9,736 research outputs found
Computation in generalised probabilistic theories
From the existence of an efficient quantum algorithm for factoring, it is
likely that quantum computation is intrinsically more powerful than classical
computation. At present, the best upper bound known for the power of quantum
computation is that BQP is in AWPP. This work investigates limits on
computational power that are imposed by physical principles. To this end, we
define a circuit-based model of computation in a class of operationally-defined
theories more general than quantum theory, and ask: what is the minimal set of
physical assumptions under which the above inclusion still holds? We show that
given only an assumption of tomographic locality (roughly, that multipartite
states can be characterised by local measurements), efficient computations are
contained in AWPP. This inclusion still holds even without assuming a basic
notion of causality (where the notion is, roughly, that probabilities for
outcomes cannot depend on future measurement choices). Following Aaronson, we
extend the computational model by allowing post-selection on measurement
outcomes. Aaronson showed that the corresponding quantum complexity class is
equal to PP. Given only the assumption of tomographic locality, the inclusion
in PP still holds for post-selected computation in general theories. Thus in a
world with post-selection, quantum theory is optimal for computation in the
space of all general theories. We then consider if relativised complexity
results can be obtained for general theories. It is not clear how to define a
sensible notion of an oracle in the general framework that reduces to the
standard notion in the quantum case. Nevertheless, it is possible to define
computation relative to a `classical oracle'. Then, we show there exists a
classical oracle relative to which efficient computation in any theory
satisfying the causality assumption and tomographic locality does not include
NP.Comment: 14+9 pages. Comments welcom
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
Strong Complementarity and Non-locality in Categorical Quantum Mechanics
Categorical quantum mechanics studies quantum theory in the framework of
dagger-compact closed categories.
Using this framework, we establish a tight relationship between two key
quantum theoretical notions: non-locality and complementarity. In particular,
we establish a direct connection between Mermin-type non-locality scenarios,
which we generalise to an arbitrary number of parties, using systems of
arbitrary dimension, and performing arbitrary measurements, and a new stronger
notion of complementarity which we introduce here.
Our derivation of the fact that strong complementarity is a necessary
condition for a Mermin scenario provides a crisp operational interpretation for
strong complementarity. We also provide a complete classification of strongly
complementary observables for quantum theory, something which has not yet been
achieved for ordinary complementarity.
Since our main results are expressed in the (diagrammatic) language of
dagger-compact categories, they can be applied outside of quantum theory, in
any setting which supports the purely algebraic notion of strongly
complementary observables. We have therefore introduced a method for discussing
non-locality in a wide variety of models in addition to quantum theory.
The diagrammatic calculus substantially simplifies (and sometimes even
trivialises) many of the derivations, and provides new insights. In particular,
the diagrammatic computation of correlations clearly shows how local
measurements interact to yield a global overall effect. In other words, we
depict non-locality.Comment: 15 pages (incl. 5 appendix). To appear: LiCS 201
Deriving Grover's lower bound from simple physical principles
Grover's algorithm constitutes the optimal quantum solution to the search
problem and provides a quadratic speed-up over all possible classical search
algorithms. Quantum interference between computational paths has been posited
as a key resource behind this computational speed-up. However there is a limit
to this interference, at most pairs of paths can ever interact in a fundamental
way. Could more interference imply more computational power? Sorkin has defined
a hierarchy of possible interference behaviours---currently under experimental
investigation---where classical theory is at the first level of the hierarchy
and quantum theory belongs to the second. Informally, the order in the
hierarchy corresponds to the number of paths that have an irreducible
interaction in a multi-slit experiment. In this work, we consider how Grover's
speed-up depends on the order of interference in a theory. Surprisingly, we
show that the quadratic lower bound holds regardless of the order of
interference. Thus, at least from the point of view of the search problem,
post-quantum interference does not imply a computational speed-up over quantum
theory.Comment: Updated title and exposition in response to referee comments. 6+2
pages, 5 figure
How to make unforgeable money in generalised probabilistic theories
We discuss the possibility of creating money that is physically impossible to
counterfeit. Of course, "physically impossible" is dependent on the theory that
is a faithful description of nature. Currently there are several proposals for
quantum money which have their security based on the validity of quantum
mechanics. In this work, we examine Wiesner's money scheme in the framework of
generalised probabilistic theories. This framework is broad enough to allow for
essentially any potential theory of nature, provided that it admits an
operational description. We prove that under a quantifiable version of the
no-cloning theorem, one can create physical money which has an exponentially
small chance of being counterfeited. Our proof relies on cone programming, a
natural generalisation of semidefinite programming. Moreover, we discuss some
of the difficulties that arise when considering non-quantum theories.Comment: 27 pages, many diagrams. Comments welcom
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
Quantum Theory is a Quasi-stochastic Process Theory
There is a long history of representing a quantum state using a
quasi-probability distribution: a distribution allowing negative values. In
this paper we extend such representations to deal with quantum channels. The
result is a convex, strongly monoidal, functorial embedding of the category of
trace preserving completely positive maps into the category of quasi-stochastic
matrices. This establishes quantum theory as a subcategory of quasi-stochastic
processes. Such an embedding is induced by a choice of minimal informationally
complete POVM's. We show that any two such embeddings are naturally isomorphic.
The embedding preserves the dagger structure of the categories if and only if
the POVM's are symmetric, giving a new use of SIC-POVM's, objects that are of
foundational interest in the QBism community. We also study general convex
embeddings of quantum theory and prove a dichotomy that such an embedding is
either trivial or faithful.Comment: In Proceedings QPL 2017, arXiv:1802.0973
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