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TABLE III: The decompositions of MUBs (0,12,16) for three-qutrit system 4 Basis Decompositions 1 F − 1 2 X 23 F − 1 1 F − 1 3 2 F − 1 1 R − 1 1 F − 1 2 R − 1 2 X 23 F − 1 3 R − 1 3 3 F − 1 1 X 13 R 3 4 F − 1 2 R 2 X − 1 23 F − 1 1 X 12 R 2 F − 1 3 5 F − 1 1 X − 1 12 F − 1 3 6 F − 1 2 R 2 X 23 F − 1 1 R 1 X − 1 13 R 3 F − 1 3 R 3 7 F − 1 2 X − 1 23 R − 1 3 F − 1 1 R − 1 1 X − 1 12 R − 1 3 8 F − 1 1 R − 1 1 X − 1 13 F − 1 2 R 2 F − 1 3 R − 1 3 9 F − 1 1 X − 1 13 F − 1 2 R 2 X − 1 23 F − 1 3 R 3 10 F − 1 1 R 2 X 12 X − 1 13 F − 1 2 11 F − 1 1 R 1 X − 1 12 F − 1 2 R 2 X − 1 23 F − 1 3 12 F − 1 1 R − 1 1 X 13 X − 1 12 F − 1 2 F − 1 3 R − 1 3 13 F − 1 1 X 12 F − 1 2 R − 1 2 X − 1 23 F − 1 3 R − 1 3 14 F − 1 1 R 1 X 12 F − 1 2 X 23 F − 1 3 R 3 Basis Decompositions 15 F − 1 2 R 2 X 23 16 F − 1 1 R 1 X 13 F − 1 2 R 2 X 23 F − 1 3 R − 1 3 17 F − 1 1 R − 1 1 X 13 F − 1 2 F − 1 3 18 F − 1 1 R − 1 1 X − 1 13 F − 1 2 R − 1 2 X − 1 23 R 3 F − 1 3 R − 1 3 19 F − 1 1 R 1 X − 1 13 F − 1 2 R − 1 2 F − 1 3 R 3 20 F − 1 1 X 12 F − 1 3 R − 1 3 21 F − 1 1 R − 1 1 X − 1 12 F − 1 2 R 2 F − 1 3 R 3 22 F − 1 1 X − 1 12 F − 1 2 X − 1 23 F − 1 3 R 3 23 F − 1 1 R − 1 1 X 13 F − 1 2 R − 1 2 X − 1 23 24 F − 1 1 R − 1 1 X − 1 $_{13}$F − 1 2 X 23 R 3 F − 1 3 25 F − 1 1 R 1 X − 1 12 X − 1 23 F − 1 2 R − 1 2 26 F − 1 1 R 1 X 12 X − 1 $_{23}$F − 1 2 R − 1 2 F − 1 3 27 F − 1 1 R − 1 1 X 12 F − 1 2 R 2 28 F − 1 1 R 1 F − 1 2 X − 1 23