Certifying the Dimension of Classical and Quantum Systems in a Prepare-and-Measure Scenario with Independent Devices
Open accessAbstract
We consider the problem of testing the dimension of uncharacterized classical and quantum systems in a prepare-and-measure setup. Here we assume the preparation and measurement devices to be independent, thereby making the problem nonconvex. We present a simple method for generating nonlinear dimension witnesses for systems of arbitrary dimension. The simplest of our witnesses is highly robust to technical imperfections, and can certify the use of qubits in the presence of arbitrary noise and arbitrarily low detection efficiency. Finally, we show that this witness can be used to certify the presence of randomness, suggesting applications in quantum information processing.
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A positive value of the dimension witness could certify randomness in the independent two-dimensional subspace scenario, similar to how a Bell inequality violation certifies randomness in self-testing QRNG.
“A positive value of this dimension witness could certify randomness in this scenario, similar to the fact that a violation of the Bell inequality could certify randomness of self-testing QRNG in Section ' Self-testing QRNG ' .”
From Quantum random number generation (2016) · cites this paper as [86]
The randomness in the independent two-dimensional subspace scenario can be estimated by adopting a dimension witness.
“The randomness can be estimated by adopting a dimension witness. 86”
From Quantum random number generation (2016) · cites this paper as [86]
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“This topic connects to several active research areas, including self-testing (see e.g. [1, 2]), dimension witnesses (see e.g. [3, 4, 5, 6]) and foundational principles of quantum theory (see e.g. [7, 8]).”
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Classical model of a delayed-choice quantum eraser
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“In the dependentdevices model [45], Alice’s (Bob’s) state could be described as the input number x(y) and hidden variable λA(λB); neither of them can be controlled or known by Eve’s device.”
MethodPractical source-independent quantum random number generation with detector efficiency mismatch
Ma, D., Wang, Y., & Wei, K. (2020) · Quantum Information Processing · doi:10.1007/s11128-020-02865-5 · Cited by 11 · see citing works
Semi device independence of the BB84 protocol
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Experimental quantum randomness generation invulnerable to the detection loopholePreprint
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Closing the detection loophole in nonlinear entanglement witnesses
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Interplays between classical and quantum entanglement-assisted communication scenarios
Vieira, C., de Gois, C., Pollyceno, L. D. S., & Rabelo, R. (2023) · New Journal of Physics · doi:10.1088/1367-2630/ad0526 · Cited by 8 · see citing works
Effect of external magnetic fields on practical quantum random number generator
Li, Y., Fei, Y., Wang, W., Meng, X., Wang, H., Duan, Q.-H., Han, Y., & Ma, Z. (2023) · EPJ Quantum Technology · doi:10.1140/epjqt/s40507-023-00206-w · Cited by 8 · see citing works
“To trade off between generation rate and practical security, the semi-device-independent QRNG (SDI-QRNG) protocols are proposed, which mainly include source-device-independent [42–44], measurement-device-independent [45, 46] and dimension witness violation based QRNG protocols [15, 47–49].”
MethodPhase-Coding Self-Testing Quantum Random Number Generator
Song, X., Li, H.-W., Yin, Z., Liang, W.-Y., Zhang, C., Han, Y.-G., Chén, W., & Han, Z. (2015) · Chinese Physics Letters · doi:10.1088/0256-307x/32/8/080302 · Cited by 8 · see citing works
A general bound for the dimension of quantum behaviours in the prepare-and-measure scenario
de Vicente, J. I. (2019) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8121/aafde7 · Cited by 8 · see citing works
Certified randomness from a remote-state-preparation dimension witness
Xing, C., Redeker, K., Garthoff, R., Rosenfeld, W., Wrachtrup, J., & Gerhardt, I. (2021) · Physical Review A · doi:10.1103/physreva.103.042211 · Cited by 8 · see citing works
Wei, S., Guo, F., Gao, F., & Wen, Q. (2021) · New Journal of Physics · doi:10.1088/1367-2630/abf614 · Cited by 8 · see citing works
“This dimension witness can be used to estimate the genuine randomness generated if the quantum random number generator system satisfied two assumptions [5, 39, 40].”
BackgroundHybrid quantum random number generator for cryptographic algorithms
Iavich, M., Kuchukhidze, T., Iashvili, G., & Gnatyuk, S. O. (2021) · RADIOELECTRONIC AND COMPUTER SYSTEMS · doi:10.32620/reks.2021.4.09 · Cited by 8 · see citing works
Optimal classical and quantum real and complex dimension witness
Batle, J. A. I., & Bednorz, A. (2022) · Physical Review A · doi:10.1103/physreva.105.042433 · Cited by 7 · see citing works
“to make it equivalent to the previous one, with all odd preparations being identical [7].”
