166 research outputs found
Hidden measurements, hidden variables and the volume representation of transition probabilities
We construct, for any finite dimension , a new hidden measurement model
for quantum mechanics based on representing quantum transition probabilities by
the volume of regions in projective Hilbert space. For our model is
equivalent to the Aerts sphere model and serves as a generalization of it for
dimensions . We also show how to construct a hidden variables scheme
based on hidden measurements and we discuss how joint distributions arise in
our hidden variables scheme and their relationship with the results of Fine.Comment: 23 pages, 1 figur
A Geometrical Representation of Entanglement as Internal Constraint
We study a system of two entangled spin 1/2, were the spin's are represented
by a sphere model developed within the hidden measurement approach which is a
generalization of the Bloch sphere representation, such that also the
measurements are represented. We show how an arbitrary tensor product state can
be described in a complete way by a specific internal constraint between the
ray or density states of the two spin 1/2. We derive a geometrical view of
entanglement as a 'rotation' and 'stretching' of the sphere representing the
states of the second particle as measurements are performed on the first
particle. In the case of the singlet state entanglement can be represented by a
real physical constraint, namely by means of a rigid rod.Comment: 10 pages, 3 figures. submitted to International Journal of
Theoretical Physic
Correlating matched-filter model for analysis and optimisation of neural networks
A new formalism is described for modelling neural networks by means of which a clear physical understanding of the network behaviour can be gained. In essence, the neural net is represented by an equivalent network of matched filters which is then analysed by standard correlation techniques. The procedure is demonstrated on the synchronous Little-Hopfield network. It is shown how the ability of this network to discriminate between stored binary, bipolar codes is optimised if the stored codes are chosen to be orthogonal. However, such a choice will not often be possible and so a new neural network architecture is proposed which enables the same discrimination to be obtained for arbitrary stored codes. The most efficient convergence of the synchronous Little-Hopfield net is obtained when the neurons are connected to themselves with a weight equal to the number of stored codes. The processing gain is presented for this case. The paper goes on to show how this modelling technique can be extended to analyse the behaviour of both hard and soft neural threshold responses and a novel time-dependent threshold response is described
Causal categories: relativistically interacting processes
A symmetric monoidal category naturally arises as the mathematical structure
that organizes physical systems, processes, and composition thereof, both
sequentially and in parallel. This structure admits a purely graphical
calculus. This paper is concerned with the encoding of a fixed causal structure
within a symmetric monoidal category: causal dependencies will correspond to
topological connectedness in the graphical language. We show that correlations,
either classical or quantum, force terminality of the tensor unit. We also show
that well-definedness of the concept of a global state forces the monoidal
product to be only partially defined, which in turn results in a relativistic
covariance theorem. Except for these assumptions, at no stage do we assume
anything more than purely compositional symmetric-monoidal categorical
structure. We cast these two structural results in terms of a mathematical
entity, which we call a `causal category'. We provide methods of constructing
causal categories, and we study the consequences of these methods for the
general framework of categorical quantum mechanics.Comment: 43 pages, lots of figure
Open System Categorical Quantum Semantics in Natural Language Processing
Originally inspired by categorical quantum mechanics (Abramsky and Coecke, LiCS'04), the categorical compositional distributional model of natural language meaning of Coecke, Sadrzadeh and Clark provides a conceptually motivated procedure to compute the meaning of a sentence, given its grammatical structure within a Lambek pregroup and a vectorial representation of the meaning of its parts. The predictions of this first model have outperformed that of other models in mainstream empirical language processing tasks on large scale data. Moreover, just like CQM allows for varying the model in which we interpret quantum axioms, one can also vary the model in which we interpret word meaning. In this paper we show that further developments in categorical quantum mechanics are relevant to natural language processing too. Firstly, Selinger's CPM-construction allows for explicitly taking into account lexical ambiguity and distinguishing between the two inherently different notions of homonymy and polysemy. In terms of the model in which we interpret word meaning, this means a passage from the vector space model to density matrices. Despite this change of model, standard empirical methods for comparing meanings can be easily adopted, which we demonstrate by a small-scale experiment on real-world data. This experiment moreover provides preliminary evidence of the validity of our proposed new model for word meaning. Secondly, commutative classical structures as well as their non-commutative counterparts that arise in the image of the CPM-construction allow for encoding relative pronouns, verbs and adjectives, and finally, iteration of the CPM-construction, something that has no counterpart in the quantum realm, enables one to accommodate both entailment and ambiguity
Probabilistic theories with purification
We investigate general probabilistic theories in which every mixed state has
a purification, unique up to reversible channels on the purifying system. We
show that the purification principle is equivalent to the existence of a
reversible realization of every physical process, namely that every physical
process can be regarded as arising from a reversible interaction of the system
with an environment, which is eventually discarded. From the purification
principle we also construct an isomorphism between transformations and
bipartite states that possesses all structural properties of the
Choi-Jamiolkowski isomorphism in quantum mechanics. Such an isomorphism allows
one to prove most of the basic features of quantum mechanics, like e.g.
