{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,19]],"date-time":"2026-02-19T05:19:04Z","timestamp":1771478344846,"version":"3.50.1"},"reference-count":58,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2025,4,14]],"date-time":"2025-04-14T00:00:00Z","timestamp":1744588800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"INFN","award":["CN00000013"],"award-info":[{"award-number":["CN00000013"]}]},{"name":"INFN","award":["PE0000023-NQSTI"],"award-info":[{"award-number":["PE0000023-NQSTI"]}]},{"name":"National Centre for HPC, Big Data, and Quantum Computing","award":["CN00000013"],"award-info":[{"award-number":["CN00000013"]}]},{"name":"National Centre for HPC, Big Data, and Quantum Computing","award":["PE0000023-NQSTI"],"award-info":[{"award-number":["PE0000023-NQSTI"]}]},{"name":"2022-PRIN Project \u201cHybrid algorithms for quantum simulators\u201d","award":["CN00000013"],"award-info":[{"award-number":["CN00000013"]}]},{"name":"2022-PRIN Project \u201cHybrid algorithms for quantum simulators\u201d","award":["PE0000023-NQSTI"],"award-info":[{"award-number":["PE0000023-NQSTI"]}]},{"name":"Italian funding within the \u201cBudget MUR\u2014Dipartimenti di Eccellenza 2023\u20132027\u201d\u2014Quantum Sensing and Modelling for One-Health (QuaSiModO)","award":["CN00000013"],"award-info":[{"award-number":["CN00000013"]}]},{"name":"Italian funding within the \u201cBudget MUR\u2014Dipartimenti di Eccellenza 2023\u20132027\u201d\u2014Quantum Sensing and Modelling for One-Health (QuaSiModO)","award":["PE0000023-NQSTI"],"award-info":[{"award-number":["PE0000023-NQSTI"]}]},{"name":"Italian National Group of Mathematical Physics (GNFM-INdAM)","award":["CN00000013"],"award-info":[{"award-number":["CN00000013"]}]},{"name":"Italian National Group of Mathematical Physics (GNFM-INdAM)","award":["PE0000023-NQSTI"],"award-info":[{"award-number":["PE0000023-NQSTI"]}]},{"name":"PNRR MUR project","award":["CN00000013"],"award-info":[{"award-number":["CN00000013"]}]},{"name":"PNRR MUR project","award":["PE0000023-NQSTI"],"award-info":[{"award-number":["PE0000023-NQSTI"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Quantum computing gives direct access to the study of the real-time dynamics of quantum many-body systems. In principle, it is possible to directly calculate non-equal-time correlation functions, from which one can detect interesting phenomena, such as the presence of quantum scars or dynamical quantum phase transitions. In practice, these calculations are strongly affected by noise, due to the complexity of the required quantum circuits. As a testbed for the evaluation of the real-time evolution of observables and correlations, the dynamics of the Zn Schwinger model in a one-dimensional lattice is considered. To control the computational cost, we adopt a quantum\u2013classical strategy that reduces the dimensionality of the system by restricting the dynamics to the Dirac vacuum sector and optimizes the embedding into a qubit model by minimizing the number of three-qubit gates. The time evolution of particle-density operators in a non-equilibrium quench protocol is both simulated in a bare noisy condition and implemented on a physical IBM quantum device. In either case, the convergence towards a maximally mixed state is targeted by means of different error mitigation techniques. The evaluation of the particle-density correlation shows a well-performing post-processing error mitigation for properly chosen coupling regimes.<\/jats:p>","DOI":"10.3390\/e27040427","type":"journal-article","created":{"date-parts":[[2025,4,15]],"date-time":"2025-04-15T03:59:06Z","timestamp":1744689546000},"page":"427","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Quantum Error Mitigation in Optimized Circuits for Particle-Density Correlations in Real-Time Dynamics of the Schwinger Model"],"prefix":"10.3390","volume":"27","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0157-3471","authenticated-orcid":false,"given":"Domenico","family":"Pomarico","sequence":"first","affiliation":[{"name":"Dipartimento di Fisica, Universit\u00e0 di Bari, I-70126 Bari, Italy"},{"name":"Istituto Nazionale di Fisica Nucleare, Sezione di Bari, I-70126 Bari, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0009-0009-6654-2746","authenticated-orcid":false,"given":"Mahul","family":"Pandey","sequence":"additional","affiliation":[{"name":"Istituto Nazionale di Fisica