We investigate the stability of quantum positivity cones under nonlinear operator means. Specifically, we examine how Kubo--Ando means interact with the separable, positive partial transpose (PPT), and Schmidt-number cones. By analyzing the curvature of operator monotone functions at the identity, we give a strict rigidity phenomenon: weighted arithmetic means are the only Kubo--Ando means that preserve the separable cone in all dimensions. We show that the strictly positive curvature of any non-arithmetic mean explicitly forces a violation of the PPT condition, even in the foundational two-qubit setting, and can strictly increase the Schmidt number of the resulting operator. Finally, using the Choi--Jamiołkowski correspondence, we translate these geometric obstructions to the map-theoretic setting, concluding that convex mixing is the uniquely permissible Kubo--Ando operation for preserving entanglement-breaking quantum channels.
We investigate the stability of quantum positivity cones under nonlinear operator means. Specifically, we examine how Kubo--Ando means interact with the separable, positive partial transpose (PPT), and Schmidt-number cones. By analyzing the curvature of operator monotone functions at the identity, we give a strict rigidity phenomenon: weighted arithmetic means are the only Kubo--Ando means that preserve the separable cone in all dimensions. We show that the strictly positive curvature of any non-arithmetic mean explicitly forces a violation of the PPT condition, even in the foundational two-qubit setting, and can strictly increase the Schmidt number of the resulting operator. Finally, using the Choi--Jamiołkowski correspondence, we translate these geometric obstructions to the map-theoretic setting, concluding that convex mixing is the uniquely permissible Kubo--Ando operation for preserving entanglement-breaking quantum channels.
A Hierarchy of Entanglement Cones via Rank-Constrained C∗-Convex Hulls
This paper systematically investigates the geometry of fundamental quantum cones, the separable cone (P+) and the Positive Partial Transpose (PPT) cone (PPPT), under generalized non-commutative convexity. We demonstrate a sharp stability dichotomy analyzing C∗-convex hulls of these cones: while P+ remains stable under local C∗-convex combinations, its global C∗-convex hull collapses entirely to the cone of all positive semidefinite matrices, MCL(P+)=P0. To gain finer control and classify intermediate structures, we introduce the concept of ``k-C∗-convexity'', by using the operator Schmidt rank of C∗-coefficients. This constraint defines a new hierarchy of nested intermediate cones, MCLk(⋅). We prove that this hierarchy precisely recovers the known Schmidt number cones for the separable case, establishing a generalized convexity characterization: MCLk(P+)=Tk. Applied to the PPT cone, this framework generates a family of conjectured non-trivial intermediate cones, CPPT,k.
Entanglement swapping is a fundamental operation in quantum repeaters for establishing entanglement between distant parties. The positive partial transpose (PPT) squared conjecture asks whether two PPT entangled links can generate terminal entanglement through entanglement swapping, or equivalently, whether the composition of two PPT maps is always entanglement breaking. Motivated by this conjecture, we investigate the map-composition problem beyond the PPT setting. For qutrit completely positive (CP) maps, we prove that the composition of any CP map whose Choi matrix is 1-undistillable with any CP map whose Choi matrix has Schmidt number at most two is entanglement breaking in either order. Moreover, we show that the cone of 1-undistillable CP maps is exactly the largest qutrit cone of CP maps whose composition with every CP map whose Choi matrix has Schmidt number at most two is entanglement breaking in both orders. Finally, although map composition captures only the standard maximally entangled outcome in entanglement swapping, we prove that any 1-undistillable two-qutrit state and any state of Schmidt number at most two cannot generate terminal entanglement under an arbitrary selective measurement on the intermediate systems.
Positive but not completely positive maps provide one of the most direct ways to detect entanglement beyond the positive-partial-transpose (PPT) criterion. We introduce and analyze an exactly solvable two-parameter family of sparse bistochastic positive maps on qutrits, in which two coherence channels are independently tuned by parameters w and z. The sparse structure makes the full phase diagram analytic: positivity holds exactly on the square 0≤w,z≤2/3, complete positivity on the smaller square 0≤w,z≤1/3, and decomposability is lost precisely outside a quarter circle in the corner w,z≥1/3. The indecomposable region is certified by explicit PPT entangled state adapted to the same witness geometry. At the endpoint W∗=W(2/3,2/3) we construct a four-parameter family of PPT edge states of rank type (5,5), derive their analytic detection region, and show that the corresponding rays are exposed faces of the PPT cone. Finally, although W∗ is not optimal, we give an explicit optimal refinement whose detection region on this family is strictly larger. The result is an analytically tractable qutrit setting in which positivity, indecomposability, PPT entanglement, optimality, and exposed convex geometry can be studied in a single framework.
The optimal error exponents of binary composite i.i.d.~state discrimination are trivially bounded by the worst-case pairwise exponents of discriminating individual elements of the sets representing the two hypotheses, and in the finite-dimensional classical case, these bounds in fact give exact single-copy expressions for the error exponents. In contrast, in the non-commutative case, the optimal exponents are only known to be expressible in terms of regularized divergences, resulting in formulas that, while conceptually relevant, are practically not very useful. In this paper, we develop further an approach initiated in [Mosonyi, Szilágyi, Weiner, IEEE Trans.~Inf.~Th.~68(2):1032--1067, 2022] to give improved single-copy bounds on the error exponents by comparing not only individual states from the two hypotheses, but also various unnormalized positive semi-definite operators associated to them. Here, we show a number of equivalent characterizations of such operators giving valid bounds, and show that in the commutative case, considering weighted geometric means of the states, and in the case of two states per hypothesis, considering weighted Kubo-Ando geometric means, are optimal for this approach. As a result, we give a new characterization of the weighted Kubo-Ando geometric means as the only 2-variable operator geometric means that are block additive, tensor multiplicative, continuous on monotone decreasing sequences, and satisfy the arithmetic-geometric mean inequality. We also extend our results to composite quantum channel discrimination, and show an analogous optimality property of the weighted Kubo-Ando geometric means of two quantum channels, a notion that seems to be new. We extend this concept to defining the notion of superoperator perspective function and establish some of its basic properties, which may be of independent interest.
We introduce and apply Hilbert's projective metric in the context of quantum information theory. The metric is induced by convex cones such as the sets of positive, separable or positive partial transpose operators. It provides bounds on measures for statistical distinguishability of quantum states and on the decrease of entanglement under protocols involving local quantum operations and classical communication or under other cone-preserving operations. The results are formulated in terms of general cones and base norms and lead to contractivity bounds for quantum channels, for instance, improving Ruskai's trace-norm contraction inequality. A new duality between distinguishability measures and base norms is provided. For two given pairs of quantum states we show that the contraction of Hilbert's projective metric is necessary and sufficient for the existence of a probabilistic quantum operation that maps one pair onto the other. Inequalities between Hilbert's projective metric and the Chernoff bound, the fidelity and various norms are proven.
The structure of cones of positive and k-positive maps acting on a finite-dimensional Hilbert space is investigated. Special emphasis is given to their duality relations to the sets of superpositive and k-superpositive maps. We characterize k-positive and k-superpositive maps with regard to their properties under taking compositions. A number of results obtained for maps are also rephrased for the corresponding cones of block positive, k-block positive, separable, and k-entangled operators due to the Jamiołkowski–Choi isomorphism. Generalizations to a situation where no such simple isomorphism is available are also made, employing the idea of mapping cones. As a side result to our discussion, we show that extreme entanglement witnesses, which are optimal, should be of special interest in entanglement studies.