{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:55:29Z","timestamp":1760241329666,"version":"build-2065373602"},"reference-count":25,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2018,1,9]],"date-time":"2018-01-09T00:00:00Z","timestamp":1515456000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"ERC-AG MEC","award":["18212"],"award-info":[{"award-number":["18212"]}]},{"name":"Swiss National Science Fondation","award":["19096"],"award-info":[{"award-number":["19096"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Quantum measurements have intrinsic properties that seem incompatible with our everyday-life macroscopic measurements. Macroscopic Quantum Measurement (MQM) is a concept that aims at bridging the gap between well-understood microscopic quantum measurements and macroscopic classical measurements. In this paper, we focus on the task of the polarization direction estimation of a system of N spins 1\/2 particles and investigate the model some of us proposed in Barnea et al., 2017. This model is based on a von Neumann pointer measurement, where each spin component of the system is coupled to one of the three spatial component directions of a pointer. It shows traits of a classical measurement for an intermediate coupling strength. We investigate relaxations of the assumptions on the initial knowledge about the state and on the control over the MQM. We show that the model is robust with regard to these relaxations. It performs well for thermal states and a lack of knowledge about the size of the system. Furthermore, a lack of control on the MQM can be compensated by repeated \u201cultra-weak\u201d measurements.<\/jats:p>","DOI":"10.3390\/e20010039","type":"journal-article","created":{"date-parts":[[2018,1,9]],"date-time":"2018-01-09T14:00:08Z","timestamp":1515506408000},"page":"39","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Robust Macroscopic Quantum Measurements in the Presence of Limited Control and Knowledge"],"prefix":"10.3390","volume":"20","author":[{"given":"Marc-Olivier","family":"Renou","sequence":"first","affiliation":[{"name":"Department of Applied Physics, University of Geneva, 1211 Geneva 4, Switzerland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0788-3494","authenticated-orcid":false,"given":"Nicolas","family":"Gisin","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, University of Geneva, 1211 Geneva 4, Switzerland"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Florian","family":"Fr\u00f6wis","sequence":"additional","affiliation":[{"name":"Department of Applied Physics, University of Geneva, 1211 Geneva 4, Switzerland"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,1,9]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1259","DOI":"10.1103\/PhysRevLett.74.1259","article-title":"Optimal Extraction of Information from Finite Quantum Ensembles","volume":"74","author":"Massar","year":"1995","journal-title":"Phys. Rev. Lett."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"432","DOI":"10.1103\/PhysRevLett.83.432","article-title":"Spin Flips and Quantum Information for Antiparallel Spins","volume":"83","author":"Gisin","year":"1999","journal-title":"Phys. Rev. Lett."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1142\/S0219749906002018","article-title":"Maximum likelihood estimation for a group of physical transformations","volume":"4","author":"Chiribella","year":"2006","journal-title":"Int. J. Quantum Inf."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1225","DOI":"10.1103\/RevModPhys.77.1225","article-title":"Quantum cloning","volume":"77","author":"Scarani","year":"2005","journal-title":"Rev. Mod. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"4344","DOI":"10.1103\/PhysRevLett.82.4344","article-title":"Optimal purification of single qubits","volume":"82","author":"Cirac","year":"1999","journal-title":"Phys. Rev. Lett."