# COMPLETE CHARACTERIZATION OF ALL QUBIT UNEXTENDIBLE PRODUCT BASES WITH 8 OR FEWER STATES
# Compiled by Nathaniel Johnston (nathaniel@njohnston.ca)
# See reference paper "N. Johnston. The structure of qubit unextendible product bases. Preprint, 2014."
#
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#
# This file gives a description of the 215 distinct qubit unextendible product bases consisting of 8 or fewer states, one on each line.
# Any qubit/size combination (with 8 or fewer states) that is not listed here does not have any qubit UPBs. For example, there are no 5-qubit UPBs consisting of 7 states.
# The format given for a UPB is the same as that of http://www.njohnston.ca/4qubitupbs.txt, so please see that file for a description of the format.
# Various comments are provided throughout this file. All comment lines begin with a hash (#) symbol. All other lines describe the UPBs themselves.
#
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#
# 1-qubit UPBs of 2 states (i.e., full product bases of 1-qubit space, 1 exists):
#
[1:2]
#
# 2-qubit UPBs of 4 states (i.e., full product bases of 2-qubit space, 2 exist):
#
[1,2:3,4], [1,3:2,4]
[1,2:3,4], [3:4 | 1:2]
#
# 3-qubit UPBs of 4 states (1 exists, this UPB is often called "Shifts"):
#
[1:2 | 3:4], [1:3 | 2:4], [1:4 | 2:3]
#
# 3-qubit UPBs of 8 states (i.e., full product bases of 3-qubit space, 17 exist):
#
[1,2,3,4:5,6,7,8], [1,2,5,6:3,4,7,8], [1,3,5,7:2,4,6,8]
[1,2,3,4:5,6,7,8], [1,2,5,6:3,4,7,8], [1,3,5:2,4,6 | 7:8]
[1,2,3,4:5,6,7,8], [1,2,5,6:3,4,7,8], [3,7:4,8 | 1,5:2,6]
[1,2,3,4:5,6,7,8], [1,2,5,6:3,4,7,8], [3,5:4,6 | 1,7:2,8]
[1,2,3,4:5,6,7,8], [1,2,5,6:3,4,7,8], [1,7:2,8 | 5:6 | 3:4]
[1,2,3,4:5,6,7,8], [1,2,5,6:3,4,7,8], [1,3:2,4 | 7:8 | 5:6]
[1,2,3,4:5,6,7,8], [1,2,5,6:3,4,7,8], [7:8 | 5:6 | 3:4 | 1:2]
[1,2,3,4:5,6,7,8], [1,6,7:2,5,8 | 3:4], [1,2,6:3,4,7 | 5:8]
[1,2,3,4:5,6,7,8], [1,2,6:3,4,7 | 5:8], [5,8:6,7 | 1,3:2,4]
[1,2,3,4:5,6,7,8], [1,2,6:3,4,7 | 5:8], [5,8:6,7 | 3:4 | 1:2]
[1,2,3,4:5,6,7,8], [5,6:7,8 | 1,2:3,4], [5,7:6,8 | 1,3:2,4]
[1,2,3,4:5,6,7,8], [5,6:7,8 | 1,2:3,4], [1,5:2,6 | 7:8 | 3:4]
[1,2,3,4:5,6,7,8], [5,6:7,8 | 1,2:3,4], [1,3:2,4 | 7:8 | 5:6]
[1,2,3,4:5,6,7,8], [5,6:7,8 | 1,2:3,4], [7:8 | 5:6 | 3:4 | 1:2]
[1,2,3,4:5,6,7,8], [5,8:6,7 | 1,4:2,3], [2,6:3,7 | 1,5:4,8]
[1,2,3,4:5,6,7,8], [5,6:7,8 | 2:3 | 1:4], [1,4:2,3 | 7:8 | 5:6]
[2,4,5:6,7,8 | 1:3], [1,3,6:2,7,8 | 4:5], [1,2,3:4,5,6 | 7:8]
#
# 4-qubit UPBs of 6 states (1 exists):
#
[1,2:3,4 | 5:6], [1,3:6 | 2,4:5], [4:6 | 3:5 | 1:2], [3:4 | 2:6 | 1:5]
#
# 4-qubit UPBs of 7 states (1 exists):
#
[1,2,3:4,5 | 6:7], [1,4:7 | 2,6:3,5], [1,4:3,6 | 2,5:7], [1,6:2 | 3:7 | 4:5]
#
# 4-qubit UPBs of 8 states (144 exist):
#
# The first 89 8-state 4-qubit UPBs arise as a form of (Shifts \otimes |0>) \cup (Shifts \otimes |1>):
#
[1,2,3,4:5,6,7,8], [3,7:4,8 | 1,5:2,6], [2,6:4,8 | 1,5:3,7], [2,6:3,7 | 1,5:4,8]
