We generalize the theory of catalytic quantum randomness to distributed and dynamical settings. First, we expand the theory of catalytic quantum randomness by calculating the amount of (R\'enyi) entropy catalytically extractable from a distributed or dynamical randomness source. We show that no entropy can be catalytically extracted when one cannot implement local projective measurement on randomness source without altering its state. As an application, we prove that quantum operation cannot be hidden in correlation between two parties without using randomness, which is the dynamical generalization of the no-hiding theorem. Moreover, the formalism of distributed catalysis is applied to develop a formal definition of semantic quantum information and it follows that utilizing semantic information is equivalent to catalysis using a catalyst already correlated with the transforming system. By doing so, we unify the utilization of semantic and nonsemantic quantum information and conclude that one can always extract more information from an incompletely depleted classical randomness source, but it is not possible for quantum randomness sources.
We generalize the theory of catalytic quantum randomness to distributed and dynamical settings. First, we expand the theory of catalytic quantum randomness by calculating the amount of (R'enyi) entropy catalytically extractable from a distributed or dynamical randomness source. We show that no entropy can be catalytically extracted when one cannot implement local projective measurement on randomness source without altering its state. As an application, we prove that quantum operation cannot be hidden in correlation between two parties without using randomness, which is the dynamical generalization of the no-hiding theorem. Moreover, the formalism of distributed catalysis is applied to develop a formal definition of semantic quantum information and it follows that utilizing semantic information is equivalent to catalysis using a catalyst already correlated with the transforming system. By doing so, we unify the utilization of semantic and nonsemantic quantum information and conclude that one can always extract more information from an incompletely depleted classical randomness source, but it is not possible for quantum randomness sources.
Catalysts are substances that assist transformation of other resourceful objects without being consumed in the process. However, the fact that their ``catalytic power'' is limited and can be depleted is often overlooked, especially in the recently developing theories on catalysis of quantum randomness utilizing building correlation with catalyst. In this work, we establish a resource theory of one-shot catalytic randomness in which uncorrelatedness is consumed in catalysis of randomness. We do so by completely characterizing bipartite unitary operators that can be used to implement catalysis of randomness using partial transpose. By doing so, we find that every catalytic channel is factorizable, and therefore there exists a unital channel that is not catalytic. We define a family of catalytic entropies that quantifies catalytically extractable R\'enyi entropies from a quantum state and show how much the degeneracy of a quantum state can boost the catalytic entropy beyond its ordinary entropy. Based on this, we demonstrate that a randomness source can be actually exhausted after a certain amount of randomness is extracted. We apply this theory to systems under superselection rules that forbids superposition of certain quantum states and find that nonmaximally mixed states can yield the maximal catalytic entropy. We discuss implications of this theory to various topics, including catalytic randomness absorption, the no-secret theorem, and the possibility of multiparty infinite catalysis.
Catalysts are substances that assist transformation of other resourceful objects without being consumed in the process. However, the fact that their `catalytic power' is limited and can be depleted is often overlooked, especially in the recently developing theories on catalysis of quantum randomness utilizing building correlation with catalyst. In this work, we establish a resource theory of one-shot catalytic randomness in which uncorrelatedness is consumed in catalysis of randomness. We do so by completely characterizing bipartite unitary operators that can be used to implement catalysis of randomness using partial transpose. By doing so, we find that every catalytic channel is factorizable, and therefore there exists a unital channel that is not catalytic. We define a family of catalytic entropies that quantifies catalytically extractable entropy within a quantum state and show how much degeneracy of quantum state can boost the catalytic entropy beyond its ordinary entropy. Based on this, we demonstrate that a randomness source can be actually exhausted after a certain amount of randomness is extracted. We apply this theory to systems under conservation law that forbids superposition of certain quantum states and find that non-maximally mixed states can yield the maximal catalytic entropy. We discuss implications of this theory to various topics including catalytic randomness absorption, the no-secret theorem and the possibility of multi-party infinite catalysis.
