45,876 research outputs found
Causality re-established
Causality never gained the status of a "law" or "principle" in physics. Some
recent literature even popularized the false idea that causality is a notion
that should be banned from theory. Such misconception relies on an alleged
universality of reversibility of laws of physics, based either on determinism
of classical theory, or on the multiverse interpretation of quantum theory, in
both cases motivated by mere interpretational requirements for realism of the
theory. Here, I will show that a properly defined unambiguous notion of
causality is a theorem of quantum theory, which is also a falsifiable
proposition of the theory. Such causality notion appeared in the literature
within the framework of operational probabilistic theories. It is a genuinely
theoretical notion, corresponding to establish a definite partial order among
events, in the same way as we do by using the future causal cone on Minkowski
space. The causality notion is logically completely independent of the
misidentified concept of "determinism", and, being a consequence of quantum
theory, is ubiquitous in physics. In addition, as classical theory can be
regarded as a restriction of quantum theory, causality holds also in the
classical case, although the determinism of the theory trivializes it. I then
conclude arguing that causality naturally establishes an arrow of time. This
implies that the scenario of the "Block Universe" and the connected "Past
Hypothesis" are incompatible with causality, and thus with quantum theory: they
both are doomed to remain mere interpretations and, as such, not falsifiable,
similar to the hypothesis of "super-determinism". This article is part of a
discussion meeting issue "Foundations of quantum mechanics and their impact on
contemporary society".Comment: Presented at the Royal Society of London, on 11/12/ 2017, at the
conference "Foundations of quantum mechanics and their impact on contemporary
society". To appear on Philosophical Transactions of the Royal Society
No-go theorem for the composition of quantum systems
Building on the Pusey-Barrett-Rudolph theorem, we derive a no-go theorem for
a vast class of deterministic hidden-variables theories, including those
consistent on their targeted domain. The strength of this result throws doubt
on seemingly natural assumptions (like the "preparation independence" of the
Pusey-Barrett-Rudolph theorem) about how "real states" of subsystems compose
for joint systems in nonentangled states. This points to constraints in
modeling tensor-product states, similar to constraints demonstrated for more
complex states by the Bell and Bell-Kochen-Specker theorems.Comment: 4 pages. v2: new title, significant revisions. v3: condensed, matches
final published versio
An invariant of topologically ordered states under local unitary transformations
For an anyon model in two spatial dimensions described by a modular tensor
category, the topological S-matrix encodes the mutual braiding statistics, the
quantum dimensions, and the fusion rules of anyons. It is nontrivial whether
one can compute the S-matrix from a single ground state wave function. Here, we
define a class of Hamiltonians consisting of local commuting projectors and an
associated matrix that is invariant under local unitary transformations. We
argue that the invariant is equivalent to the topological S-matrix. The
definition does not require degeneracy of the ground state. We prove that the
invariant depends on the state only, in the sense that it can be computed by
any Hamiltonian in the class of which the state is a ground state. As a
corollary, we prove that any local quantum circuit that connects two ground
states of quantum double models (discrete gauge theories) with non-isomorphic
abelian groups, must have depth that is at least linear in the system's
diameter.
As a tool for the proof, a manifestly Hamiltonian-independent notion of
locally invisible operators is introduced. This gives a sufficient condition
for a many-body state not to be generated from a product state by any small
depth quantum circuit; this is a many-body entanglement witness.Comment: revtex 11pt, 43 pages, (v2) minor change (v3) ref. added. To appear
in Commun. Math. Phy
Fiat-Shamir for highly sound protocols is instantiable
The Fiat–Shamir (FS) transformation (Fiat and Shamir, Crypto '86) is a popular paradigm for constructing very efficient non-interactive zero-knowledge (NIZK) arguments and signature schemes from a hash function and any three-move interactive protocol satisfying certain properties. Despite its wide-spread applicability both in theory and in practice, the known positive results for proving security of the FS paradigm are in the random oracle model only, i.e., they assume that the hash function is modeled as an external random function accessible to all parties. On the other hand, a sequence of negative results shows that for certain classes of interactive protocols, the FS transform cannot be instantiated in the standard model.