[1,2,3,4:5,6,7,8], [3,7:4,8 | 1,5:2,6], [2,6:3,7 | 1,5:4,8], [2,6:4,8 | 1,7:3,5]
[1,2,3,4:5,6,7,8], [3,7:4,8 | 1,5:2,6], [2,6:3,7 | 1,5:4,8], [2,8:4,6 | 1,7:3,5]
[1,2,3,4:5,6,7,8], [3,7:4,8 | 1,5:2,6], [2,6:3,7 | 1,5:4,8], [1,8:3,6 | 5:7 | 2:4]
[1,2,3,4:5,6,7,8], [3,7:4,8 | 1,5:2,6], [2,6:3,7 | 1,5:4,8], [1,5:3,7 | 6:8 | 2:4]
[1,2,3,4:5,6,7,8], [3,7:4,8 | 1,5:2,6], [2,6:3,7 | 1,5:4,8], [6:8 | 5:7 | 2:4 | 1:3]
[1,2,3,4:5,6,7,8], [3,8:4,7 | 1,5:2,6], [2,6:3,7 | 1,5:4,8], [6:8 | 5:7 | 2:4 | 1:3]
[1,2,3,4:5,6,7,8], [3,6:4,8 | 1,5:2,7], [2,6:3,7 | 1,5:4,8], [2,8:4,7 | 1,6:3,5]
[1,2,3,4:5,6,7,8], [3,8:4,6 | 1,5:2,7], [2,6:3,7 | 1,5:4,8], [2,7:4,8 | 1,6:3,5]
[1,2,3,4:5,6,7,8], [3,8:4,6 | 1,5:2,7], [2,6:3,7 | 1,5:4,8], [1,7:3,8 | 5:6 | 2:4]
[1,2,3,4:5,6,7,8], [3,8:4,6 | 1,5:2,7], [2,6:3,7 | 1,5:4,8], [1,6:3,5 | 7:8 | 2:4]
[1,2,3,4:5,6,7,8], [2,7:4,8 | 1,5:3,6], [2,6:3,7 | 1,5:4,8], [3,6:4,8 | 5:7 | 1:2]
[1,2,3,4:5,6,7,8], [2,6:4,8 | 1,5:3,7], [2,6:3,7 | 1,5:4,8], [3,8:4,7 | 5:6 | 1:2]
[1,2,3,4:5,6,7,8], [2,8:4,6 | 1,5:3,7], [2,6:3,7 | 1,5:4,8], [3,8:4,7 | 5:6 | 1:2]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [3,7:4,8 | 1,6:2,5], [2,6:4,8 | 1,7:3,5]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [3,7:4,8 | 1,6:2,5], [1,6:3,8 | 5:7 | 2:4]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [3,7:4,8 | 1,6:2,5], [1,8:3,6 | 5:7 | 2:4]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [3,8:4,7 | 1,6:2,5], [2,8:4,6 | 1,7:3,5]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [3,8:4,7 | 1,6:2,5], [2,7:4,5 | 1,8:3,6]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [3,8:4,7 | 1,6:2,5], [1,8:3,6 | 5:7 | 2:4]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [3,8:4,7 | 1,6:2,5], [6:8 | 5:7 | 2:4 | 1:3]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [3,5:4,7 | 1,6:2,8], [2,8:4,7 | 1,6:3,5]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [3,7:4,5 | 1,6:2,8], [2,7:4,8 | 1,6:3,5]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [3,7:4,5 | 1,6:2,8], [2,8:4,7 | 1,6:3,5]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [3,7:4,5 | 1,6:2,8], [1,8:3,7 | 5:6 | 2:4]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [2,7:4,8 | 1,6:3,5], [3,6:4,8 | 1,7:2,5]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [2,7:4,8 | 1,6:3,5], [3,8:4,6 | 1,7:2,5]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [2,8:4,7 | 1,6:3,5], [3,8:4,6 | 1,7:2,5]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [2,5:4,7 | 1,6:3,8], [3,5:4,6 | 1,7:2,8]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [2,5:4,7 | 1,6:3,8], [1,5:2,6 | 7:8 | 3:4]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [2,7:4,5 | 1,6:3,8], [1,5:2,6 | 7:8 | 