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Quantum concepts in optical polarization

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Abstract

We comprehensively review the quantum theory of the polarization properties of light. In classical optics, these traits are characterized by the Stokes parameters, which can be geometrically interpreted using the Poincaré sphere. Remarkably, these Stokes parameters can also be applied to the quantum world, but then important differences emerge: now, because fluctuations in the number of photons are unavoidable, one is forced to work in the three-dimensional Poincaré space that can be regarded as a set of nested spheres. Additionally, higher-order moments of the Stokes variables might play a substantial role for quantum states, which is not the case for most classical Gaussian states. This brings about important differences between these two worlds that we review in detail. In particular, the classical degree of polarization produces unsatisfactory results in the quantum domain. We compare alternative quantum degrees and put forth that they order various states differently. Finally, intrinsically nonclassical states are explored, and their potential applications in quantum technologies are discussed.

© 2021 Optical Society of America

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2020 (5)

X. Zheng, S.-Q. Ma, G.-F. Zhang, H. Fan, and W.-M. Liu, “Unified and exact framework for variance-based uncertainty relations,” Sci. Rep. 10, 150 (2020).
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X. F. Qian and G. S. Agarwal, “Quantum duality: a source point of view,” Phys. Rev. Res. 2, 012031 (2020).
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A. Norrman, A. T. Friberg, and G. Leuchs, “Vector-light quantum complementarity and the degree of polarization,” Optica 7, 93–97 (2020).
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U. Chabaud, D. Markham, and F. Grosshans, “Stellar representation of non-Gaussian quantum states,” Phys. Rev. Lett. 124, 063605 (2020).
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J. Martin and S. Weigert, and O. Giraud, “Optimal detection of rotations about unknown axes by coherent and anticoherent states,” Quantum 4, 285 (2020).
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J. Martin and S. Weigert, and O. Giraud, “Optimal detection of rotations about unknown axes by coherent and anticoherent states,” Quantum 4, 285 (2020).
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2019 (5)

B. Kanseri and K. R. Sethuraj, “Experimental observation of the polarization coherence theorem,” Opt. Lett. 44, 159–162 (2019).
[Crossref]

P. Sánchez, J. Gonzales, V. Avalos, F. Auccapuclla, E. Suarez, and F. D. Zela, “Experimental display of the extended polarization coherence theorem,” Opt. Lett. 44, 1052–1055 (2019).
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A. Z. Goldberg, “Quantum theory of polarimetry: from quantum operations to Mueller matrices,” Phys. Rev. Res. 2, 023038 (2019).
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N. Korolkova and G. Leuchs, “Quantum correlations in separable multi-mode states and in classically entangled light,” Rep. Prog. Phys. 82, 056001 (2019).
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A. Evrard, V. Makhalov, T. Chalopin, L. A. Sidorenkov, J. Dalibard, R. Lopes, and S. Nascimbene, “Enhanced magnetic sensitivity with non-Gaussian quantum fluctuations,” Phys. Rev. Lett. 122, 173601 (2019).
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2018 (5)

C. Sanchidrián-Vaca and A. Luis, “Entanglement between total intensity and polarization for pairs of coherent states,” Phys. Rev. A 97, 043810 (2018).
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J. Weinbub and D. K. Ferry, “Recent advances in Wigner function approaches,” Appl. Phys. Rev. 5, 041104 (2018).
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X. F. Qian, A. N. Vamivakas, and J. H. Eberly, “Entanglement limits duality and vice versa,” Optica 5, 942–947 (2018).
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C. Chryssomalakos, E. Guzmán-González, and E. Serrano-Ensástiga, “Geometry of spin coherent states,” J. Phys. A 51, 165202 (2018).
[Crossref]

A. Z. Goldberg and D. F. V. James, “Quantum-limited Euler angle measurements using anticoherent states,” Phys. Rev. A 98, 032113 (2018).
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2017 (6)

J. H. Eberly, X. F. Qian, and A. N. Vamivakas, “Polarization coherence theorem,” Optica 4, 1113–1114 (2017).
[Crossref]

M. Lahiri, A. Hochrainer, R. Lapkiewicz, G. Lemos, and A. Zeilinger, “Partial polarization by quantum distinguishability,” Phys. Rev. A 95, 033816 (2017).
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C. Chryssomalakos and H. Hernández-Coronado, “Optimal quantum rotosensors,” Phys. Rev. A 95, 052125 (2017).
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M. Żukowski, W. Laskowski, and M. Wieśniak, “Normalized Stokes operators for polarization correlations of entangled optical fields,” Phys. Rev. A 95, 042113 (2017).
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F. Bouchard, P. de la Hoz, G. Björk, R. W. Boyd, M. Grassl, Z. Hradil, E. Karimi, A. B. Klimov, G. Leuchs, J. Řeháček, and L. L. Sánchez-Soto, “Quantum metrology at the limit with extremal Majorana constellations,” Optica 4, 1429–1432 (2017).
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A. Z. Goldberg and D. F. V. James, “Perfect polarization for arbitrary light beams,” Phys. Rev. A 96, 053859 (2017).
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2016 (9)

