16,438 research outputs found
A functional quantum programming language
We introduce the language QML, a functional language for quantum computations
on finite types. Its design is guided by its categorical semantics: QML
programs are interpreted by morphisms in the category FQC of finite quantum
computations, which provides a constructive semantics of irreversible quantum
computations realisable as quantum gates. QML integrates reversible and
irreversible quantum computations in one language, using first order strict
linear logic to make weakenings explicit. Strict programs are free from
decoherence and hence preserve superpositions and entanglement - which is
essential for quantum parallelism.Comment: 15 pages. Final version, to appear in Logic in Computer Science 200
New Directions in Categorical Logic, for Classical, Probabilistic and Quantum Logic
Intuitionistic logic, in which the double negation law not-not-P = P fails,
is dominant in categorical logic, notably in topos theory. This paper follows a
different direction in which double negation does hold. The algebraic notions
of effect algebra/module that emerged in theoretical physics form the
cornerstone. It is shown that under mild conditions on a category, its maps of
the form X -> 1+1 carry such effect module structure, and can be used as
predicates. Predicates are identified in many different situations, and capture
for instance ordinary subsets, fuzzy predicates in a probabilistic setting,
idempotents in a ring, and effects (positive elements below the unit) in a
C*-algebra or Hilbert space. In quantum foundations the duality between states
and effects plays an important role. It appears here in the form of an
adjunction, where we use maps 1 -> X as states. For such a state s and a
predicate p, the validity probability s |= p is defined, as an abstract Born
rule. It captures many forms of (Boolean or probabilistic) validity known from
the literature. Measurement from quantum mechanics is formalised categorically
in terms of `instruments', using L\"uders rule in the quantum case. These
instruments are special maps associated with predicates (more generally, with
tests), which perform the act of measurement and may have a side-effect that
disturbs the system under observation. This abstract description of
side-effects is one of the main achievements of the current approach. It is
shown that in the special case of C*-algebras, side-effect appear exclusively
in the non-commutative case. Also, these instruments are used for test
operators in a dynamic logic that can be used for reasoning about quantum
programs/protocols. The paper describes four successive assumptions, towards a
categorical axiomatisation of quantitative logic for probabilistic and quantum
systems
A Categorical Framework for Quantum Theory
Underlying any theory of physics is a layer of conceptual frames. They
connect the mathematical structures used in theoretical models with physical
phenomena, but they also constitute our fundamental assumptions about reality.
Many of the discrepancies between quantum physics and classical physics
(including Maxwell's electrodynamics and relativity) can be traced back to
these categorical foundations. We argue that classical physics corresponds to
the factual aspects of reality and requires a categorical framework which
consists of four interdependent components: boolean logic, the
linear-sequential notion of time, the principle of sufficient reason, and the
dichotomy between observer and observed. None of these can be dropped without
affecting the others. However, in quantum theory the reduction postulate also
addresses the "status nascendi" of facts, i.e., their coming into being.
Therefore, quantum phyics requires a different conceptual framework which will
be elaborated in this article. It is shown that many of its components are
already present in the standard formalisms of quantum physics, but in most
cases they are highlighted not so much from a conceptual perspective but more
from their mathematical structures. The categorical frame underlying quantum
physics includes a profoundly different notion of time which encompasses a
crucial role for the present.Comment: 35 pages, 1 figur
Classical Structures Based on Unitaries
Starting from the observation that distinct notions of copying have arisen in
different categorical fields (logic and computation, contrasted with quantum
mechanics) this paper addresses the question of when, or whether, they may
coincide. Provided all definitions are strict in the categorical sense, we show
that this can never be the case. However, allowing for the defining axioms to
be taken up to canonical isomorphism, a close connection between the classical
structures of categorical quantum mechanics, and the categorical property of
self-similarity familiar from logical and computational models becomes
apparent.
The required canonical isomorphisms are non-trivial, and mix both typed
(multi-object) and untyped (single-object) tensors and structural isomorphisms;
we give coherence results that justify this approach.
We then give a class of examples where distinct self-similar structures at an
object determine distinct matrix representations of arrows, in the same way as
classical structures determine matrix representations in Hilbert space. We also
give analogues of familiar notions from linear algebra in this setting such as
changes of basis, and diagonalisation.Comment: 24 pages,7 diagram
Exponential Modalities and Complementarity (extended abstract)
The exponential modalities of linear logic have been used by various authors
to model infinite-dimensional quantum systems. This paper explains how these
modalities can also give rise to the complementarity principle of quantum
mechanics.
The paper uses a formulation of quantum systems based on dagger-linear logic,
whose categorical semantics lies in mixed unitary categories, and a formulation
of measurement therein. The main result exhibits a complementary system as the
result of measurements on free exponential modalities. Recalling that, in
linear logic, exponential modalities have two distinct but dual components, !
and ?, this shows how these components under measurement become "compacted"
into the usual notion of complementary Frobenius algebras from categorical
quantum mechanics.Comment: In Proceedings ACT 2021, arXiv:2211.01102. A full version of this
paper, containing all proofs, appears at arXiv:2103:0519
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