36 research outputs found
Relative Quantum Time
The need for a time-shift invariant formulation of quantum theory arises from
fundamental symmetry principles as well as heuristic cosmological
considerations. Such a description then leaves open the question of how to
reconcile global invariance with the perception of change, locally. By
introducing relative time observables, we are able to make rigorous the
Page-Wootters conditional probability formalism to show how local Heisenberg
evolution is compatible with global invariance
A relational perspective on the Wigner-Araki-Yanase theorem
We present a novel interpretation of the Wigner-Araki-Yanase (WAY) theorem
based on a relational view of quantum mechanics. Several models are analysed in
detail, backed up by general considerations, which serve to illustrate that the
moral of the WAY theorem may be that in the presence of symmetry, a measuring
apparatus must fulfil the dual purpose of both reflecting the statistical
behaviour of the system under investigation, and acting as a physical reference
system serving to define those quantities which must be understood as relative.Comment: Version 2 contains some corrections and improvements suggested by an
anonymous refere
A quantum reference frame size-accuracy trade-off for quantum channels
The imposition of symmetry upon the nature and structure of quantum
observables has recently been extensively studied, with quantum reference
frames playing a crucial role. In this paper, we extend this work to quantum
transformations, giving quantitative results showing, in direct analogy to the
case of observables, that a "large" reference frame is required for
non-covariant channels to be well approximated by covariant ones. We apply our
findings to the concrete setting of SU(2) symmetry
Position Measurements Obeying Momentum Conservation
We present a hitherto unknown fundamental limitation to a basic measurement:
that of the position of a quantum object when the total momentum of the object
and apparatus is conserved. This result extends the famous Wigner-Araki-Yanase
(WAY) theorem, and shows that accurate position measurements are only
practically feasible if there is a large momentum uncertainty in the apparatus
Skew Hecke Algebras
Let be a finite group, a subgroup, a commutative ring,
an -algebra, and an action of on by -algebra
automorphisms. Following Baker, we associate to this data the \emph{skew Hecke
algebra} , which is the convolution algebra of
-invariant functions from to .
In this paper we study the basic structure of these algebras, proving for
skew Hecke algebras a number of common generalisations of results about skew
group algebras and results about Hecke algebras of finite groups. We show that
skew Hecke algebras admit a certain double coset decomposition. We construct an
isomorphism from to the algebra of
-invariants in the tensor product . We show that if is a unit in , then
is isomorphic to a corner ring inside the skew
group algebra .
Alongside our main results, we show that the construction of skew Hecke
algebras is compatible with certain group-theoretic operations, restriction and
extension of scalars, certain cocycle perturbations of the action, gradings and
filtrations, and the formation of opposite algebras. The main results are
illustrated in the case where , , and is the natural
permutation action of on the polynomial algebra .Comment: 35 page
Quantum Reference Frames on Finite Homogeneous Spaces
We present an operationally motivated treatment of quantum reference frames in the setting that the frame is a covariant positive operator valued measure (POVM) on a finite homogeneous space, generalising the principal homogeneous spaces studied in previous work. We focus on the case that the reference observable is the canonical covariant projection valued measure on the given space, and show that this gives rise to a rank-one covariant POVM on the group, which can be seen as a system of coherent states
Approximating relational observables by absolute quantities : a quantum accuracy-size trade-off
The notion that any physical quantity is defined and measured relative to a reference frame is traditionally not explicitly reflected in the theoretical description of physical experiments where, instead, the relevant observables are typically represented as 'absolute' quantities. However, the emergence of the resource theory of quantum reference frames as a new branch of quantum information science in recent years has highlighted the need to identify the physical conditions under which a quantum system can serve as a good reference. Here we investigate the conditions under which, in quantum theory, an account in terms of absolute quantities can provide a good approximation of relative quantities. We find that this requires the reference system to be large in a suitable sense
Symmetry, Reference Frames, and Relational Quantities in Quantum Mechanics
We propose that observables in quantum theory are properly understood as representatives of symmetry-invariant quantities relating one system to another, the latter to be called a reference system. We provide a rigorous mathematical language to introduce and study quantum reference systems, showing that the orthodox "absolute" quantities are good representatives of observable relative quantities if the reference state is suitably localised. We use this relational formalism to critique the literature on the relationship between reference frames and superselection rules, settling a long-standing debate on the subject