Quantum bounds for inequalities involving marginal expectation values

We review and develop an algorithm to determine arbitrary quantum bounds based on the seminal work of Tsirelson [Lett. Math. Phys. 4, 93 (1980)]. The potential of this algorithm is demonstrated by both deriving marginal-involving number-valued quantum bounds and identifying a generalized class of function-valued quantum bounds. Those results facilitate an eight-dimensional volume analysis of quantum mechanics which extends the work of Cabello [Phys. Rev. A 72, 012113 (2005)]. We contrast the quantum volume defined by these bounds to that of macroscopic locality, defined by the inequalities corresponding to the first level of the hierarchy of Navascues et al. [New J. Phys. 10, 073013 (2008)], proving our function-valued quantum bounds to be more complete.