Nucleare, Sezione di Bologna, I-40127 Bologna, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0009-0001-1286-3773","authenticated-orcid":false,"given":"Riccardo","family":"Cioli","sequence":"additional","affiliation":[{"name":"Istituto Nazionale di Fisica Nucleare, Sezione di Bologna, I-40127 Bologna, Italy"},{"name":"Dipartimento di Fisica e Astronomia, Universit\u00e0 di Bologna, I-40127 Bologna, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0009-0003-7514-0075","authenticated-orcid":false,"given":"Federico","family":"Dell\u2019Anna","sequence":"additional","affiliation":[{"name":"Istituto Nazionale di Fisica Nucleare, Sezione di Bologna, I-40127 Bologna, Italy"},{"name":"Dipartimento di Fisica e Astronomia, Universit\u00e0 di Bologna, I-40127 Bologna, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7214-5685","authenticated-orcid":false,"given":"Saverio","family":"Pascazio","sequence":"additional","affiliation":[{"name":"Dipartimento di Fisica, Universit\u00e0 di Bari, I-70126 Bari, Italy"},{"name":"Istituto Nazionale di Fisica Nucleare, Sezione di Bari, I-70126 Bari, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7407-063X","authenticated-orcid":false,"given":"Francesco V.","family":"Pepe","sequence":"additional","affiliation":[{"name":"Dipartimento di Fisica, Universit\u00e0 di Bari, I-70126 Bari, Italy"},{"name":"Istituto Nazionale di Fisica Nucleare, Sezione di Bari, I-70126 Bari, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9152-6515","authenticated-orcid":false,"given":"Paolo","family":"Facchi","sequence":"additional","affiliation":[{"name":"Dipartimento di Fisica, Universit\u00e0 di Bari, I-70126 Bari, Italy"},{"name":"Istituto Nazionale di Fisica Nucleare, Sezione di Bari, I-70126 Bari, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6801-5976","authenticated-orcid":false,"given":"Elisa","family":"Ercolessi","sequence":"additional","affiliation":[{"name":"Istituto Nazionale di Fisica Nucleare, Sezione di Bologna, I-40127 Bologna, Italy"},{"name":"Dipartimento di Fisica e Astronomia, Universit\u00e0 di Bologna, I-40127 Bologna, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2025,4,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"165","DOI":"10.1140\/epjd\/e2020-100571-8","article-title":"Simulating lattice gauge theories within quantum technologies","volume":"74","author":"Blatt","year":"2020","journal-title":"Eur. Phys. J. D"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"037001","DOI":"10.1103\/PRXQuantum.5.037001","article-title":"Quantum Computing for High-Energy Physics: State of the Art and Challenges","volume":"5","author":"Jansen","year":"2024","journal-title":"PRX Quantum"},{"key":"ref_3","first-page":"020304","article-title":"Density-Matrix Renormalization Group Algorithm for Simulating Quantum Circuits with a Finite Fidelity","volume":"4","author":"Ayral","year":"2023","journal-title":"Phys. Rev. X Quantum"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2425","DOI":"10.1103\/PhysRev.128.2425","article-title":"Gauge Invariance and Mass. II","volume":"128","author":"Schwinger","year":"1962","journal-title":"Phys. Rev."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"155134","DOI":"10.1103\/PhysRevB.98.155134","article-title":"Quantum scarred eigenstates in a Rydberg atom chain: Entanglement, breakdown of thermalization, and stability to perturbations","volume":"98","author":"Turner","year":"2018","journal-title":"Phys. Rev. B"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"013288","DOI":"10.1103\/PhysRevResearch.2.013288","article-title":"Real-time-dynamics quantum simulation of (1+1)-dimensional lattice QED with Rydberg atoms","volume":"2","author":"Notarnicola","year":"2020","journal-title":"Phys. Rev. Res."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"L201105","DOI":"10.1103\/PhysRevB.107.L201105","article-title":"Weak ergodicity breaking in the Schwinger model","volume":"107","author":"Desaules","year":"2023","journal-title":"Phys. Rev. B"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"033120","DOI":"10.1103\/PhysRevResearch.4.033120","article-title":"Stabilizing lattice gauge theories through simplified local pseudogenerators","volume":"4","author":"Halimeh","year":"2022","journal-title":"Phys. Rev. Res."