# ZX calculus: What do diamond and loop mean?

Recently, I started to study practical application of ZX calculus but I am confused by meaning of "diamond" and "loop".

Issue no. 1: There are these rules:

B-rule

and D-rule

But this example seems to use the rules wrongly:

In the middle of a digram, B-rule is used, however, I do not see any loop or diamonds justifying this step (i.e. a disconection of nodes).

Similar situation occurs in this example:

Why is it possible to ignore loop and diamonds?

Issue no. 2:

Interpreation of a diamond in Hilbert space is this:

Diamond = $$\sqrt{2}$$

What does mean that diamond is $$\sqrt{2}$$? Is it a normalization constant?

Interpreation of a loop in Hilbert space is this:

Loop represent the dimension of underlying Hilbert space

Assuming D-rule, loop should represent two diamonds hence $$\sqrt{2}\sqrt{2} = 2$$ which is dimension of Hilbert space for description of single qubit states. But ZX calculus can be used for any number of qubits. What does it mean that loop represent a dimension? How is a dimension of "multi-qubits" Hilbert space represented?

2. For a multi-qubit system, you represent an identity operator with several wires. If you trace them, you get dimension equal to $$2^n$$, and in the diagram you represent this dimension as $$n$$ disjoint loops each contributing a factor of 2.