BackgroundStructure of dimension-bounded temporal correlations
Mao, Y., Spee, C., Xu, Z., & Gühne, O. (2022) · Physical Review A · doi:10.1103/physreva.105.l020201 · Cited by 7 · see citing works
Novel Certification Method for Quantum Random Number Generators
Iavich, M., Kuchukhidze, T., Gnatyuk, S. O., & Fesenko, A. (2021) · International Journal of Computer Network and Information Security · doi:10.5815/ijcnis.2021.03.03 · Cited by 7 · see citing works
“In order to obtain the final random bit string, sufficient post processing of the raw data is performed based on the entropy bound [7, 8]. pr (o | s, m) gives the conditional probability of finding the result of o (from ± 1) for a condition that is one of the defined probabilities s = 0, 1, 2, 3.”
MethodEvidence-Based Certification of Quantum Dimensions
Teo, Y. S., Shringarpure, S. U., Jeong, H., Prasannan, N., Brecht, B., Silberhorn, C., Evans, M. J., Mogilevtsev, D. S., & Sánchez-Soto, L. L. (2024) · Physical Review Letters · doi:10.1103/physrevlett.133.050204 · Cited by 7 · see citing works
Tighter bound of quantum randomness certification for independent-devices scenario
Fei, X.-W., Yin, Z., Huang, W., Xu, B., Wang, S., Chen, W., Han, Y.-G., Guo, G., & Han, Z. (2017) · Scientific Reports · doi:10.1038/s41598-017-15318-4 · Cited by 6 · see citing works
“34 proposed the so-called dimension witness to bound the quantumness of a prepare-and-measure scenario could behave, with the assumption that the state preparation and measurement devices share no correlations.”
BackgroundMethodInfluentialAlves, G. P., Gigena, N., & Kaniewski, J. (2023) · Physical Review A · doi:10.1103/physreva.108.042608 · Cited by 7 · see citing works
Generalized measurements on qubits in quantum randomness certification and expansion
Mironowicz, P., Grünfeld, M., & Bourennane, M. (2024) · Physical Review Applied · doi:10.1103/physrevapplied.22.044041 · Cited by 5 · see citing works
Measurement-Device-Independent Quantum Random-Number Generator With Source Flaws
Wu, T., Zhang, C., Zhou, X., Li, J., & Wang, Q. (2023) · IEEE photonics journal · doi:10.1109/jphot.2023.3331547 · Cited by 5 · see citing works
“Usually, there are three types of SDI-QRNGs, the source-independent QRNG (SI-QRNG) [18], [19], [20], the measurement-device-independent QRNG (MDI-QRNG) [21], [24], and the SDI-QRNG based on dimension witness [16], [17].”
BackgroundMethodDetection loophole attacks on semi-device-independent quantum and classical protocols
Dall’Arno, M., Passaro, E., Gallego, R., Pawłowski, M., & Acín, A. (2015) · Quantum Information and Computation · doi:10.26421/qic15.1-2-3 · Cited by 5 · see citing works
“[30].”
BackgroundRoy, A. K., Srivastava, V., Mahanti, S., Giarmatzi, C., & Gilchrist, A. (2024) · Physical Review A · doi:10.1103/physreva.110.012608 · Cited by 4 · see citing works
Quantum discord witness with uncharacterized devices
Wang, R., Yao, Y., & Yin, Z.-Q. (2025) · Physical Review Applied · doi:10.1103/hht6-mvyz · Cited by 4 · see citing works
Novel Quantum Random Number Generator with the Improved Certification Method
Iavich, M., Kuchukhidze, T., Iashvili, G., Gnatyuk, S. O., & Bocu, R. (2021) · International Journal of Mathematical Sciences and Computing · doi:10.5815/ijmsc.2021.03.05 · Cited by 4 · see citing works
“In order to obtain the final random bit string, sufficient post processing of the raw data is performed based on the entropy bound [18, 19].”
MethodExperimental certification of nonprojective quantum measurements under a minimum overlap assumption
Fan, Q., Ma, M.-Y., Sun, Y.-N., Su, Q.-P., & Yang, C. (2022) · Optics Express · doi:10.1364/oe.469225 · Cited by 4 · see citing works
Self-testing of an unbounded number of mutually commuting local observables
Munshi, S., & Pan, A. (2023) · Physical Review A · doi:10.1103/physreva.108.062607 · Cited by 4 · see citing works
Testing the accuracy of qubit rotations on a public quantum computer
Białecki, T., Rybotycki, T., Tworzydło, J., & Bednorz, A. (2024) · Frontiers in Physics · doi:10.3389/fphy.2024.1360080 · Cited by 4 · see citing works
Experimental test of nonclassicality with arbitrarily low detection efficiency
Hameedi, A., Marques, B., Mironowicz, P., Saha, D., Pawłowski, M., & Bourennane, M. (2020) · Physical Review A · doi:10.1103/physreva.102.032621 · Cited by 4 · see citing works
“With this added assumption, it was shown [22, 23] that any experimental setup with a non-zero detection efficiency can be used to prove nonclassicality.”
