The focus is on two parties with Hilbert spaces of dimension d , i.e. 'qudits'. In the state space of these two possibly entangled qudits an analogue to the well-known tetrahedron with the four qubit Bell states at the vertices is presented. The simplex analogue to this magic tetrahedron includes mixed states. Each of these states appears to each of the two parties as the maximally mixed state. Some studies on these states are performed, and special elements of this set are identified. A large number of them are included in the chosen simplex which fits exactly into conditions needed for teleportation and other applications. Its rich symmetry—related to that of a classical phase space—helps to study entanglement, to construct witnesses and perform partial transpositions. This simplex has been explored in detail for d = 3. In this paper, the mathematical background and extensions to arbitrary dimensions are analysed.
Focus is on two parties with Hilbert spaces of dimension d, i.e. "qudits". In the state space of these two possibly entangled qudits an analogue to the well known tetrahedron with the four qubit Bell states at the vertices is presented. The simplex analogue to this magic tetrahedron includes mixed states. Each of these states appears to each of the two parties as the maximally mixed state. Some studies on these states are performed, and special elements of this set are identified. A large number of them is included in the chosen simplex which fits exactly into conditions needed for teleportation and other applications. Its rich symmetry - related to that of a classical phase space - helps to study entanglement, to construct witnesses and perform partial transpositions. This simplex has been explored in details for d=3. In this paper the mathematical background and extensions to arbitrary dimensions are analysed.
Achieving a genuine quantum advantage relies on two distinct non-classical resources that restrict efficient classical simulation: entanglement and magic (nonstabilizerness). We investigate the interplay between these resources by characterizing the Pareto frontiers of extreme magic at fixed entanglement for systems of two qutrits (
d=3) and two ququints (
d=5). Unlike the case of two qubits, the Schmidt spectrum for two qutrits features two independent entanglement parameters, resulting in two-dimensional Pareto surfaces. For the lower frontier, we recast the minimal magic as a compact function of concurrence and negativity, with a maximal value of
ln2. For the upper frontier, we determine the maximal stabilizer Rényi entropy to be
M2=ln(81/17)≈1.561, which tightens the previous theoretical bound of
ln5≈1.609 and improves on earlier numerical estimates. The maximum magic is achieved at eighteen distinct maxima categorized into three families of six permutation-equivalent spectra. We provide analytical expressions for the maximal magic in the neighborhood of each maximum and for the corresponding maximally magical states which turn out to be Weyl-Heisenberg-covariant fiducial states for mutually unbiased bases. Finally, numerical analysis of two ququints (
d=5) reveals six permutation-inequivalent maxima with a peak magic value of
M2=ln(625/49)≈2.546. Based on these findings, we conjecture that the maximal magic for a bipartite system of two qudits with prime dimension
d is given by
ln[d4/(2d2−1)], which reproduces the previously known value for qubits, as well as the values derived here for qutrits and ququints.
We construct a simplex for multipartite qubit states of even number
n of qubits, which has the same geometry concerning separability, mixedness, kind of entanglement, amount of entanglement, and nonlocality as the bipartite qubit states. We derive the entanglement of the class of states which can be described by only three real parameters with the help of a multipartite measure for all discrete systems. We prove that the bounds on this measure are optimal for the whole class of states and that it reveals that the states possess only
n-partite entanglement and not, e.g., bipartite entanglement. We then show that this
n-partite entanglement can be increased by stochastic local operations and classical communication to the purest maximal entangled states. However, pure
n-partite entanglement cannot be distilled; consequently all entangled states in the simplex are
n-partite bound entangled. We study also Bell inequalities and find the same geometry as for bipartite qubits. Moreover, we show how the (hidden) nonlocality for all
n-partite bound entangled states can be revealed.
We characterize bound entanglement in bipartite qutrit systems by combining the Positive Partial Transpose (PPT) criterion with a structural, non-completely positive map. Focusing on a comprehensive three-parameter bipartite qutrit family
ρa,b,c, we present an exact analytical and geometric partitioning of the physical state space simplex. We derive the closed-form quadratic boundary governing the leaf-like PPT region
(a2+ab+b2−b≤0) over the domain
0<a≤31, and identify the exact linear threshold
(a>c) that characterizes the PPT bound entangled sub-region. This family strictly generalizes the well-known Horodecki bound entangled states, which appear as a single slice of the three-dimensional leaf. Applying the structural map to this decomposition, we show that it certifies bound entanglement throughout the entire threshold region, accounting for
14.76% of the total PPT leaf area.