  A photonic process named as "quantum state joining" has been recently
experimentally demonstrated [C. Vitelli et al., Nature Photon. 7, 521 (2013)]
that corresponds to the transfer of the internal two-dimensional quantum states
of two input photons, i.e., two photonic qubits, into the four-dimensional
quantum state of a single photon, i.e., a photonic ququart. A scheme for the
inverse process, namely "quantum state splitting", has also been theoretically
proposed. Both processes can be iterated in a cascaded layout, to obtain the
joining and/or splitting of more than two qubits, thus leading to a general
scheme for varying the number of photons in the system while preserving its
total quantum state, or quantum information content. Here, we revisit these
processes from a theoretical point of view. After casting the theory of the
joining and splitting processes in the more general photon occupation number
notation, we introduce some modified schemes that are in principle unitary (not
considering the implementation of the CNOT gates) and do not require projection
and feed-forward steps. This can be particularly important in the quantum state
splitting case, to obtain a scheme that does not rely on postselection.
Moreover, we formally prove that the quantum joining of two photon states with
linear optics requires the use of at least one ancilla photon. This is somewhat
unexpected, given that the demonstrated joining scheme involves the sequential
application of two CNOT quantum gates, for which a linear optical scheme with
just two photons and postselection is known to exist. Finally we explore the
relationship between the joining scheme and the generation of clusters of
multi-particle entangled states involving more than one qubit per particle.