We initiate the study of complementary positive results, namely, studying classes of interactive protocols where the FS transform does have standard-model instantiations. In particular, we show that for a class of “highly sound” protocols that we define, instantiating the FS transform via a q-wise independent hash function yields NIZK arguments and secure signature schemes. In the case of NIZK, we obtain a weaker “q-bounded” zero-knowledge flavor where the simulator works for all adversaries asking an a-priori bounded number of queries q; in the case of signatures, we obtain the weaker notion of random-message unforgeability against q-bounded random message attacks.
Our main idea is that when the protocol is highly sound, then instead of using random-oracle programming, one can use complexity leveraging. The question is whether such highly sound protocols exist and if so, which protocols lie in this class. We answer this question in the affirmative in the common reference string (CRS) model and under strong assumptions. Namely, assuming indistinguishability obfuscation and puncturable pseudorandom functions we construct a compiler that transforms any 3-move interactive protocol with instance-independent commitments and simulators (a property satisfied by the Lapidot–Shamir protocol, Crypto '90) into a compiled protocol in the CRS model that is highly sound. We also present a second compiler, in order to be able to start from a larger class of protocols, which only requires instance-independent commitments (a property for example satisfied by the classical protocol for quadratic residuosity due to Blum, Crypto '81). For the second compiler we require dual-mode commitments.
We hope that our work inspires more research on classes of (efficient) 3-move protocols where Fiat–Shamir is (efficiently) instantiable
Sequences of regressions and their independences
Ordered sequences of univariate or multivariate regressions provide
statistical models for analysing data from randomized, possibly sequential
interventions, from cohort or multi-wave panel studies, but also from
cross-sectional or retrospective studies. Conditional independences are
captured by what we name regression graphs, provided the generated distribution
shares some properties with a joint Gaussian distribution. Regression graphs
extend purely directed, acyclic graphs by two types of undirected graph, one
type for components of joint responses and the other for components of the
context vector variable. We review the special features and the history of
regression graphs, derive criteria to read all implied independences of a
regression graph and prove criteria for Markov equivalence that is to judge
whether two different graphs imply the same set of independence statements.
Knowledge of Markov equivalence provides alternative interpretations of a given
sequence of regressions, is essential for machine learning strategies and
permits to use the simple graphical criteria of regression graphs on graphs for
which the corresponding criteria are in general more complex. Under the known
conditions that a Markov equivalent directed acyclic graph exists for any given
regression graph, we give a polynomial time algorithm to find one such graph.Comment: 43 pages with 17 figures The manuscript is to appear as an invited
discussion paper in the journal TES
Propositional logic with short-circuit evaluation: a non-commutative and a commutative variant
Short-circuit evaluation denotes the semantics of propositional connectives
in which the second argument is evaluated only if the first argument does not
suffice to determine the value of the expression. Short-circuit evaluation is
widely used in programming, with sequential conjunction and disjunction as
primitive connectives.
We study the question which logical laws axiomatize short-circuit evaluation
under the following assumptions: compound statements are evaluated from left to
right, each atom (propositional variable) evaluates to either true or false,
and atomic evaluations can cause a side effect. The answer to this question
depends on the kind of atomic side effects that can occur and leads to
different "short-circuit logics". The basic case is FSCL (free short-circuit
logic), which characterizes the setting in which each atomic evaluation can
cause a side effect. We recall some main results and then relate FSCL to MSCL
(memorizing short-circuit logic), where in the evaluation of a compound
statement, the first evaluation result of each atom is memorized. MSCL can be
seen as a sequential variant of propositional logic: atomic evaluations cannot
cause a side effect and the sequential connectives are not commutative. Then we
relate MSCL to SSCL (static short-circuit logic), the variant of propositional
logic that prescribes short-circuit evaluation with commutative sequential
connectives.
We present evaluation trees as an intuitive semantics for short-circuit
evaluation, and simple equational axiomatizations for the short-circuit logics
mentioned that use negation and the sequential connectives only.Comment: 34 pages, 6 tables. Considerable parts of the text below stem from
arXiv:1206.1936, arXiv:1010.3674, and arXiv:1707.05718. Together with
arXiv:1707.05718, this paper subsumes most of arXiv:1010.367
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