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Of course I could only speak of whole states and banish the word "parts".

Mathematics enables some quantum states to be separated into partial states. When the whole state has finite dimensions, the partial states are of lower dimensionality. So "parts" are intrinsic to the maths, but what about the physics?

I'm not confining the discussion to particles but to any decomposition of whole into parts.

So how should "parts" be regarded physically? In attempting to arrive at a comprehensive notion of them, there seems to be a progressive evaporation of wholes into space time regions to pure sets of numerical parameters. As we deconstruct the whole what should we make of the sub-whole?

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I take it that by "separable" pure state, one means a quantum state that can be factorized as a tensor product of two or more lower dimensional pure quantum states, as discussed here.

The parts of separable quantum states are states of nonentangled quantum systems. What this means physically is that correlations between measurements on the subsystems are indistinguishable from any other classical correlation between classical random variables, and, in particular, heed the Bell and CHSH inequalities. The classical probability theory of correlated random variables describes the joint probability densities for measurements on the two subsystems. You can think of these states as classical mixtures of block-diagonal stripes in the full quantum state space.

There is a generalization of this notion in that if a quantum state is a classical mixture of pure quantum states, the correlations between subsystems remain classical (i.e. heed the Bell and CSCH inequalities) if and only if the state is a classical mixture of separable states only; in turn, this means that the density matrix $\rho$ can be expressed as a sum of the form $\sum_{k}p_{k}\hat{\rho}_{k}^{A}\otimes\hat{\rho}_{k}^{B}$.

For a primer on entanglement, with a detailed worked example, please see my answer here.

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