# Morphophoric POVMs, generalised qplexes, and 2-designs

Institute of Mathematics, Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland

### Abstract

We study the class of quantum measurements with the property that the image of the set of quantum states under the measurement map transforming states into probability distributions is similar to this set and call such measurements morphophoric. This leads to the generalisation of the notion of a qplex, where SIC-POVMs are replaced by the elements of the much larger class of morphophoric POVMs, containing in particular 2-design (rank-1 and equal-trace) POVMs. The intrinsic geometry of a generalised qplex is the same as that of the set of quantum states, so we explore its external geometry, investigating, inter alia, the algebraic and geometric form of the inner (basis) and the outer (primal) polytopes between which the generalised qplex is sandwiched. In particular, we examine generalised qplexes generated by MUB-like 2-design POVMs utilising their graph-theoretical properties. Moreover, we show how to extend the primal equation of QBism designed for SIC-POVMs to the morphophoric case.

### ► References

[1] Amaral, B, Terra Cunha, M, On Graph Approaches to Contextuality and their Role in Quantum Theory, Springer, 2018.
https:/​/​doi.org/​10.1007/​978-3-319-93827-1

[2] Appleby, DM, Ericsson, Å, Fuchs, CA, Properties of QBist state spaces, Found. Phys. 41 (2011), 564–579.
https:/​/​doi.org/​10.1007/​s10701-010-9458-7

[3] Appleby, M, Fuchs, CA, Stacey, BC, Zhu, H, Introducing the Qplex: a novel arena for quantum theory, Eur. Phys. J. D 71 (2017), 197.
https:/​/​doi.org/​10.1140/​epjd/​e2017-80024-y

[4] Bang, S, Hiraki, A, Koolen, JH, Delsarte clique graphs, European J. Combin. 28 (2007), 501–516.
https:/​/​doi.org/​10.1016/​j.ejc.2005.04.015

[5] Belovs, A, Smotrovs, J, A criterion for attaining the Welch bounds with applications for mutually unbiased bases, in Calmet, J, Geiselmann, W, Müller-Quade, J (Eds), Mathematical Methods in Computer Science, MMICS 2008, Karlsruhe, Germany, December 17-19, 2008 - Essays in Memory of Thomas Beth, pp. 50-69.
https:/​/​doi.org/​10.1007/​978-3-540-89994-5_6

[6] Bengtsson, I, Życzkowski, K, Geometry of Quantum States. An Introduction to Quantum Entanglement, 2nd ed., Cambridge UP, 2017.
https:/​/​doi.org/​10.1017/​9781139207010

[7] Bengtsson, I, Życzkowski, K, On discrete structures in finite Hilbert spaces, arXiv:1701.07902 [quant-ph].
arXiv:1701.07902

[8] Blokhuis, A, Brouwer, AE, Uniqueness of a Zara graph on 126 points and non-existence of a completely regular two-graph on 288 points, in de Doelder, PJ, de Graaf, J, van Lint, JH (Eds), Papers dedicated to J. J. Seidel. EUT Report 84-WSK-031. Eindhoven, Netherlands: Technische Hogeschool Eindhoven, 1984, pp. 6–19.
https:/​/​research.tue.nl/​en/​publications/​uniqueness-of-a-zara-graph-on-126-points-and-non-existence-of-a-c

[9] Bodmann, BG, Haas, JI, A short history of frames and quantum designs, in Bruillard, P, Ortiz Marrero, C, Plavnik, J (Eds), Topological Phases of Matter and Quantum Computation. Contemporary Mathematics vol. 747. Providence RI, AMS, 2020, pp. 215–226.
https:/​/​doi.org/​10.1090/​conm/​747/​15047

[10] Brandsen, S, Dall'Arno, M, Szymusiak, A, Communication capacity of mixed quantum t-designs, Phys. Rev. A 94 (2016), 022335.
https:/​/​doi.org/​10.1103/​physreva.94.022335

[11] Brouwer, AE, Cohen, AM, Neumaier, A, Distance-Regular Graphs, Springer, 1989.
https:/​/​doi.org/​10.1007/​978-3-642-74341-2

