Genuine high-dimensional entanglement, i.e., the property of having a high Schmidt number, constitutes an instrumental resource in quantum communication, overcoming limitations of low-dimensional systems. States with a positive partial transpose (PPT) are generally considered weakly entangled, as they can never be distilled into pure entangled states. This naturally raises the question of whether high Schmidt numbers are possible for PPT states. Volume estimates suggest that optimal, i.e., linear, scaling in the local dimension should be possible, albeit without providing insight into the possible slope. We provide the first explicit construction of a family of PPT states that achieves linear scaling in the local dimension and we prove that random PPT states typically share this feature. Our construction also allows us to prove a recent conjecture of Chen et al. on the existence of PPT states whose Schmidt number increases by an arbitrarily large amount upon partial transposition. Finally, we link the Schmidt number to entangled sub-block matrices of a quantum state. We use this connection to prove that quantum states that are either (i) invariant under partial transposition on the smallest of their two subsystems, or (ii) absolutely PPT cannot have a maximal Schmidt number. Overall, our findings shed new light on some fundamental problems in entanglement theory.
Genuine high-dimensional entanglement, i.e., the property of having a high Schmidt number, constitutes an instrumental resource in quantum communication, overcoming limitations of low-dimensional systems. States with a positive partial transpose (PPT) are generally considered weakly entangled, as they can never be distilled into pure entangled states. This naturally raises the question of whether high Schmidt numbers are possible for PPT states. Volume estimates suggest that optimal, i.e., linear, scaling in the local dimension should be possible, albeit without providing insight into the possible slope. We provide the first explicit construction of a family of PPT states that achieves linear scaling in the local dimension and we prove that random PPT states typically share this feature. Our construction also allows us to prove a recent conjecture of Chen et al. on the existence of PPT states whose Schmidt number increases by an arbitrarily large amount upon partial transposition. Finally, we link the Schmidt number to entangled sub-block matrices of a quantum state. We use this connection to prove that quantum states that are either (i) invariant under partial transposition on the smallest of their two subsystems, or (ii) absolutely PPT cannot have a maximal Schmidt number. Overall, our findings shed new light on some fundamental problems in entanglement theory.
Entangled states with a positive partial transpose (so-called PPT states) are central to many interesting problems in quantum theory. On the one hand, they are considered to be weakly entangled, since no pure state entanglement can be distilled from them. On the other hand, it has been shown recently that some of these PPT states exhibit genuinely high-dimensional entanglement, i.e., they have a high Schmidt number. Here we investigate the ($d\ifmmode\times\else\texttimes\fi{}d$)-dimensional PPT states for $d\ensuremath{\ge}4$ discussed recently by E. Sindici and M. Piani [Phys. Rev. A 97, 032319 (2018)]. By generalizing their methods to the calculation of Schmidt numbers, we show that a linear d/2 scaling of its Schmidt number in the local dimension d can be attained.
A deep understanding of quantum entanglement is vital for advancing quantum technologies. The strength of entanglement can be quantified by counting the degrees of freedom that are entangled, which results in a quantity called the Schmidt number. A particular challenge is to identify the strength of entanglement in quantum states that remain positive under partial transpose (PPT), otherwise recognized as undistillable states. Finding PPT states with high Schmidt numbers has become a mathematical and computational challenge. In this Letter, we introduce efficient analytical tools for calculating the Schmidt number for a class of bipartite states called grid states. Our methods improve the best-known bounds for PPT states with high Schmidt numbers. Most notably, we construct a Schmidt number 3 PPT state in five-dimensional systems and a family of states with a Schmidt number of (d+1)/2 for odd d-dimensional systems, representing the best-known scaling of the Schmidt number in a local dimension. Additionally, these states possess intriguing geometrical properties, which we utilize to construct indecomposable entanglement witnesses.
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.