3:4]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [2,7:4,5 | 1,6:3,8], [1,6:2,5 | 7:8 | 3:4]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [3,6:4,8 | 1,7:2,5], [1,5:3,6 | 7:8 | 2:4]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [3,6:4,8 | 1,7:2,5], [1,6:3,5 | 7:8 | 2:4]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [3,8:4,6 | 1,7:2,5], [1,5:3,6 | 7:8 | 2:4]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [3,8:4,6 | 1,7:2,5], [1,6:3,5 | 7:8 | 2:4]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [3,5:4,6 | 1,7:2,8], [2,6:4,8 | 1,7:3,5]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [3,5:4,6 | 1,7:2,8], [2,8:4,6 | 1,7:3,5]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [3,5:4,6 | 1,7:2,8], [1,6:3,8 | 5:7 | 2:4]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [3,5:4,6 | 1,7:2,8], [1,8:3,6 | 5:7 | 2:4]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [3,5:4,6 | 1,7:2,8], [6:8 | 5:7 | 2:4 | 1:3]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [3,6:4,5 | 1,7:2,8], [2,8:4,6 | 1,7:3,5]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [3,5:4,7 | 1,8:2,6], [7:8 | 5:6 | 2:4 | 1:3]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [3,5:4,6 | 1,8:2,7], [6:8 | 5:7 | 2:4 | 1:3]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [3,6:4,5 | 1,8:2,7], [1,6:3,8 | 5:7 | 2:4]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [3,6:4,5 | 1,8:2,7], [1,8:3,6 | 5:7 | 2:4]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [3,6:4,5 | 1,8:2,7], [1,5:3,7 | 6:8 | 2:4]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [2,6:4,5 | 7:8 | 1:3], [1,8:2,6 | 5:7 | 3:4]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [1,7:3,8 | 5:6 | 2:4], [1,6:2,8 | 5:7 | 3:4]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [1,7:3,8 | 5:6 | 2:4], [1,8:2,6 | 5:7 | 3:4]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [1,8:3,7 | 5:6 | 2:4], [1,8:2,6 | 5:7 | 3:4]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [1,6:3,8 | 5:7 | 2:4], [1,7:2,8 | 5:6 | 3:4]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [1,6:3,8 | 5:7 | 2:4], [7:8 | 5:6 | 3:4 | 1:2]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [1,8:3,6 | 5:7 | 2:4], [1,7:2,8 | 5:6 | 3:4]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [1,8:3,6 | 5:7 | 2:4], [1,8:2,7 | 5:6 | 3:4]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [1,5:3,7 | 6:8 | 2:4], [1,7:2,8 | 5:6 | 3:4]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [1,5:3,7 | 6:8 | 2:4], [1,8:2,7 | 5:6 | 3:4]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [1,5:3,7 | 6:8 | 2:4], [1,5:2,6 | 7:8 | 3:4]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [1,5:3,7 | 6:8 | 2:4], [1,6:2,5 | 7:8 | 3:4]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [1,7:3,5 | 6:8 | 2:4], [1,7:2,8 | 5:6 | 