[12] Brouwer, AE, van Maldeghem, H, Strongly Regular Graphs, https:/​/​homepages.cwi.nl/​\symbol126aeb/​ math/​srg/​rk3/​srgw.pdf.
https:/​/​homepages.cwi.nl/​~aeb/​math/​srg/​rk3/​srgw.pdf

[13] Brunner, N, Cavalcanti, D, Pironio, S, Scarani, V, Wehner, S, Bell nonlocality, Rev. Mod. Phys. 86 (2014), 419–478.
https:/​/​doi.org/​10.1103/​revmodphys.86.419

[14] Bub, J, Bananaworld. Quantum Mechanics for Primates, Oxford UP, 2016.
https:/​/​doi.org/​10.1093/​acprof:oso/​9780198718536.001.0001

[15] Byrd, MS, Khaneja, N, Characterization of the positivity of the density matrix in terms of the coherence vector representation, Phys. Rev. A 68 (2003), 062322.
https:/​/​doi.org/​10.1103/​physreva.68.062322

[16] Cabello, A, Severini, S, Winter, A, (Non-)contextuality of physical theories as an axiom, arXiv:1010.2163 [quant-ph].
arXiv:1010.2163

[17] Cabello, A, Severini, S, Winter, A, Graph-theoretic approach to quantum correlations, Phys. Rev. Lett. 112 (2014), 040401.
https:/​/​doi.org/​10.1103/​physrevlett.112.040401

[18] Conway, JH, Wales, DB, Construction of the Rudvalis group of order 145,926,144,000, J. Algebra 27 (1973), 538–548.
https:/​/​doi.org/​10.1016/​0021-8693(73)90063-x

[19] Coxeter, HSM, Regular Complex Polytopes, Cambridge UP, 1974.

[20] Coxeter, HSM, Shephard, GC, Portraits of a family of complex polytopes, Leonardo 25 (1992), 239–244.
https:/​/​doi.org/​10.2307/​1575843

[21] Crnković, D, Mikulić V, Rodrigues, BG, Some strongly regular graphs and self-orthogonal codes from the unitary group $U_{4}(3)$, Glas. Mat. Ser. III 45 (2010), 307–323.
https:/​/​doi.org/​10.3336/​gm.45.2.02

[22] Dall’Arno, M, Accessible information and informational power of quantum 2-designs, Phys. Rev. A 90 (2014), 052311.
https:/​/​doi.org/​10.1103/​physreva.90.052311

[23] Dall’Arno, M, D’Ariano, GM, Sacchi, MF, Informational power of quantum measurement, Phys. Rev. A 83 (2011), 062304.
https:/​/​doi.org/​10.1103/​physreva.83.062304

[24] DeBrota, JB, Fuchs, CA, Stacey, BC, Symmetric informationally complete measurements identify the essential difference between classical and quantum systems, Phys. Rev. Research 2 (2020), 013074.
https:/​/​doi.org/​10.1103/​physrevresearch.2.013074

[25] DeBrota, JB, Fuchs, CA, Stacey, BC, Analysis and synthesis of minimal informationally complete quantum measurements, arXiv:1812.08762 [quant-ph].
arXiv:1812.08762

[26] Delsarte, P, Goethals, JM, Seidel, JJ, Bounds for systems of lines, and Jacobi polynomials, Philips Res. Repts. 30 (1975), 91-105.
https:/​/​doi.org/​10.1016/​b978-0-12-189420-7.50020-7

[27] Egan, M, Properties of tight frames that are regular schemes, Cryptogr. Commun. 12 (2020), 499–510.
https:/​/​doi.org/​10.1007/​s12095-019-00378-2

[28] Fritz, T, Polyhedral duality in Bell scenarios with two binary observables, J. Math. Phys. 53 (2012), 072202.
https:/​/​doi.org/​10.1063/​1.4734586

[29] Fuchs, CA, My Struggles with the Block Universe. Selected Correspondence, January 2001 – May 2011, arXiv:1405.2390 [quant-ph].
arXiv:1405.2390

[30] Fuchs, CA, Schack, R, Quantum-Bayesian coherence, arXiv:0906.2187 [quant-ph].
arXiv:0906.2187

