41 research outputs found
Logical Aspects of Probability and Quantum Computation
Most of the work presented in this document can be read as a sequel to previous work of the author and collaborators, which has been published and appears in [DSZ16, DSZ17, ABdSZ17]. In [ABdSZ17], the mathematical description of quantum homomorphisms of graphs and more generally of relational structures, using the language of category theory is given. In particular, we introduced the concept of âquantumâ monad. In this thesis we show that the quantum monad fits nicely into the categorical framework of effectus theory, developed by Jacobs et al. [Jac15, CJWW15]. Effectus theory is an emergent field in categorical logic aiming to describe logic and probability, from the point of view of classical and quantum computation. The main contribution in the first part of this document prove that the Kleisli category of the quantum monad on relational structures is an effectus. The second part is rather different. There, distinct facets of the equivalence relation on graphs called cospectrality are described: algebraic, combinatorial and logical relations are presented as sufficient conditions on graphs for having the same spectrum (i.e. being âcospectralâ). Other equivalence of graphs (called fractional isomorphism) is also related using some âgameâ comonads from Abramsky et al. [ADW17, Sha17, AS18]. We also describe a sufficient condition for a pair of graphs to be cospectral using the quantum monad: two Kleisli morphisms (going in opposite directions) between them satisfying certain compatibility requirement
A Recipe for State-and-Effect Triangles
In the semantics of programming languages one can view programs as state
transformers, or as predicate transformers. Recently the author has introduced
state-and-effect triangles which capture this situation categorically,
involving an adjunction between state- and predicate-transformers. The current
paper exploits a classical result in category theory, part of Jon Beck's
monadicity theorem, to systematically construct such a state-and-effect
triangle from an adjunction. The power of this construction is illustrated in
many examples, covering many monads occurring in program semantics, including
(probabilistic) power domains
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
Categories for Me, and You?
A non-self-contained gathering of notes on category theory, including the definition of locally cartesian closed category, of the cartesian structure in slice categories, or of the âpseudo-cartesian structureâ on EilenbergâMoore categories. References and proofs are provided, sometimes, to my knowledge, for the first time
A characterisation of ordered abstract probabilities
In computer science, especially when dealing with quantum computing or other
non-standard models of computation, basic notions in probability theory like "a
predicate" vary wildly. There seems to be one constant: the only useful example
of an algebra of probabilities is the real unit interval. In this paper we try
to explain this phenomenon. We will show that the structure of the real unit
interval naturally arises from a few reasonable assumptions. We do this by
studying effect monoids, an abstraction of the algebraic structure of the real
unit interval: it has an addition which is only defined when
and an involution which make it an effect algebra, in
combination with an associative (possibly non-commutative) multiplication.
Examples include the unit intervals of ordered rings and Boolean algebras.
We present a structure theory for effect monoids that are -complete,
i.e. where every increasing sequence has a supremum. We show that any
-complete effect monoid embeds into the direct sum of a Boolean algebra
and the unit interval of a commutative unital C-algebra. This gives us from
first principles a dichotomy between sharp logic, represented by the Boolean
algebra part of the effect monoid, and probabilistic logic, represented by the
commutative C-algebra. Some consequences of this characterisation are that
the multiplication must always be commutative, and that the unique
-complete effect monoid without zero divisors and more than 2 elements
must be the real unit interval. Our results give an algebraic characterisation
and motivation for why any physical or logical theory would represent
probabilities by real numbers.Comment: 12 pages. V2: Minor change