O. Bayraktar, M. Swillo, C. Canalias, and G. Björk, “Quantum-polarization state tomography,” Phys. Rev. A 94, 020105 (2016).
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S. Shabbir and G. Björk, “SU(2) uncertainty limits,” Phys. Rev. A 93, 052101 (2016).
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C. R. Müller, L. S. Madsen, A. B. Klimov, L. L. Sánchez-Soto, G. Leuchs, C. Marquardt, and U. L. Andersen, “Parsing polarization squeezing into Fock layers,” Phys. Rev. A 93, 033816 (2016).
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U. L. Andersen, T. Gehring, C. Marquardt, and G. Leuchs, “30 years of squeezed light generation,” Phys. Scr. 91, 053001 (2016).
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R. M. A. Azzam, “Stokes-vector and Mueller-matrix polarimetry,” J. Opt. Soc. Am. A 33, 1396–1408 (2016).
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A. Luis, “Polarization in quantum optics,” Prog. Opt. 61, 283–331 (2016).
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J. A. Jones, A. J. D’Addario, B. L. Rojec, G. Milione, and E. J. Galvez, “The Poincaré-sphere approach to polarization: formalism and new labs with Poincaré beams,” Am. J. Phys. 84, 822–835 (2016).
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T. Tudor, “Generalized Lorentz transformation in polarization optics,” J. Opt. Soc. Am. B 33, 898–902 (2016).
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H. D. Liu and L. B. Fu, “Berry phase and quantum entanglement in Majorana’s stellar representation,” Phys. Rev. A 94, 022123 (2016).
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2015 (9)

C. Yang, H. Guo, L.-B. Fu, and S. Chen, “Characterization of symmetry-protected topological phases in polymerized models by trajectories of Majorana stars,” Phys. Rev. B 91, 125132 (2015).
[Crossref]

G. Björk, M. Grassl, P. de la Hoz, G. Leuchs, and L. L. Sánchez-Soto, “Stars of the quantum universe: extremal constellations on the Poincaré sphere,” Phys. Scr. 90, 108008 (2015).
[Crossref]

J. S. Brauchart and P. J. Grabner, “Distributing many points on spheres: minimal energy and designs,” J. Complexity 31, 293–326 (2015).
[Crossref]

O. Giraud, D. Braun, D. Baguette, T. Bastin, and J. Martin, “Tensor representation of spin states,” Phys. Rev. Lett. 114, 080401 (2015).
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G. R. Franssens, “Relativistic kinematics formulation of the polarization effects of Jones–Mueller matrices,” J. Opt. Soc. Am. A 32, 164–172 (2015).
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J. J. Gil, “Intrinsic Stokes parameters for 3D and 2D polarization state,” J. Eur. Opt. Soc.-Rapid 10, 15054 (2015).
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A. S. Chirkin, “Polarization-squeezed light and quantum degree of polarization (a review),” Opt. Spectrosc. 119, 371–376 (2015).
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L. Dammeier, R. Schwonnek, and R. F. Werner, “Uncertainty relations for angular momentum,” New J. Phys. 17, 093046 (2015).
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G. Björk, A. B. Klimov, P. de la Hoz, M. Grassl, G. Leuchs, and L. L. Sánchez-Soto, “Extremal quantum states and their Majorana constellations,” Phys. Rev. A 92, 031801 (2015).
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2014 (10)

G. Björk, H. de Guise, A. B. Klimov, P. de la Hoz, and L. L. Sánchez-Soto, “Classical distinguishability as an operational measure of polarization,” Phys. Rev. A 90, 013830 (2014).
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F. Töppel, A. Aiello, C. Marquardt, E. Giacobino, and G. Leuchs, “Classical entanglement in polarization metrology,” New J. Phys. 16, 073019 (2014).
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L. Maccone and A. K. Pati, “Stronger uncertainty relations for all incompatible observables,” Phys. Rev. Lett. 113, 260401 (2014).
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G. Donati, T. J. Bartley, X.-M. Jin, M.-D. Vidrighin, A. Datta, M. Barbieri, and I. A. Walmsley, “Observing optical coherence across Fock layers with weak-field homodyne detectors,” Nat. Commun. 5, 6584 (2014).
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D. Goyeneche and K. Życzkowski, “Genuinely multipartite entangled states and orthogonal arrays,” Phys. Rev. A 90, 022316 (2014).
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W. Muessel, H. Strobel, D. Linnemann, D. B. Hume, and M. K. Oberthaler, “Scalable spin squeezing for quantum-enhanced magnetometry with Bose-Einstein condensates,” Phys. Rev. Lett. 113, 103004 (2014).
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L. A. Rozema, D. H. Mahler, R. Blume-Kohout, and A. M. Steinberg, “Optimizing the choice of spin-squeezed states for detecting and characterizing quantum processes,” Phys. Rev. X 4, 041025 (2014).
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F. De Zela, “Relationship between the degree of polarization, indistinguishability, and entanglement,” Phys. Rev. A 89, 013845 (2014).
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P. de la Hoz, G. Björk, A. B. Klimov, G. Leuchs, and L. L. Sánchez-Soto, “Unpolarized states and hidden polarization,” Phys. Rev. A 90, 043826 (2014).
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L.-P. Leppänen, K. Saastamoinen, A. T. Friberg, and T. Setälä, “Interferometric interpretation for the degree of polarization of classical optical beams,” New J. Phys. 16, 113059 (2014).
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2013 (16)