[31] Fuchs, CA, Schack, R, A Quantum-Bayesian route to quantum-state space, Found. Phys. 41 (2011), 345–356.
https:/​/​doi.org/​10.1007/​s10701-009-9404-8

[32] Fuchs, CA, Schack, R, Quantum-Bayesian coherence, Rev. Mod. Phys. 85 (2013), 1693–1715.
https:/​/​doi.org/​10.1103/​revmodphys.85.1693

[33] Fuchs, CA, Hoang, MC, Stacey, BC, The SIC question: History and state of play, Axioms 6 (2017), 21.
https:/​/​doi.org/​10.3390/​axioms6030021

[34] Fuchs, CA, Stacey, BC, QBism: Quantum theory as a hero's handbook, in Rasel, EM, Schleich, WP, Wölk, S (Eds), Foundations of quantum theory, Proc. International School of Physics Enrico Fermi", IOS Press, 2019, pp. 133-202.
https:/​/​doi.org/​10.3254/​978-1-61499-937-9-133

[35] Geller, J, Piani, M, Quantifying non-classical and beyond-quantum correlations in the unified operator formalism, J. Phys. A 47 (2014), 424030.
https:/​/​doi.org/​10.1088/​1751-8113/​47/​42/​424030

[36] Goh, KT, Kaniewski, J, Wolfe, E, Vértesi, T, Wu, X, Cai, Y, Liang, Y-C, Scarani, V, Geometry of the set of quantum correlations, Phys. Rev. A 97 (2018), 022104.
https:/​/​doi.org/​10.1103/​physreva.97.022104

[37] Graydon, MA, Conical designs and categorical Jordan algebraic post-quantum theories, PhD Thesis, University of Waterloo, 2017.
arXiv:1703.06800

[38] Gross, D, Audenaert, K, Eisert, J, Evenly distributed unitaries: On the structure of unitary designs, J. Math. Phys. 48 (2007), 052104.
https:/​/​doi.org/​10.1063/​1.2716992

[39] Grötschel, M, Lovász, L, Schrijver, A, Geometric Algorithms and Combinatorial Optimization, 2nd ed., Springer, 1993.
https:/​/​doi.org/​10.1007/​978-3-642-78240-4

[40] Heinosaari, T, Jivulescu, MA, Nechita, I, Random positive operator valued measures, J. Math. Phys. 61 (2020), 042202.
https:/​/​doi.org/​10.1063/​1.5131028

[41] Heinosaari, T, Ziman, M, The Mathematical Language of Quantum Theory. From Uncertainty to Entanglement, Cambridge UP, 2011.
https:/​/​doi.org/​10.1017/​cbo9781139031103

[42] Higman, DG, Finite permutation groups of rank 3, Math. Z. 86 (1964), 145–156.
https:/​/​doi.org/​10.1007/​bf01111335

[43] Hirschfeld, JWP, On the history of generalized quadrangles, Bull. Belg. Math. Soc. Simon Stevin 3 (1994), 417–421.
https:/​/​doi.org/​10.36045/​bbms/​1103408583

[44] Hoggar, SG, $t$-designs in projective spaces, Europ. J. Combin. 3 (1982), 233–254.
https:/​/​doi.org/​10.1016/​s0195-6698(82)80035-8

[45] Hoggar, SG, A complex polytope as generalized quadrangle, Proc. Roy. Soc. Edinburgh Sect. A 95 (1983), 1–5.
https:/​/​doi.org/​10.1017/​s0308210500015754

[46] Hoggar, SG, Parameters of $t$-designs in $\mathbb{F}P^{d-1}$, European J. Combin. 5 (1984), 29–36.
https:/​/​doi.org/​10.1016/​S0195-6698(84)80015-3

[47] Hughes, D, Waldron, S, Spherical $(t,t)$-designs with a small number of vectors, Linear Algebra Appl. 608 (2021), 84–106.
https:/​/​doi.org/​10.1016/​j.laa.2020.08.010

[48] Jones, NS, Linden, N, Parts of quantum states, Phys. Rev. A 71 (2005), 012324.
https:/​/​doi.org/​10.1103/​physreva.71.012324

