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Reports on Progress in Physics

Certification of genuinely entangled subspaces of the five qubit code via robust self-testing

Yu Guo, Hao Tang, Jiaxuan Zhang, Jiale Miao, X. Q. Hu, Yu-Chun Wu, Guo‐Ping Guo, Yun‐Feng Huang, Chuan‐Feng Li, Guang‐Can Guo, Bi‐Heng Liu

Published May 1, 2025• 1 citation

Abstract

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.

DOI: 10.1088/1361-6633/add560

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Self-testing is a procedure for characterizing quantum resources with the minimal level of trust. Up to now it has been used as a device-independent certification tool for particular quantum measurements, channels, and pure entangled states. In this work we introduce the concept of self-testing more general entanglement structures. More precisely, we present the first self-tests of an entangled subspace-the five-qubit code and the toric code. We show that all quantum states maximally violating a suitably chosen Bell inequality must belong to the corresponding code subspace, which remarkably includes also mixed states.

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 is a procedure for characterising quantum resources with the minimal level of trust. Up to now it has been used as a device-independent certification tool for particular quantum measurements, channels and pure entangled states. In this work we introduce the concept of self-testing more general entanglement structures. More precisely, we present the first self-tests of an entangled subspace - the five-qubit code and the toric code. We show that all quantum states maximally violating a suitably chosen Bell inequality must belong to the corresponding code subspace, which remarkably includes also mixed states.

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.

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.

Quantum entanglement is the key resource for quantum information processing. Device-independent certification of entangled states is a long standing open question, which arouses the concept of self-testing. The central aim of self-testing is to certify the state and measurements of quantum systems without any knowledge of their inner workings, even when the used devices cannot be trusted. Specifically, utilizing Bell's theorem, it is possible to place a boundary on the singlet fidelity of entangled qubits. Here, beyond this rough estimation, we experimentally demonstrate a complete self-testing process for various pure bipartite entangled states up to four dimensions, by simply inspecting the correlations of the measurement outcomes. We show that this self-testing process can certify the exact form of entangled states with fidelities higher than 99.9% for all the investigated scenarios, which indicates the superior completeness and robustness of this method. Our work promotes self-testing as a practical tool for developing quantum techniques.