3:4]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [1,7:3,5 | 6:8 | 2:4], [1,8:2,7 | 5:6 | 3:4]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [1,7:3,5 | 6:8 | 2:4], [1,6:2,5 | 7:8 | 3:4]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [1,7:3,5 | 6:8 | 2:4], [7:8 | 5:6 | 3:4 | 1:2]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [1,5:3,6 | 7:8 | 2:4], [1,8:2,6 | 5:7 | 3:4]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [1,5:3,6 | 7:8 | 2:4], [1,7:2,5 | 6:8 | 3:4]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [1,6:3,5 | 7:8 | 2:4], [1,6:2,8 | 5:7 | 3:4]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [1,6:3,5 | 7:8 | 2:4], [1,8:2,6 | 5:7 | 3:4]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [1,6:3,5 | 7:8 | 2:4], [1,7:2,5 | 6:8 | 3:4]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [1,8:2,7 | 5:6 | 3:4], [6:8 | 5:7 | 2:4 | 1:3]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [1,5:2,7 | 6:8 | 3:4], [7:8 | 5:6 | 2:4 | 1:3]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [1,7:2,5 | 6:8 | 3:4], [7:8 | 5:6 | 2:4 | 1:3]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [1,5:2,6 | 7:8 | 3:4], [6:8 | 5:7 | 2:4 | 1:3]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [7:8 | 5:6 | 3:4 | 1:2], [6:8 | 5:7 | 2:4 | 1:3]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 1,5:4,8], [6:8 | 5:7 | 3:4 | 1:2], [7:8 | 5:6 | 2:4 | 1:3]
[1,2,3,4:5,6,7,8], [3,7:4,5 | 6:8 | 1:2], [2,6:3,7 | 5:8 | 1:4], [1,8:3,7 | 5:6 | 2:4]
[1,2,3,4:5,6,7,8], [3,5:4,6 | 7:8 | 1:2], [2,6:3,7 | 5:8 | 1:4], [1,6:3,8 | 5:7 | 2:4]
[1,2,3,4:5,6,7,8], [3,6:4,5 | 7:8 | 1:2], [2,6:3,7 | 5:8 | 1:4], [1,6:3,8 | 5:7 | 2:4]
[1,2,3,4:5,6,7,8], [3,6:4,5 | 7:8 | 1:2], [2,6:3,7 | 5:8 | 1:4], [1,8:3,6 | 5:7 | 2:4]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 5:8 | 1:4], [1,8:3,7 | 5:6 | 2:4], [1,8:2,6 | 5:7 | 3:4]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 5:8 | 1:4], [1,8:3,6 | 5:7 | 2:4], [1,6:2,5 | 7:8 | 3:4]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 5:8 | 1:4], [1,5:3,7 | 6:8 | 2:4], [1,7:2,8 | 5:6 | 3:4]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 5:8 | 1:4], [1,7:3,5 | 6:8 | 2:4], [1,5:2,6 | 7:8 | 3:4]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 5:8 | 1:4], [1,7:3,5 | 6:8 | 2:4], [1,6:2,5 | 7:8 | 3:4]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 5:8 | 1:4], [1,6:3,5 | 7:8 | 2:4], [1,7:2,5 | 6:8 | 3:4]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 5:8 | 1:4], [1,6:3,5 | 7:8 | 2:4], [6:8 | 5:7 | 3:4 | 1:2]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 5:8 | 1:4], [1,6:2,8 | 5:7 | 3:4], [7:8 | 5:6 | 2:4 | 1:3]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 5:8 | 1:4], [1,8:2,6 | 5:7 | 3:4], [7:8 | 5:6 | 2:4 | 1:3]
[1,2,3,4:5,6,7,8], [2,6:3,7 | 5:8 | 1:4], [6:8 | 5:7 | 3:4 | 1:2], [7:8 | 5:6 | 2:4 | 1:3]