X. Cui, T.-L. Lian, B. Ho, B. L. Lev, and H. Zhai, “Synthetic gauge field with highly magnetic lanthanide atoms,” Phys. Rev. A 88, 011601 (2013).
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V. D’Ambrosio, N. Spagnolo, L. Del Re, S. Slussarenko, Y. Li, L. C. Kwek, L. Marrucci, S. P. Walborn, L. Aolita, and F. Sciarrino, “Photonic polarization gears for ultra-sensitive angular measurements,” Nat. Commun. 4, 2432 (2013).
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M. M. Taddei, B. M. Escher, L. Davidovich, and R. L. de Matos Filho, “Quantum speed limit for physical processes,” Phys. Rev. Lett. 110, 050402 (2013).
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A. del Campo, I. L. Egusquiza, M. B. Plenio, and S. F. Huelga, “Quantum speed limits in open system dynamics,” Phys. Rev. Lett. 110, 050403 (2013).
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L. Arnaud and N. J. Cerf, “Exploring pure quantum states with maximally mixed reductions,” Phys. Rev. A 87, 012319 (2013).
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G. Puentes, G. Colangelo, R. J. Sewell, and M. W. Mitchell, “Planar squeezing by quantum non-demolition measurement in cold atomic ensembles,” New J. Phys. 15, 103031 (2013).
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O. Civitarese, M. Reboiro, and D. Tielas, “Spin observables in an atomic CQE system,” J. Phys. B 46, 065502 (2013).
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A. Rivas and A. Luis, “SU(2)-invariant depolarization of quantum states of light-invariant depolarization of quantum states of light,” Phys. Rev. A 88, 052120 (2013).
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R. S. Singh and H. Prakash, “On the polarization of non-Gaussian optical quantum field: higher-order optical-polarization,” Ann. Phys. 333, 198–205 (2013).
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K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photonics 7, 72–78 (2013).
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L. L. Sánchez-Soto, A. B. Klimov, P. de la Hoz, and G. Leuchs, “Quantum versus classical polarization states: when multipoles count,” J. Phys. B 46, 104011 (2013).
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I. Rigas, A. B. Klimov, L. L. Sánchez-Soto, and G. Leuchs, “Nonlinear cross-Kerr quasiclassical dynamics,” New J. Phys. 15, 043038 (2013).
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C. Kothe, L. Madsen, U. L. Andersen, and G. Björk, “Experimental determination of the degree of polarization of quantum states,” Phys. Rev. A 87, 043814 (2013).
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R. S. Singh and H. Prakash, “Degree of polarization in quantum optics through the second generalization of intensity,” Phys. Rev. A 87, 025802 (2013).
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P. de la Hoz, A. B. Klimov, G. Björk, Y. H. Kim, C. Müller, C. Marquardt, G. Leuchs, and L. L. Sánchez-Soto, “Multipolar hierarchy of efficient quantum polarization measures,” Phys. Rev. A 88, 063803 (2013).
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J. Z. Salvail, M. Agnew, A. S. Johnson, E. Bolduc, J. Leach, and R. W. Boyd, “Full characterization of polarization states of light via direct measurement,” Nat. Photonics 7, 316–321 (2013).
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2012 (8)

H. de Guise, A. Vourdas, and L. L. Sánchez-Soto, “Complementarity and phases in SU(3),” J. Phys. A 45, 244030 (2012).
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H. You and J. D. Franson, “Theoretical comparison of quantum Zeno gates and logic gates based on the cross-Kerr nonlinearity,” Quantum Inf. Process. 11, 1627–1651 (2012).
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G. Björk, J. Söderholm, Y. S. Kim, Y. S. Ra, H. T. Lim, C. Kothe, Y. H. Kim, L. L. Sánchez-Soto, and A. B. Klimov, “Central-moment description of polarization for quantum states of light,” Phys. Rev. A 85, 053835 (2012).
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C. R. Müller, B. Stoklasa, C. Peuntinger, C. Gabriel, J. Řeháček, Z. Hradil, A. B. Klimov, G. Leuchs, C. Marquardt, and L. L. Sánchez-Soto, “Quantum polarization tomography of bright squeezed light,” New J. Phys. 14, 085002 (2012).
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R. J. Sewell, M. Koschorreck, M. Napolitano, B. Dubost, N. Behbood, and M. W. Mitchell, “Magnetic sensitivity beyond the projection noise limit by spin squeezing,” Phys. Rev. Lett. 109, 253605 (2012).
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P. Bruno, “Quantum geometric phase in Majorana’s stellar representation: mapping onto a many-body Aharonov-Bohm phase,” Phys. Rev. Lett. 108, 240402 (2012).
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B. Lian, T.-L. Ho, and H. Zhai, “Searching for non-Abelian phases in the Bose-Einstein condensate of dysprosium,” Phys. Rev. A 85, 051606 (2012).
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A. R. U. Devi Sudha and A. K. Rajagopal, “Majorana representation of symmetric multiqubit states,” Quantum Inf. Process. 11, 685–710 (2012).
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2011 (10)

E. Bannai and M. Tagami, “A note on anticoherent spin states,” J. Phys. A 44, 342002 (2011).
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G. K. Gupta, A. Kumar, and R. S. Singh, “Generation of hidden optical-polarization: squeezing and non-classicality,” Opt. Commun. 284, 4951–4956 (2011).
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M. Lahiri, “Wave-particle duality and polarization properties of light in single-photon interference experiments,” Phys. Rev. A 83, 045803 (2011).
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V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222–229 (2011).
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Q. Y. He, S.-G. Peng, P. D. Drummond, and M. D. Reid, “Planar quantum squeezing and atom interferometry,” Phys. Rev. A 84, 022107 (2011).
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J. Ma, X. Wang, C. P. Sun, and F. Nori, “Quantum spin squeezing,” Phys. Rep. 509, 89–165 (2011).
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R. Prakash and N. Shukla, “Polarization squeezing in polarized light,” Opt. Commun. 284, 3568–3570 (2011).
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T. G. Brown, “Unconventional polarization states: beam propagation, focusing, and imaging,” Prog. Opt. 56, 81–129 (2011).
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J. B. Altepeter, N. N. Oza, M. Medi, E. R. Jeffrey, and P. Kumar, “Entangled photon polarimetry,” Opt. Express 19, 26011–26016 (2011).
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X.-F. Qian and J. H. Eberly, “Entanglement and classical polarization states,” Opt. Lett. 36, 4110–4112 (2011).
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2010 (21)

