Rules For Mists
Following are nine rules determining how mists behave and evolve as they interact
(Rule 1) A mist contains a series of entries separated by commas, where each entry is a collection of white and black marbles (one for each qubit), possibly with a minus sign in front.
Each entry in a mist must contain the same number of marbles.
This number is equal to the number of qubits.
Here is an example of a mist with 3 entries describing 2 qubits: [WB,BB,BW].
(Rule 2) The order of the different entries within a mist does not matter.
For example, [WB,BB,BW] is the same as [BB,WB,BW].
However, the order of the marbles in a single entry does matter since each marble is associated with a different qubit.
For example, [WB,BB,BW] is not the same as [BW,BB,BW].
(Rule 3) If two entries contain exactly the same sequence of marbles but with opposite signs in front of them, then these entries cancel and can be removed from the mist.
For example, consider the following mist: [WB,BB,BW,-BB].
This is the same as [WB,BW].
(Rule 4) If a mist contains several identical entries (including same sign), then redundant entries can be deleted so long as the ratios of distinct entries remain the same.
For example, [WW,WW,BB,BB] can be reduced to [WW,BB] because the ratio of WW to BB entries remains 1:1, but [WW,WW,WW,BB,BB] cannot be reduced because the 3:2 ratio would change.
Note that this rule implies that if every entry in a mist is the same, then all but one entry can be deleted, e.g., [WB,WB,WB,WB] → [WB].
(Rule 5) If there is only one entry in the mist, then the mist can be dropped, e.g., [WB] → WB.
(Rule 6) Mists within mists can be eliminated.
(Rule 7) If the first few qubits in every entry have the same sequence of marble colors, then these qubits can be factored out of the mist on the left.
Similarly, if the last few qubits in every entry have the same color sequence, these qubits can be factored out to the right.
This is similar to factoring out ‘a’ in an equation such as ax + ay = b.
Note, this is analogous with the associative property of multiplication.
(Rule 8) Two or more mists can be combined into a single mist identical to the FOIL (‘Firsts, Outers, Inners, Lasts’) English-language mnemonic in algebra, e.g.,
(a + b)(c + d) = ac + ad + bc + bd.
(Rule 9) Two mists related by an overall minus sign are equivalent, e.g.,
[-WB,-BB,BW] = -[WB,BB,-BW] = [WB,BB,-BW].