Randomness can help one to implement quantum maps that cannot be realized in a deterministic fashion.Recently, it was discovered that explicitly treating a randomness source as a quantum system could double the efficiency as a catalyst for some tasks.In this work, we first show that every quantum channel that can be implemented with a randomness source without leaking information to it must be a catalysis.For that purpose, we prove a new no-go theorem that generalizes the no-hiding theorem, the no-secret theorem that states no quantum information can be shared with other systems as a secret without leaking some information.Second, we show that nondegenerate catalysts should be used classically when no extra dimension is allowed, which leads to the fact that the quantum advantage of a catalytic process strictly comes from the uniformness of the randomness source.Finally, we discuss a method to circumvent the previous result that achieves quantum advantage with nondegenerate catalyst uniformized by employing extra work space.
Randomness can help one to implement quantum maps that cannot be realized in a deterministic fashion. Recently, it was discovered that explicitly treating a randomness source as a quantum system could double the efficiency as a catalyst for some tasks. In this work, we generalize this result to general randomness-utilizing processes and show that the gap between the efficiencies of classical and quantum catalytic randomness is generic. To explore the achievability of catalysis of quantum randomness, we first show that the second law of thermodynamics holds for general randomness-utilizing processes: the amount of randomness never decreases. Especially, for two important randomization processes, dephasing and erasure maps, randomness sources are forced to be used catalytically, i.e., in such a way that the randomness is preserved exactly. Second, we show that non-degenerate catalysts should be used classically, which leads to the fact that the quantum advantage of a catalytic process strictly comes from the uniformness of the randomness source. Finally, we discuss a method to circumvent the previous result that achieves quantum advantage with non-degenerate catalyst uniformized by employing extra work space.
In this work, we address fundamental limitations of quantum teleportation-the process of transferring quantum information using classical communication and preshared entanglement. We develop a new teleportation protocol based upon the idea of using ancillary entanglement catalytically, i.e., without depleting it. This protocol is then used to show that catalytic entanglement allows for a noiseless quantum channel to be simulated with a quality that could never be achieved using only entanglement from the shared state, even for catalysts with a small dimension. On the one hand, this allows for a more faithful transmission of quantum information using generic states and fixed amount of consumed entanglement. On the other hand, this shows, for the first time, that entanglement catalysis provides a genuine advantage in a generic quantum-information processing task. Finally, we show that similar ideas can be directly applied to study quantum catalysis for more general problems in quantum mechanics. As an application, we show that catalysts can activate so-called passive states, a concept that finds widespread application, e.g., in quantum thermodynamics.
Quantum thermodynamics is a blossoming research program that uses tools and ideas from quantum information theory to extend the laws of thermodynamics to the domain of systems to which the laws of quantum mechanics apply and the thermodynamic limit does not necessarily apply. The contributions of this program are both foundational, in providing an approach to constructively derive the laws of phenomenological thermodynamics from the postulates of unitary quantum mechanics in a rigorous and bottom-up fashion, but also prac- tical, in providing the theory for an increasing number of experiments that attempt to build thermodynamic machines at scales that were previously inaccessible. One key mathematical tool used in quantum thermodynamics is the framework of resource theories, specifically the resource theory of thermal operations, in which the thermodynamic interaction of a system with a heat bath and various additional components such as work batteries, clocks or catalysts are modeled. In this cumulative thesis, various extensions and modifications of this framework are studied in order to derive novel results and insights about the possible thermodynamic evolution of quantum systems. In particular, following a system- atic exposition of the resource theory of thermal operations from first principles, I develop answers to the following questions: i) What is the smallest heat bath required to provide the necessary randomness for a system to undergo a given stochastic or thermodynamical evolution? ii) Does this size differ depending on whether the interaction between bath and system is quantum or classical? iii) How can quantum systems thermodynamically evolve in the presence of catalytic bystander systems and how can this be put to use? iv) What are the thermodynamic state transitions that an agent can operationally affect when she only has access to partial information about the underlying states of system and bath? v) How can we understand the emergence of the canonical ensemble in statistical thermodynamics from quantum mechanics? These questions cover a wide ground, but it will become clear that they can in fact all be discussed using the same formal tools. As such, the answers to the above questions provided in this thesis are both interesting in their own right — showing for example that there exists a gap in how efficiently randomness can be exploited quantumly as compared to classically or that catalytic bystander systems can be used to extract finite amounts of work per particle from macroscopic systems with non-vanishing probability — as well as illustrate the power of quantum thermodynamics as a set of tools to connect quantum mechanics and the theory of thermodynamics more generally.