Sum-of-squares decompositions for a family of Clauser-Horne-Shimony-Holt-like inequalities and their application to self-testing

Bamps, Cédric; Pironio, Stefano

doi:10.1103/physreva.91.052111

Cited by 156 publications

(252 citation statements)

References 13 publications

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“…This is similar to the works of [11,[36][37][38][39][40][41][42] and in fact we adopt a similar notation to that of [36]. The main difference with respect to those works, is that in our case we trust Alice's measurements.…”

Section: State and Strategy Certification Via Steeringsupporting

confidence: 49%

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Rigidity of quantum steering and one-sided device-independent verifiable quantum computation

Gheorghiu¹,

Wallden²,

Kashefi³

2017

Quantum Information and Measurement (QIM) 2017

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The relationship between correlations and entanglement has played a major role in understanding quantum theory since the work of Einstein et al (1935 Phys. Rev. 47 777-80). Tsirelson proved that Bell states, shared among two parties, when measured suitably, achieve the maximum non-local correlations allowed by quantum mechanics (Cirel'son 1980 Lett. Math. Phys. 4 93-100). Conversely, Reichardt et al showed that observing the maximal correlation value over a sequence of repeated measurements, implies that the underlying quantum state is close to a tensor product of maximally entangled states and, moreover, that it is measured according to an ideal strategy (Reichardt et al 2013 Nature 496 456-60). However, this strong rigidity result comes at a high price, requiring a large number of entangled pairs to be tested. In this paper, we present a significant improvement in terms of the overhead by instead considering quantum steering where the device of the one side is trusted. We first demonstrate a robust one-sided device-independent version of self-testing, which characterises the shared state and measurement operators of two parties up to a certain bound. We show that this bound is optimal up to constant factors and we generalise the results for the most general attacks. This leads us to a rigidity theorem for maximal steering correlations. As a key application we give a onesided device-independent protocol for verifiable delegated quantum computation, and compare it to other existing protocols, to highlight the cost of trust assumptions. Finally, we show that under reasonable assumptions, the states shared in order to run a certain type of verification protocol must be unitarily equivalent to perfect Bell states.

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Section: State and Strategy Certification Via Steeringsupporting

confidence: 49%

“…Thus, a similar theorem can be proven in the case where both Alice and Bob are untrusted. In that case, one could simply use the self-testing results of [12,36,41,42] for the i.i.d. setting, and then obtain a statement about the closeness of a typical state to the ideal one in the non-i.i.d.…”

Section: Removing the Independence Assumptionmentioning

confidence: 99%

Rigidity of quantum steering and one-sided device-independent verifiable quantum computation

Gheorghiu¹,

Wallden²,

Kashefi³

2017

Quantum Information and Measurement (QIM) 2017

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“…To prove these relations, let   · stand for the vector norm defined as y yy ñ = á ñ   | | . Then, the following reasoning applies [20]      y y y y y…”

Section: Exact Casementioning

confidence: 99%

“…Exploiting equations (18) and (20) to convertZ B to ¢ Z B and then ¢ Z B to ¢ Z A , and the fact that ¢ Z A has eigenvalues ±1, meaning that  + ¢ Z A ( ) and  -¢ Z A ( ) are projectors onto orthogonal subspaces, one finds that the terms in equation (27) containing the ancillary vectors ñ 01 | and ñ 10 | simply vanish, and the whole expression simplifies to…”

Section: Exact Casementioning

confidence: 99%

Self-testing protocols based on the chained Bell inequalities

et al. 2016

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Self-testing is a device-independent technique based on non-local correlations whose aim is to certify the effective uniqueness of the quantum state and measurements needed to produce these correlations. It is known that the maximal violation of some Bell inequalities suffices for this purpose. However, most of the existing self-testing protocols for two devices exploit the well-known Clauser-Horne-Shimony-Holt Bell inequality or modifications of it, and always with two measurements per party. Here, we generalize the previous results by demonstrating that one can construct self-testing protocols based on the chained Bell inequalities, defined for two devices implementing an arbitrary number of two-output measurements. On the one hand, this proves that the quantum state and measurements leading to the maximal violation of the chained Bell inequality are unique. On the other hand, in the limit of a large number of measurements, our approach allows one to self-test the entire plane of measurements spanned by the Pauli matrices X and Z. Our results also imply that the chained Bell inequalities can be used to certify two bits of perfect randomness.As mentioned, the concept of self-testing was introduced by Mayers and Yao in [4], where the procedure to self-test a maximally entangled pair of qubits is described. This protocol was made robust in subsequent works, see [7,11]. In the following years new self-testing protocols for more complicated states such as graph states were described [10], as well as protocols for self-testing more complicated operations, such as entire quantum circuits [11]. A general numerical method for self-testing, known as the SWAP method, was introduced in [12], providing much better estimations of robustness than the analytical proofs. This numerical method can also be used to self-test three-qubit states such as GHZ states [13] and W states [14].Despite its importance, we lack general techniques to construct and prove self-testing protocols. Most of the existing examples are built from the maximal violation of a Bell inequality. Based on geometrical considerations, see for instance [15,16], one expects that generically there is a unique way, state and measurements, of producing the extremal correlations attaining the maximal quantum violation of a Bell inequality. This is not always the case, but whenever it is, we say that the corresponding Bell inequality is useful for self-testing. Following this approach, it is possible to prove that the state and measurements maximally violating the Clauser-Horne-Shimony-Holt (CHSH) inequality [25] are unique [5,7], and the corresponding state is a maximally entangled two-qubit state. More recently, a self-testing protocol for any two-qubit entangled states has been derived in [20] using the Bell inequalities introduced in [8], and all the self-testing configurations for a maximally entangled state of two qubits using two measurements of two outputs have been identified in [9]. From a general perspective, it is an interesting question to understand whic...

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“…The NS principle is just IC for m = 0 6. As pointed out by one referee, robust self-testing conditions for any partially entangled two-qubit states have been derived by Bamps and Pironio[34] (see also Ref [35]. for closely related work and references therein).…”

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confidence: 97%

Chained Clauser–Horne–Shimony–Holt inequality for Hardy’s ladder test of nonlocality

Cereceda

2015

Quantum Inf Process

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Relativistic causality forbids superluminal signaling between distant observers. By exploiting the non-signaling principle, we derive the exact relationship between the chained Clauser-Horne-Shimony-Holt sum of correlations CHSH K and the success probability P K associated with Hardy's ladder test of nonlocality for two qubits and K + 1 observables per qubit. Then, by invoking the Tsirelson bound for CHSH K , the derived relationship allows us to establish an upper limit on P K . In addition, we draw the connection between CHSH K and the chained version of the Clauser-Horne (CH) inequality.

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Sum-of-squares decompositions for a family of Clauser-Horne-Shimony-Holt-like inequalities and their application to self-testing

Cited by 156 publications

References 13 publications

Rigidity of quantum steering and one-sided device-independent verifiable quantum computation

Rigidity of quantum steering and one-sided device-independent verifiable quantum computation

Self-testing protocols based on the chained Bell inequalities

Chained Clauser–Horne–Shimony–Holt inequality for Hardy’s ladder test of nonlocality

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