We study bipartite entanglement from the convex geometry of quantum states. Taking the maximally mixed state as a reference point, we define a geometric entangled-space quantity $G(ρ)=[1-L(ρ)]/L(ρ)$ from the relative positions of the separable and quantum-state boundaries, and introduce a relative entanglement degree $Q(ρ)=[(1-p)/p]/G(ρ)$ by comparing it with the robustness relative to the maximally mixed state.
We apply this geometric construction to the Best Separable Approximation (BSA). For the optimal decomposition, we derive the general bound $(1-p)L_B/[p(1-L_B)]\leq p_0\leq Q(ρ)$ and show that the entangled component of the BSA has an entangled-space size no smaller than that of the original mixed state, namely $[1-L_R]/L_R\leq[1-L_B]/L_B$.
For two-qubit states, the BSA entangled component is a pure entangled state. Using the PPT criterion, its geometric parameter can be evaluated explicitly, giving the bound $(1-p)/(2p)\leq p_0\leq Q(ρ)$. These results provide a simple geometric description of the relation between BSA, robustness, and the entangled region of the quantum-state space.
We study bipartite entanglement from the convex geometry of quantum states. Taking the maximally mixed state as a reference point, we define a geometric entangled-space quantity G(ρ)=[1−L(ρ)]/L(ρ) from the relative positions of the separable and quantum-state boundaries, and introduce a relative entanglement degree Q(ρ)=[(1−p)/p]/G(ρ) by comparing it with the robustness relative to the maximally mixed state.
We apply this geometric construction to the Best Separable Approximation (BSA). For the optimal decomposition, we derive the general bound (1−p)LB/[p(1−LB)]≤p0≤Q(ρ) and show that the entangled component of the BSA has an entangled-space size no smaller than that of the original mixed state, namely [1−LR]/LR≤[1−LB]/LB.
For two-qubit states, the BSA entangled component is a pure entangled state. Using the PPT criterion, its geometric parameter can be evaluated explicitly, giving the bound (1−p)/(2p)≤p0≤Q(ρ). These results provide a simple geometric description of the relation between BSA, robustness, and the entangled region of the quantum-state space.
Quantum states are the key mathematical objects in quantum mechanics, and entanglement lies at the heart of the nascent fields of quantum information processing and computation. However, there has not been a general, necessary and sufficient, and operational separability condition to determine whether an arbitrary quantum state is entangled or separable. In this paper, we show that whether a quantum state is entangled or not is determined by a threshold within the quantum state. We first introduce the concept of finer and optimal separable states based on the properties of separable states in the role of higher-level witnesses. Then we show that any bipartite quantum state can be decomposed into a convex mixture of its optimal entangled state and its optimal separable state. Furthermore, we show that whether an arbitrary quantum state is entangled or separable, as well as positive partial transposition (PPT) or not, is determined by the robustness of its optimal entangled state to its optimal separable state with reference to a crucial threshold. Moreover, for an arbitrary quantum state, we provide operational algorithms to obtain its optimal entangled state, its optimal separable state, its best separable approximation (BSA) decomposition, and its best PPT approximation decomposition. And the open question of how to calculate the BSA in high-dimension systems is partially done.
We investigate the geometric structure of the set of spectra of bipartite absolute separable states (ASEPm,n) and absolute positive partial transpose states (APPTm,n), \emph{i.e.}, bipartite quantum states that remain separable or PPT, respectively, under all global unitary transformations. First, we establish general geometric properties of absolute convex sets of matrices, their spectra and extreme points. Regarding absolute separability, we present a permutation-symmetric reformulation of the absolute PPT criterion and use it to demonstrate that APPTm,n is a spectrahedron for all m≤n: in particular, all its faces are exposed. While ASEP2,n is a spectrahedron, too, for ASEPm,n we can in general only show that it is a semialgebraic set for all m,n≥3. Furthermore, we provide a complete characterization of the faces and extreme points of APPTm,n and demonstrate that the dimension of a face is determined by the rank of a certain matrix, with maximal proper faces having dimension mn−m−1.
In the quantitative setting, we provide a rigorous lower bound on the maximal attainable purity of APPTm,n via an inscribed polytope Pm,n and conjecture that the maximal purity of APPTm,n (along with its spectra) coincides with the polytope for arbitrary dimensions except when m=n=2. Additionally, we also provide a rigorous upper bound on the minimal von Neumann entropy of APPTm,n and demonstrate numerically that the minimum entropy eventually coincides with the polytope Pm,n as the local system dimension n increases. Finally, we show that the relative spectral volume of APPTm,n decays exponentially in n by a constant multiplicative factor of the relative volume of the inscribed polytope Pm,n.
