Quantum resource theories provide a mathematically rigorous way of understanding the nature of various quantum resources. An important problem in any quantum resource theory is to determine how quantum states can be converted into each other within the physical constraints of the theory. The standard approach to this problem is to study approximate or probabilistic transformations. Here, we investigate the intermediate regime, providing limits on both, the fidelity and the probability of state transformations. We derive limitations on the transformations, which are valid in all quantum resource theories, by providing bounds on the maximal transformation fidelity for a given transformation probability. As an application, we show that these bounds imply an upper bound on the asymptotic rates for various classes of states under probabilistic transformations. We also show that the deterministic version of the single copy bounds can be applied for drawing limitations on the manipulation of quantum channels, which goes beyond the previously known bounds of channel manipulations. Furthermore, we completely solve the question of stochastic-approximate state conversion via local operations and classical communication in the following two cases: (i) Both initial and target states are pure bipartite entangled states of arbitrary dimensions. (ii) The target state is a two-qubit entangled state and the initial state is a pure bipartite state.
Quantum resource theories provide a mathematically rigorous way of understanding the nature of various quantum resources. An important problem in any quantum resource theory is to determine how quantum states can be converted into each other within the physical constraints of the theory. The standard approach to this problem is to study approximate or probabilistic transformations. Here, we investigate the intermediate regime, providing limits on both, the fidelity and the probability of state transformations. We derive limitations on the transformations, which are valid in all quantum resource theories, by providing bounds on the maximal transformation fidelity for a given transformation probability. As an application, we show that these bounds imply an upper bound on the asymptotic rates for various classes of states under probabilistic transformations. We also show that the deterministic version of the single copy bounds can be applied for drawing limitations on the manipulation of quantum channels, which goes beyond the previously known bounds of channel manipulations. Furthermore, we completely solve the question of stochastic-approximate state conversion via local operations and classical communication in the following two cases: (i) Both initial and target states are pure bipartite entangled states of arbitrary dimensions. (ii) The target state is a two-qubit entangled state and the initial state is a pure bipartite state.
Among the most fundamental questions in the manipulation of quantum resources such as entanglement is the possibility of reversibly transforming all resource states. The key consequence of this would be the identification of a unique entropic resource measure that exactly quantifies the limits of achievable transformation rates. Remarkably, previous results claimed that such asymptotic reversibility holds true in very general settings; however, recently those findings have been found to be incomplete, casting doubt on the conjecture. Here we show that it is indeed possible to reversibly interconvert all states in general quantum resource theories, as long as one allows protocols that may only succeed probabilistically. Although such transformations have some chance of failure, we show that their success probability can be ensured to be bounded away from zero, even in the asymptotic limit of infinitely many manipulated copies. As in previously conjectured approaches, the achievability here is realised through operations that are asymptotically resource non-generating, and we show that this choice is optimal: smaller sets of transformations cannot lead to reversibility. Our methods are based on connecting the transformation rates under probabilistic protocols with strong converse rates for deterministic transformations, which we strengthen into an exact equivalence in the case of entanglement distillation.
Quantum resource theories provide a unified theoretical framework for understanding inherent quantum-mechanical properties that serve as resources in quantum information processing. State conversion can be implemented by free operations. The partial trace and the incoherent operation, similarly to the local operation and classical communication, are all free operations in the corresponding quantum resource theory. This work mainly studies the limitations of these two types of free operations: (1) Masking information is a protocol that encodes quantum information into a bipartite correlated state while the information is completely unknown to local systems. While the no-masking theorem turns out that it is not possible to determinately mask arbitrary quantum states, quantum information in certain restricted sets of states can be masked. We consider the new regime of probabilistic and approximate masking quantum information. Here, the goal is to mask a set of quantum states with maximal probability, allowing for a small error in the masking. And we provide limits on both the fidelity and the probability of masking, and derive limitations on the masking of quantum information by providing bounds on the maximal masking fidelity. (2) From quantum sensors to quantum computers, many technologies require quantum resources. The properties of such states are coherent or correlated in a way that would not be possible in classical physics. Ideally, for technology applications, these states should be in the so-called maximally resource state, one in which all possible measures of quantum resource are maximized. In a quantum system with noise, an important question is whether there exists maximal resource states. This work answered this important quantum question with a negative answer, i.e., there are no maximal resource states in general. In order to do so, this work considers the case of rank-2 states and shows that for particular values of the eigenvalues, there exists no state that can be transformed to all other states under free operations. This in particular implies that in these cases the state that maximizes a given resource measure among all states depends on the choice of resource measure, i.e., it cannot be that the aforementioned family of states maximizes all resource measures.
