Correlated Knowledge: an Epistemic-Logic View on Quantum Entanglement

Abstract

In this paper we give a logical analysis of both classical and quantum correlations. We propose a new logical system to reason about the information carried by a complex system composed of several parts. Our formalism is based on an extension of epistemic logic with operators for "group knowledge" (the logic GEL), further extended with atomic sentences describing the results of "joint observations" (the logic LCK). As models we introduce correlation models, as a generalization of the standard representation of epistemic models as vector models. We give sound and complete axiomatizations for our logics, and we use this setting to investigate the relationship between the information carried by each of the parts of a complex system and the information carried by the whole system. In particular we distinguish between the "distributed information", obtainable by simply pooling together all the information that can be separately observed in any of the parts, and "correlated information", obtainable only by doing joint observations of the parts (and pooling together the results). Our formalism throws a new light on the difference between classical and quantum information and gives rise to an informational-logical characterization of the notion of "quantum entanglement".

Logics of Informational Interactions

“…This view equips quantum logic with an operational dimension, linking every physical property or proposition about a physical system to the experimental procedures that can be performed on those systems. Inspired by the work of C. Piron, [33,34], the operational view has lead to new axiomatic systems that are now studied in the context of Dynamic Quantum Logic [7][8][9][12][13][14]. In contrast to the traditional work on quantum logic (following [24,39]), the dynamic logical approach has several advantages.…”

“…However, in our dynamic approach we capture entanglement in a different way, by using pure qualitative tools. In [8][9][10]12], we added (qualitative) spatial features to dynamic quantum logic, allowing us to talk about the local properties of given subsystems of a quantum system. We will illustrate the formal details of this in the next section.…”

“…In [7], we first illustrate how Hilbert spaces can be structured as QTS's. The work in [7] was the start of a new semantic approach, which was later further developed in a series of papers [8,9,[12][13][14]. In the next paragraph we introduce the setting of Quantum Transition Systems (QTS) as part of our quantum semantics for the language of P DL.…”

“…In [9,12], we added (qualitative) spatial features to dynamic quantum logic, allowing us to express local properties of given subsystems of a quantum system. Starting from the above (generalized) QTS semantics, we now extend the setting in order to express that the actions can be of various types, depending on their location.…”

Quantum logic as a dynamic logic

“…All that Quantum Mechanics shows (and Quantum Dynamic Logic formalizes) is the fact that certain informational interactions change the systems involved; and moreover, they change them non-locally. (But this last aspect can only be formalized in the multi-partite version of our Quantum Dynamic Logic, see Baltag and Smets 2004, 2006a, 2008, 2010.) As a dynamic logic, quantum logic may still be in some sense "non-classical": its tests do not behave like classical PDL tests, since they change the state.…”