14,404 research outputs found
Higher-order interference and single-system postulates characterizing quantum theory
We present a new characterization of quantum theory in terms of simple
physical principles that is different from previous ones in two important
respects: first, it only refers to properties of single systems without any
assumptions on the composition of many systems; and second, it is closer to
experiment by having absence of higher-order interference as a postulate, which
is currently the subject of experimental investigation. We give three
postulates -- no higher-order interference, classical decomposability of
states, and strong symmetry -- and prove that the only non-classical
operational probabilistic theories satisfying them are real, complex, and
quaternionic quantum theory, together with 3-level octonionic quantum theory
and ball state spaces of arbitrary dimension. Then we show that adding
observability of energy as a fourth postulate yields complex quantum theory as
the unique solution, relating the emergence of the complex numbers to the
possibility of Hamiltonian dynamics. We also show that there may be interesting
non-quantum theories satisfying only the first two of our postulates, which
would allow for higher-order interference in experiments while still respecting
the contextuality analogue of the local orthogonality principle.Comment: 21 + 6 pages, 1 figure. v4: published version (includes several minor
corrections
On quantum vs. classical probability
Quantum theory shares with classical probability theory many important
properties. I show that this common core regards at least the following six
areas, and I provide details on each of these: the logic of propositions,
symmetry, probabilities, composition of systems, state preparation and
reductionism. The essential distinction between classical and quantum theory,
on the other hand, is shown to be joint decidability versus smoothness; for the
latter in particular I supply ample explanation and motivation. Finally, I
argue that beyond quantum theory there are no other generalisations of
classical probability theory that are relevant to physics.Comment: Major revision: key results unchanged, but derivation and discussion
completely rewritten; 33 pages, no figure
Oracles and query lower bounds in generalised probabilistic theories
We investigate the connection between interference and computational power
within the operationally defined framework of generalised probabilistic
theories. To compare the computational abilities of different theories within
this framework we show that any theory satisfying three natural physical
principles possess a well-defined oracle model. Indeed, we prove a subroutine
theorem for oracles in such theories which is a necessary condition for the
oracle to be well-defined. The three principles are: causality (roughly, no
signalling from the future), purification (each mixed state arises as the
marginal of a pure state of a larger system), and strong symmetry existence of
non-trivial reversible transformations). Sorkin has defined a hierarchy of
conceivable interference behaviours, where the order in the hierarchy
corresponds to the number of paths that have an irreducible interaction in a
multi-slit experiment. Given our oracle model, we show that if a classical
computer requires at least n queries to solve a learning problem, then the
corresponding lower bound in theories lying at the kth level of Sorkin's
hierarchy is n/k. Hence, lower bounds on the number of queries to a quantum
oracle needed to solve certain problems are not optimal in the space of all
generalised probabilistic theories, although it is not yet known whether the
optimal bounds are achievable in general. Hence searches for higher-order
interference are not only foundationally motivated, but constitute a search for
a computational resource beyond that offered by quantum computation.Comment: 17+7 pages. Comments Welcome. Published in special issue
"Foundational Aspects of Quantum Information" in Foundations of Physic
Ruling out higher-order interference from purity principles
As first noted by Rafael Sorkin, there is a limit to quantum interference.
The interference pattern formed in a multi-slit experiment is a function of the
interference patterns formed between pairs of slits, there are no genuinely new
features resulting from considering three slits instead of two. Sorkin has
introduced a hierarchy of mathematically conceivable higher-order interference
behaviours, where classical theory lies at the first level of this hierarchy
and quantum theory theory at the second. Informally, the order in this
hierarchy corresponds to the number of slits on which the interference pattern
has an irreducible dependence. Many authors have wondered why quantum
interference is limited to the second level of this hierarchy. Does the
existence of higher-order interference violate some natural physical principle
that we believe should be fundamental? In the current work we show that such
principles can be found which limit interference behaviour to second-order, or
"quantum-like", interference, but that do not restrict us to the entire quantum
formalism. We work within the operational framework of generalised
probabilistic theories, and prove that any theory satisfying Causality, Purity
Preservation, Pure Sharpness, and Purification---four principles that formalise
the fundamental character of purity in nature---exhibits at most second-order
interference. Hence these theories are, at least conceptually, very "close" to
quantum theory. Along the way we show that systems in such theories correspond
to Euclidean Jordan algebras. Hence, they are self-dual and, moreover,
multi-slit experiments in such theories are described by pure projectors.Comment: 18+8 pages. Comments welcome. v2: Minor correction to Lemma 5.1, main
results are unchange
A universe of processes and some of its guises
Our starting point is a particular `canvas' aimed to `draw' theories of
physics, which has symmetric monoidal categories as its mathematical backbone.
In this paper we consider the conceptual foundations for this canvas, and how
these can then be converted into mathematical structure. With very little
structural effort (i.e. in very abstract terms) and in a very short time span
the categorical quantum mechanics (CQM) research program has reproduced a
surprisingly large fragment of quantum theory. It also provides new insights
both in quantum foundations and in quantum information, and has even resulted
in automated reasoning software called `quantomatic' which exploits the
deductive power of CQM. In this paper we complement the available material by
not requiring prior knowledge of category theory, and by pointing at
connections to previous and current developments in the foundations of physics.
This research program is also in close synergy with developments elsewhere, for
example in representation theory, quantum algebra, knot theory, topological
quantum field theory and several other areas.Comment: Invited chapter in: "Deep Beauty: Understanding the Quantum World
through Mathematical Innovation", H. Halvorson, ed., Cambridge University
Press, forthcoming. (as usual, many pictures
Symmetry and Self-Duality in Categories of Probabilistic Models
This note adds to the recent spate of derivations of the probabilistic
apparatus of finite-dimensional quantum theory from various axiomatic packages.
We offer two different axiomatic packages that lead easily to the Jordan
algebraic structure of finite-dimensional quantum theory. The derivation relies
on the Koecher-Vinberg Theorem, which sets up an equivalence between order-unit
spaces having homogeneous, self-dual cones, and formally real Jordan algebras.Comment: In Proceedings QPL 2011, arXiv:1210.029
Operational formulation of time reversal in quantum theory
The symmetry of quantum theory under time reversal has long been a subject of
controversy because the transition probabilities given by Born's rule do not
apply backward in time. Here, we resolve this problem within a rigorous
operational probabilistic framework. We argue that reconciling time reversal
with the probabilistic rules of the theory requires a notion of operation that
permits realizations via both pre- and post-selection. We develop the
generalized formulation of quantum theory that stems from this approach and
give a precise definition of time-reversal symmetry, emphasizing a previously
overlooked distinction between states and effects. We prove an analogue of
Wigner's theorem, which characterizes all allowed symmetry transformations in
this operationally time-symmetric quantum theory. Remarkably, we find larger
classes of symmetry transformations than those assumed before. This suggests a
possible direction for search of extensions of known physics.Comment: 17 pages, 5 figure
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