The main purpose of this paper is to show that the notions of frames and fusion frames introduced in non-harmonic Fourier analysis are also very natural in discussion of some basic problems in theory of open quantum systems and, in particular, in quantum optics. Frames are collections of vectors in a Hilbert space which assure a natural representation of each vector in the space, but may have infinitely many different representations for any given vector. For a given quantum system represented in a Hilbert space the question of minimal number η of observables Q 1 ,..., Q η , whose expectation values at some instants t 1 ,...,t p determine the statistical state of the system is discussed. We assume that the time evolution of the system in question is governed by a semigroup of linear transformations with generator .
The construction of Parseval frames with special, rigid geometric properties has left many open problems even after decades of efforts. The construction of similar types of fusion frames is even less developed. We construct a large family of equi-isoclinic Parseval fusion frames by taking the Naimark complement of the union of orthonormal bases. If these bases are chosen to be mutually unbiased, then the resulting fusion frame subspaces are spanned by mutually unbiased basic sequences. By giving an explicit representation for Naimark complements, we are able to construct concrete fusion frames in their respective Hilbert spaces.
When reference frames are treated quantum mechanically, the subsystem structure of quantum systems is no longer absolute, but depends on the choice of the quantum reference frame (QRF). This raises a basic question: which dynamical properties are preserved across QRFs, and which depend on the physical reference used to define the system? We study this question in the general setting of open quantum systems. At the operational level, after a QRF transformation, the old reference frame and environmental degrees of freedom may be inaccessible and must therefore be traced out. This motivates the definition of reduced quantum-reference-frame channels: maps that connect the joint description in one frame to the accessible subsystem in another. We characterize their symmetry-constrained structure and define a regime in which a reduced entropy-coherence conservation law holds. We also identify when the induced reduced action on the open system admits a classical interpretation as random frame misalignment, and when it instead reflects quantum reduced-frame effects. We then apply the framework to pure-dephasing dynamics and derive a necessary and sufficient compatibility condition for population preservation. When the frame symmetry commutes with the open system's free Hamiltonian, coherences acquire a multiplicative frame factor, so that locally inferred decoherence rates split into environmental and reference-induced contributions. Ramsey interferometry gives this split a direct operational meaning. Finally, a gravity-motivated dephasing model illustrates how degradation of a phase reference can mimic signatures usually attributed to intrinsic decoherence mechanisms.