[1,2,3,4:5,6,7,8], [6:8 | 5:7 | 3:4 | 1:2], [7:8 | 5:6 | 2:4 | 1:3], [6:7 | 5:8 | 2:3 | 1:4]
#
# The remaining 55 8-state 4-qubit UPBs are different from (Shifts \otimes |0>) \cup (Shifts \otimes |1>):
#
[1,7:2,3,6 | 4:5,8], [2,3,5:6,8 | 1,4:7], [1,2,3:4,5 | 6,7:8], [2,5:3,7 | 4:6 | 1:8]
[1,7:2,3,6 | 4:5,8], [2,3,5:6,8 | 1,4:7], [1,2,3:4,5 | 6,7:8], [5:7 | 4:6 | 2:3 | 1:8]
[5,6,7:8 | 1,2,3:4], [2,5:3,6 | 1,4:7,8], [3,4:6,8 | 1,7:2,5], [2,4:5,8 | 1,7:3,6]
[1,2,4,6:3,8 | 5:7], [1,2,3,6:4,7 | 5:8], [1,2,3,4:5,6 | 7:8], [1,7:2,4 | 5:6 | 3:8]
[1,2,4,6:3,8 | 5:7], [1,2,3,6:4,7 | 5:8], [1,2,3,4:5,6 | 7:8], [5:6 | 4:7 | 3:8 | 1:2]
[1,2,3,6:4,7 | 5:8], [1,2,3,4:5,6 | 7:8], [1,5:3,7 | 2,4,6:8], [1,3:2,8 | 5:6 | 4:7]
[1,2,3,6:4,7 | 5:8], [1,2,3,4:5,6 | 7:8], [2,4,6:8 | 5:7 | 1:3], [1,3:2,8 | 5:6 | 4:7]
[1,2,3:4,5,6 | 7:8], [2,5,7:3,6 | 1,4:8], [1,4:2,5,7 | 3,6:8], [2,5:7,8 | 1,6:3,4]
[1,2,3:4,5,6 | 7:8], [2,5,7:3,6 | 1,4:8], [1,4:2,5,7 | 3,6:8], [2,5:7,8 | 4:6 | 1:3]
[1,2,3:4,5,6 | 7:8], [3,6:7,8 | 1,4:2,5], [2,5:7,8 | 1,4:3,6], [2,5:3,6 | 1,4:7,8]
[1,2,3:4,5,6 | 7:8], [3,6:7,8 | 1,4:2,5], [2,5:7,8 | 1,4:3,6], [1,4:7,8 | 5:6 | 2:3]
[1,2,3:4,5,6 | 7:8], [3,6:7,8 | 1,4:2,5], [2,5:3,6 | 1,4:7,8], [2,5:7,8 | 1,6:3,4]
[1,2,3:4,5,6 | 7:8], [3,5:7,8 | 1,4:2,6], [2,5:3,6 | 1,4:7,8], [2,6:7,8 | 1,5:3,4]
[1,2,3:4,5,6 | 7:8], [3,5:6,8 | 1,4:2,7], [2,7:3,5 | 1,4:6,8], [2,6:7,8 | 1,5:3,4]
[1,2,3:4,5,6 | 7:8], [2,6:7,8 | 1,4:3,5], [2,5:3,6 | 1,4:7,8], [3,5:7,8 | 4:6 | 1:2]
[1,2,3:4,5,6 | 7:8], [2,5:7,8 | 1,4:3,6], [3,6:7,8 | 4:5 | 1:2], [1,4:7,8 | 5:6 | 2:3]
[1,2,3:4,5,6 | 7:8], [2,5:7,8 | 1,4:3,6], [3,4:7,8 | 5:6 | 1:2], [1,6:7,8 | 4:5 | 2:3]
[1,2,3:4,5,6 | 7:8], [2,5:6,8 | 1,4:3,7], [2,5:3,7 | 1,4:6,8], [3,6:7,8 | 4:5 | 1:2]
[1,2,3:4,5,6 | 7:8], [2,6:5,8 | 1,4:3,7], [2,7:3,5 | 1,4:6,8], [3,5:4,8 | 1,7:2,6]
[1,2,3:4,5,6 | 7:8], [2,7:3,5 | 1,4:6,8], [2,5:4,8 | 1,6:3,7], [3,6:5,8 | 1,7:2,4]
[1,2,3:4,5,6 | 7:8], [2,5:3,6 | 1,4:7,8], [2,6:7,8 | 1,5:3,4], [3,5:7,8 | 1,6:2,4]
[1,2,3:4,5,6 | 7:8], [2,5:3,6 | 1,4:7,8], [3,5:7,8 | 1,6:2,4], [2,6:7,8 | 4:5 | 1:3]
[1,2,3:4,5,6 | 7:8], [2,5:3,6 | 1,4:7,8], [3,6:4,8 | 1,7:2,5], [2,5:4,8 | 1,7:3,6]
[1,2,3:4,5,6 | 7:8], [3,6:7,8 | 4:5 | 1:2], [2,5:7,8 | 4:6 | 1:3], [1,4:7,8 | 5:6 | 2:3]
[1,2,3:6,7 | 4,5:8], [1,2,3:4,5 | 6,7:8], [4,6:5,7 | 1,2:3,8], [4,7:5,6 | 1,8:2,3]
[1,2,3:6,7 | 4,5:8], [1,2,3:4,5 | 6,7:8], [4,6:5,7 | 1,2:3,8], [4,7:5,6 | 3:8 | 1:2]
[1,2,3:6,7 | 4,5:8], [1,2,3:4,5 | 6,7:8], [4,6:5,7 | 1,2:3,8], [1,8:2,3 | 5:7 | 4:6]
[1,2,3:6,7 | 4,5:8], [1,2,3:4,5 | 6,7:8], [4,6:5,7 | 1,8:2,3], [2,7:3,5 | 4:6 | 1:8]
[1,2,3:6,7 | 4,5:8], [1,2,3:4,5 | 6,7:8], [4,6:5,7 | 1,8:2,3], [5:7 | 4:6 | 2:3 | 1:8]
[1,2,3:6,7 | 4,5:8], [1,2,3:4,5 | 6,7:8], [4,6:5,7 | 2:8 | 1:3], [1,3:2,8 | 5:7 | 4:6]