[49] Kiktenko, EO, Malyshev, AO, Mastiukova, AS, Man'ko, VI, Fedorov, AK, Chruściński D, Probability representation of quantum dynamics using pseudostochastic maps, Phys. Rev. A 101 (2020), 052320.
https:/​/​doi.org/​10.1103/​physreva.101.052320

[50] Klappenecker, A, Rötteler, M, Mutually unbiased bases are complex projective $2$-designs, in Proc IEEE International Symposium on Information Theory, Adelaide, Australia, 4-9 September, IEEE, 2005, pp. 1740–1744.
https:/​/​doi.org/​10.1109/​isit.2005.1523643

[51] Knuth, DE, The sandwich theorem, Electron. J. Combin. 1 (1994), A1.
https:/​/​doi.org/​10.37236/​1193

[52] Kwapisz, J, Trace optimality of SIC POVMs, J. Phys. A: Math. Theor. 52 (2019), 115203.
https:/​/​doi.org/​10.1088/​1751-8121/​ab0067

[53] Lehrer GI, Taylor, DE, Unitary Reflection Groups, Cambridge UP, 2009.

[54] Levenshtein, VI, Designs as maximum codes in polynomial metric spaces, Acta Appl. Math., 29 (1992), 1–82.
https:/​/​doi.org/​10.1007/​bf00053379

[55] Massad JE, Aravind, PK, The Penrose dodecahedron revisited, Am. J. Phys. 67 (1999), 631–638.
https:/​/​doi.org/​10.1119/​1.19336

[56] Mitchell, HH, Determination of all primitive collineation groups in more than four variables which contain homologies, Amer. J. Math. 36 (1914), 1–12.
https:/​/​doi.org/​10.2307/​2370513

[57] Neumaier, A, Combinatorial configurations in terms of distances, Memorandum 81-09 (Dept. of Mathematics), Eindhoven University of Technology, 1981.
https:/​/​www.mat.univie.ac.at/​~neum/​scan/​combcon.pdf

[58] Neumaier, A, Regular cliques in graphs and special $1\frac12$ designs, in Cameron, PJ, Hirschfeld, JWP, Hughes, DR (Eds), Finite Geometries and Designs, Proc. Second Isle of Thorns Conference 1980, Cambridge UP, 1981, pp. 244–259.
https:/​/​doi.org/​10.1017/​cbo9781107325579.027

[59] Oreshkov, O, Calsamiglia, J, Muñoz-Tapia, R, Bagan, E, Optimal signal states for quantum detectors, New J. Phys. 13 (2011), 073032.
https:/​/​doi.org/​10.1088/​1367-2630/​13/​7/​073032

[60] Payne, SE, All generalized quadrangles of order $3$ are known, J. Combin. Theory Ser. A 18 (1975), 203–206.
https:/​/​doi.org/​10.1016/​0097-3165(75)90009-6

[61] Payne, SE, Thas, JA, Finite Generalized Quadrangles, 2nd ed., EMS, 2009.
https:/​/​doi.org/​10.4171/​066

[62] Penrose, R, On Bell non-locality without probabilities: some curious geometry, preprint, Mathematical Institute, Oxford, 1992, published as Ellis J, Amati D (Eds), Quantum Reflections (in honour of J.S. Bell), Cambridge UP, 2000, pp. 1–27.

[63] Penrose, R, Shadows of the Mind, Oxford UP, 1994.

[64] Pitowsky, I, The range of quantum probability, J. Math. Phys. 27 (1986), 1556–1565.
https:/​/​doi.org/​10.1063/​1.527066

[65] Polster, B, A Geometrical Picture Book, Springer, 1998.
https:/​/​doi.org/​10.1007/​978-1-4419-8526-2

[66] Popescu, S, Nonlocality beyond quantum mechanics, Nature Physics 10 (2014), 264–270.
https:/​/​doi.org/​10.1038/​nphys2916

[67] Popescu, S, Rohrlich, D, Quantum nonlocality as an axiom, Found. Phys. 24 (1994), 379–385.
https:/​/​doi.org/​10.1007/​bf02058098

