[Paper Review] Decoherent histories on graphs

Richard Blute, Ivan T. Ivanov+1 more|ArXiv.org|Nov 7, 2001

Quantum Mechanics and Applications7 references3 citations

TL;DR

This paper extends the consistent/decoherent histories approach in quantum mechanics by replacing linear time ordering with directed acyclic graphs (DAGs) to represent causal structures. It enables local, causal descriptions of quantum systems without requiring global time, allowing probabilistic reasoning about spatially separated events while preserving entanglement and consistency via a generalized decoherence functional on graphs.

ABSTRACT

The consistent histories approach to quantum mechanics is traditionally based on linearly ordered sequences of events. We extend the histories formalism to sets of events whose causal ordering is described by directed acyclic graphs. The need for a global time is eliminated and our construction reflects the causal structure faithfully.

Motivation & Objective

• To generalize the consistent histories formalism beyond linearly ordered sequences of events to accommodate complex causal structures in quantum systems.
• To eliminate the need for a global time parameter by representing causal relations through directed acyclic graphs (DAGs).
• To maintain the consistency and decoherence conditions of quantum histories while allowing for non-linear, spatially distributed events.
• To enable probabilistic reasoning about quantum systems where events are causally related but not temporally ordered.
• To provide a framework for analyzing entangled systems and quantum field theory on curved spacetime without assuming global hyperbolicity.

Proposed method

• Represent quantum events as vertices in a directed acyclic graph (DAG), with edges labeled by Hilbert spaces and density matrices encoding local quantum information.
• Associate each vertex with either a unitary evolution operator or a projection operator to describe local dynamics or property realization.
• Define incoming and outgoing Hilbert spaces at each vertex, identified as $\mathcal{H}_i$, to track quantum degrees of freedom across causal connections.
• Generalize the decoherence functional to histories described by DAGs, ensuring consistency conditions are preserved under causal graph structure.
• Introduce framework refinement by 'blowing up' edges into subgraphs with initial and final edges, allowing for localized questioning of the system.
• Compute probabilities for refined histories using conditional probabilities derived from the original framework, with histories extended by identity operators where necessary.

Experimental results

Research questions

• RQ1 How can the consistent histories formalism be generalized to describe quantum systems with non-linear causal structures?
• RQ2 Can a global time parameter be eliminated while preserving the consistency and probabilistic interpretation of quantum histories?
• RQ3 How can spatially separated quantum events be consistently described and assigned probabilities without assuming temporal ordering?
• RQ4 What is the mathematical structure underlying the composition of quantum events in a causal graph framework?
• RQ5 How does framework refinement affect the probabilities assigned to individual histories in a non-linear causal setting?

Key findings

• The decoherence functional can be generalized to histories represented by directed acyclic graphs, preserving the consistency condition without requiring linear time ordering.
• The framework allows for the description of entangled quantum systems with spatially separated events, where causal relations are explicitly encoded in the graph structure.
• Probabilities for histories in a refined framework are computed via conditional probabilities, with the original probabilities preserved for histories in the initial framework.
• The scheme supports local quantum evolution and causal reasoning, enabling questions about properties at specific spacetime points without assuming a global time.
• The approach avoids the Kochen-Specker contextuality problem by treating each framework as context-dependent, with probabilities dependent on the chosen refinement.
• The method provides a natural setting for quantum field theory on curved spacetime and quantum cosmology by replacing global hyperbolicity with causal graph structure.