[1,2,3:4,5 | 6,7:8], [2,5:3,6 | 1,4:7,8], [3,4:6,8 | 1,7:2,5], [2,5:4,8 | 1,7:3,6]
[5,6:7,8 | 1,2:3,4], [3,4:7,8 | 1,2:5,6], [3,4:5,6 | 1,2:7,8], [1,3:2,4 | 7:8 | 5:6]
[5,6:7,8 | 1,2:3,4], [3,4:7,8 | 1,2:5,6], [3,4:5,6 | 1,2:7,8], [7:8 | 5:6 | 3:4 | 1:2]
[5,6:7,8 | 1,2:3,4], [3,4:7,8 | 1,2:5,6], [3,5:4,6 | 1,2:7,8], [1,6:2,4 | 7:8 | 3:5]
[5,6:7,8 | 1,2:3,4], [3,4:7,8 | 1,2:5,6], [2,5:4,7 | 1,3:6,8], [3,6:4,5 | 1,8:2,7]
[5,6:7,8 | 1,2:3,4], [3,4:7,8 | 1,2:5,6], [3,4:5,6 | 1,7:2,8], [1,8:2,7 | 5:6 | 3:4]
[5,6:7,8 | 1,2:3,4], [3,4:7,8 | 1,2:5,6], [3,5:4,6 | 1,7:2,8], [1,2:7,8 | 4:6 | 3:5]
[5,6:7,8 | 1,2:3,4], [3,4:7,8 | 1,2:5,6], [3,5:4,6 | 7:8 | 1:2], [1,2:7,8 | 4:6 | 3:5]
[5,6:7,8 | 1,2:3,4], [3,4:6,8 | 1,2:5,7], [3,4:5,7 | 1,2:6,8], [1,3:2,4 | 7:8 | 5:6]
[5,6:7,8 | 1,2:3,4], [3,4:5,6 | 1,2:7,8], [3,7:4,8 | 2:5 | 1:6], [1,6:2,5 | 4:8 | 3:7]
[5,6:7,8 | 1,2:3,4], [5,7:6,8 | 1,3:2,4], [2,4:6,8 | 1,3:5,7], [2,4:5,7 | 1,3:6,8]
[5,6:7,8 | 1,2:3,4], [2,4:7,8 | 1,3:5,6], [2,4:5,6 | 1,3:7,8], [7:8 | 5:6 | 3:4 | 1:2]
[5,6:7,8 | 1,2:3,4], [2,4:6,8 | 1,3:5,7], [2,4:5,7 | 1,3:6,8], [5,7:6,8 | 1,4:2,3]
[5,6:7,8 | 1,2:3,4], [2,4:6,8 | 1,3:5,7], [2,4:5,7 | 1,3:6,8], [5,8:6,7 | 1,4:2,3]
[5,6:7,8 | 1,2:3,4], [2,4:6,8 | 1,3:5,7], [2,4:5,7 | 1,3:6,8], [1,7:2,8 | 5:6 | 3:4]
[5,6:7,8 | 1,2:3,4], [2,4:6,8 | 1,3:5,7], [2,4:5,7 | 1,3:6,8], [1,4:2,3 | 7:8 | 5:6]
[5,6:7,8 | 1,2:3,4], [2,4:6,8 | 1,3:5,7], [2,4:5,7 | 1,3:6,8], [7:8 | 5:6 | 3:4 | 1:2]
[5,6:7,8 | 1,2:3,4], [2,4:5,7 | 1,3:6,8], [2,6:4,8 | 1,5:3,7], [3,8:4,7 | 1,6:2,5]
[5,6:7,8 | 1,2:3,4], [2,4:5,7 | 1,3:6,8], [3,8:4,7 | 1,6:2,5], [1,2:7,8 | 4:6 | 3:5]
[5,6:7,8 | 1,2:3,4], [2,4:5,7 | 1,3:6,8], [3,6:5,7 | 1,8:2,4], [2,4:3,6 | 1,8:5,7]
[5,6:7,8 | 1,2:3,4], [2,4:5,7 | 1,3:6,8], [3,6:4,5 | 1,8:2,7], [2,7:3,6 | 1,8:4,5]
[5,6:7,8 | 1,2:3,4], [2,4:5,7 | 1,3:6,8], [3,6:4,5 | 1,8:2,7], [1,4:5,8 | 3:7 | 2:6]
[5,6:7,8 | 1,2:3,4], [2,4:5,7 | 1,3:6,8], [3,6:4,5 | 1,8:2,7], [4:8 | 3:7 | 2:6 | 1:5]
[5,6:7,8 | 1,2:3,4], [2,4:5,6 | 1,3:7,8], [2,4:7,8 | 1,5:3,6], [7:8 | 3:4,6 | 1:2,5]
[5,6:7,8 | 1,2:3,4], [2,4:5,6 | 1,3:7,8], [2,7:4,8 | 1,5:3,6], [3,8:4,6 | 1,7:2,5]
#
# 5-qubit UPBs of 6 states (1 exists):
#
[1:2 | 3:4 | 5:6], [1:3 | 2:6 | 4:5], [1:4 | 2:5 | 3:6], [1:5 | 2:3 | 4:6], [1:6 | 2:4 | 3:5]
#
# 5-qubit UPBs of 8 states (32 exist):
#
[1,2,3:4 | 5,6,7:8], [2,5:3,6 | 4:7 | 1:8], [1,7:3,6 | 4:5 | 2:8], [1,5:2,7 | 4:8 | 3:6], [4:6 | 3:8 | 2:7 | 1:5]
[1,2,3:4 | 5,6,7:8], [2,5:3,6 | 4:7 | 1:8], [1,7:3,5 | 4:6 | 2:8], [1,5:2,7 | 4:8 | 3:6], [1,7:2,6 | 4:5 | 3:8]
[1,2,3:4 | 5,6,7:8], [2,5:3,6 | 4:7 | 1:8], [1,7:3,5 | 4:6 | 2:8], [1,7:2,6 | 4:5 | 3:8], [4:8 | 3:6 | 2:5 | 1:7]
[1,2:3,4 | 5,6:7,8], [3,4:5,6 | 1,2:7,8], [7:8 | 5:6 | 3:4 | 1:2], [4:8 | 3:7 | 2:6 | 1:5], [4:7 | 3:8 | 2:5 | 1:6]
