14 research outputs found
Locality and information transfer in quantum operations
We investigate the situation in which no information can be transferred from
a quantum system B to a quantum system A, even though both interact with a
common system C
Quantum Causal Graph Dynamics
Consider a graph having quantum systems lying at each node. Suppose that the
whole thing evolves in discrete time steps, according to a global, unitary
causal operator. By causal we mean that information can only propagate at a
bounded speed, with respect to the distance given by the graph. Suppose,
moreover, that the graph itself is subject to the evolution, and may be driven
to be in a quantum superposition of graphs---in accordance to the superposition
principle. We show that these unitary causal operators must decompose as a
finite-depth circuit of local unitary gates. This unifies a result on Quantum
Cellular Automata with another on Reversible Causal Graph Dynamics. Along the
way we formalize a notion of causality which is valid in the context of quantum
superpositions of time-varying graphs, and has a number of good properties.
Keywords: Quantum Lattice Gas Automata, Block-representation,
Curtis-Hedlund-Lyndon, No-signalling, Localizability, Quantum Gravity, Quantum
Graphity, Causal Dynamical Triangulations, Spin Networks, Dynamical networks,
Graph Rewriting.Comment: 8 pages, 1 figur
The physical Church-Turing thesis and the principles of quantum theory
Notoriously, quantum computation shatters complexity theory, but is innocuous
to computability theory. Yet several works have shown how quantum theory as it
stands could breach the physical Church-Turing thesis. We draw a clear line as
to when this is the case, in a way that is inspired by Gandy. Gandy formulates
postulates about physics, such as homogeneity of space and time, bounded
density and velocity of information --- and proves that the physical
Church-Turing thesis is a consequence of these postulates. We provide a quantum
version of the theorem. Thus this approach exhibits a formal non-trivial
interplay between theoretical physics symmetries and computability assumptions.Comment: 14 pages, LaTe
Unitarity plus causality implies localizability
We consider a graph with a single quantum system at each node. The entire
compound system evolves in discrete time steps by iterating a global evolution
. We require that this global evolution be unitary, in accordance with
quantum theory, and that this global evolution be causal, in accordance
with special relativity. By causal we mean that information can only ever be
transmitted at a bounded speed, the speed bound being quite naturally that of
one edge of the underlying graph per iteration of . We show that under these
conditions the operator can be implemented locally; i.e. it can be put into
the form of a quantum circuit made up with more elementary operators -- each
acting solely upon neighbouring nodes. We take quantum cellular automata as an
example application of this representation theorem: this analysis bridges the
gap between the axiomatic and the constructive approaches to defining QCA.
KEYWORDS: Quantum cellular automata, Unitary causal operators, Quantum walks,
Quantum computation, Axiomatic quantum field theory, Algebraic quantum field
theory, Discrete space-time.Comment: V1: 5 pages, revtex. V2: Generalizes V1. V3: More precisions and
reference
Quantum Common Causes and Quantum Causal Models
Reichenbach’s principle asserts that if two observed variables are found to be correlated, then there should be a causal explanation of these correlations. Furthermore, if the explanation is in terms of a common cause, then the conditional probability distribution over the variables given the complete common cause should factorize. The principle is generalized by the formalism of causal models, in which the causal relationships among variables constrain the form of their joint probability distribution. In the quantum case, however, the observed correlations in Bell experiments cannot be explained in the manner Reichenbach’s principle would seem to demand. Motivated by this, we introduce a quantum counterpart to the principle. We demonstrate that under the assumption that quantum dynamics is fundamentally unitary, if a quantum channel with input
A
and outputs
B
and
C
is compatible with
A
being a complete common cause of
B
and
C
, then it must factorize in a particular way. Finally, we show how to generalize our quantum version of Reichenbach’s principle to a formalism for quantum causal models and provide examples of how the formalism works
When is a quantum cellular automaton (QCA) a quantum lattice gas automaton (QLGA)?
Quantum cellular automata (QCA) are models of quantum computation of
particular interest from the point of view of quantum simulation. Quantum
lattice gas automata (QLGA - equivalently partitioned quantum cellular
automata) represent an interesting subclass of QCA. QLGA have been more deeply
analyzed than QCA, whereas general QCA are likely to capture a wider range of
quantum behavior. Discriminating between QLGA and QCA is therefore an important
question. In spite of much prior work, classifying which QCA are QLGA has
remained an open problem. In the present paper we establish necessary and
sufficient conditions for unbounded, finite Quantum Cellular Automata (QCA)
(finitely many active cells in a quiescent background) to be Quantum Lattice
Gas Automata (QLGA). We define a local condition that classifies those QCA that
are QLGA, and we show that there are QCA that are not QLGA. We use a number of
tools from functional analysis of separable Hilbert spaces and representation
theory of associative algebras that enable us to treat QCA on finite but
unbounded configurations in full detail.Comment: 37 pages, 7 figures, with changes to explanatory text and updated
figures, J. Math. Phys. versio