28 research outputs found
Quantum coin tossing and bit-string generation in the presence of noise
We discuss the security implications of noise for quantum coin tossing
protocols. We find that if quantum error correction can be used, so that noise
levels can be made arbitrarily small, then reasonable security conditions for
coin tossing can be framed so that results from the noiseless case will
continue to hold. If, however, error correction is not available (as is the
case with present day technology), and significant noise is present, then
tossing a single coin becomes problematic. In this case, we are led to consider
random n-bit string generation in the presence of noise, rather than
single-shot coin tossing. We introduce precise security criteria for n-bit
string generation and describe an explicit protocol that could be implemented
with present day technology. In general, a cheater can exploit noise in order
to bias coins to their advantage. We derive explicit upper bounds on the
average bias achievable by a cheater for given noise levels.Comment: REVTeX. 6 pages, no figures. Early versions contained errors in
statements of security conditions, although results were correct. v4: PRA
versio
Tight bounds for classical and quantum coin flipping
Coin flipping is a cryptographic primitive for which strictly better
protocols exist if the players are not only allowed to exchange classical, but
also quantum messages. During the past few years, several results have appeared
which give a tight bound on the range of implementable unconditionally secure
coin flips, both in the classical as well as in the quantum setting and for
both weak as well as strong coin flipping. But the picture is still incomplete:
in the quantum setting, all results consider only protocols with perfect
correctness, and in the classical setting tight bounds for strong coin flipping
are still missing. We give a general definition of coin flipping which unifies
the notion of strong and weak coin flipping (it contains both of them as
special cases) and allows the honest players to abort with a certain
probability. We give tight bounds on the achievable range of parameters both in
the classical and in the quantum setting.Comment: 18 pages, 2 figures; v2: published versio
Modal quantum theory
We present a discrete model theory similar in structure to ordinary quantum
mechanics, but based on a finite field instead of complex amplitudes. The
interpretation of this theory involves only the "modal" concepts of possibility
and necessity rather than quantitative probability measures. Despite its
simplicity, our model theory includes entangled states and has versions of both
Bell's theorem and the no cloning theorem.Comment: Presented at the 7th Workshop on Quantum Physics and Logic, Oxford
University (29-30 May 2010). Revised 1 Aug 2011 in response to referee
comment
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
Bell nonlocality, signal locality and unpredictability (or What Bohr could have told Einstein at Solvay had he known about Bell experiments)
The 1964 theorem of John Bell shows that no model that reproduces the
predictions of quantum mechanics can simultaneously satisfy the assumptions of
locality and determinism. On the other hand, the assumptions of \emph{signal
locality} plus \emph{predictability} are also sufficient to derive Bell
inequalities. This simple theorem, previously noted but published only
relatively recently by Masanes, Acin and Gisin, has fundamental implications
not entirely appreciated. Firstly, nothing can be concluded about the
ontological assumptions of locality or determinism independently of each other
-- it is possible to reproduce quantum mechanics with deterministic models that
violate locality as well as indeterministic models that satisfy locality. On
the other hand, the operational assumption of signal locality is an empirically
testable (and well-tested) consequence of relativity. Thus Bell inequality
violations imply that we can trust that some events are fundamentally
\emph{unpredictable}, even if we cannot trust that they are indeterministic.
This result grounds the quantum-mechanical prohibition of arbitrarily accurate
predictions on the assumption of no superluminal signalling, regardless of any
postulates of quantum mechanics. It also sheds a new light on an early stage of
the historical debate between Einstein and Bohr.Comment: Substantially modified version; added HMW as co-autho
Einstein, incompleteness, and the epistemic view of quantum states
Does the quantum state represent reality or our knowledge of reality? In
making this distinction precise, we are led to a novel classification of hidden
variable models of quantum theory. Indeed, representatives of each class can be
found among existing constructions for two-dimensional Hilbert spaces. Our
approach also provides a fruitful new perspective on arguments for the
nonlocality and incompleteness of quantum theory. Specifically, we show that
for models wherein the quantum state has the status of something real, the
failure of locality can be established through an argument considerably more
straightforward than Bell's theorem. The historical significance of this result
becomes evident when one recognizes that the same reasoning is present in
Einstein's preferred argument for incompleteness, which dates back to 1935.
This fact suggests that Einstein was seeking not just any completion of quantum
theory, but one wherein quantum states are solely representative of our
knowledge. Our hypothesis is supported by an analysis of Einstein's attempts to
clarify his views on quantum theory and the circumstance of his otherwise
puzzling abandonment of an even simpler argument for incompleteness from 1927.Comment: 18 pages, 8 figures, 1 recipe for cupcakes; comments welcom
Revisiting consistency conditions for quantum states of systems on closed timelike curves: an epistemic perspective
There has been considerable recent interest in the consequences of closed
timelike curves (CTCs) for the dynamics of quantum mechanical systems. A vast
majority of research into this area makes use of the dynamical equations
developed by Deutsch, which were developed from a consistency condition that
assumes that mixed quantum states uniquely describe the physical state of a
system. We criticise this choice of consistency condition from an epistemic
perspective, i.e., a perspective in which the quantum state represents a state
of knowledge about a system. We demonstrate that directly applying Deutsch's
condition when mixed states are treated as representing an observer's knowledge
of a system can conceal time travel paradoxes from the observer, rather than
resolving them. To shed further light on the appropriate dynamics for quantum
systems traversing CTCs, we make use of a toy epistemic theory with a strictly
classical ontology due to Spekkens and show that, in contrast to the results of
Deutsch, many of the traditional paradoxical effects of time travel are
present.Comment: 10 pages, 6 figures, comments welcome; v2 added references and
clarified some points; v3 published versio
Coupled-mode theory for Bose-Einstein condensates
We apply the concepts of nonlinear guided-wave optics to a Bose-Einstein
condensate (BEC) trapped in an external potential. As an example, we consider a
parabolic double-well potential and derive coupled-mode equations for the
complex amplitudes of the BEC macroscopic collective modes. Our equations
describe different regimes of the condensate dynamics, including the nonlinear
Josephson effect for any separation between the wells. We demonstrate
macroscopic self-trapping for both repulsive and attractive interactions, and
confirm our results by numerical simulations.Comment: 4 pages, 5 figures; typos removed, figures amended; submitted to PR
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