existence of pure bipartite states giving perfect correlations in independent
experiments, no information without disturbance, no joint discrimination of all
pure states, no cloning, teleportation, no programming, no bit commitment,
complementarity between correctable channels and deletion channels,
characterization of entanglement-breaking channels as measure-and-prepare
channels, and others, without resorting to the mathematical framework of
Hilbert spaces.Comment: Differing from the journal version, this version includes a table of
contents and makes extensive use of boldface type to highlight the contents
of the main theorems. It includes a self-contained introduction to the
framework of general probabilistic theories and a discussion about the role
of causality and local discriminabilit
Stabilizer notation for Spekkens' toy theory
Spekkens has introduced a toy theory [Phys. Rev. A, 75, 032110 (2007)] in
order to argue for an epistemic view of quantum states. I describe a notation
for the theory (excluding certain joint measurements) which makes its
similarities and differences with the quantum mechanics of stabilizer states
clear. Given an application of the qubit stabilizer formalism, it is often
entirely straightforward to construct an analogous application of the notation
to the toy theory. This assists calculations within the toy theory, for example
of the number of possible states and transformations, and enables
superpositions to be defined for composite systems.Comment: 7+4 pages, 5 tables. v2: Clarifications added and typos fixed in
response to referee comment
Quantum picturalism for topological cluster-state computing
Topological quantum computing is a way of allowing precise quantum
computations to run on noisy and imperfect hardware. One implementation uses
surface codes created by forming defects in a highly-entangled cluster state.
Such a method of computing is a leading candidate for large-scale quantum
computing. However, there has been a lack of sufficiently powerful high-level
languages to describe computing in this form without resorting to single-qubit
operations, which quickly become prohibitively complex as the system size
increases. In this paper we apply the category-theoretic work of Abramsky and
Coecke to the topological cluster-state model of quantum computing to give a
high-level graphical language that enables direct translation between quantum
processes and physical patterns of measurement in a computer - a "compiler
language". We give the equivalence between the graphical and topological
information flows, and show the applicable rewrite algebra for this computing
model. We show that this gives us a native graphical language for the design
and analysis of topological quantum algorithms, and finish by discussing the
possibilities for automating this process on a large scale.Comment: 18 pages, 21 figures. Published in New J. Phys. special issue on
topological quantum computin
Completeness of dagger-categories and the complex numbers
The complex numbers are an important part of quantum theory, but are
difficult to motivate from a theoretical perspective. We describe a simple
formal framework for theories of physics, and show that if a theory of physics
presented in this manner satisfies certain completeness properties, then it
necessarily includes the complex numbers as a mathematical ingredient. Central
to our approach are the techniques of category theory, and we introduce a new
category-theoretical tool, called the dagger-limit, which governs the way in
which systems can be combined to form larger systems. These dagger-limits can
be used to characterize the dagger-functor on the category of
finite-dimensional Hilbert spaces, and so can be used as an equivalent
definition of the inner product. One of our main results is that in a
nontrivial monoidal dagger-category with all finite dagger-limits and a simple
tensor unit, the semiring of scalars embeds into an involutive field of
characteristic 0 and orderable fixed field.Comment: 39 pages. Accepted for publication in the Journal of Mathematical
Physic
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