. This product spans the alternating block structure: it is in these operators that the crucial connection between blocks is encoded. It is not difficult, finally, to extend the proof of self-testing to the ideal measurements (see the Supplementary Methods).