1,779 research outputs found
Pure Maps between Euclidean Jordan Algebras
We propose a definition of purity for positive linear maps between Euclidean
Jordan Algebras (EJA) that generalizes the notion of purity for quantum
systems. We show that this definition of purity is closed under composition and
taking adjoints and thus that the pure maps form a dagger category (which sets
it apart from other possible definitions.) In fact, from the results presented
in this paper, it follows that the category of EJAs with positive contractive
linear maps is a dagger-effectus, a type of structure originally defined to
study von Neumann algebras in an abstract categorical setting. In combination
with previous work this characterizes EJAs as the most general systems allowed
in a generalized probabilistic theory that is simultaneously a dagger-effectus.
Using the dagger structure we get a notion of dagger-positive maps of the form
f = g*g. We give a complete characterization of the pure dagger-positive maps
and show that these correspond precisely to the Jordan algebraic version of the
sequential product that maps (a,b) to sqrt(a) b sqrt(a). The notion of
dagger-positivity therefore characterizes the sequential product.Comment: In Proceedings QPL 2018, arXiv:1901.0947
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
Ockham on Divine Concurrence
The focus of this paper is Ockham's stance on the question of divine concurrence---the question whether God is causally active in the causal happenings of the created world, and if so, what God's causal activity amounts to and what place that leaves for created causes. After discussing some preliminaries, I turn to presenting what I take to be Ockham's account. As I show, Ockham, at least in this issue, is rather conservative: he agrees with the majority of medieval thinkers (including Aquinas, Giles of Rome, Duns Scotus, and others) that both God and created agents are causally active in the causal happenings of the world. Then I turn to some texts that may suggest otherwise; I argue that reading Ockham as either an occasionalist or a mere conservationist based on these texts originates from a misunderstanding of his main concern. I conclude with raising and briefly addressing some systematic worries regarding Ockham's account of concurrence
Lower and Upper Conditioning in Quantum Bayesian Theory
Updating a probability distribution in the light of new evidence is a very
basic operation in Bayesian probability theory. It is also known as state
revision or simply as conditioning. This paper recalls how locally updating a
joint state can equivalently be described via inference using the channel
extracted from the state (via disintegration). This paper also investigates the
quantum analogues of conditioning, and in particular the analogues of this
equivalence between updating a joint state and inference. The main finding is
that in order to obtain a similar equivalence, we have to distinguish two forms
of quantum conditioning, which we call lower and upper conditioning. They are
known from the literature, but the common framework in which we describe them
and the equivalence result are new.Comment: In Proceedings QPL 2018, arXiv:1901.0947
Condition/Decision Duality and the Internal Logic of Extensive Restriction Categories
In flowchart languages, predicates play an interesting double role. In the
textual representation, they are often presented as conditions, i.e.,
expressions which are easily combined with other conditions (often via Boolean
combinators) to form new conditions, though they only play a supporting role in
aiding branching statements choose a branch to follow. On the other hand, in
the graphical representation they are typically presented as decisions,
intrinsically capable of directing control flow yet mostly oblivious to Boolean
combination. While categorical treatments of flowchart languages are abundant,
none of them provide a treatment of this dual nature of predicates. In the
present paper, we argue that extensive restriction categories are precisely
categories that capture such a condition/decision duality, by means of
morphisms which, coincidentally, are also called decisions. Further, we show
that having these categorical decisions amounts to having an internal logic:
Analogous to how subobjects of an object in a topos form a Heyting algebra, we
show that decisions on an object in an extensive restriction category form a De
Morgan quasilattice, the algebraic structure associated with the (three-valued)
weak Kleene logic . Full classical propositional logic can be
recovered by restricting to total decisions, yielding extensive categories in
the usual sense, and confirming (from a different direction) a result from
effectus theory that predicates on objects in extensive categories form Boolean
algebras. As an application, since (categorical) decisions are partial
isomorphisms, this approach provides naturally reversible models of classical
propositional logic and weak Kleene logic.Comment: 19 pages, including 6 page appendix of proofs. Accepted for MFPS XXX
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