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arXiv:1203.2976 · Journal of Physics A Mathematical and Theoretical

Robust self-testing of the singlet

Matthew McKague, T. H. Yang, Valerio Scarani

Published Oct 19, 2012• 227 citations
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Abstract

In this paper, we introduce a general framework to study the concept of robust self-testing which can be used to self-test maximally entangled pairs of qubits (EPR pairs) and local measurement operators. The result is based only on probabilities obtained from the experiment, with tolerance to experimental errors. In particular, we show that if the results of an experiment approach the Cirel'son bound, or approximate the Mayers–Yao-type correlations, then the experiment must contain an approximate EPR pair. More specifically, there exist local bases in which the physical state is close to an EPR pair, possibly encoded in a larger environment or ancilla. Moreover, in these bases the measurements are close to the qubit operators used to achieve the Cirel'son bound or the Mayers–Yao results.

DOI
10.1088/1751-8113/45/45/455304

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In this paper, we introduce a general framework to study the concept of robust self-testing which can be used to self-test maximally entangled pairs of qubits (EPR pairs) and local measurement operators. The result is based only on probabilities obtained from the experiment, with tolerance to experimental errors. In particular, we show that if the results of an experiment approach the Cirel'son bound, or approximate the Mayers–Yao-type correlations, then the experiment must contain an approximate EPR pair. More specifically, there exist local bases in which the physical state is close to an EPR pair, possibly encoded in a larger environment or ancilla. Moreover, in these bases the measurements are close to the qubit operators used to achieve the Cirel'son bound or the Mayers–Yao results.
Self-testing refers to the possibility of characterizing uniquely (up to local isometries) the state and measurements contained in quantum devices, based only on the observed input-output statistics. Already in the basic case of the two-qubit singlet, self-testing is not unique: the two known criteria (the maximal violation of the CHSH inequality and the Mayers–Yao correlations) are not equivalent. It is unknown how many criteria there are. In this paper, we find the whole set of criteria for the ideal self-testing of a singlet with two measurements and two outcomes on each side; it coincides with all the extremal points of the quantum set that can be obtained by measuring the singlet.
Self-testing is a method with which a classical user can certify the state and measurements of quantum systems in a device-independent way. In particular, self-testing of entangled states is of great importance in quantum information processing. An understandable example is that the maximal violation of the Clauser-Horne-Shimony-Holt inequality necessarily implies that the bipartite system shares a singlet. One essential question in self-testing is that, when one observes a nonmaximum violation, how far is the tested state from the target state (which maximally violates a certain Bell inequality)? The answer to this question describes the robustness of the used self-testing criterion, which is highly important in a practical sense. Recently, J. Kaniewski derived two analytic self-testing bounds for bipartite and tripartite systems. In this Letter, we experimentally investigate these two bounds with high-quality two-qubit and three-qubit entanglement sources. The results show that these bounds are valid for various entangled states that we prepared. Thereby, a proof-of-concept demonstration of robust self-testing is achieved, which improves on the previous results significantly.
Self-testing refers to a way of uniquely identifying an uncharacterized quantum device based solely on certain extremal quantum correlations. It is known that in the ideal case of a two-qubit singlet all the extremal points that can be obtained by measuring the singlet form the whole criteria for self-testing the singlet and the associated measurements. To make these self-testing criteria more suitable for practical applications, relevant robustness bounds are necessary to discuss. However, the previous robustness analyses for these criteria were obtained by the swap method, which may not be guaranteed to be tight. In this paper, we construct a general extraction map applying to all self-testing criteria of the singlet state with two binary measurements for each party and derive analytic robustness bounds for a self-testing singlet in different regions of the boundary of the extremal quantum correlation set. The comparison shows that our robustness bounds are better than the known results and close to optimal for these self-testing criteria of a singlet state with two binary measurements.
The verification of quantum devices is an important aspect of quantum information, especially with the emergence of more advanced experimental implementations of quantum computation and secure communication. Within this, the theory of device-independent robust self-testing via Bell tests has reached a level of maturity now that many quantum states and measurements can be verified without direct access to the quantum systems: interaction with the devices is solely classical. However, the requirements for this robust level of verification are daunting and require high levels of experimental accuracy. In this paper we discuss the possibility of self-testing where we only have direct access to one part of the quantum device. This motivates the study of self-testing via EPR-steering, an intermediate form of entanglement verification between full state tomography and Bell tests. Quantum non-locality implies EPR-steering so results in the former can apply in the latter, but we ask what advantages may be gleaned from the latter over the former given that one can do partial state tomography? We show that in the case of self-testing a maximally entangled two-qubit state, or ebit, EPR-steering allows for simpler analysis and better error tolerance than in the case of full device-independence. On the other hand, this improvement is only a constant improvement and (up to constants) is the best one can hope for. Finally, we indicate that the main advantage in self-testing based on EPR-steering could be in the case of self-testing multi-partite quantum states and measurements. For example, it may be easier to establish a tensor product structure for a particular party’s Hilbert space even if we do not have access to their part of the global quantum system.
Bell state measurements, of which the eigenvectors are in an entangled form, are crucial resources in the construction of quantum networks. Therefore, device-independent certification of a Bell state measurement has significance in the quantum information process because it satisfies the exact demand on security. In this study, we implement a proof-of-concept experiment to certify a Bell state measurement device independently in an entanglement swapping process, namely, self-testing. Instead of preparing a tensor product of two singlets with four photons, multiplex encoding in polarization and spatial modes is utilized to produce two pairs of entangled qubits. As a result, we implement a full Bell state measurement and achieve a high degree of Bell violation on the remaining two qubits, which are required for nontrivial self-testing of a Bell state measurement. Furthermore, our results combine the correlations before and after the swapping; thus, the quality of the performed Bell state measurement can be precisely inferred.
Device-independent self-testing offers the possibility of certifying the quantum state and measurements, up to local isometries, using only the statistics observed by querying uncharacterized local devices. In this paper we study parallel self-testing of two maximally entangled pairs of qubits; in particular, the local tensor product structure is not assumed but derived. We prove two criteria that achieve the desired result: a double use of the Clauser-Horne-Shimony-Holt inequality and the $3\ifmmode\times\else\texttimes\fi{}3$ magic square game. This demonstrate that the magic square game can only be perfectly won by measuring a two-singlet state. The tolerance to noise is well within reach of state-of-the-art experiments.