[Paper Review] Dynamical Resources

Gilad Gour, Carlo Maria Scandolo|arXiv (Cornell University)|Dec 31, 2020

Quantum Information and Cryptography5 references4 citations

TL;DR

This paper establishes a rigorous framework for dynamical resource theories, where quantum channels are the primary resources rather than quantum states. It introduces convex resource monotones and shows that channel convertibility under free superchannels reduces to conic linear programming, enabling complete characterization of asymptotic distillation and cost in convex theories.

ABSTRACT

Quantum channels are quintessential to quantum information, being used in all protocols, and describing how systems evolve in space and time. As such, they play a key role in the manipulation of quantum resources, and they are often resources themselves, called dynamical resources. This forces us to go beyond standard resource theories of quantum states. Here we provide a rigorous foundation for dynamical resource theories, where the resources into play are quantum channels, explaining how to manipulate dynamical resources with free superchannels. In particular, when the set of free superchannels is convex, we present a novel construction of an infinite and complete family of convex resource monotones, giving necessary and sufficient conditions for convertibility under free superchannels. After showing that the conversion problem in convex dynamical resource theories can be solved with conic linear programming, we define various resource-theoretic protocols for dynamical resources. These results serve as the framework for the study of concrete examples of theories of dynamical resources, such as dynamical entanglement theory.

Motivation & Objective

• To develop a general mathematical framework for resource theories of quantum processes, focusing on quantum channels as dynamical resources.
• To address the limitations of standard resource theories based on quantum states by extending the formalism to include dynamical operations and superchannels.
• To characterize the convertibility of quantum channels under free superchannels, particularly in convex resource theories.
• To define and analyze adaptive protocols for resource distillation and cost in the context of dynamical resources.
• To investigate the realizability of free superchannels via free pre- and post-processing channels, and to identify potential limitations in their physical implementation.

Proposed method

• Introduces the concept of free superchannels as supermaps that map free channels to free channels, generalizing free operations in state-based theories.
• Defines a novel construction of an infinite and complete family of convex resource monotones for convex dynamical resource theories.
• Reduces the channel convertibility problem to a conic linear program, leveraging the convexity of the free superchannel set.
• Introduces adaptive protocols where multiple copies of a channel are inserted into a free quantum comb in a causal, sequential order.
• Uses the notion of a (possibly non-free) comb to model the action of multiple channel copies in a circuit, enabling general adaptive manipulation.
• Defines asymptotic resource cost and distillation via limits over sequences of approximate conversions, with error thresholds approaching zero.

Experimental results

Research questions

• RQ1 How can resource theories be generalized from quantum states to quantum channels as the fundamental resources?
• RQ2 What conditions ensure that a free superchannel is physically realizable using only free pre- and post-processing channels?
• RQ3 Can the problem of channel convertibility under free superchannels be reduced to a computationally tractable form, such as conic programming?
• RQ4 What is the role of adaptivity in resource conversion protocols when multiple copies of a channel are involved?
• RQ5 Are there fundamental limitations in realizing all free superchannels through free pre- and post-processing, and what are the implications for resource theory?

Key findings

• The paper constructs a complete and infinite family of convex resource monotones that fully characterize convertibility in convex dynamical resource theories.
• The problem of determining whether one channel can be converted into another under free superchannels is equivalent to solving a conic linear program.
• Asymptotic resource distillation and cost are defined via limits of approximate conversions, with the distillation rate always less than or equal to the cost rate.
• The framework allows for adaptive protocols where the order of channel insertion into a free comb matters, generalizing the standard tensor product approach.
• The paper conjectures that not all free superchannels are physically realizable via free pre- and post-processing, suggesting a fundamental gap between abstract and physical realizability.
• The results provide a complete mathematical foundation for dynamical entanglement theory and similar frameworks, extending the reach of resource theory beyond static states.