Abstract Different properties of a state may make it resourceful for different purposes, and the value taken by one or more of these properties may constrain the values that can be taken by the rest. For example, the phenomenon known as the monogamy of entanglement, which plays a crucial role in quantum cryptography and communication, describes a trade-off among different entanglement properties of a multipartite state. In this article, we lay out the conceptual and formal groundwork for the study of resource dependence relations, and we illustrate how to fully characterize them with a concrete example. We consider three resource theories concerning the about-face asymmetry properties of a qubit along three mutually orthogonal axes on the Bloch ball, where the about-face symmetry group consists of the identity map and a π rotation about the given axis. We derive a complete set of monotones for each of these resource theories and determine all of the relations that hold among them, thereby fully characterizing the resource dependence relations. In spite of the relative simplicity of the individual resource orderings in this example, these relations are already quite nuanced.
Abstract Different properties of a state may make it resourceful for different purposes, and the value taken by one or more of these properties may constrain the values that can be taken by the rest. For example, the phenomenon known as the monogamy of entanglement, which plays a crucial role in quantum cryptography and communication, describes a trade-off among different entanglement properties of a multipartite state. In this article, we lay out the conceptual and formal groundwork for the study of resource dependence relations, and we illustrate how to fully characterize them with a concrete example. We consider three resource theories concerning the about-face asymmetry properties of a qubit along three mutually orthogonal axes on the Bloch ball, where the about-face symmetry group consists of the identity map and a π rotation about the given axis. We derive a complete set of monotones for each of these resource theories and determine all of the relations that hold among them, thereby fully characterizing the resource dependence relations. In spite of the relative simplicity of the individual resource orderings in this example, these relations are already quite nuanced.
Quantum physics currently provides our deepest description of nature. To understand precisely how it differs from classical physics is not only of foundational interest, but also important to advance technology: the advent of quantum information has led to the insight that certain quantum phenomena, e.g., entanglement and coherence, are resources which enable us to solve practical tasks that are unsolvable within the realm of classical physics. But how can we quantify quantum resources and how do we decide which resource is relevant for a specific operational advantage in quantum cryptography, quantum computation, or quantum metrology? How can we employ these resources optimally? Addressing these questions is the motivation for the development of quantum resource theories, a mathematically rigorous framework that allows us to investigate a specific quantum phenomenon as a resource. Formally, such theories emerge from restrictions which are imposed in addition to the laws of quantum mechanics. A prominent example is the restriction to local operations and classical communication, which singles out entanglement as a resource. Typically, such theories describe the value of quantum states, i.e., static resources. An important question in such theories concerns their structure as determined by the restriction under consideration, which is analyzed for coherence theory on qubits in this thesis. Only recently, so-called dynamical resource theories that quantify the value of operations were proposed, which are advantageous for several reasons: they include resource theories of states as special cases, allow us to quantify additional quantum properties such as the detection or usage of static resources, which will be a main topic of this thesis, and are more natural from an operational perspective. But are different types of dynamical resources present in quantum operations related? For the case of dynamical coherence and entanglement, it will be shown that there is not only a quantitative, but also a qualitative connection.
Many quantum information processing tasks require the consumption of various types of informational resource. Entanglement is a well-studied example of this type of resource; frameness is another. Both types of resource may be quantified through the use of realvalued functions known as monotones, a collection of which can fully characterize the resourcefulness of a state. A complete characterization of the entanglement or frameness of even a finite-dimensional quantum state might require an infinite collection of monotones, despite finiteness of the descriptions of the states in question. We propose a new framework of relative monotones. A relative monotone quantifies the resourcefulness of a quantum state relative to another quantum state. We give an example of the efficacy of relative monotones by characterizing a simple type of frameness with a single relative monotone. This type of frameness is unlikely to yield to analysis with 'absolute' monotones.
Entanglement quantification aims to assess the value of quantum states for quantum information processing tasks. A closely related problem is state convertibility, asking whether two remote parties can convert a shared quantum state into another one without exchanging quantum particles. Here, we explore this connection for quantum entanglement and for general quantum resource theories. For any quantum resource theory which contains resource-free pure states, we show that there does not exist a finite set of resource monotones which completely determines all state transformations. We discuss how these limitations can be surpassed, if discontinuous or infinite sets of monotones are considered, or by using quantum catalysis. We also discuss the structure of theories which are described by a single resource monotone and show equivalence with totally ordered resource theories. These are theories where a free transformation exists for any pair of quantum states. We show that totally ordered theories allow for free transformations between all pure states. For single-qubit systems, we provide a full characterization of state transformations for any totally ordered resource theory.
