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 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.
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.
Quantum entanglement lies at the heart of quantum information science, yet its reliable detection in high-dimensional or noisy systems remains a fundamental computational challenge. Semidefinite programming (SDP) hierarchies, such as the Doherty-Parrilo-Spedalieri (DPS) and Extension (EXT) hierarchies, offer complete methods for entanglement detection, but it is well known that their practical use is limited by exponential growth in problem size if implemented naively. We make three contributions. First, we introduce a new SDP hierarchy, PST, that is sandwiched between EXT and DP -- offering a tighter approximation to the set of separable states than EXT, while incurring significantly lower computational overhead than DPS. Second, we explicitly construct compact, polynomially-scalable descriptions of EXT and PST using partition mappings and operators. These descriptions in turn yield formulations that satisfy desirable properties such as the Slater condition and are well-suited to both first-order methods (FOMs) and interior-point methods (IPMs). Third, we design a suite of entanglement detection algorithms: three FOMs (Frank-Wolfe, projected gradient, and fast projected gradient) based on a least-squares formulation, and a custom primal-dual IPM based on a conic programming formulation. These methods are numerically stable and capable of producing entanglement witnesses or proximity measures, even in cases where states lie near the boundary of separability. Numerical experiments on benchmark quantum states demonstrate that our algorithms improve the ability to solve deeper levels of the SDP hierarchy.
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.
Quantum channel discrimination is a fundamental task in quantum information processing. In the one-shot regime, discrimination between two candidate channels is characterized by the diamond norm. Beyond this basic setting, however, many scenarios in distributed quantum information processing remain unresolved, motivating notions of distinguishability that capture the power of the available resources. In this work, we formulate a theory of testers for bipartite channel discrimination, leading to the concept of the entanglement cost of bipartite channel discrimination: the minimum Schmidt rank k of a shared maximally entangled state required for local protocols to achieve the globally optimal success probability. We introduce k-injectable testers as a tester-based description of entanglement-assisted local discrimination and, in particular, study the class of k-injectable positive-partial-transpose (PPT) testers, which constitutes a numerically tractable relaxation of the practically relevant class of LOCC testers.
For every k, we derive a semidefinite program (SDP) for the optimal success probability, which in turn yields an efficiently computable one-shot PPT entanglement cost. To render these optimization problems numerically feasible, we prove a symmetry-reduction principle for covariant channel pairs, thereby reducing the effective dimension of the associated SDPs. Finally, by dualizing the SDP, we derive bounds on the composite channel-discrimination problem and illustrate our framework with proof-of-principle examples based on the depolarizing channel, the depolarized SWAP channel, and the Werner--Holevo channels.
Quantum resource theory under different classes of quantum operations advances multiperspective understandings of inherent quantum-mechanical properties, such as quantum coherence and quantum entanglement. We establish hierarchies of different operations for manipulating coherence and entanglement in distributed settings, where at least one of the two spatially separated parties are restricted from generating coherence. In these settings, we introduce new classes of operations and also characterize those maximal, i.e. , the resource-non-generating operations, progressing beyond existing studies on incoherent versions of local operations and classical communication and those of separable operations. The maximal operations admit a semidefinite-programming formulation useful for numerical algorithms, whereas the existing operations not. To establish the hierarchies, we prove a sequence of inclusion relations among the operations by clarifying tasks where separation of the operations appears. We also demonstrate an asymptotically non-surviving separation of the operations in the hierarchy in terms of performance of the task of assisted coherence distillation, where a separation in a one-shot scenario vanishes in the asymptotic limit. Our results serve as fundamental analytical and numerical tools to investigate interplay between coherence and entanglement under different operations in the resource theory.