Neven et al. have explored an unexpected alliance between the mathematical insights of Sir Isaac Newton and René Descartes which culminates in the reduction of the Positive Partial Transpose (PPT) criterion to an equivalent hierarchy of entanglement tests based on the moments of the partial transpose. By repurposing these classical results in the context of modern quantum theory, they illuminate new pathways for entanglement verification. Here, we expand on this work by providing a closed form for the inequalities defining these entanglement tests and producing an equivalent set of graph theoretic conditions on the weighted graph induced by the partial transpose.
Neven et al. have explored an unexpected alliance between the mathematical insights of Sir Isaac Newton and René Descartes which culminates in the reduction of the Positive Partial Transpose (PPT) criterion to an equivalent hierarchy of entanglement tests based on the moments of the partial transpose. By repurposing these classical results in the context of modern quantum theory, they illuminate new pathways for entanglement verification. Here, we expand on this work by providing a closed form for the inequalities defining these entanglement tests and producing an equivalent set of graph theoretic conditions on the weighted graph induced by the partial transpose.
The p3-PPT criterion is an experimentally viable relaxation of the well-known positive partial transposition (PPT) criterion for the certification of quantum entanglement. Recently, it has been generalized to various families of entanglement criteria based on the PT moments pk=Tr[(ρΓ)k], where ρΓ denotes the partially transposed density matrix of a quantum state ρ. While most of these generalizations are strictly more powerful than the p3-PPT criterion, their m-th level versions usually rely on the availability of pk for all moment orders k≤m. Here, we show that one can alternatively compare any three PT moments of orders k<l<m, which can significantly reduce experimental overheads. More precisely, we show that any state satisfying $p_l>p_k^xp_m^{1-x} $ must be entangled, where x=(m−l)/(m−k). Using the example of locally depolarized GHZ states, we identify the most promising versions of these three-moment criteria and compare their performance with a broad range of entanglement criteria. In the case of globally depolarized stabilizer states, we prove that having access to pk for k≤5 is sufficient to reproduce the full PPT criterion. More generally, we show that the Stieltjes-m criterion is as powerful as the PPT criterion whenever ρΓ has no more than (m+1)/2 distinct eigenvalues. Finally, we introduce a notion of quantum weight enumerators that capture the decay of pk under local white noise for arbitrary quantum states and illustrate this concept for an AME state. Our results contribute to the growing body of literature on higher-moment PPT relaxations and modern applications of weight enumerators in quantum error correction and information theory.
Equivalence between positive partial transpose (PPT) entanglement and bound entanglement is a long-standing open problem in quantum information theory. Limited progress has been made so far, even on the seemingly simple case of the bound entanglement of Werner states. The primary challenge is to give a concise mathematical representation of undistillability. To this end, we propose a decomposition of the N-undistillability verification into log(N) repeated steps of 1-undistillability verification. For Werner states' N-undistillability verification, a parameter interval for N-undistillability is given, which is independent of the dimensionality of Werner states. Equivalent forms of inequalities for both rank 1 and 2 matrices are presented, before transforming the 2-undistillability case into a matrix analysis problem. A new perspective is also attempted by seeing it as a nonconvex multivariable function, proving its critical points and conjecturing Hessian positivity, which would make them local minimums.
We propose a Partial Lorentz Transformation (PLT) test for detecting entanglement in a two qubit system. One can expand the density matrix of a two qubit system in terms of a tensor product of (I,σ). The matrix A of the coefficients that appears in such an expansion can be "squared" to form a 4×4 matrix B. It can be shown that the eigenvalues λ0,λ1,λ2,λ3 of B are positive. With the choice of λ0 as the dominant eigenvalue, the separable states satisfy λ1+λ2+λ3≤λ0. Violation of this inequality is a test of entanglement. Thus, this condition is both necessary and sufficient and serves as an alternative to the celebrated Positive Partial Transpose (PPT) test for entanglement detection. We illustrate this test by considering some explicit examples.
Moment-based relaxations of the positive partial transpose (PPT) criterion have been recently introduced, as a hierarchy of entanglement criteria involving only experimentally accessible quantities of a given bipartite state. The goal of this work is to study their typical detection performance on high-dimensional bipartite systems. Concretely, we investigate whether random bipartite mixed states on Cd⊗Cd, obtained as the marginal over an environment Cs of a uniformly distributed pure state, generically satisfy or violate them. For each fixed level m∈N in this hierarchy of moment-based PPT criteria, we are able to identify a threshold environment dimension s=λmd2 at which the behavior of the associated random state switches from violating to satisfying it, with probability going to 1 as d grows. The proof combines combinatorics of permutations techniques to estimate the average value of moments of partially transposed random states and concentration of measure arguments to bound the probability of deviating from such average, when the underlying local dimension d is large. We additionally need tools from the theory of Hankel determinant evaluation via orthogonal polynomials.
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.
We study entanglement distillation under operations that remain completely positive under partial-transpose conjugation, a.k.a. PPT channels, in the standard quantum Shannon theory regime of asymptotically vanishing but non-zero error. Our main results are: (a) a regularised formula for the PPT distillable entanglement in terms of a measured-relative-entropy-like quantity; and (b) two different single-letter converses, one based on operator quadratic forms and the other on Hirschman's strengthening of the Hadamard three-line theorem. As an immediate application of (b), we show that the PPT distillable entanglement can be strictly smaller than the regularised Rains bound, thereby resolving an open problem in the theory of entanglement manipulation [Regula et al., NJP 21:103017, 2019]. A gap appears already for 3×3 Werner states: at antisymmetric weight 25/26, a certified upper bound of 0.62107 ebits lies below the (regularised) Rains bound, which equals 2625log25−log23≈0.64766 ebits. On a different note, (a) implies a faithful lower bound on the PPT distillable entanglement in terms of the entanglement negativity, which provides a quantitative counterpart to the qualitatively known fact that any NPT state is PPT distillable [Eggeling et al., PRL 87:257902, 2001].