[1,2:3,4 | 5,6:7,8], [3,4:5,6 | 1,2:7,8], [5:6 | 4:8 | 3:7 | 1:2], [7:8 | 3:4 | 2:6 | 1:5], [4:7 | 3:8 | 2:5 | 1:6]
[1,2:3,4 | 5,6:7,8], [5,7:6,8 | 1,3:2,4], [1,4:7 | 3:5,8 | 2:6], [1,4:5,8 | 3:6 | 2:7], [1,4:6 | 3:7 | 2:5,8]
[1,2:3,4 | 5,6:7,8], [2,6:3,7 | 1,5:4,8], [3,8:4,7 | 2:5 | 1:6], [1,6:2,5 | 4:7 | 3:8], [4:6 | 3:5 | 2:8 | 1:7]
[1,2:3,4 | 5,6:7,8], [3,4:5,6 | 1,7:2,8], [3,5:4,6 | 2:8 | 1:7], [4:8 | 3:7 | 2:6 | 1:5], [4:7 | 3:8 | 2:5 | 1:6]
[1,2:3,4 | 5,6:7,8], [3,4:5,6 | 1,7:2,8], [3,5:4,6 | 2:8 | 1:7], [4:7 | 3:8 | 2:6 | 1:5], [4:8 | 3:7 | 2:5 | 1:6]
[1,2:3,4 | 5,6:7,8], [3,4:5,6 | 1,7:2,8], [4:7 | 3:8 | 2:6 | 1:5], [4:8 | 3:7 | 2:5 | 1:6], [5:6 | 3:4 | 2:8 | 1:7]
[1,2:3,4 | 5,6:7,8], [3,5:4,6 | 1,7:2,8], [3,6:4,5 | 1,8:2,7], [4:8 | 3:7 | 2:6 | 1:5], [4:7 | 3:8 | 2:5 | 1:6]
[1,2:3,4 | 5,6:7,8], [3,5:4,6 | 1,7:2,8], [3,6:4,5 | 1,8:2,7], [4:7 | 3:8 | 2:6 | 1:5], [4:8 | 3:7 | 2:5 | 1:6]
[1,2:3,4 | 5,6:7,8], [3,5:4,6 | 1,7:2,8], [1,8:4,5 | 3:7 | 2:6], [4:7 | 3:8 | 2:5 | 1:6], [4:6 | 3:5 | 2:8 | 1:7]
[1,2:3,4 | 5,6:7,8], [3,5:4,6 | 1,7:2,8], [1,8:4,5 | 3:7 | 2:6], [4:7 | 3:5 | 2:8 | 1:6], [4:6 | 3:8 | 2:5 | 1:7]
[1,2:3,4 | 5,6:7,8], [3,5:4,6 | 1,7:2,8], [1,8:2,7 | 4:6 | 3:5], [4:8 | 3:7 | 2:6 | 1:5], [4:7 | 3:8 | 2:5 | 1:6]
[1,2:3,4 | 5,6:7,8], [3,5:4,6 | 1,7:2,8], [1,8:2,7 | 4:6 | 3:5], [4:7 | 3:8 | 2:6 | 1:5], [4:8 | 3:7 | 2:5 | 1:6]
[1,2:3,4 | 5,6:7,8], [3,5:4,6 | 1,7:2,8], [4:8 | 3:7 | 2:6 | 1:5], [4:7 | 3:8 | 2:5 | 1:6], [4:6 | 3:5 | 2:8 | 1:7]
[1,2:3,4 | 5,6:7,8], [3,5:4,6 | 1,7:2,8], [4:7 | 3:8 | 2:6 | 1:5], [4:8 | 3:7 | 2:5 | 1:6], [4:6 | 3:5 | 2:8 | 1:7]
[1,2:3,4 | 5,6:7,8], [3,5:4,6 | 1,7:2,8], [4:6 | 3:7 | 2:8 | 1:5], [4:7 | 3:8 | 2:5 | 1:6], [4:8 | 3:5 | 2:6 | 1:7]
[1,2:3,4 | 5,6:7,8], [2,8:3,5 | 1,7:4,6], [3,6:4,5 | 1,8:2,7], [4:8 | 3:7 | 2:6 | 1:5], [4:5 | 3:6 | 2:7 | 1:8]
[1,2:3,4 | 5,6:7,8], [2,8:3,5 | 1,7:4,6], [3,6:4,5 | 1,8:2,7], [4:8 | 3:6 | 2:7 | 1:5], [4:5 | 3:7 | 2:6 | 1:8]
[1,2:3,4 | 5,6:7,8], [4:5,7 | 3:6,8 | 1:2], [3:4 | 2:5,8 | 1:6,7], [1,4:8 | 2,3:7 | 5:6], [7:8 | 2,4:6 | 1,3:5]
[1,2:3,4 | 5,6:7,8], [2,7:8 | 3:4,5 | 1:6], [2,4:5,7 | 3:6 | 1:8], [4,5:6 | 3:8 | 1:2,7], [4:8 | 3:7 | 2:6 | 1:5]
[1,2:3,4 | 5,6:7,8], [2,4:8 | 3:5,7 | 1:6], [2,4:5,7 | 3:6 | 1:8], [4,5:6 | 3:8 | 1:2,7], [7:8 | 3:4 | 2:6 | 1:5]
[1,2:3,4 | 5,6:7,8], [2,4:8 | 3:5,7 | 1:6], [2,4:5,7 | 3:6 | 1:8], [2,4:6 | 3:8 | 1:5,7], [1,4:2,3 | 7:8 | 5:6]
[1,2:3,4 | 5,6:7,8], [2,4:8 | 3:5,7 | 1:6], [2,4:5,7 | 3:6 | 1:8], [2,4:6 | 3:8 | 1:5,7], [7:8 | 5:6 | 3:4 | 1:2]
[1,2:3,4 | 5,6:7,8], [3,5:4,6 | 2:7 | 1:8], [1,7:4,5 | 3:8 | 2:6], [1,8:2,7 | 4:6 | 3:5], [4:8 | 3:7 | 2:5 | 1:6]
[1,2:3,4 | 5,6:7,8], [3,5:4,6 | 2:7 | 1:8], [1,8:2,7 | 4:6 | 3:5], [4:8 | 3:7 | 2:6 | 1:5], [4:7 | 3:8 | 2:5 | 1:6]
[1,2:3,4 | 5,6:7,8], [3,5:4,6 | 2:7 | 1:8], [1,8:2,7 | 4:6 | 3:5], [4:7 | 3:8 | 2:6 | 1:5], [4:8 | 3:7 | 2:5 | 1:6]
