We study a conditional state on a quantum logic using Renyi's approach (or Bayesian principle). This approach helps us to define independence of events and differently from the situation in the classical theory of probability, if an event $a$ is independent of an event $b$ , then the event $b$ can be dependent on the event $a$ . We will show that we can define a $s$ -map (function for simultaneous measurements on a quantum logic). It can be shown that if we have the conditional state we can define the $s$ -map and conversely. By using the $s$ -map we can introduce joint distribution also for noncompatible observables on a quantum logic.We study a conditional state on a quantum logic using Renyi's approach (or Bayesian principle). This approach helps us to define independence of events and differently from the situation in the classical theory of probability, if an event $a$ is independent of an event $b$, then the event $b$ can be dependent on the event $a$. We will show that we can define a $s$-map (function for simultaneous measurements on a quantum logic). It can be shown that if we have the conditional state we can define the $s$-map and conversely. By using the $s$-map we can introduce joint distribution also for noncompatible observables on a quantum logic.

arXiv:quant-ph/0302053Feb 7, 20030 citations

Events in context

Samson Abramsky

Theoretical Computer ScienceMar 17, 20140 citations

On orthoposets of numerical events in quantum logic

Dietmar Dorninger, Helmut Länger

Let S be a set of states of a physical system and p(s) the probability of the occurrence of an event when the system is in state s in S. Such a function p from S to [0,1] is known as a numerical event or more accurately an S-probability. A set P of numerical events including the constant functions 0 and 1 and 1-p with every p in P becomes a poset when ordered by the order of real functions and can serve as a general setting for quantum logics. We call such a poset P a general set of events (GSE). The thoroughly investigated algebras of S-probabilities (including Hilbert logics), concrete logics and Boolean algebras can all be represented within this setting. In this paper we study various classes of GSEs, in particular those that are orthoposets and their interrelations and connections to known logics. Moreover, we characterize GSEs as posets by means of states and discuss the situation for GSEs to be lattices.Let S be a set of states of a physical system and p(s) the probability of the occurrence of an event when the system is in state s in S. Such a function p from S to [0,1] is known as a numerical event or more accurately an S-probability. A set P of numerical events including the constant functions 0 and 1 and 1-p with every p in P becomes a poset when ordered by the order of real functions and can serve as a general setting for quantum logics. We call such a poset P a general set of events (GSE). The thoroughly investigated algebras of S-probabilities (including Hilbert logics), concrete logics and Boolean algebras can all be represented within this setting. In this paper we study various classes of GSEs, in particular those that are orthoposets and their interrelations and connections to known logics. Moreover, we characterize GSEs as posets by means of states and discuss the situation for GSEs to be lattices.

quant-pharXiv:2601.00772Reports on Mathematical PhysicsJan 5, 20260 citations

The Probabilistic Argument for a Non-Classical Logic of Quantum Mechanics

Patrick Suppes

The aim of this paper is to state the single most powerful argument for use of a non-classical logic in quantum mechanics. In outline the argument is the following. The working logic of a science is the logic of the events and propositions to which probabilities are assigned. A probability should be assigned to every element of the algebra of events. In the case of quantum mechanics probabilities may be assigned to events but not, without restriction, to the conjunction of two events. The conclusion is that the working logic of quantum mechanics is not classical. The nature of the logic that is appropriate for quantum mechanics is examined.The aim of this paper is to state the single most powerful argument for use of a non-classical logic in quantum mechanics. In outline the argument is the following. The working logic of a science is the logic of the events and propositions to which probabilities are assigned. A probability should be assigned to every element of the algebra of events. In the case of quantum mechanics probabilities may be assigned to events but not, without restriction, to the conjunction of two events. The conclusion is that the working logic of quantum mechanics is not classical. The nature of the logic that is appropriate for quantum mechanics is examined.

Philosophy of ScienceMar 1, 196686 citations

Time and Events

Ph. Blanchard, Arkadiusz Jadczyk

arXiv:quant-ph/9702018International Journal of Theoretical PhysicsJan 1, 199813 citations

Observations and events

Lachlan Mackinnon

Physica B+CJul 1, 19880 citations

a

is independent of an event

b

, then the event

b

can be dependent on the event

a

. We will show that we can define a

s

-map (function for simultaneous measurements on a quantum logic). It can be shown that if we have the conditional state we can define the

s

-map and conversely. By using the

s

-map we can introduce joint distribution also for noncompatible observables on a quantum logic.