ABSTRACT In the context of quantum resource theories (QRTs), free states are defined as those that can be obtained at no cost under a certain restricted set of conditions. However, when taking a free state from one QRT and evaluating it through the optics of another QRT, it might well turn out that the state is now extremely resourceful. Such realization has recently prompted numerous works characterizing states across several QRTs. In this work, we contribute to this body of knowledge by analyzing the resourcefulness in free states for—and across witnesses of—the QRTs of multipartite entanglement, fermionic non‐Gaussianity, imaginarity, realness, spin coherence, Clifford non‐stabilizerness, ‐equivariance, and non‐uniform entanglement. We provide rigorous theoretical results as well as present numerical studies that showcase the rich and complex behavior that arises in this type of cross‐examination.
ABSTRACT In the context of quantum resource theories (QRTs), free states are defined as those that can be obtained at no cost under a certain restricted set of conditions. However, when taking a free state from one QRT and evaluating it through the optics of another QRT, it might well turn out that the state is now extremely resourceful. Such realization has recently prompted numerous works characterizing states across several QRTs. In this work, we contribute to this body of knowledge by analyzing the resourcefulness in free states for—and across witnesses of—the QRTs of multipartite entanglement, fermionic non‐Gaussianity, imaginarity, realness, spin coherence, Clifford non‐stabilizerness, ‐equivariance, and non‐uniform entanglement. We provide rigorous theoretical results as well as present numerical studies that showcase the rich and complex behavior that arises in this type of cross‐examination.
It was recently realized that many quantum phenomena, such as entanglement or non-locality, can be expressed in the framework of quantum resource theories (QRTs) as resources that can be expended to perform tasks. This approach has revolutionized our understanding of these phenomena and has provided a framework to rigorously characterize them, quantify them, and study the question of convertibility of resources within them. Moreover, this framework has enabled the discovery of similar structures common to many QRTs, and thus offered a better and deeper understanding of the phenomena they describe. Most of the research that has been done so far in the area of QRTs has been concerned with resource theories of states, that is, where the objects in the QRT are static resources that are divided among free and resource states. Free operations, defined as quantum channels that do not generate resources, are used to transform these states. However, this concept can be generalized to include dynamical resources, or non-free operations. We call the resource theories that arise from this, QRTs of quantum processes. This thesis introduces different QRTs of quantum processes that we call QRTs of non-projectiveness. More precisely, these are QRTs where projective measurements are considered free. There are different resource theories that arise based on exactly which additional operations are allowed and whether quantum measurements are described with a classical output, a quantum output, or both. This research looks into four such QRTs of non-projectiveness.
In quantum resource theories (QRTs) certain quantum states and operations are deemed more valuable than others. While the determination of the ``free'' elements is usually guided by the constraints of some experimental setup, this can make it difficult to study similarities and differences between QRTs. In this work, we argue that QRTs follow from the choice of a preferred algebraic structure E to be preserved, thus setting the free operations as the automorphisms of E. We illustrate our finding by determining E for the QRTs of entanglement, Clifford stabilizerness, purity, imaginarity, fermionic Gaussianity, reference frames, thermodynamics and coherence; showing instances where E is a Lie algebra, group, ring, or even a simple set. This unified understanding allows us to generalize the concept of stochastic local operations and classical communication (SLOCC) to identify novel resource non-increasing operations for Lie-algebra based QRTs, thus finding a new solution to an open problem in the literature. We showcase the sanity of our new set of operations by rigorously proving that they map free states to free states, as well as determine more general situations where these transformations strictly do not increase the resource of a state.