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
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R. B. A. Adamson and A. M. Steinberg, “Improving quantum state estimation with mutually unbiased bases,” Phys. Rev. Lett. 105, 030406 (2010).
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L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi, R. Ramponi, and R. Osellame, “Polarization entangled state measurement on a chip,” Phys. Rev. Lett. 105, 200503 (2010).
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T. Durt, B.-G. Englert, I. Bengtsson, and K. Życzkowski, “On mutually unbiased bases,” Int. J. Quantum Inform. 08, 535–640 (2010).
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S. N. Filippov and V. I. Man’ko, “Inverse spin-s portrait and representation of qudit states by single probability vectors,” J. Russ. Las. Res. 31, 32–54 (2010).
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U. Schilling, J. von Zanthier, and G. S. Agarwal, “Measuring arbitrary-order coherences: Tomography of single-mode multiphoton polarization-entangled states,” Phys. Rev. A 81, 013826 (2010).
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G. Björk, J. Söderholm, L. L. Sánchez-Soto, A. B. Klimov, I. Ghiu, P. Marian, and T. A. Marian, “Quantum degrees of polarization,” Opt. Commun. 283, 4440–4447 (2010).
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A. B. Klimov, G. Björk, J. Söderholm, L. S. Madsen, M. Lassen, U. L. Andersen, J. Heersink, R. Dong, C. Marquardt, G. Leuchs, and L. L. Sánchez-Soto, “Assessing the polarization of a quantum field from Stokes fluctuations,” Phys. Rev. Lett. 105, 153602 (2010).
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I. Ghiu, G. Björk, P. Marian, and T. A. Marian, “Probing light polarization with the quantum Chernoff bound,” Phys. Rev. A 82, 023803 (2010).
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C. Brosseau, “Polarization and coherence optics: historical perspective, status, and future directions,” Prog. Opt. 54, 149–208 (2010).
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D. Mahler, P. Joanis, R. Vilim, and H. de Guise, “Squeezing in SU(2) intelligent states,” New J. Phys. 12, 033037 (2010).
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B. N. Simon, S. Simon, N. Mukunda, F. Gori, M. Santarsiero, R. Borghi, and R. Simon, “A complete characterization of pre-Mueller and Mueller matrices in polarization optics,” J. Opt. Soc. Am. A 27, 188–199 (2010).
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S. Wehner and A. Winter, “Entropic uncertainty relations—a survey,” New J. Phys. 12, 025009 (2010).
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A. Polkovnikov, “Phase space representation of quantum dynamics,” Ann. Phys. 325, 1790–1852 (2010).
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W. Wasilewski, K. Jensen, H. Krauter, J. J. Renema, M. V. Balabas, and E. S. Polzik, “Quantum noise limited and entanglement-assisted magnetometry,” Phys. Rev. Lett. 104, 133601 (2010).
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O. Giraud, P. Braun, and D. Braun, “Quantifying quantumness and the quest for Queens of Quantumness,” New J. Phys. 12, 063005 (2010).
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M. Aulbach, D. Markham, and M. Murao, “The maximally entangled symmetric state in terms of the geometric measure,” New J. Phys 12, 073025 (2010).
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M. Jakob and J. A. Bergou, “Quantitative complementarity relations in bipartite systems: entanglement as a physical reality,” Opt. Commun. 283, 827–830 (2010).
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J. Crann, R. Pereira, and D. W. Kribs, “Spherical designs and anticoherent spin states,” J. Phys. A 43, 255307 (2010).
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H. Mäkelä and A. Messina, “N-qubit states as points on the Bloch sphere,” Phys. Scr. T140, 014054 (2010).
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A. Lamacraft, “Low-energy dynamics of spinor condensates,” Phys. Rev. B 81, 184526 (2010).
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2009 (7)

L. K. Shalm, R. B. A. Adamson, and A. M. Steinberg, “Squeezing and over-squeezing of triphotons,” Nature 457, 67–70 (2009).
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T. Iskhakov, M. V. Chekhova, and G. Leuchs, “Generation and direct detection of broadband mesoscopic polarization-squeezed vacuum,” Phys. Rev. Lett. 102, 183602 (2009).
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J. Tervo, T. Setälä, A. Roueff, P. Réfrégier, and A. T. Friberg, “Two-point Stokes parameters: interpretation and properties,” Opt. Lett. 34, 3074–3076 (2009).
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M. Rådmark, M. Żukowski, and M. Bourennane, “Experimental high fidelity six-photon entangled state for telecloning protocols,” New J. Phys. 11, 103016 (2009).
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P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, “Ultrasensitive beam deflection measurement via interferometric weak value amplification,” Phys. Rev. Lett. 102, 173601 (2009).
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J. Řeháček, S. Olivares, D. Mogilevtsev, Z. Hradil, M. G. A. Paris, S. Fornaro, V. D’Auria, A. Porzio, and S. Solimeno, “Effective method to estimate multidimensional Gaussian states,” Phys. Rev. A 79, 032111 (2009).
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A. I. Lvovsky and M. G. Raymer, “Continuous-variable optical quantum-state tomography,” Rev. Mod. Phys. 81, 299–322 (2009).
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2008 (9)

G. Björk, A. B. Klimov, and L. L. Sánchez-Soto, “The discrete Wigner function,” Prog. Opt. 51, 469–516 (2008).
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Y. F. Xiao, Ş. K. Özdemir, V. Gaddam, C.-H. Dong, N. Imoto, and L. Yang, “Quantum nondemolition measurement of photon number via optical Kerr effect in an ultra-high-Q microtoroid cavity,” Opt. Express 16, 21462–21475 (2008).
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J. P. Dowling, “Quantum optical metrology–the lowdown on high-N00N states,” Contemp. Phys. 49, 125–143 (2008).
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A. B. Klimov and J. L. Romero, “A generalized Wigner function for quantum systems with the SU(2) dynamical symmetry group,” J. Phys. A 41, 055303 (2008).
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M. Yao, Y. Cai, H. T. Eyyuboğlu, Y. Baykal, and O. Korotkova, “Evolution of the degree of polarization of an electromagnetic Gaussian Schell-model beam in a Gaussian cavity,” Opt. Lett. 33, 2266–2268 (2008).
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J. F. Corney, J. Heersink, R. Dong, V. Josse, P. D. Drummond, G. Leuchs, and U. L. Andersen, “Simulations and experiments on polarization squeezing in optical fiber,” Phys. Rev. A 78, 023831 (2008).
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A. B. Klimov, J. L. Romero, L. L. Sánchez-Soto, A. Messina, and A. Napoli, “Quantum light depolarization: the phase-space perspective,” Phys. Rev. A 77, 033853 (2008).
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A. Rivas and A. Luis, “Characterization of quantum angular-momentum fluctuations via principal components,” Phys. Rev. A 77, 022105 (2008).
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P. Kolenderski and R. Demkowicz-Dobrzanski, “Optimal state for keeping reference frames aligned and the platonic solids,” Phys. Rev. A 78, 052333 (2008).
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2007 (17)

