Metrics

The quantum superposition neighborhood can be generated according to different criteria. Here, we have the two main groups of available metrics:

Distance is a quantitative measure representing the magnitude of the difference between two states. The greater the distance, the higher the dissimilarity between the states.
Similarity is a measure that quantifies the degree of likeness or relationship between two states. The greater the similarity, the closer or more concordant the states are.

The following nomenclature will be followed when referring to binary strings:

Binary strings will be represented as x and y, each consisting of binary digits (0s and 1s).
\(a\): Number of bits where both x and y have the value 1 (presence), indicating positive matches.
\(b\): Number of bits where the value of x is 0 and y is 1, indicating x absence mismatches.
\(c\): Number of bits where the value of x is 1 and y is 0, indicating y absence mismatches.
\(d\): Number of bits where both x and y have the value 0 (absence), indicating negative matches.

The binary metrics discussed here offer insights into the dissimilarity or similarity between binary strings. The formulas for these distances are presented with respect to the mentioned nomenclature.

Additional considerations:

The diagonal sum \(a+d\) represents the total number of matches between x and y.
The other diagonal sum \(b+c\) represents the total number of mismatches between x and y.
The total sum of \(a+b+c+d\), is always equal to \(n\) (the total number of bits).

Below is a table summarizing the nomenclature used:

	1 (Presence in x)	0 (Absence in x)
1 (Presence in y)	\(a = x\cdot y\)	\(b = \overline{x}\cdot y\)
0 (Absence in y)	\(c = x\cdot \overline{y}\)	\(d = \overline{x}\cdot \overline{y}\)

These conventions are adapted from the guidelines presented in the article [CCT09].