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"062318","DOI":"10.1103\/PhysRevA.71.062318","article-title":"Comprehensive analysis of quantum pure-state estimation for two-level systems","volume":"71","author":"Bagan","year":"2005","journal-title":"Phys. Rev. A"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"032301","DOI":"10.1103\/PhysRevA.73.032301","article-title":"Optimal full estimation of qubit mixed states","volume":"73","author":"Bagan","year":"2006","journal-title":"Phys. Rev. A"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"012306","DOI":"10.1103\/PhysRevA.75.012306","article-title":"Information-disturbance tradeoff for spin coherent state e stimation","volume":"75","author":"Sacchi","year":"2007","journal-title":"Phys. Rev. A"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"012128","DOI":"10.1103\/PhysRevA.88.012128","article-title":"Optimal parameter estimation with a fixed rate of abstention","volume":"88","author":"Gendra","year":"2013","journal-title":"Phys. Rev. A"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1351","DOI":"10.1103\/PhysRevLett.81.1351","article-title":"Minimal optimal generalized quantum measurement","volume":"81","author":"Latorre","year":"1998","journal-title":"Phys. Rev. Lett."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"190403","DOI":"10.1103\/PhysRevLett.98.190403","article-title":"How Continuous Quantum Measurements in Finite Dimensions Are Actually Discrete","volume":"98","author":"Chiribella","year":"2007","journal-title":"Phys. Rev. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"012111","DOI":"10.1103\/PhysRevA.96.012111","article-title":"Macroscopic quantum measurements of non-commuting observables","volume":"96","author":"Barnea","year":"2017","journal-title":"Phys. Rev. A"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"022102","DOI":"10.1103\/PhysRevA.71.022102","article-title":"Macroscopic observables","volume":"71","author":"Poulin","year":"2005","journal-title":"Phys. Rev. A"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1016\/S0375-9601(01)00809-X","article-title":"A quantum measurement of the spin direction","volume":"292","author":"Presti","year":"2002","journal-title":"Phys. Lett. A"},{"key":"ref_15","unstructured":"Holevo, A. (1982). Probabilistic and Statistical Aspects of Quantum Theory, Edizioni della Normale."},{"key":"ref_16","unstructured":"Often, the considered score is \n          \n            \n              \n                F\n\t\t\t\t=\n\t\t\t\t\u222b\n                d\n\t\t\t\tr\n\t\t\t\t\u222b\n\t\t\t\td\n\t\t\t\t\n\t\t\t\t  u\n\t\t\t\t  \u2192\n\t\t\t\t\n\t\t\t\tp\n\t\t\t\t(\n\t\t\t\tr\n\t\t\t\t|\n\t\t\t\t\n\t\t\t\t  u\n\t\t\t\t  \u2192\n\t\t\t\t\n\t\t\t\t)\n\t\t\t\tf\n\t\t\t\t(\n\t\t\t\t\n\t\t\t\t  u\n\t\t\t\t  \u2192\n\t\t\t\t\n\t\t\t\t,\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t  v\n\t\t\t\t  \u2192\n\t\t\t\t\n\t\t\t\tr\n\t\t\t\t\n\t\t\t\t)\n              \n            \n          \n        , where \n          \n            \n              \n\t\t\t\tf\n\t\t\t\t(\n\t\t\t\t\n\t\t\t\t  u\n\t\t\t\t  \u2192\n\t\t\t\t\n\t\t\t\t,\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t  v\n\t\t\t\t  \u2192\n\t\t\t\t\n\t\t\t\tr\n\t\t\t\t\n\t\t\t\t)\n\t\t\t\t=\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t|\n\t\t\t\t\u2329\n\t\t\t\t\n\t\t\t\t  u\n\t\t\t\t  \u2192\n\t\t\t\t\n\t\t\t\t|\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t  v\n\t\t\t\t  \u2192\n\t\t\t\t\n\t\t\t\tr\n\t\t\t\t\n\t\t\t\t\u232a\n\t\t\t\t|\n\t\t\t\t\n\t\t\t\t2\n\t\t\t\t\n              \n            \n          \n         can be seen as the fidelity between qubits \n          \n            \n              \n\t\t\t\t|\n\t\t\t\t\n\t\t\t\t  u\n\t\t\t\t  \u2192\n\t\t\t\t\n\t\t\t\t\u232a\n              \n            \n          \n         and \n          \n            \n              \n\t\t\t\t|\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t  v\n\t\t\t\t  \u2192\n\t\t\t\t\n\t\t\t\tr\n\t\t\t\t\n\t\t\t\t\u232a\n              \n            \n          \n        , where a unit vector is associated with the corresponding qubit via the Bloch sphere identification. As \n          \n            \n              \n\t\t\t\tF\n\t\t\t\t=\n\t\t\t\t\n\t\t\t\t  1\n\t\t\t\t  2\n\t\t\t\t\n\t\t\t\t(\n\t\t\t\t1\n\t\t\t\t+\n\t\t\t\tG\n\t\t\t\t)\n              \n            \n          \n        , this is equivalent. We chose this formulation for practical reason."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"725","DOI":"10.1002\/j.1538-7305.1965.tb01684.x","article-title":"On the Simultaneous Measurement of a Pair of Conjugate Observables","volume":"44","author":"Arthurs","year":"1965","journal-title":"Bell Syst. Tech. J."