Abstract We develop explicit device-independent self-testing constructions for logical subspaces of quantum error-correcting codes. By constructing multipartite Bell inequalities tailored to stabilizer code structures, we show that maximal quantum violations uniquely certify that the underlying shared state resides within the logical subspace of a given code, up to local isometries. These constructions extend earlier stabilizer-subspace self-testing results beyond the five-qubit code to several representative stabilizer codes, including the six-qubit code, the Steane code, and Shor’s nine-qubit code. The tilted sum-of-squares variants further impose algebraic constraints on the relative weights of the logical sectors. Together, these results provide explicit examples of device-independent self-testing for representative stabilizer-code logical subspaces and suggest possible directions for verification and benchmarking of encoded quantum information.
Abstract We develop explicit device-independent self-testing constructions for logical subspaces of quantum error-correcting codes. By constructing multipartite Bell inequalities tailored to stabilizer code structures, we show that maximal quantum violations uniquely certify that the underlying shared state resides within the logical subspace of a given code, up to local isometries. These constructions extend earlier stabilizer-subspace self-testing results beyond the five-qubit code to several representative stabilizer codes, including the six-qubit code, the Steane code, and Shor’s nine-qubit code. The tilted sum-of-squares variants further impose algebraic constraints on the relative weights of the logical sectors. Together, these results provide explicit examples of device-independent self-testing for representative stabilizer-code logical subspaces and suggest possible directions for verification and benchmarking of encoded quantum information.
Self-testing provides a device-independent framework for certifying quantum properties based solely on input-output statistics while treating quantum devices as black boxes. It has evolved significantly from its origins in bipartite systems to applications in multipartite entanglement and, more recently, genuinely entangled subspaces. Notably, It has been revealed that the logical subspaces of numerous stabilizer quantum error correction codes are exclusively composed of genuinely multipartite entangled states, opening new avenues for developing device-independent tools to characterize these subspaces. In this work, we leverage the self-testing technique to certify genuinely entangled logical subspaces within the five-qubit code using both photonic and superconducting platforms. This is achieved by preparing informationally complete logical states, simulating Pauli errors on a physical qubit, and testing several stabilizer-formalized Bell inequalities. Our certification is supported by an extractability measure of at least0.828±0.006and0.621±0.007for the photonic and superconducting systems, respectively. Our results demonstrate the feasibility of device-independent certification of general entangled quantum structures in experimental settings, extending beyond quantum states and quantum measurements.
Abstract Self-testing was originally introduced as a device-independent method of certification of entangled quantum states and local measurements performed on them. Recently, in (Baccari et al 2020 Phys. Rev. Lett. 125 260507) the notion of state self-testing has been generalized to entangled subspaces and the first self-testing strategies for exemplary genuinely entangled subspaces have been given. The main aim of our work is to pursue this line of research and to address the question how ‘large’ (in terms of dimension) are genuinely entangled subspaces that can be self-tested, concentrating on the multiqubit stabilizer formalism. To this end, we first introduce a framework allowing to efficiently check whether a given stabilizer subspace is genuinely entangled. Building on it, we then determine the maximal dimension of genuinely entangled subspaces that can be constructed within the stabilizer subspaces and provide an exemplary construction of such maximally-dimensional subspaces for any number of qubits. Third, we construct Bell inequalities that are maximally violated by any entangled state from those subspaces and thus also any mixed states supported on them, and we show these inequalities to be useful for self-testing. Interestingly, our Bell inequalities allow for identification of higher-dimensional face structures in the boundaries of the sets of quantum correlations in the simplest multipartite Bell scenarios in which every observer performs two dichotomic measurements.
Quantum nonlocality based on the violation of Bell-like inequalities constitutes a fundamental feature of quantum physics and drives the development of device-independent (DI) quantum information technologies. Existing studies of Bell nonlocality have mainly focused on physical qubit systems, where the observed nonlocal correlations are directly encoded in physical degrees of freedom. The decoherence sensitivity of Bell nonlocality largely limits the performance and security of its DI applications. Here, we investigate the robust logical Bell nonlocality based on quantum error correction codes. We construct the general logical Bell inequality in the stabilizer coding subspace and prove its violation indicates the global nonlocal feature of the logical system. Then, we indicate that the logical Bell nonlocality is robust against decoherence. Comparing with the physical qubit system, the fidelity thresholds for the logical Bell inequality violation based on the [[3,1,1]] and [[7,1,1]] repetition codes under the bit-flip error model can be reduced from 82.8% to 73.10% and 66.35%, increasing DI QKD's bit-flip noise threshold from 10.64% to 14.42% and 23.36%, respectively. Such stabilizer-based framework can be also used to characterize the multipartite logical Bell nonlocality in principle. Finally, a logical Bell test implementation circuit based on the [[3,1,1]] repetition code is presented. This work provides a feasible avenue for unlocking robust Bell nonlocality in scalable logical quantum systems and facilitates its applications in future scalable quantum network.
We present a generalization of the tilted Bell inequality for quantum [[n,k,d]] error-correcting codes and explicitly utilize the simplest perfect code, the [[5,1,3]] code, the Steane [[7,1,3]] code, and Shor's [[9,1,3]] code, to demonstrate the self-testing property of their respective codespaces. Additionally, we establish a framework for the proof of self-testing, as detailed in \cite{baccari2020device}, which can be generalized to the codespace of CSS stabilizers. Our method provides a self-testing scheme for cosθ∣0ˉ⟩+sinθ∣1ˉ⟩, where θ∈[0,2π], and also discusses its experimental application. We also investigate whether such property can be generalized to qudit and show one no-go theorem. We then define a computational problem called ISSELFTEST and describe how this problem formulation can be interpreted as a statement that maximal violation of a specific Bell-type inequality can self-test a particular entanglement subspace. We also discuss the computational complexity of ISSELFTEST in comparison to other classical complexity challenges and some related open problems.
Abstract We address the task of verifying whether a quantum computer, designed to be protected by a specific stabilizer code, correctly encodes the corresponding logical qubits. To achieve this, we develop a general framework for subspace verification and explore several practically significant stabilizer code subspaces. First, we present two efficient verification strategies for general stabilizer code subspaces, using measurements of their stabilizer generators and stabilizer groups, respectively. Then, building on the observation that certain tests can be conducted in parallel due to specific structural properties of the subspace, we propose a coloring strategy tailored to graph code subspaces and an XZ strategy for Calderbank–Shor–Steane code subspaces. Compared to stabilizer-based strategies, these new strategies require significantly fewer measurement settings and consume fewer state copies, achieving near-global optimality. Notably, all the strategies employ a limited number of Pauli measurements, are non-adaptive, and work for mixed states, enabling efficient experimental certification of whether quantum states remain within the target code subspace in noisy quantum computers. This work presents the first systematic study of efficient verification of stabilizer code subspaces with local measurements.
Protecting information in systems that have more than two basis states (qudits) not only offers a promising route for reducing the number of individual quantum locations that must be protected, while more accurately reflecting the structure of realistic quantum hardware, but also has some possibly enticing foundational strengths. While work in the past has largely focused on protecting information in quantum devices with locations that are some consistent local structure, this work considers coding-theoretic constraints on devices constructed from locations which may vary in their local structures -- these are mixed-register quantum devices. In this work we provide some general results for mixed-register Pauli operators, then identify some stabilizer encoded information forms that are forbidden. Building on these insights, we construct coding-theoretically optimal mixed-register stabilizer codes from sets of codes defined on coprime local-dimensions. The construction of such codes results in codes with logical subspaces that do not directly correspond to any of the constituent local-dimensions.