246 research outputs found
Quantum Picturalism
The quantum mechanical formalism doesn't support our intuition, nor does it
elucidate the key concepts that govern the behaviour of the entities that are
subject to the laws of quantum physics. The arrays of complex numbers are kin
to the arrays of 0s and 1s of the early days of computer programming practice.
In this review we present steps towards a diagrammatic `high-level' alternative
for the Hilbert space formalism, one which appeals to our intuition. It allows
for intuitive reasoning about interacting quantum systems, and trivialises many
otherwise involved and tedious computations. It clearly exposes limitations
such as the no-cloning theorem, and phenomena such as quantum teleportation. As
a logic, it supports `automation'. It allows for a wider variety of underlying
theories, and can be easily modified, having the potential to provide the
required step-stone towards a deeper conceptual understanding of quantum
theory, as well as its unification with other physical theories. Specific
applications discussed here are purely diagrammatic proofs of several quantum
computational schemes, as well as an analysis of the structural origin of
quantum non-locality. The underlying mathematical foundation of this high-level
diagrammatic formalism relies on so-called monoidal categories, a product of a
fairly recent development in mathematics. These monoidal categories do not only
provide a natural foundation for physical theories, but also for proof theory,
logic, programming languages, biology, cooking, ... The challenge is to
discover the necessary additional pieces of structure that allow us to predict
genuine quantum phenomena.Comment: Commissioned paper for Contemporary Physics, 31 pages, 84 pictures,
some colo
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
Equational reasoning with context-free families of string diagrams
String diagrams provide an intuitive language for expressing networks of
interacting processes graphically. A discrete representation of string
diagrams, called string graphs, allows for mechanised equational reasoning by
double-pushout rewriting. However, one often wishes to express not just single
equations, but entire families of equations between diagrams of arbitrary size.
To do this we define a class of context-free grammars, called B-ESG grammars,
that are suitable for defining entire families of string graphs, and crucially,
of string graph rewrite rules. We show that the language-membership and
match-enumeration problems are decidable for these grammars, and hence that
there is an algorithm for rewriting string graphs according to B-ESG rewrite
patterns. We also show that it is possible to reason at the level of grammars
by providing a simple method for transforming a grammar by string graph
rewriting, and showing admissibility of the induced B-ESG rewrite pattern.Comment: International Conference on Graph Transformation, ICGT 2015. The
final publication is available at Springer via
http://dx.doi.org/10.1007/978-3-319-21145-9_
Three qubit entanglement within graphical Z/X-calculus
The compositional techniques of categorical quantum mechanics are applied to
analyse 3-qubit quantum entanglement. In particular the graphical calculus of
complementary observables and corresponding phases due to Duncan and one of the
authors is used to construct representative members of the two genuinely
tripartite SLOCC classes of 3-qubit entangled states, GHZ and W. This nicely
illustrates the respectively pairwise and global tripartite entanglement found
in the W- and GHZ-class states. A new concept of supplementarity allows us to
characterise inhabitants of the W class within the abstract diagrammatic
calculus; these method extends to more general multipartite qubit states.Comment: In Proceedings HPC 2010, arXiv:1103.226
Early Greek Thought and Perspectives for the Interpretation of Quantum Mechanics: Preliminaries to an Ontological Approach
It will be shown in this article that an ontological approach for some
problems related to the interpretation of Quantum Mechanics could emerge from a
re-evaluation of the main paradox of early Greek thought: the paradox of Being
and non-Being, and the solutions presented to it by Plato and Aristotle.
Plato's and Aristotle's systems are argued here to do on the ontological level
essentially the same: to introduce stability in the world by introducing the
notion of a separable, stable object, for which a principle of contradiction is
valid: an object cannot be and not-be at the same place at the same time. After
leaving Aristotelian metaphysics, early modern science had to cope with these
problems: it did so by introducing ``space'' as the seat of stability, and
``time'' as the theater of motion. But the ontological structure present in
this solution remained the same. Therefore the fundamental notion `separable
system', related to the notions observation and measurement, themselves related
to the modern concepts of space and time, appears to be intrinsically
problematic, because it is inextricably connected to classical logic on the
ontological level. We see therefore the problems dealt with by quantum logic
not as merely formal, and the problem of `non-locality' as related to it,
indicating the need to re-think the notions `system', `entity', as well as the
implications of the operation `measurement', which is seen here as an
application of classical logic (including its ontological consequences) on the
material world.Comment: 18 page
The GHZ/W-calculus contains rational arithmetic
Graphical calculi for representing interacting quantum systems serve a number
of purposes: compositionally, intuitive graphical reasoning, and a logical
underpinning for automation. The power of these calculi stems from the fact
that they embody generalized symmetries of the structure of quantum operations,
which, for example, stretch well beyond the Choi-Jamiolkowski isomorphism. One
such calculus takes the GHZ and W states as its basic generators. Here we show
that this language allows one to encode standard rational calculus, with the
GHZ state as multiplication, the W state as addition, the Pauli X gate as
multiplicative inversion, and the Pauli Z gate as additive inversion.Comment: In Proceedings HPC 2010, arXiv:1103.226
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
Time-asymmetry of probabilities versus relativistic causal structure: an arrow of time
There is an incompatibility between the symmetries of causal structure in
relativity theory and the signaling abilities of probabilistic devices with
inputs and outputs: while time-reversal in relativity will not introduce the
ability to signal between spacelike separated regions, this is not the case for
probabilistic devices with space-like separated input-output pairs. We
explicitly describe a non-signaling device which becomes a perfect signaling
device under time-reversal, where time-reversal can be conceptualized as
playing backwards a videotape of an agent manipulating the device. This leads
to an arrow of time that is identifiable when studying the correlations of
events for spacelike separated regions. Somewhat surprisingly, although
time-reversal of Popuscu-Roerlich boxes also allows agents to signal, it does
not yield a perfect signaling device. Finally, we realize time-reversal using
post-selection, which could lead experimental implementation.Comment: 4 pages, some figures; replaces arXiv:1010.4572 [quant-ph
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
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