Abstract
Because we can treat quantum states as probability distributions, we can apply the concepts and methods of probability theory to them, including Shannon’s theory of information. The structures that I will discuss in the following sections came to my attention thanks to Shannon theory. In particular, the question of recurring interest is, “Out of all the extremal states of quantum state space—i.e., the ‘pure’ states \(\rho = {\left| \psi \big \rangle \!\big \langle \psi \right| }\)—which minimize the Shannon entropy of their probabilistic representation?” I will focus on the cases of dimensions 2, 3 and 8, where the so-called sporadic SICs occur. In these cases, the information-theoretic question of minimizing Shannon entropy leads to intricate geometrical structures.
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References
G.N.M. Tabia, Experimental scheme for qubit and qutrit symmetric informationally complete positive operator-valued measurements using multiport devices. Phys. Rev. A 86, 062107 (2012). https://doi.org/10.1103/PhysRevA.86.062107
G.N.M. Tabia, D.M. Appleby, Exploring the geometry of qutrit state space using symmetric informationally complete probabilities. Phys. Rev. A 88(1), 012131 (2013). https://doi.org/10.1103/PhysRevA.88.012131
B.C. Stacey, SIC-POVMs and compatibility among quantum states. Mathematics 4(2), 36 (2016). https://doi.org/10.3390/math4020036
H. Zhu, Super-symmetric informationally complete measurements. Ann. Phys. (NY) 362, 311–326 (2015). https://doi.org/10.1016/j.aop.2015.08.005
A. Streltsov, G. Adesso, M.B. Plenio, Colloquium: quantum coherence as a resource. Rev. Mod. Phys. 89, 041003 (2017). https://doi.org/10.1103/RevModPhys.89.041003
C.H. Bennett, C.A. Fuchs, J.A. Smolin, Entanglement-enhanced classical communication on a noisy quantum channel, in Quantum Communication, Computing, and Measurement (Springer, Berlin, 1997)
H. Zhu, M. Hayashi, L. Chen, Axiomatic and operational connections between the \(l_1\)-norm of coherence and negativity. Phys. Rev. A 97, 022342 (2018). https://doi.org/10.1103/PhysRevA.97.022342
M. Appleby, H.B. Dang, C.A. Fuchs, Symmetric informationally-complete quantum states as analogues to orthonormal bases and minimum uncertainty states. Entropy 16, 1484 (2014)
C.A. Fuchs, R. Schack, Quantum-Bayesian coherence. Rev. Mod. Phys. 85, 1693–1715 (2013). https://doi.org/10.1103/RevModPhys.85.1693
B. Bodmann, J. Haas, A short history of frames and quantum designs (2017). arXiv:1709.01958
C.A. Fuchs, B.C. Stacey, QBism: quantum theory as a hero’s handbook (2016). arXiv:1612.07308
W.K. Wootters, D.M. Sussman, Discrete phase space and minimum-uncertainty states (2007). arXiv:0704.1277
M. Appleby, I. Bengtsson, H.B. Dang, Galois unitaries, Mutually Unbiased Bases, and MUB-balanced states (2014). arXiv:1409.7887
I. Bengtsson, K. Blanchfield, A. Cabello, A Kochen-Specker inequality from a SIC. Phys. Lett. A 376, 374–376 (2012). https://doi.org/10.1016/j.physleta.2011.12.011
V. Veitch, S.A.H. Mousavian, D. Gottesman, J. Emerson, The resource theory of stabilizer computation. New J. Phys. 16, 013009 (2014). https://doi.org/10.1088/1367-2630/16/1/013009
S.G. Hoggar, 64 lines from a quaternionic polytope. Geom. Dedicata. 69, 287–289 (1998). https://doi.org/10.1023/A:1005009727232
A. Szymusiak, W. Słomczyński, Informational power of the Hoggar symmetric informationally complete positive operator-valued measure. Phys. Rev. A 94, 012122 (2015). https://doi.org/10.1103/PhysRevA.94.012122
E. Campbell, M. Howard, Application of a resource theory for magic states to fault-tolerant quantum computing. Phys. Rev. Lett. 118, 090501 (2017). https://doi.org/10.1103/PhysRevLett.118.090501
J. Lawrence, C. Brukner, A. Zeilinger, Mutually unbiased binary observable sets on \(n\) qubits. Phys. Rev. A 65, 032320 (2002). https://doi.org/10.1103/PhysRevA.65.032320
J.L. Romero, G. Björk, A.B. Klimov, L.L. Sánchez-Soto, On the structure of the sets of mutually unbiased bases for \(n\) qubits. Phys. Rev. A 72, 062310 (2005). https://doi.org/10.1103/PhysRevA.72.062310
H. Zhu, Quantum state estimation and symmetric informationally complete POMs. Ph.D. thesis, National University of Singapore (2012). http://scholarbank.nus.edu.sg/bitstream/handle/10635/35247/ZhuHJthesis.pdf
B.C. Stacey, Sporadic SICs and the normed division algebras. Found. Phys. 47, 1060–64 (2017). https://doi.org/10.1007/s10701-017-0087-2
W.K. Wootters, B.D. Fields, Optimal state-determination by mutually unbiased measurements. Ann. Phys. 191, 363–81 (1989). https://doi.org/10.1016/0003-4916(89)90322-9
H. Zhu, M. Hayashi, L. Chen, Coherence and entanglement measures based on Rényi relative entropies. J. Phys. A 50, 475303 (2017). https://doi.org/10.1088/1751-8121/aa8ffc
R.W. Spekkens, Evidence for the epistemic view of quantum states: a toy theory. Phys. Rev. A 75(3), 032110 (2007). https://doi.org/10.1103/PhysRevA.75.032110
R.W. Spekkens, Quasi-quantization: classical statistical theories with an epistemic restriction, in Quantum Theory: Informational Foundations and Foils, eds. by G. Chiribella, R.W. Spekkens (eds.) (Springer, Berlin, 2016), pp. 83–135. https://doi.org/10.1007/978-94-017-7303-4_4
R.W. Spekkens, Reassessing claims of nonclassicality for quantum interference phenomena (2016). http://pirsa.org/16060102/
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Stacey, B.C. (2021). Geometry and Information Theory for Qubits and Qutrits. In: A First Course in the Sporadic SICs. SpringerBriefs in Mathematical Physics, vol 41. Springer, Cham. https://doi.org/10.1007/978-3-030-76104-2_3
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