We introduce a hierarchy of entanglement measures Ek based on k-symmetric PPT extensions. Each Ek, defined via a minimal eigenvalue shift and computed by semidefinite programming, is faithful, convex, and monotone under free operations. The hierarchy strictly refines PPT-robustness at k = 1, detects bound entanglement at k = 2, and converges exactly to the separability measure as k -> infinity. Numerical experiments on Horodecki, Werner, UPB, and random states demonstrate practical scalability. Our framework unifies computational efficiency with operational fidelity in a single tunable family -- a combination previously believed to be fundamentally incompatible in entanglement quantification. It supplies, for the first time, a systematically improvable resource-theoretic yardstick that accounts for all entangled states, including the bound entangled ones that have long resisted quantitative treatment.
We introduce a hierarchy of entanglement measures Ek based on k-symmetric PPT extensions. Each Ek, defined via a minimal eigenvalue shift and computed by semidefinite programming, is faithful, convex, and monotone under free operations. The hierarchy strictly refines PPT-robustness at k = 1, detects bound entanglement at k = 2, and converges exactly to the separability measure as k -> infinity. Numerical experiments on Horodecki, Werner, UPB, and random states demonstrate practical scalability. Our framework unifies computational efficiency with operational fidelity in a single tunable family -- a combination previously believed to be fundamentally incompatible in entanglement quantification. It supplies, for the first time, a systematically improvable resource-theoretic yardstick that accounts for all entangled states, including the bound entangled ones that have long resisted quantitative treatment.
Abstract We introduce a framework unifying the mathematical characterisation of different measures of general quantum resources and allowing for a systematic way to define a variety of faithful quantifiers for any given convex quantum resource theory. The approach allows us to describe many commonly used measures such as matrix norm-based quantifiers, robustness measures, convex roof-based measures, and witness-based quantifiers together in a common formalism based on the convex geometry of the underlying sets of resource-free states. We establish easily verifiable criteria for a measure to possess desirable properties such as faithfulness and strong monotonicity under relevant free operations, and show that many quantifiers obtained in this framework indeed satisfy them for any considered quantum resource. We derive various bounds and relations between the measures, generalising and providing significantly simplified proofs of results found in the resource theories of quantum entanglement and coherence. We also prove that the quantification of resources in this framework simplifies for pure states, allowing us to obtain more easily computable forms of the considered measures, and show that many of them are in fact equal on pure states. Further, we investigate the dual formulation of resource quantifiers, which provide a characterisation of the sets of resource witnesses. We present an explicit application of the results to the resource theories of multi-level coherence, entanglement of Schmidt number k , multipartite entanglement, as well as magic states, providing insight into the quantification of the four resources by establishing novel quantitative relations and introducing new quantifiers, such as a measure of entanglement of Schmidt number k which generalises the convex roof–extended negativity, a measure of k -coherence which generalises the ℓ 1 norm of coherence, and a hierarchy of norm-based quantifiers of k -partite entanglement generalising the greatest cross norm.
A k-entanglement measure complying with the many-body resource theory and its evaluation
k-entanglement provides a hierarchical characterization of multipartite correlations according to their entanglement structure, ranging from multipartite entanglement (k=n) to genuine multipartite entanglement (k=2). Its quantitative treatment for mixed states faces three related obstacles: the standard axioms for such quantifications does not incorporate the requirements from many-body resource theory, as it
does not regulate how to quantify the entanglement assigned to part systems of the same multipartite state and how to describe the rule that the entanglement possessed by a part system never exceeds the entanglement possed by whole system;
no explicit measures are known that satisfy the above requirements from many-body resource theory; and the extreme difficulty in numerical evaluation at mixed states of known quantifications. In this work, we first formulate an axiomatic framework for \emph{true} k-entanglement measures that obeys the many-body resource theory, by supplementing usual faithfulness and LOCC monotonicity with two new ones: unification condition and hierarchy condition. Then, for every 2≤k≤n, we construct a witness-based true k-entanglement measure Ew(k,n) without the convex-roof extension, which is valid for both finite- and infinite-dimensional n-partite systems. Finally, we derive a method of constructing finite-dimensional k-entanglement witnesses from positive operator and k-separable-state optimizations, design an evaluation scheme of Ew(k,n) based on pre-established database of k-entanglement witnesses sampled by positive operators. To check the feasibility of this computing scheme, we develop a software tool, which implements the computation of the approximation of Ew(k,n) for systems up to four qubits. Numerical tests show that the software
is sufficiently accurate and reliable.
In the realm of quantum information theory, the detection and quantification of quantum entanglement stand as paramount tasks. The relative entropy of entanglement (REE) serves as a prominent measure of entanglement, with extensive applications spanning numerous related fields. The positive partial transpose (PPT) criterion, while providing an efficient method for the computation of REE, unfortunately, falls short when dealing with bound entanglement. In this study, we propose a method termed “pure bosonic extension” to enhance the practicability of k -bosonic extensions, which approximates the set of separable states from the “outside”, through a hierarchical structure. It enables efficient characterization of the set of k -bosonic extendible states, facilitating the derivation of accurate lower bounds for REE. Compared to the semi-definite programming (SDP) approach, such as the symmetric/bosonic extension function in QETLAB, our algorithm supports much larger dimensions and higher values of extension k . Published by the American Physical Society 2024
Quantifying quantum entanglement is a pivotal challenge in quantum information science, particularly for high-dimensional systems, due to its computational complexity. This thesis extends the geometric measure of entanglement (GME) to introduce and investigate a suite of GME-based entanglement monotones tailored for diverse quantum contexts, including pure states, subspaces, and mixed states. These monotones are applicable to both bipartite and multipartite systems, offering a unified framework for characterizing entanglement across various scenarios. Notably, the proposed monotones are adept at identifying entanglement with varying entanglement dimensionalities, making them particularly effective for detecting high-dimensional entanglement. To support practical computation, we develop a non-convex optimization framework that yields accurate upper bounds, complemented by semidefinite programming techniques to establish robust lower bounds. Together, these approaches provide a consistent and efficient computational methodology. This work advances both the theoretical understanding and algorithmic tools for entanglement quantification, contributing to the study of complex quantum correlations in entangled systems.
Quantifying the minimum entanglement needed to prepare quantum states and implement quantum processes is a key challenge in quantum information theory. In this Letter, we develop computable and faithful lower bounds on the entanglement cost under quantum operations that completely preserve the positivity of partial transpose (PPT operations), by introducing the generalized divergence of k-negativity, a generalization of logarithmic negativity. Our bounds are efficiently computable via semidefinite programming and provide nontrivial values for all states that are non-PPT (NPT), establishing their faithfulness for the resource theory of NPT entanglement. Notably, we find and affirm the irreversibility of asymptotic entanglement manipulation under PPT operations for full-rank entangled states. Furthermore, we extend our methodology to derive lower bounds on the entanglement cost of both point-to-point and bipartite quantum channels. Our bound demonstrates improvements over previously known computable bounds for a wide range of quantum states and channels. These findings push the boundaries of understanding the structure of entanglement and the fundamental limits of entanglement manipulation.
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.