Entanglement is a central resource in quantum information science, yet its structure in high dimensions remains notoriously difficult to characterize. One of the few general results on high-dimensional entanglement is given by peel-off theorems, which relate the entanglement of a state to that of its lower-dimensional local projections. We build on this idea by introducing , the inverse process to peel-off projections, which provide a systematic way to construct higher-dimensional entangled states from lower-dimensional ones. This dual perspective leads to general bounds on how the Schmidt number can change under projections and extensions, and reveals mechanisms for generating bound entangled states of higher dimensionality. As a concrete application, we construct a positive-partial-transpose state of Schmidt number three in local dimensions 4 × 5 , the smallest system known to host such entanglement. We further extend this approach to identify an elegant family of with increasing Schmidt number, including explicit examples of a 7 × 7 state with Schmidt number 4 and a 9 × 9 state with Schmidt number 5, suggesting ( d + 1 ) / 2 scaling in odd local dimensions d × d . Taken together, our results provide a constructive toolkit for probing the scaling of bound entanglement in high dimensions.
The Schmidt number quantifies the dimensionality of entanglement in bipartite quantum states. We investigate when the spectrum of a state on Cd⊗Cd alone guarantees that its Schmidt number is not maximal. By deriving spectral bounds for (d−1)-block positive operators, we prove that LSp⊆ASNd−1 for p≤⌊d2/2⌋, where LSp denotes the set of states whose largest eigenvalue does not exceed the sum of their p smallest eigenvalues, and ASNd−1 denotes the set of states whose Schmidt number remains at most d−1 under all global unitaries. This also yields a simple sufficient condition involving only the largest eigenvalue. As an application, we show that states satisfying the reduction criterion under arbitrary global unitaries belong to ASNd−1 in every dimension. This subsumes the earlier result that absolutely positive partial transpose states belong to ASNd−1. The global unitary requirement is nevertheless essential: for every d≥3, we construct states of maximal Schmidt number satisfying the reduction criterion on both subsystems.
Summary: \nEntanglement is a strange feature contained in the quantum mechanical framework, first observed theoretically by Erwin Schrödinger and Albert Einstein. An inherent attribute of entanglement is non-locality, and in the continuation of this, several non-reductionist features, which are conceptually challenging and philosophically interesting. \n \nA considerable growth of the interest in quantum entanglement has occurred since the early 1990s, and much of the reason for this is that the tools of quantum optics and modern instrumentation have made it possible to measure and manipulate entangled quantum systems. Today, several applications of entanglement exist, including quantum computers and several areas in quantum communications. \n \nA quantum state is either separable or entangled, it cannot be none or both. A separable state do not exhibit the non-local features found in entangled states, and are in this sense comparable to classic states. An important issue here is the separability problem, which is the problem of determining whether a given state is entangled or not. \n \nWe study these quantum states as Hermitian matrices with positive eigenvalues in convex spaces. The PPT criterion is based on the partial transposition operation, which essentially transposes one of the two subsystems in the quantum system. For some states the result is another quantum (positive) state, and these are PPT (Positive Partial Transpose) states, while others that gain negative eigenvalues are called NPT states. \n \nFor systems of dimensions 2 × 2 and 2 × 3, the PPT states are identical to the separable states, and the NPT states are identical to the entangled states, so the PPT criterion is always conclusive. But for larger dimensions, such as for instance the 3 × 3 system, there always exist PPT states that are entangled, so for this case the PPT criterion can only be used in a contrapositive way. Since entangled PPT states in this sense seem to have great significance, their characteristics are worth studying. \nAs mentioned, the PPT criterion uses the transposition map on one of the two subsystems of a composite quantum system. Transposition is an example of a positive map, in that it maps all quantum states into quantum states. It is possible to use positive maps to formulate a much more general criterion for the separability problem.There is a link (an isomorphism) between positive maps and a subset of Hermitian operators called entanglement witnesses. \n \nThe so-called Horodecki criterion states that a quantum state is separable if and only if the expectation value of all entanglement witnesses in the state are non-negative. If the expectation value of one (or possibly several) entanglement witnesses in a given quantum state is negative, then the state is always entangled. This criterion is both necessary and sufficient, but demands that we can characterize the entire set of entanglement witnesses. \n \nThe approach in this thesis is through geometry. Of fundamental importance is the convex nature of the sets in question. The knowledge and understanding of the extremal points in these sets is very relevant, since they describe the sets completely. Undertaking completely analytic studies of the geometry at hand is in general very hard, especially as the dimension of the systems grow. The development and use of numerical tools is therefore essential.