Schlosshauer (in Decoherence and the Quantum-to-Classical Transition) defines ideal quantum measurement as a von-Neumann measurement in which
(1) the apparatus states correspond 1-to-1 to given system states (so in the measurement scheme we would have $(\sum |s_i \rangle)|a_r \rangle \to \sum |s_i \rangle|a_i \rangle$ where $|a_r \rangle$ is the ready state of the apparatus and $s$ denotes the system of interest.
(2) the measurement process does not change the system states.
My question is with respect to (1). Do we need also to require that the apparatus states are orthogonal (else we don't truly have 1-to-1 correspondence)?