Abstract Self-testing is a promising approach to certifying quantum states or measurements. Originally, it relied solely on the outcome statistics of the measurements involved in a device-independent (DI) setup. Extra physical assumptions about the system make the setup semi-DI. In the latter approach, we consider a prepare-and-measure scenario in which the dimension of the mediating particle is assumed to be two. In a setup involving four (three) preparations and three (two) projective measurements in addition to the target, we exemplify how to self-test any four- (three-) outcome extremal positive operator-valued measure using a linear witness. One of our constructions also achieves self-testing of any number of states with the help of as many projective measurements as the dimensionality of the space spanned by the corresponding Bloch vectors. These constructions are conjectured to be minimal in terms of the number of preparations and measurements required. In addition, we implement one of our prepare-and-measure constructionson IBM and IonQ quantum processors and certify the existence of a complex qubit Hilbert space based on the data obtained from these experiments.
Abstract Self-testing is a promising approach to certifying quantum states or measurements. Originally, it relied solely on the outcome statistics of the measurements involved in a device-independent (DI) setup. Extra physical assumptions about the system make the setup semi-DI. In the latter approach, we consider a prepare-and-measure scenario in which the dimension of the mediating particle is assumed to be two. In a setup involving four (three) preparations and three (two) projective measurements in addition to the target, we exemplify how to self-test any four- (three-) outcome extremal positive operator-valued measure using a linear witness. One of our constructions also achieves self-testing of any number of states with the help of as many projective measurements as the dimensionality of the space spanned by the corresponding Bloch vectors. These constructions are conjectured to be minimal in terms of the number of preparations and measurements required. In addition, we implement one of our prepare-and-measure constructionson IBM and IonQ quantum processors and certify the existence of a complex qubit Hilbert space based on the data obtained from these experiments.
Self-testing represents the strongest form of certification of a quantum system. Here, we theoretically and experimentally investigate self-testing of nonprojective quantum measurements. That is, how can one certify, from observed data only, that an uncharacterized measurement device implements a desired nonprojective positive-operator valued measure (POVM). We consider a prepare-and-measure scenario with a bound on the Hilbert space dimension and develop methods for (i) robustly self-testing extremal qubit POVMs and (ii) certifying that an uncharacterized qubit measurement is nonprojective. Our methods are robust to noise and thus applicable in practice, as we demonstrate in a photonic experiment. Specifically, we show that our experimental data imply that the implemented measurements are very close to certain ideal three- and four-outcome qubit POVMs and hence non-projective. In the latter case, the data certify a genuine four-outcome qubit POVM. Our results open interesting perspective for semi-device-independent certification of quantum devices.
Self-testing is a powerful method for certifying quantum systems. Initially proposed in the device-independent (DI) setting, self-testing has since been relaxed to the semi-device-independent (semi-DI) setting. In this study, we focus on the self-testing of a specific type of nonprojective qubit measurements belonging to a one-parameter family, using the semi-DI prepare-and-measure (PM) scenario. Remarkably, we identify the simplest PM scenario discovered so far, involving only four preparations and four measurements, for self-testing the fourth measurement. This particular measurement is a four-outcome nonprojective positive operator-valued measure (POVM) and falls in the class of semisymmetric informationally complete POVMs introduced by Geng et al. [Phys. Rev. Lett. 126, 100401 (2021)]. To achieve this, we develop analytic techniques for semi-DI self-testing in the PM scenario. Our results pave the way toward self-testing any extremal qubit POVM within a potentially minimal PM scenario.
The number of outcomes is a defining property of a quantum measurement, in particular, if the measurement cannot be decomposed into simpler measurements with fewer outcomes. Importantly, the number of outcomes of a quantum measurement can be irreducibly higher than the dimension of the system. The certification of this property is possible in a semi-device-independent way either based on a Bell-like scenario or by utilizing the simpler prepare-and-measure scenario. Here we show that in the latter scenario the minimal scheme for certifying an irreducible three-outcome qubit measurement requires three state preparations and only two measurements and we provide experimentally feasible examples for this minimal certification scheme. We also discuss the dimension assumption characteristic of the semi-device-independent approach and to what extent it can be mitigated.
Self-testing is the high-level security verification of a claimed quantum device, confirming the quantum states prepared in the device and the measurements performed based solely on the observed statistics. The statistical correlations can realize the self-testing of the quantum system in the preparing-and-measuring scenario. However, most of previous studies focused on the self-testing of shared entangled states between devices, at present only a few researches are presented and the existing work can only simultaneously self-test the states and measurements when some witness inequalities reach a maximum violation. We focus on four-state preparation and the selected scenarios of two measurements. In this scenario, Armin Tavakoli et al. [Tavakoli A, Kaniewski J, Vértesi T, Rosset D, Brunner N 2018 Phys. Rev. A 98 062307] have put forward a criterion based on the dimensional witness violation inequality which can achieve BB84 particles and corresponding Pauli measurements. However, in addition to the maximum violation of the inequality, any statistics with deviation from the maximum deviation cannot be self-tested. Besides, only the BB84 particle preparation and measurements system can be self-tested with that criterion, resulting in a large number of four-state preparation and two measurement systems that cannot be self-tested. Therefore, in this work, in addition to the maximum violation of that dimension inequality, we directly focus on the full observed statistics and further propose some new criteria for self-testing qubit quantum systems in the preparing-and-measiuring scenarios. And the self-testing criteria are proven in an ideal case. We construct a local isometry by using the constructions commonly used in device-independent cases, exchange the target system with the additional system, and realize the self-testing of more qubit state sets and measurement sets than BB84 particles. This meets the requirements for practical experiments to realize various tasks by different quantum state sets. In addition, we perform a robust analysis of the proposed criteria and use fidelity to describe the closeness of the state to the ideal state of the auxiliary system. Finally, an improved dimensional-dependent NPA method is used to optimize the lower bound of the robustness, making the new criteria practical under experimental noise. We use the YALIMP software package in MATLAB and the solver SEDUMI to solve this optimization problem. The present research increases the diversity of qubit state preparations and self-testing of measurement system, which is beneficial to the actual self-testing of different non-entangled single quantum systems.
We consider the problem of certification of arbitrary ensembles of pure states and projective measurements solely from the experimental statistics in the prepare-and-measure scenario assuming the upper bound on the dimension of the Hilbert space. To this aim, we propose a universal and intuitive scheme based on establishing perfect correlations between target states and suitably-chosen projective measurements. The method works in all finite dimensions and allows for robust certification of the overlaps between arbitrary preparation states and between the corresponding measurement operators. Finally, we prove that for qubits, our technique can be used to robustly self-test arbitrary configurations of pure quantum states and projective measurements. These results pave the way towards the practical application of the prepare-and-measure paradigm to certification of quantum devices.
Characterizing large quantum systems with minimal assumptions is a central challenge in quantum information science. Self-testing provides the strongest form of certification by identifying the underlying quantum state solely from observed measurement statistics. However, existing self-testing methods for generic n-partite states face a scalability barrier, requiring exponentially many samples in the system size. In this work, we overcome this barrier by introducing a protocol that robustly self-tests almost all n-qubit states with only polynomial sample complexity. The key ingredient is an efficient scheme for device-independently evaluating multipartite Pauli measurements, which can be implemented using only a linear number of ancillary Bell pairs together with standard projective and Bell measurements, well within the reach of current quantum technology. Beyond self-testing states, our scheme provides a general framework for implementing a wide range of learning and certification protocols in the device-independent setting, thereby opening a scalable route to device-independent quantum information processing in large-scale quantum networks.