P. Ribeiro, J. Vidal, and R. Mosseri, “Thermodynamical limit of the Lipkin-Meshkov-Glick model,” Phys. Rev. Lett. 99, 050402 (2007).
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C. Marquardt, J. Heersink, R. Dong, M. V. Chekhova, A. B. Klimov, L. L. Sánchez-Soto, U. L. Andersen, and G. Leuchs, “Quantum reconstruction of an intense polarization squeezed optical state,” Phys. Rev. Lett. 99, 220401 (2007).
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V. P. Karassiov, “Polarization of light in classical and quantum optics: concepts and applications,” Opt. Spectrosc. 103, 137–144 (2007).
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R. Dong, J. Heersink, J.-I. Yoshikawa, O. Glöckl, U. L. Andersen, and G. Leuchs, “An efficient source of continuous variable polarization entanglement,” New J. Phys. 9, 410 (2007).
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A. Al-Qasimi, O. Korotkova, D. James, and E. Wolf, “Definitions of the degree of polarization of a light beam,” Opt. Lett. 32, 1015–1016 (2007).
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V. P. Karassiov and S. P. Kulik, “Polarization transformations of multimode light fields,” JETP 104, 30–46 (2007).
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K. J. Resch, K. L. Pregnell, R. Prevedel, A. Gilchrist, G. J. Pryde, J. L. O’Brien, and A. G. White, “Time-reversal and super-resolving phase measurements,” Phys. Rev. Lett. 98, 223601 (2007).
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J. J. Gil, “Polarimetric characterization of light and media,” Eur. Phys. J. Appl. Phys. 40, 1–47 (2007).
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P. Réfrégier and A. Roueff, “Intrinsic coherence: a new concept in polarization and coherence theory,” Opt. Photon. News 18, 30–35 (2007).
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B. Schaefer, E. Collett, R. Smyth, D. Barrett, and B. Fraher, “Measuring the Stokes polarization parameters,” Am. J. Phys. 75, 163–168 (2007).
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N. L. Harshman and S. Wickramasekara, “Tensor product structures, entanglement, and particle scattering,” Open Syst. Inf. Dyn. 14, 341–351 (2007).
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M. Barbieri, G. Vallone, P. Mataloni, and F. De Martini, “Complete and deterministic discrimination of polarization Bell states assisted by momentum entanglement,” Phys. Rev. A 75, 042317 (2007).
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N. Matsuda, Y. Mitsumori, H. Kosaka, K. Edamatsu, and R. Shimizu, “Lossless all-optical phase gate using a polarization-division Sagnac interferometer applicable to a waveguide-type Kerr medium,” Appl. Phys. Lett. 91, 171119 (2007).
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A. Vourdas, “Quantum systems with finite Hilbert space: Galois fields in quantum mechanics,” J. Phys. A 40, R285–R331 (2007).
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A. Luis, “Polarization distributions and degree of polarization for quantum Gaussian light fields,” Opt. Commun. 273, 173–181 (2007).
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L. L. Sánchez-Soto, E. C. Yustas, G. Björk, and A. B. Klimov, “Maximally polarized states for quantum light fields,” Phys. Rev. A 76, 043820 (2007).
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A. Luis, “Degree of polarization of type-II unpolarized light,” Phys. Rev. A 75, 053806 (2007).
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2006 (10)

L. L. Sánchez-Soto, J. Söderholm, E. C. Yustas, A. B. Klimov, and G. Björk, “Degrees of polarization for a quantum field,” J. Phys. Conf. Series 36, 177–182 (2006).
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D. Elser, U. L. Andersen, A. Korn, O. Glöckl, S. Lorenz, C. Marquardt, and G. Leuchs, “Reduction of guided acoustic wave Brillouin scattering in photonic crystal fibers,” Phys. Rev. Lett. 97, 133901 (2006).
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A. B. Klimov, J. Delgado, and L. L. Sánchez-Soto, “Quantum phase-space description of light polarization,” Opt. Commun. 258, 210–218 (2006).
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C. Brosseau and A. Dogariu, “Symmetry properties and polarization descriptors for an arbitrary electromagnetic wavefield,” Prog. Opt. 49, 315–380 (2006).
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A. Luis and N. Korolkova, “Polarization squeezing and nonclassical properties of light,” Phys. Rev. A 74, 043817 (2006).
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E. V. Moreva, G. A. Maslennikov, S. S. Straupe, and S. P. Kulik, “Realization of four-level qudits using biphotons,” Phys. Rev. Lett. 97, 023602 (2006).
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S. Kryszewski and M. Zachciał, “Positivity of the N × N density matrix expressed in terms of polarization operators,” J. Phys. A 39, 5921–5931 (2006).
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V. P. Karassiov, “P-quasispin formalism in polarization optics,” JETP Lett. 84, 640–644 (2006).
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S. Chaturvedi, G. Marmo, and N. Mukunda, “The Schwinger representation of a group: concept and applications,” Rev. Math. Phys. 18, 887–912 (2006).
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J. Zimba, “‘Anticoherent’ spin states via the Majorana representation,” Electron. J. Theor. Phys. 3, 143–156 (2006).

