On SIC-POVMs in Prime Dimensions

Abstract

The generalized Pauli group and its normalizer, the Clifford group, have a rich mathematical structure which is relevant to the problem of constructing symmetric informationally complete POVMs (SIC-POVMs). To date, almost every known SIC-POVM fiducial vector is an eigenstate of a "canonical" unitary in the Clifford group. I show that every canonical unitary in prime dimensions p > 3 lies in the same conjugacy class of the Clifford group and give a class representative for all such dimensions. It follows that if even one such SIC-POVM fiducial vector is an eigenvector of such a unitary, then all of them are (for a given such dimension). I also conjecture that in all dimensions d, the number of conjugacy classes is bounded above by 3 and depends only on d mod 9, and I support this claim with computer computations in all dimensions < 48.

6 pages, no figures. v3 Refs added, improved discussion of previous work. Ref to a proof of the main conjecture also added

References in corpus (15)

• Symmetric Informationally Complete Quantum Measurements
• Experimental Quantum State Tomography of Optical Fields and Ultrafast Statistical Sampling
• Unknown Quantum States and Operations, a Bayesian View
• Optimal Quantum Cloning Machines
• Quantum probabilities as Bayesian probabilities
• Unknown Quantum States: The Quantum de Finetti Representation
• Unambiguous Discrimination Between Linearly-Independent Quantum States
• SIC-POVMs and the Extended Clifford Group
• Optimum Unambiguous Discrimination Between Linearly Independent Symmetric States
• Informationally complete measurements and groups representation
• Minimal Informationally Complete Measurements for Pure States
• On PSI-complete and PSIR-complete measurements
• Spherical Code Key Distribution Protocols for Qubits
• Equiangular Spherical Codes in Quantum Cryptography
• Simple Minimal Informationally Complete Measurements for Qudits

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• SIC-POVMs: A new computer study
• The SIC Question: History and State of Play
• Distinguishability of quantum states under restricted families of measurements with an application to quantum data hiding
• Optimizing quantum process tomography with unitary 2-designs
• SIC~POVMs and Clifford groups in prime dimensions
• Weighted complex projective 2-designs from bases: optimal state determination by orthogonal measurements
• An angular momentum approach to quadratic Fourier transform, Hadamard matrices, Gauss sums, mutually unbiased bases, unitary group and Pauli group
• Minimum tomography of two entangled qutrits using local measurements of one-qutrit SIC-POVM
• The Lie Algebraic Significance of Symmetric Informationally Complete Measurements
• Variations on a theme of Heisenberg, Pauli and Weyl
• Scalable Quantum Tomography with Fidelity Estimation
• Linear Dependencies in Weyl-Heisenberg Orbits
• Spectra of phase point operators in odd prime dimensions and the extended Clifford group
• Maximally informative ensembles for SIC-POVMs in dimension 3
• Order 3 Symmetry in the Clifford Hierarchy
• Sequential measurements of conjugate observables
• Magic informationally complete POVMs with permutations
• Discrimination of POVMs with rank-one effects
• Phase-space representations of SIC-POVM fiducial states
• Generalized spin bases for quantum chemistry and quantum information
• Excluding false negative error in certification of quantum channels
• SIC-POVMs from Stark Units: Dimensions n^2+3=4p, p prime
• SICs and the Triangle Group $(3,3,3)$