A Dynamic-Logical Perspective on Quantum Behavior

Abstract

In this paper we show how recent concepts from Dynamic Logic, and in particular from Dynamic Epistemic logic, can be used to model and interpret quantum behavior. Our main thesis is that all the non-classical properties of quantum systems are explainable in terms of the non-classical flow of quantum information. We give a logical analysis of quantum measurements (formalized using modal operators) as triggers for quantum information flow, and we compare them with other logical operators previously used to model various forms of classical information flow: the "test" operator from Dynamic Logic, the "announcement" operator from Dynamic Epistemic Logic and the "revision" operator from Belief Revision theory. The main points stressed in our investigation are the following: (1) The perspective and the techniques of "logical dynamics" are useful for understanding quantum information flow. (2) Quantum mechanics does not require any modification of the classical laws of "static" propositional logic, but only a non-classical dynamics of information. (3) The main such non-classical feature is that, in a quantum world, all informationgathering actions have some ontic side-effects. (4) This ontic impact can affect in its turn the flow of information, leading to non-classical epistemic side-effects (e.g. a type of non-monotonicity) and to states of "objectively imperfect information". (5) Moreover, the ontic impact is non-local: an information-gathering action on one part of a quantum system can have ontic side-effects on other, far-away parts of the system.

Correlated Knowledge: an Epistemic-Logic View on Quantum Entanglement

“…For groups {i} of one individual agent, we use the simplified notation K i instead of K {i} . The use of K I -operators to capture qualitative spatial features of complex systems, extends our previous approach in [8] which in its turn is based on [5,7] and inspired by [2,16].…”

“…Implicit Knowledge. We adopt the distinction in the CS literature between implicit and explicit knowledge (see e.g [21]), and use the notion of "information carried by a system" interchangeably with the notion of "implicit knowledge" (in line also with [8,10]). Explicit knowledge is the actual information possessed by a real agent, the information stored in its "database" or its subjective internal state.…”

“…A third way to define observational equivalence is via invariance under changes that do not affect the information carried by subsystem I (see also [8]) : An evolution (unitary map) U is said to be I -remote (or "remote from I ") if it corresponds to applying only a local unitary map on the subsystem N \ I (the "non-I " part of the system, also known as I 's "environment"): i.e., if U is of the form I d I ⊗ U N\I , where I d I is the identity map on subsystem I and U N\I is a unitary map on the subsystem N \ I . In other words, U is I -remote if it is N \ I -local.…”

Johan van Benthem on Logic and Information Dynamics

“…In our own work, we gave a similar deconstruction of quantum logic [11,12,[18][19][20], showing that the so-called quantum implication (known as ''Sasaki Hook'') P ! Q is best understood semantically as a PDL-style dynamic modality ½?P Q for a ''quantum-test'' action ?P, representing a successful measurement of the quantum property P. In contrast to classical PDL tests (and classical idealized measurements), quantum tests are ''really dynamic'': they change the state of the system under observation, which explains the non-classical behavior of quantum ''implication''.…”

Categorical Quantum Models and Logics

“…4.6.8 One can define a weakest precondition modality [f ] from dynamic logic [14,15,68] in the setting of the previous example. For a morphism f : X → Y in OMLatGal and y ∈ Y , define…”