It is claimed that empty space is full of quantum fields which their waves perfectly cancel out each other, but this can only happens when all the waves are entangled. Is this claim valid because I thought as long as there is no excitation in the field it remains empty? What about non-zero higgs field?
When you draw the Cartesian 3 axis of (x,y,z), each point in space described by accurate numbers, do these numbers change anything in space? They are a model , a mathematical map of space, useful for describing , for example, gravitational interactions as well as all of classical physics.
Quantum field theory uses the plane wave mathematical solutions of the quantum mechanical equations describing the particles in the standard model of physics, conceptually as a "coordinate system" through which interactions of these particles can be modeled, by the use of Feynman diagrams.
It is the Feynman diagrams that hold the information of entanglement, a meta-level on the fields, not the hypothesized fields. It is similar to the meta level of written language based on the alphabet, the meaning is in the combinations of the alphabet, not in the 24 letters.
The non zero vacuum expectation value, VEV, of the Higgs field is similar, in the coordinate system analogy, to picking another $(0,0,0)$ with a different potential energy, and one has to go into the mathematics of the model to understand what is happening . The fields in the table of particle physics, excepting the Higgs, have $0$ VEV.
The speculative idea that spacetime may emerge out of quantum entanglement has its basis in the mathematics of tensor networks. A tensor network provides a way of describing a highly entangled quantum state.
Surprisingly, the graph geometry of the connections in the tensor network corresponds to the spatial geometry of a spacelike slice of a particular spacetime called anti-deSitter space (ADS).
This particular spacetime is important in string theory, but physicists are also trying to understand whether there are connections between entanglement and other spacetimes.