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"035153","DOI":"10.1103\/PhysRevB.107.035153","article-title":"Reliability of lattice gauge theories in the thermodynamic limit","volume":"107","author":"Damme","year":"2023","journal-title":"Phys. Rev. B"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1004","DOI":"10.22331\/q-2023-05-15-1004","article-title":"Robust quantum many-body scars in lattice gauge theories","volume":"7","author":"Halimeh","year":"2023","journal-title":"Quantum"},{"key":"ref_11","unstructured":"Halimeh, J.C., and Hauke, P. (2022). Stabilizing Gauge Theories in Quantum Simulators: A Brief Review. arXiv."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"144306","DOI":"10.1103\/PhysRevB.107.144306","article-title":"Scrambling in quantum cellular automata","volume":"107","author":"Kent","year":"2023","journal-title":"Phys. Rev. B"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"030323","DOI":"10.1103\/PRXQuantum.4.030323","article-title":"Quantum Computation of Dynamical Quantum Phase Transitions and Entanglement Tomography in a Lattice Gauge Theory","volume":"4","author":"Mueller","year":"2023","journal-title":"PRX Quantum"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"140602","DOI":"10.1103\/PhysRevLett.115.140602","article-title":"Scaling and Universality at Dynamical Quantum Phase Transitions","volume":"115","author":"Heyl","year":"2015","journal-title":"Phys. Rev. Lett."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"054001","DOI":"10.1088\/1361-6633\/aaaf9a","article-title":"Dynamical quantum phase transitions: A review","volume":"81","author":"Heyl","year":"2018","journal-title":"Rep. Prog. Phys."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"245110","DOI":"10.1103\/PhysRevB.106.245110","article-title":"Dynamical quantum phase transitions in spin-S\u00a0U(1) quantum link models","volume":"106","author":"Zache","year":"2022","journal-title":"Phys. Rev. B"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"033090","DOI":"10.1103\/PhysRevResearch.5.033090","article-title":"Anatomy of Dynamical Quantum Phase Transitions","volume":"5","author":"Desaules","year":"2023","journal-title":"Phys. Rev. Res."},{"key":"ref_18","unstructured":"Osborne, J., McCulloch, I.P., and Halimeh, J.C. (2023). Probing Confinement Through Dynamical Quantum Phase Transitions: From Quantum Spin Models to Lattice Gauge Theories. arXiv."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"235103","DOI":"10.1103\/PhysRevB.103.235103","article-title":"Lattice gauge theory and dynamical quantum phase transitions using noisy intermediate-scale quantum devices","volume":"103","author":"Pedersen","year":"2021","journal-title":"Phys. Rev. B"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"224309","DOI":"10.1103\/PhysRevB.105.224309","article-title":"Dynamical quantum phase transitions in a noisy lattice gauge theory","volume":"105","author":"Jensen","year":"2022","journal-title":"Phys. Rev. B"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"103020","DOI":"10.1088\/1367-2630\/aa89ab","article-title":"U(1) Wilson lattice gauge theories in digital quantum simulators","volume":"19","author":"Muschik","year":"2017","journal-title":"New J. Phys."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"664","DOI":"10.1103\/PhysRev.82.664","article-title":"On Gauge Invariance and Vacuum Polarization","volume":"82","author":"Schwinger","year":"1951","journal-title":"Phys. Rev."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"516","DOI":"10.1038\/nature18318","article-title":"Real-time dynamics of lattice gauge theories with a few-qubit quantum computer","volume":"534","author":"Martinez","year":"2016","journal-title":"Nature"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"032331","DOI":"10.1103\/PhysRevA.98.032331","article-title":"Quantum-classical computation of Schwinger model dynamics using quantum computers","volume":"98","author":"Klco","year":"2018","journal-title":"Phys. Rev. A"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1168","DOI":"10.1038\/s41567-019-0649-7","article-title":"Floquet approach to \u21242 lattice gauge theories with ultracold atoms in optical lattices","volume":"15","author":"Schweizer","year":"2019","journal-title":"Nat. Phys."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"392","DOI":"10.1038\/s41586-020-2910-8","article-title":"Observation of gauge invariance in a 71-site Bose\u2013Hubbard quantum simulator","volume":"587","author":"Yang","year":"2020","journal-title":"Nature"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"815","DOI":"10.1038\/s41567-024-02400-8","article-title":"Observation of Josephson harmonics in tunnel junctions","volume":"20","author":"Willsch","year":"2024","journal-title":"Nat. Phys."