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Abstract

A short introduction to the concepts in quantum resource theories.

Introduction

Quantum resource theories quantify the usefulness of certain quantum phenomena in information process protocols (Chitambar & Gour (2019)). They provide a rigorous mathematical framework with which to compare the ability of quantum objects to provide an advantage in information processing tasks, subject to certain physical constraints. By tailoring these constraints, quantum resource theories can be applicable to tasks of both a pragmatic and fundamental nature. Thus far they have proved a fruitful way to approach problems concerning a number of different quantum phenomena, with resource theories of entanglement (Horodecki et al. (2009)), athermality (Goold et al. (2016), Horodecki & Oppenheim (2013), Gour et al. (2015), Lostaglio (2019)), Bell-non locality (Brunner et al. (2014)), measurement (Skrzypczyk & Linden (2019)), and non-Gaussianity (Weedbrook et al. (2012)) to name a few.

Framework

Overview

Value emerges from limitation. The study of these limitations, that arise from some set of physical constraints, is the central goal of resource theories. Within these theories, some agent is allowed to perform some operations, whilst other operations are prohibited by the physical constraints. This restricts what it is possible for an agent to generate within the framework, with anything the agent is unable to generate considered a resource. The agents inability to access these resources arbitrarily then ascribes them value. These resource can then be utilised for a variety of tasks that would not be possible solely using allowed operations.

Within a given resource theory, one therefore has allowed operations and free objects.

a) The free objects of a resource theory inside the set of all objects. A resourceful and a resourceless object are shown b) The set of allowed operations of a resource theory inside the set of all operations.

a) The free objects of a resource theory inside the set of all objects. A resourceful and a resourceless object are shown b) The set of allowed operations of a resource theory inside the set of all operations.

Allowed Operations

Allowed operations are resource non-increasing operations. Hence, if there exists an allowed operation E\mathcal{E}, between two objects ρ \rho and σ \sigma , such that E(ρ)=σ\mathcal{E}(\rho)=\sigma, then it must be the case that σ \sigma has less than or equal to the resource content of ρ \rho .

The allowed operations define a preorder on the space of states 💭. If an allowed operation exists mapping a state ρ\rho to σ\sigma then we write that

ρ≺σ.\rho \prec \sigma.
(1)

The preorder determines how resourceful objects are compared to each other and finding it is one of the central goals of any resource theory.

Allowed operations are typically defined by the physical constraints of the system one is interested in investigating, such as in the resource theory of entanglement. However, they can be mathematically motivated, such as in the resource theory of symmetry.

Free Objects

Free objects are the set of objects upon which the allowed operations act invariantly. Hence, if an agent - who can only perform allowed operations - has only objects from the free set, they are only able to reach other objects in the free set.

It is assumed that an agent can generate any arbitrarily large amount of free objects at no cost. If they have an object not in the free set, they can also throw that object away and prepare an object in the free set. Therefore, all elements of the free set are reachable from any other object.

Objects in the free set are said to be resourceless. Any object not in the set of free objects is said to contain resource.

Resource Theory Features

Simulation of Non-allowed Operations

A resourceful object allows an agent to overcome the physical constraints of their system; if the agent has access to a resourceful object and allowed operations, they will able to perform operations they would not be able to do with allowed operations only.

Let P\mathcal{P} be a non-allowed operation. It could be possible that given some resourceful state ρ \rho , and some allowed operation E\mathcal{E}, it could be possible to simulate the action of P\mathcal{P} on some state ψ \psi ,

E(ψ⊗ρ)=P(ψ).\mathcal{E}(\psi \otimes \rho) = \mathcal{P}(\psi).
(2)

The most notable example of this would be within the resource theory of entanglement (Horodecki et al. (2009)). Here, the allowed operations are local operations and classical communication (LOCC) meaning there is no allow operation that can send quantum information between two spatially separated parties. However, using LOCC (the allow operations) and a shared entangled state between the parties (an state with entanglement is a resource in the resource theory of entanglement) a channel that sends quantum information between the two spatially separated parties can be simulated via the quantum teleportation protocol.

A central aim of resource theories is to find operations that can be simulated given a particular resourceful object.

Operational Interpretations

Once one has defined something to be a resource, how resourceful an object is can be quantified by how well it can perform at some useful task. Typically, this is done by finding different physically meaningful task for which different resource measures limit the agents success in the task.