Quantifying entanglement for multipartite quantum state is a crucial task in many aspects of quantum information theory. Among all the entanglement measures, relative entropy of entanglement ER is an important quantity due to its clear geometric meaning, easy compatibility with different system sizes, and various applications in many other related quantity calculations. Lower bounds of ER were previously found based on distance to the set of positive partial transpose states. We propose a method to calculate upper bounds of ER based on the convex hull approximation of the set of separable states. We apply our method to calculate ER for composite systems of various sizes, and compare with the previous known lower bounds, obtaining promising results. Our method adds a new tool for entanglement measure calculation and deepens our understanding for the structure of separable states.
Quantifying entanglement in composite systems is a fundamental challenge, yet exact results are available in only a few special cases. This is because hard optimization problems are routinely involved, such as finding the convex decomposition of a mixed state with the minimal average pure-state entanglement, the so-called convex roof. We show that under certain conditions such a problem becomes trivial. Precisely, we prove by a geometric argument that polynomial entanglement measures of degree 2 are independent of the choice of pure-state decomposition of a mixed state, when the latter has only one pure unentangled state in its range. This allows for the analytical evaluation of convex roof extended entanglement measures in classes of rank-2 states obeying such a condition. We give explicit examples for the square root of the three-tangle in three-qubit states, and we show that several representative classes of four-qubit pure states have marginals that enjoy this property.
Entanglement for pure bipartite states is most commonly quantified in a state-by-state manner to each pure state of a bipartite system a scalar quantity, such as the von Neumann entropy of a reduced density matrix. This provides a precise local characterization of how entangled a given state is. At the same time, this local description naturally invites a set of complementary, more global questions about the structure of the space of pure states: How abundant are the states with a given amount of entanglement within the full state space? Do the manifolds of constant entanglement exhibit distinct geometric regimes? These questions shift the focus from assigning an entanglement value to a single state to understanding the global organization and geometry of entanglement across the entire manifold of pure states.
In this work, we develop a geometric framework in which these questions become natural. We regard the projective Hilbert space of pure states, endowed with the Fubini-Study metric, as a Riemannian manifold and promote bipartite entanglement to a macroscopic functional on this manifold. Its level sets stratify the space of pure states into hypersurfaces of constant entanglement, and we define a geometric entanglement entropy as the log-volume of these hypersurfaces, weighted by the Fubini-Study gradient of entanglement. This quantity plays the role of a microcanonical entropy in entanglement space: it measures the degeneracy of a given entanglement value in the natural quantum geometry.
The framework is illustrated first in the simplest case of a single spin-1/2 and then for bipartite entanglement of spin systems, including a two-qubit example where explicit calculations can be carried out, along with a sketch of the extension to spin chains.
Randomized measurement protocols give experimental access to low-degree polynomial invariants of a quantum state rather than to the state itself, which raises the question of what the best entanglement criterion is that can be built from finitely many such invariants. In this work we answer this question through convex duality.
For each degree we introduce the cone of local unitary trace inequalities that are valid for separable bipartite states in all local dimensions, together with its one-matrix counterparts for Hermitian and positive semidefinite matrices. Each cone is the polar dual of a moment cone; we also determine the minimal combinatorial parametrization of the separable cone. The framework organizes the known moment criteria: the PPT criterion using the first three moments of the partial transposition is the image under an embedding of a Hankel determinant from the positive semidefinite cone to the bipartite cone, which we show to be an extremal ray, and we determine the first nontrivial separable cone completely, finding exactly four extremal rays.
The framework allows us to analyze the realignment criterion and its centered (or enhanced) variant. We prove that the even singular value moments of the (centered) realignment matrix are trace invariants, associated with explicit fixed-point-free involutions. Extracting the optimal trace-norm bound from m such moments thus becomes a dimension-free truncated moment problem, which we solve in closed form for m=2 and relax to a semidefinite program of size O(m) for arbitrary m. The dual solutions are polynomial minorants of x on [0,1], and two explicit families, binomial and L1-weighted, yield hierarchies of separability inequalities. Finally, we compare the criteria against one another and on PPT entangled states.