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"040338","DOI":"10.1103\/PRXQuantum.3.040338","article-title":"Quasiparticles in Superconducting Qubits with Asymmetric Junctions","volume":"3","author":"Marchegiani","year":"2022","journal-title":"PRX Quantum"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"217001","DOI":"10.1103\/PhysRevLett.132.217001","article-title":"Coexistence of Nonequilibrium Density and Equilibrium Energy Distribution of Quasiparticles in a Superconducting Qubit","volume":"132","author":"Connolly","year":"2024","journal-title":"Phys. Rev. Lett."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"044063","DOI":"10.1103\/PhysRevApplied.22.044063","article-title":"Quasiparticle effects in magnetic-field-resilient three-dimensional transmons","volume":"22","author":"Krause","year":"2024","journal-title":"Phys. Rev. Appl."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"024519","DOI":"10.1103\/PhysRevB.110.024519","article-title":"Modeling phonon-mediated quasiparticle poisoning in superconducting qubit arrays","volume":"110","author":"Yelton","year":"2024","journal-title":"Phys. Rev. B"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"070","DOI":"10.21468\/SciPostPhys.17.3.070","article-title":"Nonequilibrium quasiparticle distribution in superconducting resonators: Effect of pair-breaking photons","volume":"17","author":"Fischer","year":"2024","journal-title":"SciPost Phys."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"240601","DOI":"10.1103\/PhysRevLett.133.240601","article-title":"Resisting High-Energy Impact Events through Gap Engineering in Superconducting Qubit Arrays","volume":"133","author":"McEwen","year":"2024","journal-title":"Phys. Rev. Lett."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"6425","DOI":"10.1038\/s41467-022-33997-0","article-title":"Phonon downconversion to suppress correlated errors in superconducting qubits","volume":"13","author":"Iaia","year":"2022","journal-title":"Nat. Commun."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Pomarico, D., Cosmai, L., Facchi, P., Lupo, C., Pascazio, S., and Pepe, F.V. (2023). Dynamical Quantum Phase Transitions of the Schwinger Model: Real-Time Dynamics on IBM Quantum. Entropy, 25.","DOI":"10.3390\/e25040608"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"035046","DOI":"10.1088\/2058-9565\/ad5228","article-title":"Quantum kernels for classifying dynamical singularities in a multiqubit system","volume":"9","author":"Tancara","year":"2024","journal-title":"Quantum Sci. Technol."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"045020","DOI":"10.1088\/2058-9565\/ac1dff","article-title":"Indexed improvements for real-time trotter evolution of a (1+1) field theory using NISQ quantum computers","volume":"6","author":"Gustafson","year":"2021","journal-title":"Quantum Sci. Technol."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"094502","DOI":"10.1103\/PhysRevD.106.094502","article-title":"Toward the real-time evolution of gauge-invariant \u21242 and U(1) quantum link models on noisy intermediate-scale quantum hardware with error mitigation","volume":"106","author":"Huffman","year":"2022","journal-title":"Phys. Rev. D"},{"key":"ref_39","first-page":"020324","article-title":"Digital Quantum Simulation of the Schwinger Model and Symmetry Protection with Trapped Ions","volume":"3","author":"Nguyen","year":"2022","journal-title":"Phys. Rev. X Quantum"},{"key":"ref_40","unstructured":"Javanmard, Y., Liaubaite, U., Osborne, T.J., and Santos, L. (2022). Quantum simulation of dynamical phase transitions in noisy quantum devices. arXiv."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"020315","DOI":"10.1103\/PRXQuantum.5.020315","article-title":"Scalable Circuits for Preparing Ground States on Digital Quantum Computers: The Schwinger Model Vacuum on 100 Qubits","volume":"5","author":"Farrell","year":"2024","journal-title":"PRX Quantum"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"076004","DOI":"10.1103\/PhysRevD.109.076004","article-title":"Quantum simulations for strong-field QED","volume":"109","author":"Hidalgo","year":"2024","journal-title":"Phys. Rev. D"},{"key":"ref_43","unstructured":"Filippov, S., Leahy, M., Rossi, M.A.C., and Garc\u00eda-P\u00e9rez, G. (2023). Scalable tensor-network error mitigation for near-term quantum computing. arXiv."