Abstract In the study of distributed quantum information processing, it is crucial to minimize the entanglement consumption by optimizing local operations and classical communications (LOCC). We develop a framework based on algebraic geometry to systematically simplify the optimization over separable (SEP) channels, providing a good approximation of LOCC. As a first application, we generalize previous results on the one-shot entanglement cost of non-local operations in a unified way. Via the generalization, we also resolve an open problem posed by Yu et al. regarding the entanglement cost of local state discrimination. As a second application, we strengthen the Doherty--Parrilo--Spedalieri hierarchy and numerically determine the trade-off between the entanglement cost and the success probability of implementing various non-local operations under separable channels---such as entanglement distillation, non-local unitary channels, measurements, and state verification.
Abstract In the study of distributed quantum information processing, it is crucial to minimize the entanglement consumption by optimizing local operations and classical communications (LOCC). We develop a framework based on algebraic geometry to systematically simplify the optimization over separable (SEP) channels, providing a good approximation of LOCC. As a first application, we generalize previous results on the one-shot entanglement cost of non-local operations in a unified way. Via the generalization, we also resolve an open problem posed by Yu et al. regarding the entanglement cost of local state discrimination. As a second application, we strengthen the Doherty--Parrilo--Spedalieri hierarchy and numerically determine the trade-off between the entanglement cost and the success probability of implementing various non-local operations under separable channels---such as entanglement distillation, non-local unitary channels, measurements, and state verification.
In the study of distributed quantum information processing, it is a fundamental problem to optimize local operations in the implementation of non-local quantum operations assisted by limited entanglement. We develop an algebraic–geometric framework that systematically simplifies optimization over separable (SEP) channels—widely used as approximations of local operations—and strengthens the Doherty–Parrilo–Spedalieri (DPS) hierarchy for solving such problems. We apply this framework to computing maximum success probability for exactly implementing a broad range of different non-local operations under SEP channels. First, we present a unified generalization of previous analytical results on the entanglement cost. Via the generalization, we resolve an open problem posed by Yu et al. regarding the entanglement cost of local state discrimination. Second, we numerically determine the trade-off between the strength of entanglement and the success probability of implementing various operations—such as entanglement distillation, non-local unitary channels, measurements, and state verification.
In this thesis we study the subset of quantum operations that can be implemented using only local quantum operations and classical communication (LOCC). This restricted paradigm serves as a tool to study not only quantum correlations and other nonlocal quantum effects, but also resource transformations such as channel capacities. \n \nThe mathematical structure of LOCC is complex and difficult to characterize. In the first part of this thesis we provide a precise description of LOCC and related operational classes in terms of quantum instruments. Our formalism captures both finite round protocols as well as those that utilize an unbounded number of communication rounds. This perspective allows us to measure the distance between two LOCC instruments and hence discuss the closure of LOCC in a rigorous way. While the set of LOCC is not topologically closed, we show that the operations that can be implemented using some fixed number rounds of communication constitute a compact subset of all quantum operations. We also exhibit a subset of LOCC measurements that is closed. Additionally we establish the existence of an open ball around the completely depolarizing map consisting entirely of LOCC implementable maps. \n \nIn the second part of this thesis we focus on the task of discriminating states from some known set S by LOCC. Building on the work in the paper "Quantum nonlocality without entanglement", we provide a framework for lower bounding the error probability of any LOCC protocol aiming at discriminating the states from S. We apply our framework to an orthonormal product basis known as the domino states. This gives an alternative and simplified bound quantifying how well these states can be discriminated using LOCC. We generalize this result for similar bases in larger dimensions, as well as the "rotated" domino states, resolving a long-standing open question. These results give new examples of quantitative gaps between the classes of separable and LOCC operations. \n \nIn the last part of this thesis, we ask what differentiates separable from LOCC operations. Both of these classes play a key role in the study of entanglement. Separable operations are known to be strictly more powerful than LOCC ones, but no simple explanation of this phenomenon is known. We show that, in the case of bipartite von Neumann measurements, the ability to interpolate is an operational principle that separates LOCC and separable operations.
