Weak form of self-testing

Abstract

The concept of self-testing (or rigidity) refers to the fact that for certain Bell inequalities the maximal violation can be achieved in an essentially unique manner. In this work we present a family of Bell inequalities which are maximally violated by multiple inequivalent quantum realizations. We completely characterize the quantum realizations achieving the maximal violation and we show that each of them requires a maximally entangled state of two qubits. This implies the existence of a new, weak form of self-testing in which the maximal violation allows us to identify the state, but does not fully determine the measurements. From the geometric point of view the set of probability points that saturate the quantum bound is a line segment. We then focus on a particular member of the family and show that the self-testing statement is robust, i.e., that observing a nonmaximal violation allows us to make a quantitative statement about the unknown state. To achieve this we present a new construction of extraction channels and analyze their performance. For completeness we provide two independent approaches: analytical and numerical. The noise robustness, i.e., the amount of white noise at which the bound becomes trivial, of the analytical bound is rather small (≈0.06%), but the numerical method takes us into an experimentally relevant regime (≈5%). We conclude by investigating the amount of randomness that can be certified using these Bell violations. Perhaps surprisingly, we find that the qualitative behavior resembles the behavior of rigid inequalities such as the Clauser-Horne-Shimony-Holt inequality. This shows that rigidity is not strictly necessary for device-independent applications.

Near-Maximal Two-Photon Entanglement for Optical Quantum Communication at 2.1μm

“…Most importantly, we give an experimental demonstration of a state corresponding to a positive secure key rate in the mid-infrared region in a DIQKD setting. We have shown that our source is capable of violating a Bell inequality for which a weak form of self-testing has recently been proven [28]. This type of self-testing allows the certification of the entangled state without certifying the implemented measurements.…”

“…We further exploit the quality of the entangled state and the measurement capabilities to demonstrate a previously unexplored quantum application-a weak form of self-testing recently derived in Ref. [28]. Self-testing [44][45][46] is a means for DI characterization of quantum devices by a classical user solely on the basis of observed non-local correlations, without requiring any assumptions about the devices under test [47].…”

“…Self-testing with such inequalities is rigid in the sense that it certifies both the quantum state measured and the measurement implemented by the device. The weak form of self-testing [28], which has not yet been addressed experimentally, now makes it possible to certify the quantum state without full determination of the measurements. This is an important result for DI quantum applications.…”

Self-testing quantum states via nonmaximal violation in Hardy's test of nonlocality

“…Then only from the input-output statistics termed as correlation or behavior of the box one can find the quantum state of the device. Such a certification is referred to as a self-test as it enables a user to verify the device without knowing details of its inner-workings [16][17][18][19][20][21][22][23][24][25][26][27][28].…”

“…Bell inequalities naturally fit into the deviceindependent paradigm for certification of quantum systems since their derivations, based on the assumptions of local-realism [29], are independent of a quantum description of the physical state of the system or measurements applied to it [30][31][32][33][34][35]. Therefore, various kind of Bell inequalities play a central role in the construction of self-testing protocols [16][17][18][19][20][21][22][23][24][25][26][27][28]. In a Bell test only some special type of input-output statistics can self-test a quantum state; the one which can be realized (up to local isometry) by a unique quantum state and measurements [28].…”