A useful and unifying concept in quantum information theory is the idea of a consumable "resource". The basic idea behind quantum resource theories is that whenever we face restrictions on the set of operations we can perform, some states (resource states) enable us to perform particular tasks that otherwise would have been impossible. In this thesis we first review a resource theory that recently has gained a lot of attention, namely the resource theory of asymmetry in which one is restricted to symmetric operations and therefore asymmetric states are the resource states. The asymmetry properties of a state relative to some symmetry group specify how and to what extent the given symmetry is broken by the state. In the remainder of the thesis we explain how the ideas developed in this resource theory are related to three different areas in quantum theory: theory of quantum reference frames, the Wigner-Araki-Yanase(WAY) theorem and the conditional probability interpretation of time in quantum mechanics. Theory of quantum reference frames treats reference frames like any other physical system within the formalism of quantum theory, which causes the measurements to be an approximation of the measurements against their classical counterparts. We consider the dynamics of a quantum directional reference frame undergoing repeated interactions. These interactions induce a back-action on the reference which is the central focus of our study. The effect of a precise sequence of measurement outcomes on the reference frame is studied by looking at both the case that the measurement record is averaged over and the case wherein it is retained. We find, in particular, that there is interesting dynamics in the latter situation, which cannot be revealed by considering the average case. We then consider in detail how a sequence of rotationally invariant unitary interactions affects the reference frame, a situation, which leads to quite different dynamics than the case of repeated measurements. Different strategies for correcting reference frame drift is considered given that we have access to a set of particles with polarization opposite to the direction of drift. In particular, we find that, by implementing a suitably chosen unitary interaction after every two measurements, we can eliminate the rotational drift of the reference frame. The WAY theorem establishes an important constraint that conservation laws impose on quantum mechanical measurements. We formulate the WAY theorem in the broader context of resource theories, where one is constrained to a subset of quantum mechanical operations described by a symmetry group. Establishing connections with the theory of quantum state discrimination we obtain optimal unitaries describing the measurement of arbitrary observables, explain how prior information can permit perfect measurements that circumvent the WAY constraint, and provide a framework that establishes a natural ordering on measurement apparatuses through a decomposition into asymmetry and charge subsystems. Finally we review two di erent schemes in the conditional probability interpretation (CPI) of time in quantum mechanics. In these schemes parameter time "t" as it appears in Schrodinder equation is considered to be unobservable, yet one can use an extension of conditional probabilities in order to study dynamics of a system relative to another system, namely the quantum clock. We phrase CPI as a quantum communication protocol which enables us to use the machinery developed in the resource theory of asymmetry. This provides us with a deeper understanding of the decoherence effect caused due to the inaccessibility of the parameter time "t" and that how this effect can be suppressed by choosing an optimal initial state for the quantum clock.
One of the main goals of any resource theory such as entanglement, quantum thermodynamics, quantum coherence, and asymmetry, is to find necessary and sufficient conditions that determine whether one resource can be converted to another by the set of free operations. Here we find such conditions for a large class of quantum resource theories which we call affine resource theories. Affine resource theories include the resource theories of athermality, asymmetry, and coherence, but not entanglement. Remarkably, the necessary and sufficient conditions can be expressed as a family of inequalities between resource monotones (quantifiers) that are given in terms of the conditional min-entropy. The set of free operations is taken to be (1) the maximal set (i.e., consists of all resource nongenerating quantum channels) or (2) the self-dual set of free operations (i.e., consists of all resource nongenerating maps for which the dual map is also resource nongenerating). As an example, we apply our results to quantum thermodynamics with Gibbs preserving operations, and several other affine resource theories. Finally, we discuss the applications of these results to resource theories that are not affine and, along the way, provide the necessary and sufficient conditions that a quantum resource theory consists of a resource destroying map.
Considerable effort has been devoted to developing techniques for witnessing and characterizing quantum resources that emerge from collective properties of a set of states. In this context, Bargmann invariants play a central role: they witness coherence and related resources, and underpin important applications. In this work, we introduce a unified formalism that fully characterizes and organizes the capability of Bargmann invariants to witness different manifestations of coherence in sets of states. It is formulated around the construction of Bargmann scenarios, which specify relevant tuples of Bargmann invariants, and Bargmann polytopes, which bound the values that said invariants can have when the states are incoherent. We study their basic geometry, connect them to existing formalisms, and illustrate their physical relevance. Our construction opens new opportunities for the certification of quantum devices and lays the path toward a full quantum resource theory based entirely on multivariate traces of states.