[68] Rosado, JI, Representation of quantum states as points in a probability simplex associated to a SIC-POVM, Found. Phys. 41 (2011), 1200–1213.
https:/​/​doi.org/​10.1007/​s10701-011-9540-9

[69] Saniga, M, On the Veldkamp space of GQ(4,2), Int. J. Geom. Methods M. 8 (2011), 39–47.
https:/​/​doi.org/​10.1142/​s0219887811004951

[70] Scott, AJ, Tight informationally complete quantum measurements, J. Phys. A: Math. Gen. 39 (2006), 13507–13530.
https:/​/​doi.org/​10.1088/​0305-4470/​39/​43/​009

[71] Słomczyński, W, Szymusiak, A, Highly symmetric POVMs and their informational power, Quantum Inf. Process. 15 (2016), 565–606.
https:/​/​doi.org/​10.1007/​s11128-015-1157-z

[72] Soicher, LH, On cliques in edge-regular graphs, J. Algebra 421 (2015), 260–267.
https:/​/​doi.org/​10.1016/​j.jalgebra.2014.08.028

[73] Stacey, BC, Quantum theory as symmetry broken by vitality, arXiv:1907.02432v3 [quant-ph].
arXiv:1907.02432

[74] Szymusiak, A, Maximally informative ensembles for SIC-POVMs in dimension 3, J. Phys. A 47 (2014), 445301.
https:/​/​doi.org/​10.1088/​1751-8113/​47/​44/​445301

[75] Szymusiak, A, Pure states of maximum uncertainty with respect to a given POVM, Open Syst. Inf. Dyn. 27 (2020), 2050002.
https:/​/​doi.org/​10.1142/​s123016122050002x

[76] Szymusiak, A, Słomczyński, W, Informational power of the Hoggar symmetric informationally complete positive operator-valued measure, Phys. Rev. A 94 (2016), 012122.
https:/​/​doi.org/​10.1103/​physreva.94.012122

[77] Tabia, GNM, Appleby, DM, Exploring the geometry of qutrit state space using symmetric informationally complete probabilities, Phys. Rev. A 88 (2013), 012131.
https:/​/​doi.org/​10.1103/​physreva.88.012131

[78] Talata, I, A volume formula for medial sections of simplices, Discrete Comput. Geom. 30 (2003), 343–353.
https:/​/​doi.org/​10.1007/​s00454-003-0015-6

[79] Waegell, M, Aravind, PK, The Penrose dodecahedron and the Witting polytope are identical in $\mathbb{CP}^{3}$, Phys. Lett. A 381 (2017), 1853–1857.
https:/​/​doi.org/​10.1016/​j.physleta.2017.03.039

[80] Waldron, SFD, An Introduction to Finite Tight Frames, Birkhäuser, 2018.
https:/​/​doi.org/​10.1007/​978-0-8176-4815-2

[81] Weis, S, Quantum convex support, Linear Alg. Appl. 435 (2011), 3168–3188.
https:/​/​doi.org/​10.1016/​j.laa.2011.06.004

[82] Welch, LR, Lower bounds on the maximum cross correlations of signals, IEEE Trans. Inf. Theory, 20 (1974), 397–399.
https:/​/​doi.org/​10.1109/​tit.1974.1055219

[83] Zhu, H, Multiqubit Clifford groups are unitary $3$-designs, Phys. Rev. A 96 (2017), 062336.
https:/​/​doi.org/​10.1103/​physreva.96.062336

[84] Zimba, J, Penrose, R, On Bell non-locality without probabilities: More curious geometry, Stud. Hist. Philos. Sci. 24 (1993), 697–720.
https:/​/​doi.org/​10.1016/​0039-3681(93)90061-n

### Cited by

[1] Xin Yan, Ye‐Chao Liu, and Jiangwei Shang, "Operational Detection of Entanglement via Quantum Designs", Annalen der Physik 534 5, 2100594 (2022).

[2] Christopher A. Fuchs and Blake C. Stacey, "QBians Do Not Exist", arXiv:2012.14375, (2020).

[3] Gerhard Zauner, "The finite Fourier Transform and projective 2-designs", arXiv:2207.09922, (2022).

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