[1,2,3:4,5 | 6,7:8], [2,6:3,7 | 4:5 | 1:8], [2,4:8 | 5:7 | 1:3,6], [1,4:7 | 2,5:6 | 3:8], [5:8 | 4:6 | 3:7 | 1:2]
[1:6 | 2:8 | 3,4:5,7], [3,5:8 | 2,4:6 | 1:7], [1,4:8 | 3,7:6 | 2:5], [1,7:2,5 | 6:8 | 3:4], [1,2:3,4 | 7:8 | 5:6]
[1:8 | 2:6 | 3,7:4,5], [1,8:2,6 | 4:5 | 3:7], [1,2:3,4 | 7:8 | 5:6], [1,2:7 | 3,4:6 | 5:8], [3,4:8 | 6:7 | 1,2:5]
#
# 6-qubit UPBs of 8 states (9 exist):
#
[5,6:7,8 | 1,2:3,4], [7:8 | 5:6 | 3:4 | 1:2], [4:8 | 3:7 | 2:6 | 1:5], [4:7 | 3:8 | 2:5 | 1:6], [4:6 | 3:5 | 2:8 | 1:7], [4:5 | 3:6 | 2:7 | 1:8]
[5,6:7,8 | 1,2:3,4], [7:8 | 5:6 | 3:4 | 1:2], [4:7 | 3:8 | 2:6 | 1:5], [4:8 | 3:7 | 2:5 | 1:6], [4:6 | 3:5 | 2:8 | 1:7], [4:5 | 3:6 | 2:7 | 1:8]
[5,6:7,8 | 1,2:3,4], [7:8 | 5:6 | 3:4 | 1:2], [4:6 | 3:8 | 2:7 | 1:5], [4:5 | 3:7 | 2:8 | 1:6], [4:8 | 3:6 | 2:5 | 1:7], [4:7 | 3:5 | 2:6 | 1:8]
[5,6:7,8 | 1,2:3,4], [7:8 | 5:6 | 3:4 | 1:2], [4:7 | 3:6 | 2:8 | 1:5], [4:8 | 3:5 | 2:7 | 1:6], [4:6 | 3:8 | 2:5 | 1:7], [4:5 | 3:7 | 2:6 | 1:8]
[5,6:7,8 | 1,2:3,4], [7:8 | 5:6 | 3:4 | 1:2], [4:6 | 3:7 | 2:8 | 1:5], [4:7 | 3:8 | 2:5 | 1:6], [4:8 | 3:5 | 2:6 | 1:7], [4:5 | 3:6 | 2:7 | 1:8]
[5,6:7,8 | 1,2:3,4], [5:6 | 4:7 | 3:8 | 1:2], [7:8 | 3:4 | 2:6 | 1:5], [4:8 | 3:7 | 2:5 | 1:6], [4:6 | 3:5 | 2:8 | 1:7], [4:5 | 3:6 | 2:7 | 1:8]
[5,6:7,8 | 1,2:3,4], [5:6 | 4:7 | 3:8 | 1:2], [4:8 | 3:7 | 2:6 | 1:5], [7:8 | 3:4 | 2:5 | 1:6], [4:6 | 3:5 | 2:8 | 1:7], [4:5 | 3:6 | 2:7 | 1:8]
[5,6:7,8 | 1,2:3,4], [5:6 | 4:7 | 3:8 | 1:2], [4:6 | 3:7 | 2:8 | 1:5], [7:8 | 3:4 | 2:5 | 1:6], [4:8 | 3:5 | 2:6 | 1:7], [4:5 | 3:6 | 2:7 | 1:8]
[1,2:3,4 | 7:8 | 5:6], [1,3:7 | 2,4:6 | 5:8], [1,4:8 | 6:7 | 2,3:5], [6:8 | 5:7 | 3:4 | 1:2], [4:7 | 3:6 | 2:8 | 1:5], [4:5 | 3:8 | 2:7 | 1:6]
#
# 7-qubit UPBs of 8 states (6 exist):
#
[1:2 | 2:3 | 5:6 | 7:8], [6:8 | 5:7 | 2:4 | 1:3], [6:7 | 5:8 | 2:3 | 1:4], [4:8 | 3:7 | 2:6 | 1:5], [4:7 | 3:8 | 2:5 | 1:6], [4:6 | 3:5 | 2:8 | 1:7], [4:5 | 3:6 | 2:7 | 1:8]
[1:2 | 2:3 | 5:6 | 7:8], [5:7 | 4:6 | 2:8 | 1:3], [6:7 | 5:8 | 2:3 | 1:4], [6:8 | 3:7 | 2:4 | 1:5], [4:7 | 3:8 | 2:5 | 1:6], [4:8 | 3:5 | 2:6 | 1:7], [4:5 | 3:6 | 2:7 | 1:8]
[1:2 | 2:3 | 5:6 | 7:8], [5:7 | 4:6 | 2:8 | 1:3], [6:8 | 3:5 | 2:7 | 1:4], [6:7 | 3:8 | 2:4 | 1:5], [5:8 | 4:7 | 2:3 | 1:6], [4:8 | 3:6 | 2:5 | 1:7], [4:5 | 3:7 | 2:6 | 1:8]
[1:2 | 2:3 | 5:6 | 7:8], [5:7 | 4:6 | 2:8 | 1:3], [5:8 | 3:6 | 2:7 | 1:4], [6:7 | 3:8 | 2:4 | 1:5], [4:8 | 3:7 | 2:5 | 1:6], [6:8 | 4:5 | 2:3 | 1:7], [4:7 | 3:5 | 2:6 | 1:8]
[1:2 | 2:3 | 5:6 | 7:8], [5:7 | 4:6 | 2:8 | 1:3], [5:8 | 3:6 | 2:7 | 1:4], [4:8 | 3:7 | 2:6 | 1:5], [4:7 | 3:8 | 2:5 | 1:6], [6:8 | 4:5 | 2:3 | 1:7], [6:7 | 3:5 | 2:4 | 1:8]
[1:2 | 2:3 | 5:6 | 7:8], [5:7 | 4:6 | 2:8 | 1:3], [5:8 | 3:6 | 2:7 | 1:4], [4:8 | 3:7 | 2:6 | 1:5], [4:7 | 3:8 | 2:5 | 1:6], [6:8 | 3:5 | 2:4 | 1:7], [6:7 | 4:5 | 2:3 | 1:8]