  Relational formulation of quantum mechanics is based on the idea that
relational properties among quantum systems, instead of the independent
properties of a quantum system, are the most fundamental elements to construct
quantum mechanics. In a recent paper (J. M. Yang, Sci. Rep. 8:13305, 2018),
basic relational quantum mechanics framework is formulated to derive quantum
probability, Born's Rule, Schr\"{o}dinger Equations, and measurement theory.
This paper further extends the reformulation effort in three aspects. First, it
gives a clearer explanation of the key concepts behind the framework to
calculate measurement probability. Second, we provide a concrete implementation
of the relational probability amplitude by extending the path integral
formulation. The implementation not only clarifies the physical meaning of the
relational probability amplitude, but also allows us to elegantly explain the
double slit experiment, to describe the interaction history between the
measured system and a series of measuring systems, and to calculate
entanglement entropy based on path integral and influence functional. In
return, the implementation brings back new insight to path integral itself by
completing the explanation on why measurement probability can be calculated as
modulus square of probability amplitude. Lastly, we clarify the connection
between our reformulation and the quantum reference frame theory. A complete
relational formulation of quantum mechanics needs to combine the present works
with the quantum reference frame theory.