2005 (9)

J. Heersink, V. Josse, G. Leuchs, and U. L. Andersen, “Efficient polarization squeezing in optical fibers,” Opt. Lett. 30, 1192–1194 (2005).
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O. Korotkova and E. Wolf, “Generalized Stokes parameters of random electromagnetic beams,” Opt. Lett. 30, 198–200 (2005).
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A. Luis, “Quantum polarization distributions via marginals of quadrature distributions,” Phys. Rev. A 71, 053801 (2005).
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A. Luis, “Properties of spatial-angular Stokes parameters,” Opt. Commun. 251, 243–253 (2005).
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F. L. Semião and A. Vidiella-Barranco, “Effective cross-Kerr nonlinearity and robust phase gates with trapped ions,” Phys. Rev. A 72, 064305 (2005).
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A. Gilchrist, N. K. Langford, and M. A. Nielsen, “Distance measures to compare real and ideal quantum processes,” Phys. Rev. A 71, 062310 (2005).
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A. B. Klimov, L. L. Sánchez-Soto, E. C. Yustas, J. Söderholm, and G. Björk, “Distance-based degrees of polarization for a quantum field,” Phys. Rev. A 72, 033813 (2005).
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V. Karassiov, “Polarization states of light and their quantum tomography,” J. Russ. Las. Res. 26, 484–513 (2005).
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A. Sehat, J. Söderholm, G. Björk, P. Espinoza, A. B. Klimov, and L. L. Sánchez-Soto, “Quantum polarization properties of two-mode energy eigenstates,” Phys. Rev. A 71, 033818 (2005).
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2004 (7)

A. Vourdas, “Quantum systems with finite Hilbert space,” Rep. Prog. Phys. 67, 267–320 (2004).
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V. P. Karassiov and A. V. Masalov, “The method of polarization tomography of radiation in quantum optics,” JETP 99, 51–60 (2004).
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H. F. Hofmann and S. Takeuchi, “Quantum-state tomography for spin-1 systems,” Phys. Rev. A 69, 042108 (2004).
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Y. I. Bogdanov, M. V. Chekhova, S. P. Kulik, G. A. Maslennikov, A. A. Zhukov, C. H. Oh, and M. K. Tey, “Qutrit state engineering with biphotons,” Phys. Rev. Lett. 93, 230503 (2004).
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P. Zanardi, D. A. Lidar, and S. Lloyd, “Quantum tensor product structures are observable induced,” Phys. Rev. Lett. 92, 060402 (2004).
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J. Ellis and A. Dogariu, “Complex degree of mutual polarization,” Opt. Lett. 29, 536–538 (2004).
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P. Vahimaa and J. Tervo, “Unified measures for optical fields: degree of polarization and effective degree of coherence,” J. Opt. A 6, S41–S44 (2004).
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2003 (7)

J. Heersink, T. Gaber, S. Lorenz, O. Glöckl, N. Korolkova, and G. Leuchs, “Polarization squeezing of intense pulses with a fiber-optic Sagnac interferometer,” Phys. Rev. A 68, 013815 (2003).
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O. Glöckl, J. Heersink, N. Korolkova, G. Leuchs, and S. Lorenz, “A pulsed source of continuous variable polarization entanglement,” J. Opt. B 5, S492–S496 (2003).
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R. Schnabel, W. P. Bowen, N. Treps, T. C. Ralph, H.-A. Bachor, and P. K. Lam, “Stokes-operator-squeezed continuous-variable polarization states,” Phys. Rev. A 67, 012316 (2003).
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H. K. Urbantke, “The Hopf fibration—seven times in physics,” J. Geom. Phys. 46, 125–150 (2003).
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M. Barbieri, F. De Martini, G. Di Nepi, P. Mataloni, G. M. D’Ariano, and C. Macchiavello, “Detection of entanglement with polarized photons: experimental realization of an entanglement witness,” Phys. Rev. Lett. 91, 227901 (2003).
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G. M. D’Ariano, L. Maccone, and M. Paini, “Spin tomography,” J. Opt. B 5, 77–84 (2003).
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P. M. Alberti, “Playing with fidelities,” Rep. Math. Phys. 51, 87–125 (2003).
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2002 (11)

G. Björk, J. Söderholm, A. Trifonov, P. A. Usachev, L. L. Sánchez-Soto, and A. B. Klimov, “Applications of entangled-state interference,” Proc. SPIE 4750,1–12 (2002).
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A. B. Klimov, O. V. Man’ko, V. I. Man’ko, Y. F. Smirnov, and V. N. Tolstoy, “Tomographic representation of spin and quark states,” J. Phys. A 35, 6101–6123 (2002).
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A. Luis, “Degree of polarization in quantum optics,” Phys. Rev. A 66, 013806 (2002).
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P. Marian, T. A. Marian, and H. Scutaru, “Quantifying nonclassicality of one-mode Gaussian states of the radiation field,” Phys. Rev. Lett. 88, 153601 (2002).
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R. T. Thew, K. Nemoto, A. G. White, and W. J. Munro, “Qudit quantum-state tomography,” Phys. Rev. A 66, 012303 (2002).
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H. Lee, P. Kok, and J. P. Dowling, “A quantum Rosetta stone for interferometry,” J. Mod. Opt. 49, 2325–2338 (2002).
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A. B. Klimov, “Exact evolution equations for SU(2) quasidistribution functions,” J. Math. Phys. 43, 2202–2213 (2002).
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N. L. Manakov, A. V. Meremianin, and A. F. Starace, “Multipole expansions of irreducible tensor sets and some applications,” J. Phys. B 35, 77 (2002).
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W. P. Bowen, R. Schnabel, H.-A. Bachor, and P. K. Lam, “Polarization squeezing of continuous variable Stokes parameters,” Phys. Rev. Lett. 88, 093601 (2002).
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M. Abdel-Aty, M. S. Abdalla, and A.-S. F. Obada, “Entropy squeezing of a two-mode multiphoton Jaynes-Cummings model in the presence of a nonlinear medium,” J. Opt. B 4, 134 (2002).
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N. Korolkova, G. Leuchs, R. Loudon, T. C. Ralph, and C. Silberhorn, “Polarization squeezing and continuous-variable polarization entanglement,” Phys. Rev. A 65, 052306 (2002).
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2001 (7)