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"485303","DOI":"10.1088\/1751-8113\/44\/48\/485303","article-title":"Approximate joint measurement of qubit observables through an Arthur\u2013Kelly model","volume":"44","author":"Pal","year":"2011","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1007\/BF00734524","article-title":"On the simultaneous measurement of spin components using spin-1\/2 meters: Naimark embedding and projections","volume":"19","author":"Levine","year":"1989","journal-title":"Found. Phys."},{"key":"ref_20","unstructured":"Von Neumann, J. (1955). Mathematical Foundations of Quantum Mechanics, Princeton University Press."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Busch, P., Lahti, P.J., and Mittelstaedt, P. (1991). The Quantum Theory of Measurement, Springer.","DOI":"10.1007\/978-3-662-13844-1"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Peres, A. (2002). Quantum Theory: Concepts and Methods, Kluwer Academic Publishers.","DOI":"10.1007\/0-306-47120-5"},{"key":"ref_23","unstructured":"This method is equivalent to the one previously used in [12]. There, \n          \n            \n              \n\t\t\t    \n\t\t\t    \n\t\t\t\t\u2329\n                z\n\t\t\t\t\n\t\t\t\t  |\n\t\t\t\t  \n\t\t\t\t    \u2297\n\t\t\t\t\tN\n\t\t\t\t  \n\t\t\t\t\n\t\t\t\t\n\t\t\t\t  \u03a9\n\t\t\t\t  \n\t\t\t\t    r\n\t\t\t\t    \u2192\n\t\t\t\t  \n\t\t\t\t\n\t\t\t\t|\n\t\t\t\tz\n\t\t\t\t\u232a\n\t\t\t\t\n\t\t\t\t\n\t\t\t\t  \u2297\n\t\t\t\t  N\n\t\t\t\t\n\t\t\t\t\n              \n            \n          \n         is lower bounded by \n          \n            \n              \n\t\t\t    \n\t\t\t      \n\t\t\t\t    |\n\t\t\t\t\t\n\t\t\t\t\t  \n\t\t\t\t      \u2329\n\t\t\t\t\t  z\n\t\t\t\t      |\n\t\t\t\t\t  \n\t\t\t\t\t  \n\t\t\t\t\t    \u2297\n\t\t\t\t\t\tN\n\t\t\t\t\t  \n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t  E\n\t\t\t\t\t  \n\t\t\t\t        r\n\t\t\t\t        \u2192\n\t\t\t\t      \n\t\t\t\t\t\n\t\t\t\t\t|\n\t\t\t\t\t\n\t\t\t\t      r\n\t\t\t\t      \u2192\n\t\t\t\t    \n\t\t\t\t\t\u232a\n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t  \u2297\n\t\t\t\t\t  N\n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t|\n\t\t\t\t  \n\t\t\t\t  2\n\t\t\t\t\n              \n            \n          \n        , but as the Krauss operator is diagonal, this last term is nothing else than \n          \n            \n              \n\t\t\t    \n\t\t\t\t  o\n\t\t\t\t  J\n\t\t\t\t\n\t\t\t    \n\t\t\t      \n\t\t\t\t    |\n\t\t\t\t\t\n\t\t\t\t\t  \n\t\t\t\t      \u2329\n\t\t\t\t      z\n\t\t\t\t      |\n\t\t\t\t\t  \n\t\t\t\t\t  \n\t\t\t\t\t    \u2297\n\t\t\t\t\t\tN\n\t\t\t\t\t  \n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t|\n\t\t\t\t\t\n\t\t\t\t      r\n\t\t\t\t      \u2192\n\t\t\t\t    \n\t\t\t\t\t\u232a\n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t  \u2297\n\t\t\t\t\t  N\n\t\t\t\t\t\n\t\t\t\t\t\n\t\t\t\t\t|\n\t\t\t\t  \n\t\t\t\t  2\n\t\t\t\t\n              \n            \n          \n        ."},{"key":"ref_24","unstructured":"We can interpret this physically. We see from Section 2.3 that the best covariant measurement is obtained from Oz \u221d |J,J\u232a \u2329J,J|. Other covariant measurements can be obtained with Oz \u221d |m,J\u232a \u2329m,J| for 0 \u2264 m < J. The coefficients om can be interpreted as how much each of these measurements is done. The term |m,J\u232a \u2329m,J| can also be thought as the physical system used to measure. When it is highly polarized (m = J), the measurement is efficient. However, when the polarization is low, the information gain is weak, e.g., m = 0, and we clearly see that all POVM elements are \u221d 11."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"491","DOI":"10.1038\/nature19762","article-title":"Quantum dynamics of simultaneously measured non-commuting observables","volume":"538","author":"Martin","year":"2016","journal-title":"Nature"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/1\/39\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T14:50:40Z","timestamp":1760194240000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/1\/39"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,1,9]]},"references-count":25,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2018,1]]}},"alternative-id":["e20010039"],"URL":"https:\/\/doi.org\/10.3390\/e20010039","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2018,1,9]]}}}