},{"key":"ref_44","unstructured":"Fischer, L.E., Leahy, M., Eddins, A., Keenan, N., Ferracin, D., Rossi, M.A.C., Kim, Y., He, A., Pietracaprina, F., and Sokolov, B. (2024). Dynamical simulations of many-body quantum chaos on a quantum computer. arXiv."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"033281","DOI":"10.1103\/PhysRevResearch.2.033281","article-title":"Calculation of the Green\u2019s function on near-term quantum computers","volume":"2","author":"Endo","year":"2020","journal-title":"Phys. Rev. Res."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"043038","DOI":"10.1103\/PhysRevResearch.4.043038","article-title":"Effective calculation of the Green\u2019s function in the time domain on near-term quantum processors","volume":"4","author":"Libbi","year":"2022","journal-title":"Phys. Rev. Res."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"043011","DOI":"10.1103\/PhysRevResearch.4.043011","article-title":"One-particle Green\u2019s functions from the quantum equation of motion algorithm","volume":"4","author":"Rizzo","year":"2022","journal-title":"Phys. Rev. Res."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"032620","DOI":"10.1103\/PhysRevA.105.032620","article-title":"Model-free readout-error mitigation for quantum expectation values","volume":"105","author":"Minev","year":"2022","journal-title":"Phys. Rev. A"},{"key":"ref_49","doi-asserted-by":"crossref","unstructured":"Giurgica-Tiron, T., Hindy, Y., LaRose, R., Mari, A., and Zeng, W.J. (2020, January 12\u201316). Digital zero noise extrapolation for quantum error mitigation. Proceedings of the 2020 IEEE International Conference on Quantum Computing and Engineering (QCE), Denver, CO, USA.","DOI":"10.1109\/QCE49297.2020.00045"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"074503","DOI":"10.1103\/PhysRevD.98.074503","article-title":"Phase transitions in \u2124n gauge models: Towards quantum simulations of the Schwinger-Weyl QED","volume":"98","author":"Ercolessi","year":"2018","journal-title":"Phys. Rev. D"},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"395","DOI":"10.1103\/PhysRevD.11.395","article-title":"Hamiltonian formulation of Wilson\u2019s lattice gauge theories","volume":"11","author":"Kogut","year":"1975","journal-title":"Phys. Rev. D"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"30FT01","DOI":"10.1088\/1751-8113\/48\/30\/30FT01","article-title":"Discrete Abelian gauge theories for quantum simulations of QED","volume":"48","author":"Notarnicola","year":"2015","journal-title":"J. Phys. A: Math. Theor."},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"2845","DOI":"10.1103\/PhysRevLett.85.2845","article-title":"Geometric Phases for Mixed States in Interferometry","volume":"85","author":"Pati","year":"2000","journal-title":"Phys. Rev. Lett."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"160402","DOI":"10.1103\/PhysRevLett.90.160402","article-title":"Geometric Phase in Open Systems","volume":"90","author":"Carollo","year":"2003","journal-title":"Phys. Rev. Lett."},{"key":"ref_55","unstructured":"IBM Quantum (2025, April 10). Available online: https:\/\/quantum-computing.ibm.com\/."},{"key":"ref_56","doi-asserted-by":"crossref","unstructured":"Pomarico, D. (2023). Multiscale Entanglement Renormalization Ansatz: Causality and Error Correction. Dynamics, 3.","DOI":"10.3390\/dynamics3030033"},{"key":"ref_57","unstructured":"Cormen, T.H., Leiserson, C.E., Rivest, R.L., and Stein, C. (2022). Introduction to Algorithms, 4 ed., The MIT Press."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"217901","DOI":"10.1103\/PhysRevLett.88.217901","article-title":"Direct Estimations of Linear and Nonlinear Functionals of a Quantum State","volume":"88","author":"Ekert","year":"2002","journal-title":"Phys. Rev. Lett."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/4\/427\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:14:30Z","timestamp":1760030070000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/4\/427"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,14]]},"references-count":58,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2025,4]]}},"alternative-id":["e27040427"],"URL":"https:\/\/doi.org\/10.3390\/e27040427","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,4,14]]}}}