Understanding multipartite entanglement is a key goal in quantum information. Entanglement in pure states can be characterized by considering transformations under local operations assisted by classical communication (LOCC). However, it has been shown that, for $n\ensuremath{\ge}5$ parties, multipartite pure states are generically isolated, i.e., they can be neither reached nor transformed under LOCC. Nonetheless, in any real laboratory, one never deterministically transforms a pure initial state exactly to a pure target state. Instead, one transforms a mixed state near the initial state to an ensemble that is on average close to the target state. This motivates studying approximate LOCC transformations. After reviewing in detail the known results in the bipartite case, we present the gaps that remain open in the multipartite case. While the analysis of the multipartite setting is much more technically involved due to the existence of different stochastic LOCC (SLOCC) classes, certain features simplify in the approximate setting. In particular, we show that it is sufficient to consider pure initial states, that it is sufficient to consider LOCC protocols with finitely many rounds of communication, and that approximate transformations can be approximated by ensemble transformations within an SLOCC class. Then we formally define a hierarchy of different forms of approximate transformations that are relevant from a physical point of view. Whereas this hierarchy collapses in the bipartite case, we show that this is not the case for the multipartite setting, which is fundamentally richer. To wit, we show that optimal multipartite approximate transformations are not generally deterministic, that ensemble transformations within an SLOCC class can achieve a higher fidelity than deterministic transformations within an SLOCC class, and that there are approximate transformations with no deterministic transformations nearby.
Abstract Resource theory of imaginarity provides a useful framework to understand the role of complex numbers, which are essential in the formulation of quantum mechanics, in a mathematically rigorous way. In the first part of this article, we study the properties of ‘real’ (quantum) operations both in single-party and bipartite settings. As a consequence, we provide necessary and sufficient conditions for state transformations under real operations and show the existence of ‘real entanglement’ monotones. In the second part of this article, we focus on the problem of single copy state transformation via real quantum operations. When starting from pure initial states, we completely solve this problem by finding an analytical expression for the optimal fidelity of transformation, for a given probability of transformation and vice versa. Moreover, for state transformations involving arbitrary initial states and pure final states, we provide a semidefinite program to compute the optimal achievable fidelity, for a given probability of transformation.
Quantum theory provides us with a mathematical framework to understand nature at the microscopic level.It explains phenomena that classical physics could not account for.A deeper understanding of these non-classical phenomena (e.g., entanglement and quantum coherence) is not only of a foundational interest, but also paves the way towards potential technological advancements.Developments in the field of quantum information showed that such quantum features help us to perform operational tasks which would be impossible in the framework of classical physics.Therefore, these quantum phenomena can be viewed as resources which help us to go beyond the restrictions imposed by classical physics.Inspired by this, the mathematical framework of quantum resource theories was developed.Each resource theory singles out a particular quantum resource by setting the states which contain no resource as free states.Furthermore, the set of free operations are defined such that they do not generate resource states out of free states.An important problem in any resource theory is to understand how different resources transform into each other under the action of free operations.This problem of state transformation will be the main focus of this thesis.We will start by studying the notion of deterministic transformations, where one aims to achieve a target state without a chance of failure.We will explore the connection between deterministic transformations and the problem of quantification of resources.We will then go further and concentrate on transformations which allow for a probability of failure, and study the fundamental restrictions imposed on the achievable error and the probability of success.Finally we will explore the idea of catalytic transformations, where we allow for an ancillary quantum system which increases the transformation power while remaining invariant in the process.For the resource theory of entanglement, we will give a full solution for this problem when the target state is pure and the initial state is distillable.Finally, we discuss potential applications of catalysis to the problem of sending quantum information through noisy channels.
In this paper, we present a general numerical framework for both deterministic and probabilistic quantum state transformations, under locality constraints. For a given arbitrary bipartite initial state and a desired bipartite target state, we construct an optimized local quantum channel that transforms the initial state into the target state with high fidelity. To achieve this goal, local quantum channels are parametrized on a complex Stiefel manifold and optimized using gradient-based methods. We demonstrate that this approach significantly enhances entanglement distillation for weakly entangled states via two complementary strategies: optimized local state transformation and probabilistic local transformation. These results establish our method as a powerful and versatile tool for a broad class of quantum information processing tasks.