A useful and unifying concept in quantum information theory is the idea of a
consumable "resource". The basic idea behind quantum resource theories is that
whenever we face restrictions on the set of operations we can perform, some states
(resource states) enable us to perform particular tasks that otherwise would have
been impossible. In this thesis we first review a resource theory that recently
has gained a lot of attention, namely the resource theory of asymmetry in which
one is restricted to symmetric operations and therefore asymmetric states are the
resource states. The asymmetry properties of a state relative to some symmetry
group specify how and to what extent the given symmetry is broken by the state.
In the remainder of the thesis we explain how the ideas developed in this resource
theory are related to three different areas in quantum theory: theory of quantum
reference frames, the Wigner-Araki-Yanase(WAY) theorem and the conditional
probability interpretation of time in quantum mechanics.
Theory of quantum reference frames treats reference frames like any other physical
system within the formalism of quantum theory, which causes the measurements
to be an approximation of the measurements against their classical counterparts.
We consider the dynamics of a quantum directional reference frame undergoing
repeated interactions. These interactions induce a back-action on the reference
which is the central focus of our study. The effect of a precise sequence of measurement
outcomes on the reference frame is studied by looking at both the case
that the measurement record is averaged over and the case wherein it is retained.
We find, in particular, that there is interesting dynamics in the latter situation,
which cannot be revealed by considering the average case. We then consider in
detail how a sequence of rotationally invariant unitary interactions affects the reference
frame, a situation, which leads to quite different dynamics than the case of
repeated measurements. Different strategies for correcting reference frame drift
is considered given that we have access to a set of particles with polarization
opposite to the direction of drift. In particular, we find that, by implementing a suitably chosen unitary interaction after every two measurements, we can eliminate
the rotational drift of the reference frame.
The WAY theorem establishes an important constraint that conservation laws
impose on quantum mechanical measurements. We formulate the WAY theorem
in the broader context of resource theories, where one is constrained to a subset
of quantum mechanical operations described by a symmetry group. Establishing
connections with the theory of quantum state discrimination we obtain optimal
unitaries describing the measurement of arbitrary observables, explain how
prior information can permit perfect measurements that circumvent the WAY
constraint, and provide a framework that establishes a natural ordering on measurement
apparatuses through a decomposition into asymmetry and charge subsystems.
Finally we review two di erent schemes in the conditional probability interpretation
(CPI) of time in quantum mechanics. In these schemes parameter time "t"
as it appears in Schrodinder equation is considered to be unobservable, yet one
can use an extension of conditional probabilities in order to study dynamics of a
system relative to another system, namely the quantum clock. We phrase CPI
as a quantum communication protocol which enables us to use the machinery
developed in the resource theory of asymmetry. This provides us with a deeper
understanding of the decoherence effect caused due to the inaccessibility of the
parameter time "t" and that how this effect can be suppressed by choosing an
optimal initial state for the quantum clock.