The unique features of entanglement and non-locality in quantum systems, where there are pairs of bipartite states perfectly distinguishable by general entangled measure-ments yet indistinguishable by local operations and classical communication, hold significant importance in quantum entanglement theory, distributed quantum information processing, and quantum data hiding. This paper delves into the entanglement cost for discriminating two quantum states, employing positive operator-valued measures (POVMs) with positive partial trans-pose (PPT) to achieve optimal success probability through general entangled measurements. First, we introduce an efficiently com-putable quantity called the spectral PPT-distance of a POVM to quantify the localness of a general measurement. We show that it can be a lower bound for the entanglement cost of optimal discrimination by PPT POVMs. Second, we establish an upper bound on the entanglement cost of optimal discrimination by PPT POVMs for any pair of states. Leveraging this result, we show that a pure state can be optimally discriminated against any other state with the assistance of a single Bell state. This study advances our understanding of the pivotal role played by entanglement in quantum state discrimination, serving as a crucial element in unlocking quantum data hiding against locally constrained measurements.
The unique features of entanglement and non-locality in quantum systems, where there are pairs of bipartite states perfectly distinguishable by general entangled measurements yet indistinguishable by local operations and classical communication, hold significant importance in quantum entanglement theory, distributed quantum information processing, and quantum data hiding. This paper delves into the entanglement cost for discriminating two bipartite quantum states, employing positive operator-valued measures (POVMs) with positive partial transpose (PPT) to achieve optimal success probability through general entangled measurements. First, we introduce an efficiently computable quantity called the spectral PPT-distance of a POVM to quantify the localness of a general measurement. We show that it can be a lower bound for the entanglement cost of optimal discrimination by PPT POVMs. Second, we establish an upper bound on the entanglement cost of optimal discrimination by PPT POVMs for any pair of states. Leveraging this result, we show that a pure state can be optimally discriminated against any other state with the assistance of a single Bell state. This study advances our understanding of the pivotal role played by entanglement in quantum state discrimination, serving as a crucial element in unlocking quantum data hiding against locally constrained measurements.
The unique features of entanglement and non-locality in quantum systems, where there are pairs of bipartite states perfectly distinguishable by general entangled measurements yet indistinguishable by local operations and classical communication, hold significant importance in quantum entanglement theory, distributed quantum information processing, and quantum data hiding. This paper delves into the entanglement cost for discriminating two bipartite quantum states, employing positive operator-valued measures (POVMs) with positive partial transpose (PPT) to achieve optimal success probability through general entangled measurements. We first introduce two quantities called the spectral PPT-distance and relative spectral PPT-distance of a POVM to quantify the localness of a general measurement. We show these quantities are related to the entanglement cost of optimal discrimination by PPT POVMs. Following this, we establish bounds and develop SDP hierarchies to estimate the entanglement cost of optimal discrimination by PPT POVMs for any pair of states. Leveraging these results, we show that a pure state can be optimally discriminated against any other state with the assistance of a single Bell state. This study advances our understanding of the pivotal role played by entanglement in quantum state discrimination, serving as a crucial element in unlocking quantum data hiding against locally constrained measurements.
Entanglement and nonlocality play a fundamental role in quantum computing. To understand the interplay between these phenomena, researchers have considered the model of local operations and classical communication, or LOCC for short, which is a restricted subset of all possible operations that can be performed on a multipartite quantum system. The task of distinguishing states from a set that is known a priori to all parties is one of the most basic problems among those used to test the power of LOCC protocols, and it has direct applications to quantum data hiding, secret sharing and quantum channel capacity. The focus of this thesis is on state distinguishability problems for classes of quantum operations that are more powerful than LOCC, yet more restricted than global operations, namely the classes of separable and positive-partial-transpose (PPT) measurements. We \nbuild a framework based on convex optimization (on cone programming, in particular) to study such problems. Compared to previous approaches to the problem, the method described in this thesis provides precise numerical bounds and quantitative analytic results. By combining the duality theory of cone programming with the channel-state duality, we also establish a novel connection between the state distinguishability problem and the study of positive linear maps, which is a topic of independent interest in quantum information theory. \n \nWe apply our framework to several questions that were left open in previous works regarding the distinguishability of maximally entangled states and unextendable product sets. First, we exhibit small sets of orthogonal maximally entangled states in that are not perfectly distinguishable by LOCC. As a consequence of this, we show a gap between the power of PPT and separable measurements for the task of distinguishing sets consisting only of maximally entangled states. Furthermore, we prove tight bounds on the entanglement cost that is necessary to distinguish any sets of Bell states, thus showing that quantum teleportation is optimal for this task. Finally, we provide an easily checkable characterization of when an unextendable product set is perfectly discriminated by separable measurements, along with the first known example of an unextendable product set that cannot be perfectly discriminated by separable measurements.