Undecidability in resource theory: can you tell theories apart?

Abstract: A central question in resource theory is whether one can construct a set of monotones that completely characterise the allowed transitions dictated by a set of free operations. A similar question is whether two distinct sets of free operations generate the same class of transitions. These questions are part of the more general problem of whether it is possible to pass from one characterisation of a resource theory to another. In the present letter we prove that in the context of quantum resource theories this class of problems is undecidable in general. This is done by proving the undecidability of the membership problem for CPTP maps, which subsumes all the other results.

• The paper demonstrates that determining whether a transformation belongs to a set of operations within a quantum resource theory is generally undecidable.
• This undecidability implies that algorithmically constructing a complete set of monotones to fully describe a resource theory from its free operations is impossible.
• These findings reveal fundamental computational limits in characterizing and distinguishing different quantum resource theories and their capabilities.

Undecidability in Quantum Resource Theory

The paper "Undecidability in Resource Theory: Can You Tell Theories Apart?" by Matteo Scandi and Jacopo Surace investigates the decision problems in quantum resource theories, exploring the computational limitations in characterizing the capabilities and transformations within these frameworks. The core contribution lies in demonstrating that certain problems within the field of quantum resource theories are undecidable, relying on an extensive theoretical foundation involving Completely Positive Trace Preserving (CPTP) maps.

Key Contributions and Results

The core problem the paper addresses is the characterization of quantum resource theories, specifically tackling the decidability of whether transformations are part of a given resource theory. The authors focus on determining if a transition between states is caused by free operations, and under what conditions distinct sets of these operations yield the same transformations. Key results are as follows:

• Undecidability of Membership for CPTP Maps: The paper establishes that, in general, the problem of determining whether a specific CPTP map is within a semigroup generated by a set of such maps is undecidable. This shows inherent computational limits in resolving whether certain transformations fit within a specified set of operations.
• Implications for Quantum Resource Theories: A significant corollary of the main theorem is the undecidability of determining whether a given transformation can be achieved using a particular resource theory’s set of free operations. This undecidability extends to other problems, such as determining if two sets of free operations can generate the same transformations and transitions.
• Non-existence of a Complete Constructive Set of Monotones: The authors prove that it is impossible to construct algorithmically a complete set of monotones that describe the resource theory from its free operations. This highlights the limitations when attempting to abstract resource theories solely based on functions that quantify state resources.

Implications for Quantum Information Science

The implications of this work are profound for quantum information science, particularly in bridging the theoretical framework of quantum operations and practical implementations:

• Infeasibility of Decisional Completeness: The undecidability results underscore that, despite knowing a set of operations, fully characterizing the transformations it can yield lacks a comprehensive algorithmic solution. This challenges efforts to exhaustively document quantum protocols within specific theoretical frameworks.
• Complexity in Resource Theories: Resource theories, quintessential for understanding quantum operational limits under constrained conditions, face fundamental characterizability limitations as demonstrated by the paper. These intrinsic computational barriers call for alternative approaches in quantifying and utilizing resources within such frameworks.
• Future Directions in Quantum Computation and Communication: The work evidences the lurking complexity in seemingly well-defined quantum settings. Hence, researchers exploring quantum computation and communication must consider structural limitations conveyed by this undecidability while designing protocols or developing frameworks.

Speculative Future Directions

While this paper primarily contributes a negative result in terms of computational tractability, it opens up intriguing avenues for research:

• Alternative Frameworks for Resource Theory: Given the undecidability in classical approaches, research may pivot towards more nuanced frameworks or heuristics potentially sidestepping undecidability issues.
• Exploration in Specific Theories: Applying similar undecidability investigations in particular, natural quantum theories, such as LOCC, could clarify why these frameworks resist simple characterization.
• Intersection with Quantum Complexity Theory: Integrating undecidability results with ongoing developments in quantum complexity could reshape understanding in both fields, leading to integrated solutions or insights.

In conclusion, Scandi and Surace offer a compelling perspective on the theoretical limits within quantum resource theories, an endeavor essential for understanding the theoretical underpinnings of quantum mechanics applied to information science. By situating resource theories within computational decidability challenges, their work poses essential questions about the limits and possibilities within quantum theoretical constructs.