J. Söderholm, G. Björk, and A. Trifonov, “Unpolarized light in quantum optics,” Opt. Spectrosc. 91, 532–534 (2001).
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P. Usachev, J. Söderholm, G. Björk, and A. Trifonov, “Experimental verification of differences between classical and quantum polarization properties,” Opt. Commun. 193, 161–173 (2001).
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Figures (13)

Figure 1.
Figure 1. (Left) Polarization ellipse showing the orientation angle $\psi$ and the ellipticity $\chi$, which are functions of the semi-major and semi-minor axes, $a$ and $b$. (Right) Poincaré sphere and the parametrization of Stokes parameters for polarized light in the form of spherical coordinates.
Figure 2.
Figure 2. Typical Stokes polarimetric setups. (Left) Rotating wave plate polarimeter consisting of a rotating quarter-wave plate, a linear polarizer and a photodetector. (Right) Division-of-amplitude polarimeter, which splits the beam into three separate beams. Each of these beams is analyzed to provide a Stokes parameter. The three identical boxes in the right setup correspond to polarizing beam splitters.
Figure 3.
Figure 3. Density plots of the Husimi $ Q $-function for: (left) an SU(2) coherent state centered in the north pole; (right) a NOON state. In both cases, $ S = 4 $. In all the density plots in the paper, we use the same colormap that ranges from dark blue (corresponding to the numerical value 0) to bright red (corresponding to the numerical value 1).
Figure 4.
Figure 4. Density plots of the Husimi $ Q $-function for: (left) a two-mode coherent state in which only the $ + $ mode is excited; (right) a two-mode squeezed vacuum state with squeezing parameter $ r = 0.9 $.
Figure 5.
Figure 5. Density plot of Husimi $ Q $-function for the state $ |1,0\rangle $. Here, the dipolar component ($ K = 1 $ vanishes), so this state lacks first-order information. The features of the Husimi function are then due to the quadrupolar ($ K = 2 $) components of the state.
Figure 6.
Figure 6. Schematic representation of the effect of a Kerr medium in the phase space of a single mode, assuming an initial coherent state. Reprinted with permission from I. Rigas et al., New J. Phys. 15, 043038 (2013) [229]. Licensed under a Creative Commons CC BY 3.0 license.
Figure 7.
Figure 7. Typical setup for an efficient Stokes tomography. The modes are combined on a polarizing beam splitter (PBS), and interference between the modes can be adjusted with the combination of a half-wave plate ($ \lambda /2,\theta $) and a quarter-wave plate ($ \lambda /4,\phi $). The polarization states are spatially separated into orthogonal components that can be processed independently.
Figure 8.
Figure 8. Experimental scheme for the multipole reconstruction of a three-photon state. The detection of a single photon at detector D1 heralds the state. Fourfold coincidences with detectors D1, D2, D3, and D4 allow for the required projection measurements. The scheme also works for states with a higher photon number.
Figure 9.
Figure 9. (Top left) Experimental setup for the characterization of two-mode fields. The polarization states are separated into orthogonal components followed by homodyne tomography. (Top right) Measured polarization sector (black blocks) for the case of a polarization squeezed state with coherent amplitude $ \alpha = 1.13 $ and squeezing parameter $ r = 0.41 $. (Bottom) Reconstructed Husimi functions $ Q(S,{\textbf{n}}) $ of the Fock layers indicated in the insets for the same state. Adapted from Figs. 1 and 4 from C. Müller et al., Phys. Rev. A 93, 033816 (2016) [136]. Copyright 2016 by the American Physical Society. https://journals.aps.org/pra/abstract/10.1103/PhysRevA.93.033816.
Figure 10.
Figure 10. (Left) Isocontour surface (of the level $ 1/e $ from the maximum) of the Husimi function $ Q(S,{\textbf{n}}) $ for a bright squeezed state. We also show cross sections of the $ Q $-function through the three coordinate planes. In blue we show the isotropic section for a coherent state, which we scale to unity for all the plots. (Right) Reconstructed total Husimi function $ Q({\textbf{n}}) $ obtained by summing over all the (continuous) Fock layers. Reprinted from C. R. Müller, et al., New J. Phys. 14, 085002 (2012) [109]. © IOP Publishing and Deutsche Physikalische Gesellschaft. Reproduced by permission of IOP Publishing. CC BY-NC-SA.
Figure 11.
Figure 11. Left, degree of polarization $ {{\mathbb P}_M} $ as a function of the multipole order $ M $ for the state $ |S,0\rangle $; right, a quadrature coherent state $ |{\alpha _ + },{\alpha _ - }\rangle $ with average number of photons $ \bar N = |{\alpha _ + }{|^2} + |{\alpha _ - }{|^2} $.
Figure 12.
Figure 12. Second-order degree of polarization as a function of purity, for the first-order unpolarized states [Eq. (163)].
Figure 13.
Figure 13. Density plots of the SU(2) $ Q $-functions for the optimal states in Table 1 for the cases $ S = 3, 5, 6 $, and 10 (from left to right, blue indicates the zero values and red maximal ones). On top, we sketch the Majorana constellation for each of them.