Quantum resource theories (QRTs) provide a unified theoretical framework for understanding inherent quantum-mechanical properties that serve as resources in quantum information processing, but resources motivated by physics may possess intractable mathematical structure to analyze, such as non-uniqueness of maximally resourceful states, lack of convexity, and infinite dimension. We investigate state conversion and resource measures in general QRTs under minimal assumptions to figure out universal properties of physically motivated quantum resources that may have such intractable mathematical structure. In the general setting, we prove the existence of maximally resourceful states in one-shot state conversion. Also analyzing asymptotic state conversion, we discover catalytic replication of quantum resources, where a resource state is infinitely replicable by free operations. In QRTs without assuming uniqueness of maximally resourceful states, we formulate the tasks of distillation and formation of quantum resources, and introduce distillable resource and resource cost based on the distillation and the formation, respectively. Furthermore, we introduce consistent resource measures that quantify the amount of quantum resources without contradicting the rate of state conversion even in QRTs with non-unique maximally resourceful states. Progressing beyond the previous work showing a uniqueness theorem for additive resource measures, we prove the corresponding uniqueness inequality for the consistent resource measures; that is, consistent resource measures of a quantum state take values between the distillable resource and the resource cost of the state. These formulations and results establish a foundation of QRTs applicable to mathematically intractable but physically motivated quantum resources in a unified way.
Quantum resource theories (QRTs) provide a unified theoretical framework for understanding inherent quantum-mechanical properties that serve as resources in quantum information processing, but resources motivated by physics may possess structure whose analysis is mathematically intractable, such as non-uniqueness of maximally resourceful states, lack of convexity, and infinite dimension. We investigate state conversion and resource measures in general QRTs under minimal assumptions to figure out universal properties of physically motivated quantum resources that may have such mathematical structure whose analysis is intractable. In the general setting, we prove the existence of maximally resourceful states in one-shot state conversion. Also analyzing asymptotic state conversion, we discover catalytic replication of quantum resources, where a resource state is infinitely replicable by free operations. In QRTs without assuming the uniqueness of maximally resourceful states, we formulate the tasks of distillation and formation of quantum resources, and introduce distillable resource and resource cost based on the distillation and the formation, respectively. Furthermore, we introduce consistent resource measures that quantify the amount of quantum resources without contradicting the rate of state conversion even in QRTs with non-unique maximally resourceful states. Progressing beyond the previous work showing a uniqueness theorem for additive resource measures, we prove the corresponding uniqueness inequality for the consistent resource measures; that is, consistent resource measures of a quantum state take values between the distillable resource and the resource cost of the state. These formulations and results establish a foundation of QRTs applicable in a unified way to physically motivated quantum resources whose analysis can be mathematically intractable.
Abstract The use of imaginary numbers in modelling quantum mechanical systems encompasses the wave-like nature of quantum states. Here we introduce a resource theoretic framework for imaginarity, where the free states are taken to be those with density matrices that are real with respect to a fixed basis. This theory is closely related to the resource theory of coherence, as it is basis dependent, and the imaginary numbers appear in the off-diagonal elements of the density matrix. Unlike coherence however, the set of physically realizable free operations is identical to both completely resource non-generating (RNG) operations, and stochastically RNG operations. Moreover, the resource theory of imaginarity does not have a self-adjoint resource destroying map. After introducing and characterizing the free operations, we provide several measures of imaginarity, and give necessary and sufficient conditions for pure state transformations via physically consistent free operations in the single shot regime.
Recently, the resource theory of imaginarity was proposed, viewing the imaginary part of a quantum state as a basic resource leading to quantum advantages. As with other quantum resources such as correlations, coherence, and asymmetry, a natural question arises: Can imaginarity resource be broadcast by a real operation which is free in the resource theory of imaginarity? In this work, we introduce two concepts of imaginarity broadcasting. The first pertains to nonreal states, while the second is related to the imaginarity resource. For nonreal states, we demonstrate that a specific type of nonreal states can be broadcast. However, when it comes to broadcasting the imaginarity resource, we get a surprising result that any imaginarity resource can be perfectly broadcast via a real operation. By comparing different (no-)broadcasting theorems, we distinguish the resource theory of imaginarity from the resource theories of correlations, coherence, and asymmetry, and reveal certain distinctive features of the resource theory of imaginarity.