For general convex resource theories, state discrimination and state exclusion have been related to the resource measures of the robustness (Takagi & Regula (2019)) and weight respectively (Uola et al. (2020)).

Catalysis

It might be the case that there does not exists an allowed operation, E\mathcal{E}, such that σ=E(ρ)\sigma = \mathcal{E}(\rho) where ρ,σ \rho, \sigma are not free state. However, there could exists a non-free state, ω \omega , so that there does exists an allowed operation, E′\mathcal{E}', such that

E′(ρ⊗ω)=σ⊗ω.\mathcal{E}'(\rho \otimes \omega) = \sigma \otimes \omega.
(3)

The state ω \omega is called a catalyst as it facilitates a state change without itself being consumed.

Within the literature, the above condition for catalyst have been relaxed and studied. For example, the catalyst has been allowed to be returned close but not exactly to its original state or correlations have been allowed to build up between the output state and catalysts such that

E′(ρ⊗ω)=σSC,  where  trC(σSC)=σ  and  trS(σSC)=ω,\mathcal{E}'(\rho \otimes \omega) = \sigma_{SC}, ~ ~ \textrm{where} ~~ \textrm{tr}_{C}( \sigma_{SC}) = \sigma ~ ~ \textrm{and} ~ ~ \textrm{tr}_S(\sigma_{SC}) = \omega,
(4)

where SS labels the system and CC the catalyst.

Further Reading

References
  1. Chitambar, E., & Gour, G. (2019). Quantum resource theories. Reviews of Modern Physics, 91(2). 10.1103/revmodphys.91.025001
  2. Horodecki, R., Horodecki, P., Horodecki, M., & Horodecki, K. (2009). Quantum entanglement. Reviews of Modern Physics, 81(2), 865–942. 10.1103/revmodphys.81.865
  3. Goold, J., Huber, M., Riera, A., Rio, L. del, & Skrzypczyk, P. (2016). The role of quantum information in thermodynamics—a topical review. Journal of Physics A: Mathematical and Theoretical, 49(14), 143001. 10.1088/1751-8113/49/14/143001
  4. Horodecki, M., & Oppenheim, J. (2013). Fundamental limitations for quantum and nanoscale thermodynamics. Nature Communications, 4(1). 10.1038/ncomms3059
  5. Gour, G., Müller, M. P., Narasimhachar, V., Spekkens, R. W., & Yunger Halpern, N. (2015). The resource theory of informational nonequilibrium in thermodynamics. Physics Reports, 583, 1–58. 10.1016/j.physrep.2015.04.003
  6. Lostaglio, M. (2019). An introductory review of the resource theory approach to thermodynamics. Reports on Progress in Physics, 82(11), 114001. 10.1088/1361-6633/ab46e5
  7. Brunner, N., Cavalcanti, D., Pironio, S., Scarani, V., & Wehner, S. (2014). Bell nonlocality. Reviews of Modern Physics, 86(2), 419–478. 10.1103/revmodphys.86.419
  8. Skrzypczyk, P., & Linden, N. (2019). Robustness of Measurement, Discrimination Games, and Accessible Information. Physical Review Letters, 122(14). 10.1103/physrevlett.122.140403
  9. Weedbrook, C., Pirandola, S., García-Patrón, R., Cerf, N. J., Ralph, T. C., Shapiro, J. H., & Lloyd, S. (2012). Gaussian quantum information. Reviews of Modern Physics, 84(2), 621–669. 10.1103/revmodphys.84.621
  10. Takagi, R., & Regula, B. (2019). General Resource Theories in Quantum Mechanics and Beyond: Operational Characterization via Discrimination Tasks. Physical Review X, 9(3). 10.1103/physrevx.9.031053
  11. Uola, R., Bullock, T., Kraft, T., Pellonpää, J.-P., & Brunner, N. (2020). All Quantum Resources Provide an Advantage in Exclusion Tasks. Physical Review Letters, 125(11). 10.1103/physrevlett.125.110402
  12. Gour, G. (2024). Resources of the Quantum World. arXiv. 10.48550/ARXIV.2402.05474
  13. Lipka-Bartosik, P., Wilming, H., & Ng, N. H. Y. (2023). Catalysis in Quantum Information Theory. 10.48550/ARXIV.2306.00798
Quantum Information 1
Glossary
Resource Theories
Resource Theories Criteria