What is the minimum CHSH score certifying that a state resembles the singlet?

Xavier Valcarce1, Pavel Sekatski1, Davide Orsucci1, Enky Oudot1,2, Jean-Daniel Bancal1,2, and Nicolas Sangouard1

1Departement Physik, Universität Basel, Klingelbergstraße 82, 4056 Basel, Schweiz
2Département de Physique Appliquée, Université de Genève, 1211 Genève, Suisse

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A quantum state can be characterized from the violation of a Bell inequality. The well-known CHSH inequality for example can be used to quantify the fidelity (up to local isometries) of the measured state with respect to the singlet state. In this work, we look for the minimum CHSH violation leading to a non-trivial fidelity. In particular, we provide a new analytical approach to explore this problem in a device-independent framework, where the fidelity bound holds without assumption about the internal working of devices used in the CHSH test. We give an example which pushes the minimum CHSH threshold from $\approx2.0014$ to $\approx2.05,$ far from the local bound. This is in sharp contrast with the device-dependent (two-qubit) case, where entanglement is one-to-one related to a non-trivial singlet fidelity. We discuss this result in a broad context including device-dependent/independent state characterizations with various classical resources.

Alice and Bob found a company selling sources producing entanglement states. They decide to buy the one producing two-qubit maximally entangled states. A few weeks later, they receive a black box and they want to convince themselves that this black box indeed produces the desired state. On paper, they could open the box, find out about the physics describing their functioning and deduce the structure of produced states. The problem is that the company has placed a warranty seal on the box. On the bright side, however, the company provided two measurement boxes with the source, each with two buttons and a screen displaying either 0 or 1 when a state is received and a button is pressed. How can Alice and Bob use them to satisfy their curiosity?

In quantum physics, a method known as self-testing can be used to know the functioning of a black box, and relies on Bell non-locality. In particular, when the goal is to certify that the box produces two-qubit maximally entangled states, Alice and Bob can perform the famous CHSH test. To do so Alice and Bob repeatedly measure their part of the state by randomly pressing one of the two buttons on their measurement boxes. Then they meet and combine all their measurement results in a single number called the CHSH score. If this score is larger than 2, Alice and Bob can easily conclude that the box produces some entanglement. Moreover, when the CHSH score is equal to $2\sqrt{2}$, the state shared by Alice and Bob as well as the actions performed locally by their measurement boxes can be fully determined. In particular, it can be concluded that the state produced by the entanglement box is indeed the two-qubit maximally entangled state -- up to local unitaries.

In the real world however, even if the boxes are designed for it, achieving $2\sqrt{2}$ is impossible due to experimental imperfections, noise, finite number of experimental runs, and so on... In this case, Alice and Bob can use the observed CHSH score to infer the fidelity of the produced state with respect to a two-qubit maximally entangled state. It has been recently proven that a non-trivial self-testing fidelity can be obtained as long as the CHSH score is greater or equal to $\approx2.11$. It is not clear however, if a self-testing statement can be obtained for lower CHSH scores. What is the minimum CHSH score certifying that a given source produces a state that resembles a two-qubit maximally entangled state?

By constructing an example of a state that does not resemble the singlet but leads to a CHSH score of $\approx2.05,$ this manuscript partially answers this natural question and raises many more.

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Cited by

[1] Xavier Valcarce, Julian Zivy, Nicolas Sangouard, and Pavel Sekatski, "Self-testing two-qubit maximally entangled states from generalized Clauser-Horne-Shimony-Holt tests", Physical Review Research 4 1, 013049 (2022).

[2] Jean-Daniel Bancal, Kai Redeker, Pavel Sekatski, Wenjamin Rosenfeld, and Nicolas Sangouard, "Self-testing with finite statistics enabling the certification of a quantum network link", Quantum 5, 401 (2021).

[3] M. Ho, P. Sekatski, E. Y.-Z. Tan, R. Renner, J.-D. Bancal, and N. Sangouard, "Noisy Preprocessing Facilitates a Photonic Realization of Device-Independent Quantum Key Distribution", Physical Review Letters 124 23, 230502 (2020).

[4] Pavel Sekatski, Jean-Daniel Bancal, Xavier Valcarce, Ernest Y.-Z. Tan, Renato Renner, and Nicolas Sangouard, "Device-independent quantum key distribution from generalized CHSH inequalities", Quantum 5, 444 (2021).

[5] George Cowperthwaite and Adrian Kent, "Comparing Singlet Testing Schemes", arXiv:2211.13750, (2022).

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