MethodModified quantum delayed-choice experiment without quantum control or entanglement assistance
Guo, Q., Zhang, W.-J., Li, G., Zhang, T., Wang, H.-F., & Zhang, S. (2021) · Physical Review A · doi:10.1103/physreva.104.022210 · Cited by 4 · see citing works
“In that work, the authors treated the WDC experiment as a device-independent prepare-and-measure (PAM) scenario [40], and suggested a slight modification for the WDC experiment can exclude any two-dimensional nonretrocausal hidden classical model in a device-independent manner by violating the dimension witness inequa…”
BackgroundMethodInfluentialDetection Efficiency Bounds in (Semi-)Device-Independent Scenarios
Sarubi, T. S., Zamora, S., Alves, M., Alves, V. F., Viswanathan, G. M., & Chaves, R. (2026) · Brazilian Journal of Physics · doi:10.1007/s13538-026-02008-8 · Cited by 3 · see citing works
“This leads to a nonconvex set of correlations, making the analysis of such models more subtle [118, 119].”
Optimality of quantum randomness certification with independent devices
Fei, X.-W., Yin, Z., Cui, C., Huang, W., Xu, B., Wang, S., Chen, W., Han, Y.-G., Guo, G., & Han, Z. (2018) · Journal of the Optical Society of America B · doi:10.1364/josab.35.002186 · Cited by 3 · see citing works
An, X.-B., Han, Y.-G., Yin, Z., Huang, W., Chen, W., Wang, S., Guo, G., & Han, Z. (2017) · Optics Letters · doi:10.1364/ol.42.004139 · Cited by 3 · see citing works
Practical decoy-state quantum random number generator with weak coherent sources
Han, S.-S., Ding, H.-J., Zhang, C., Zhou, X., Zhang, C.-M., & Wang, Q. (2020) · Quantum Information Processing · doi:10.1007/s11128-020-02902-3 · Cited by 3 · see citing works
Jebarathinam, C., & Das, D. (2023) · Quantum Information and Computation · doi:10.26421/qic23.5-6-2 · Cited by 4 · see citing works
“[41] serve as the certification of quantumness of sequential correlations arising in the prepare and measure scenario in a semi-device-independent way.”
BackgroundPrecise certification of a qubit space
Białecki, T., Rybotycki, T., Batle, J. A. I., Tworzydło, J., & Bednorz, A. (2024) · EPJ Quantum Technology · doi:10.1140/epjqt/s40507-024-00230-4 · Cited by 3 · see citing works
“A good witness certifies the linear independence of specific dichotomic outcome probability p ( M | N ) for the preparation N and measurement M and so it is expressed by a suitable determinant [23–25].”
BackgroundTowards minimal self-testing of qubit states and measurements in prepare-and-measure scenarios
Drótos, G., Pál, K. F., Taoutioui, A., & Vértesi, T. (2024) · New Journal of Physics · doi:10.1088/1367-2630/ad4e5c · Cited by 3 · see citing works
“[47] for how to generate them.”
Estimating the privacy of quantum-random numbers
Seiler, J., Strohm, T., & Schleich, W. P. (2020) · OPen Access Repositorium der Universität Ulm (OPARU) (Ulm University) · doi:10.18725/oparu-34102 · Cited by 3 · see citing works
The role of entanglement in energy-restricted communication and randomness generation
Carceller, C. R. I., & Tavakoli, A. (2026) · Quantum Science and Technology · doi:10.1088/2058-9565/ae4d80 · Cited by 2 · see citing works
“Common to all these frameworks for semi-device-independent PM scenarios is that they suppose that Alice and Bob do not share entanglement, and sometimes not even classical random variables [ 23–25 ] .”
Semi-device-independent randomness certification on discretized continuous-variable platforms
Alves, M., Sena, V. L. O., Zamora, S., Sarubi, T. S., de Oliveira, A., Tacla, A. B., & Chaves, R. (2026) · Physical Review A · doi:10.1103/z3s5-bhb7 · Cited by 2 · see citing works
Theory-independent randomness generation from spatial symmetries
Jones, C. L., Ludescher, S. L., Aloy, A., & Müller, M. P. (2026) · Quantum · doi:10.22331/q-2026-01-16-1966 · Cited by 2 · see citing works
“This has often been implemented with a bound on the dimension of the Hilbert space of the transmitted sys-tem, restricting the communication to qubits or qutrits, as in [14, 21, 27, 28], although this is arguably not very well-motivated for non-idealized physical scenarios.”