Tables (1)

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Table 1. States for Which A M Vanishes for the Indicated Values of S a,b

Equations (199)

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E H ( z , t ) = E 0 H exp [ i ( ω t k z + δ H ) ] , E V ( z , t ) = E 0 V exp [ i ( ω t k z + δ V ) ] .
( E H E 0 H ) 2 + ( E V E 0 V ) 2 2 ( E H E 0 H ) ( E V E 0 V ) cos δ = sin 2 δ ,
tan ( 2 ψ ) = 2 E 0 H E 0 V E 0 H 2 E 0 V 2 cos δ , sin ( 2 χ ) = 2 E 0 H E 0 V E 0 H 2 + E 0 V 2 sin δ .
E H ( z , t ) = E 0 a H exp [ i ( ω t k z ) ] , E V ( z , t ) = E 0 a V exp [ i ( ω t k z ) ] ,
A HV = ( a H a V ) ,
A ± = 1 2 ( 1 i 1 i ) A HV .
J ± = A ± A ± = ( a + a + a + a a + a a a ) ,
S μ = 1 2 Tr ( J ± σ μ ) ,
S μ = 1 2 A ± σ μ A ± .
S 0 = 1 2 ( a + a + + a a ) , S 1 = 1 2 ( a + a + a + a ) , S 2 = i 2 ( a + a a + a ) , S 3 = 1 2 ( a + a + a a ) .
S 0 2 S 1 2 + S 2 2 + S 3 2 ,
S 1 = S 0 cos ( 2 χ ) sin ( 2 ψ ) , S 2 = S 0 cos ( 2 χ ) cos ( 2 ψ ) , S 3 = S 0 sin ( 2 χ ) .
= A 2 = A A = | a + | 2 + | a | 2 = 2 S 0
S 0 = 1 2 ( | a + | 2 + | a | 2 ) , S 1 = 1 2 ( | a H | 2 | a V | 2 ) , S 2 = 1 2 ( | a 45 | 2 | a 135 | 2 ) , S 3 = 1 2 ( | a + | 2 | a | 2 ) ,
A = T A .
U ( n , ϑ ) = exp ( i ϑ n σ / 2 ) = ( cos ( ϑ / 2 ) i sin ( ϑ / 2 ) cos Θ i sin ( ϑ / 2 ) sin Θ e i Φ i sin ( ϑ / 2 ) sin Θ e i Φ cos ( ϑ / 2 ) + i sin ( ϑ / 2 ) cos Θ ) .
S = R ( U ) S ,
U ( S σ ) U = ( R ( U ) S ) σ ,
R ( U ) j k = 1 2 Tr ( σ j U σ k U 1 ) .
U ( e 3 , ϕ ) = e i ϕ σ 3 / 2 = ( e i ϕ / 2 0 0 e i ϕ / 2 ) R ( e 3 , ϕ ) = ( cos ϕ sin ϕ 0 sin ϕ cos ϕ 0 0 0 1 ) , [ 24 p t ] U ( e 2 , θ ) = e i θ σ 2 / 2 = ( cos ( θ / 2 ) sin ( θ / 2 ) sin ( θ / 2 ) cos ( θ / 2 ) ) R ( e 2 , θ ) = ( cos θ 0 sin θ 0 1 0 sin θ 0 cos θ ) .
H ( n , η ) = e ρ exp ( η n σ / 2 ) = e ρ ( cosh ( η / 2 ) + sinh ( η / 2 ) cos Θ sinh ( η / 2 ) sin Θ e i Φ sinh ( η / 2 ) sin Θ e i Φ cosh ( η / 2 ) sinh ( η / 2 ) cos Θ ) ,
H ( n , η ) = U ( n , 0 ) e η σ 3 / 2 U ( n , 0 ) ,
S μ = M μ ν ( H ) S ν ,
M ( H ) μ ν = 1 2 Tr ( σ μ H σ ν H ) ,
T = U H ,
M 2 = S 0 2 S 1 2 S 2 2 S 3 2
J = T i J T i .
J pol 2 = J pol .
J unpol = 1 2 𝟙 ,
J = [ P J pol + ( 1 P ) J unpol ] ,
P = 2 Tr ( J 2 ) Tr 2 ( J ) 1 = 1 4 det ( J ) Tr 2 ( J ) ,
P = λ + λ λ + + λ .
P = S 1 2 + S 2 2 + S 3 2 S 0 ,
S = Tr ( J ln J ) .
S = i = ± λ i ln λ i ,
S = { 1 2 ( 1 + P ) ln [ 1 2 ( 1 + P ) ] + 1 2 ( 1 P ) ln [ 1 2 ( 1 P ) ] } .
[ a ^ s , a ^ s ] = δ s s 𝟙 , s , s { + , } ,
S ^ 0 = 1 2 ( a ^ + a ^ + + a ^ a ^ ) , S ^ 1 = 1 2 ( a ^ + a ^ + a ^ + a ^ ) , S ^ 2 = i 2 ( a ^ + a ^ a ^ + a ^ ) , S ^ 3 = 1 2 ( a ^ + a ^ + a ^ a ^ ) ,
[ S ^ k , S ^ ] = i ϵ k m S ^ m , [ S ^ 0 , S ^ ] = 0 ,
S ^ 2 = S ^ 1 2 + S ^ 2 2 + S ^ 3 2 = S ( S + 1 ) 𝟙 ^ ,
| S , m | n + = S + m , n = S m .
S ^ 3 | S , m = m | S , m , S ^ ± | S , m = S ( S + 1 ) m ( m ± 1 ) | S , m ± 1 ,
ϱ ^ pol = S w S ϱ ^ ( S ) = S = 0 m , m = S S w S ϱ ^ m m ( S ) | S , m S , m | .
Var ( S ^ k ) Var ( S ^ ) | ϵ k m S ^ m | 2 ,
Var ( S ^ ) = Var ( S ^ 1 ) + Var ( S ^ 2 ) + Var ( S ^ 3 ) 2 S ^ 0 .
| S , n = m = S S c m ( n ) | S , m ,
c m ( n ) = ( 2 S S + m ) 1 / 2 [ sin ( θ / 2 ) ]