17 research outputs found
Gauge invariant surface holonomy and monopoles
There are few known computable examples of non-abelian surface holonomy. In
this paper, we give several examples whose structure 2-groups are covering
2-groups and show that the surface holonomies can be computed via a simple
formula in terms of paths of 1-dimensional holonomies inspired by earlier work
of Chan Hong-Mo and Tsou Sheung Tsun on magnetic monopoles. As a consequence of
our work and that of Schreiber and Waldorf, this formula gives a rigorous
meaning to non-abelian magnetic flux for magnetic monopoles. In the process, we
discuss gauge covariance of surface holonomies for spheres for any 2-group,
therefore generalizing the notion of the reduced group introduced by Schreiber
and Waldorf. Using these ideas, we also prove that magnetic monopoles have an
abelian group structure.Comment: 99 pages, 31 figures (2 are new), v2 is published version, updates
include: several points clarified, added Defn 2.33 and 3.37 for markings,
statement of smoothness removed from Thm 2.39 and 3.41, proof of Thm 4.13
corrected, proof of Lem 3.46 has been enhanced, appendix on 2-categories
removed, index of notation adde
From time-reversal symmetry to quantum Bayes' rules
Bayes' rule is
one of the simplest yet most profound, ubiquitous, and far-reaching results of
classical probability theory, with applications in any field utilizing
statistical inference. Many attempts have been made to extend this rule to
quantum systems, the significance of which we are only beginning to understand.
In this work, we develop a systematic framework for defining Bayes' rule in the
quantum setting, and we show that a vast majority of the proposed quantum
Bayes' rules appearing in the literature are all instances of our definition.
Moreover, our Bayes' rule is based upon a simple relationship between the
notions of state over time and a time-reversal symmetry map, both of which are
introduced here.Comment: Some adjustments and organizational changes, typos fixed, new tables
added; 24 pages tota
On dynamical measures of quantum information
In this work, we use the theory of quantum states over time to define an
entropy associated with quantum processes
, where is a state and is a quantum
channel responsible for the dynamical evolution of . The entropy
is a generalization of the von Neumann entropy in the
sense that (where denotes the
identity channel), and is a dynamical analogue of the quantum joint entropy for
bipartite states. Such an entropy is then used to define dynamical formulations
of the quantum conditional entropy and quantum mutual information, and we show
such information measures satisfy many desirable properties, such as a quantum
entropic Bayes' rule. We also use our entropy function to quantify the
information loss/gain associated with the dynamical evolution of quantum
systems, which enables us to formulate a precise notion of information
conservation for quantum processes.Comment: Comments welcome
SVD Entanglement Entropy
In this paper, we introduce a new quantity called SVD entanglement entropy.
This is a generalization of entanglement entropy in that it depends on two
different states, as in pre- and post-selection processes. This SVD
entanglement entropy takes non-negative real values and is bounded by the
logarithm of the Hilbert space dimensions. The SVD entanglement entropy can be
interpreted as the average number of Bell pairs distillable from intermediates
states. We observe that the SVD entanglement entropy gets enhanced when the two
states are in the different quantum phases in an explicit example of the
transverse-field Ising model. Moreover, we calculate the R{\'e}nyi SVD entropy
in various field theories and examine holographic calculations using the
AdS/CFT correspondence.Comment: 42 pages, 23 figure
Sufficient conditions for two-dimensional localization by arbitrarily weak defects in periodic potentials with band gaps
We prove, via an elementary variational method, 1d and 2d localization within
the band gaps of a periodic Schrodinger operator for any mostly negative or
mostly positive defect potential, V, whose depth is not too great compared to
the size of the gap. In a similar way, we also prove sufficient conditions for
1d and 2d localization below the ground state of such an operator. Furthermore,
we extend our results to 1d and 2d localization in d dimensions; for example, a
linear or planar defect in a 3d crystal. For the case of D-fold degenerate band
edges, we also give sufficient conditions for localization of up to D states.Comment: 9 pages, 3 figure