Single- and two-photon wave-particle superpositions: Theory and experiment
Xue, Q.-F., Zhuang, X.-C., An, N. B., Duan, D., Ma, R., Pan, W., Wang, Q., Xia, Y., & Man, Z. (2023) · Physical Review A · doi:10.1103/physreva.108.022223 · Cited by 2 · see citing works
Null dimension witness based on single measurements
Batle, J. A. I., & Bednorz, A. (2023) · Physical Review A · doi:10.1103/physreva.108.062202 · Cited by 2 · see citing works
“In previous works, a witness of dimension d needed 2k preparations and k measurements with binary outcomes, with d ≤ k for the classical system and d(2) ≤ k for the quantum system [6].”
BackgroundMethodInfluentialQuantum null-hypothesis device-independent Schmidt number witness
Batle, J. A. I., Białecki, T., Rybotycki, T., Tworzydło, J., & Bednorz, A. (2024) · EPJ Quantum Technology · doi:10.1140/epjqt/s40507-024-00273-7 · Cited by 2 · see citing works
“The quantum dimension can also be tested by null hypotheses [30–32], demonstrated experimentally [33, 34].”
BackgroundNoise-resilient quantum random access codes
Karthik, H. S., Gómez, S., Quinteros, F., Akshata Shenoy, H., Pawłowski, M., Walborn, S. P., Lima, G., & Gómez, E. S. (2025) · Physical Review A · doi:10.1103/physreva.111.032613 · Cited by 2 · see citing works
Experimental Implementation of Dimension-Dependent Contextuality Inequality
Håkansson, E., Piveteau, A., Seguinard, A., Muhammad, S., Bourennane, M., Gühne, O., & Plávala, M. (2025) · Physical Review Letters · doi:10.1103/physrevlett.134.200202 · Cited by 2 · see citing works
Experimental device-independent tests of classical and quantum entropy
Zhu, F., Zhang, W., Chen, S., You, L., Wang, Z., & Huang, Y. (2016) · Physical Review A · doi:10.1103/physreva.94.062340 · Cited by 2 · see citing works
Quantum dimension witness with a single repeated operation
Białecki, T., Rybotycki, T., Batle, J. A. I., & Bednorz, A. (2024) · Physical Review A · doi:10.1103/physreva.109.l020201 · Cited by 2 · see citing works
Characterizing high-dimensional quantum contextuality
Yu, X., Veeren, I., & Gühne, O. (2024) · Physical Review A · doi:10.1103/physreva.109.l030201 · Cited by 2 · see citing works
“[ 36 – 39 ] for some examples and their applications.”
BackgroundMethodSun, Y., Ma, M.-Y., Su, Q.-P., Sun, Z., Yang, C.-P., & Nori, F. (2026) · Physical Review Letters · doi:10.1103/q85z-wl38 · Cited by 2 · see citing works
“As discussed in [36, 37], it is more natural to assume that all devices are not correlated in a practical setup, i.e., they are independent.”
- Authors
- Bowles, J., Quintino, M. T., & Brunner, N.
- Affiliations
- University of Geneva (CH)
- Venue
- Physical Review Letters
- Publisher
- American Physical Society (APS)
- Published
- 2014-04-11
- Volume
- 112
- Issue
- 14
- Pages
- 140407-140407
- ISSN
- 0031-9007 (ISSN-L)
- 1079-7114
- OpenAlex
- W2070059557
- Cited by
- 116
- References
- 22
- Citations by year
- 2026: 7 · 2025: 4 · 2024: 13 · 2023: 16 · 2022: 7 · 2021: 13 · 2020: 13 · 2019: 9 · 2018: 5 · 2017: 9 · 2016: 10 · 2015: 8 · 2014: 1 · 2012: 1
- FWCI
- 9.30
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Physical Review Letters (publishedVersion)
- PubMed (publishedVersion)
- Archive ouverte UNIGE (University of Geneva) (publishedVersion, other-oa)
- arXiv (Cornell University) (submittedVersion)
- Archive ouverte UNIGE (University of Geneva) (submittedVersion, cc-by)
- arXiv (Cornell University)
- Funders
- National Science Foundation (PP00P2_138917)
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Keywords
- dimension witness, prepare-and-measure scenario, noise robustness, randomness certification, non-convex optimization
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