We investigate the relationship between partial traces and their dilations for general complex matrices, focusing on two main aspects: the existence of (joint) dilations and norm inequalities relating partial traces and their dilations. Throughout our analysis, we pay particular attention to rank constraints. We find that every pair of matrices of equal size and trace admits dilations of any rank larger than one. We generalize Audenaert’s subadditivity inequality to encompass general matrices, multiple tensor factors, and different norms. A central ingredient for this is a novel majorization relation for Kronecker sums. As an application, we extend the interval of Werner states in which they are provably two-undistillable in any dimension d ≥ 4. We also prove new Schmidt-number witnesses and k-positive maps.
We solve the two-copy distillability problem for Werner states in every local dimension. Our main matrix result is a sharp, dimension-free inequality: for every rank-at-most-two operator, the sum of the squared Hilbert--Schmidt norms of its two partial traces is bounded by twice its squared Hilbert--Schmidt norm plus one half of the squared modulus of its trace. This implies that a Werner state
ρα is two-copy distillable if and only if
α<−1/2. In particular, the two-ququart state
ρ−1/2(4) is two-copy undistillable, resolving Problem 5 of Horodecki, Rudnicki, and Życzkowski. For an arbitrary finite number
k of copies, we give three exact formulations of the remaining problem. At the endpoint
α=−1/2, undistillability is equivalent to nonnegativity of the endpoint partial-trace form on every rank-at-most-two operator. We also derive an equivalent hierarchy of operator inequalities
Hk(ψ)⪰0 for pure states with a maximally mixed qubit marginal. The two-copy proof does not formally induct, because partial trace can increase rank and
2-positivity is not generally preserved by tensor products. We prove two rigorous many-copy extensions. First, the quadratic form factorizes exactly on tensor-factorized witnesses; for any such decomposition, the endpoint inequality holds if its possible rank-two factor is supported on a block containing at most two copies. Second, we construct explicit constants
γk>0 such that
α≥−γk implies
k-copy undistillability in every dimension. The initial proofs were generated by ChatGPT 5.6 Sol; the authors have verified and rewritten them to enhance readability and provide additional context.
Motivated by the equivalent partial-trace formulations of Werner-state distillability [P. Costa Rico, Lett. Math. Phys. 115, 47 (2025); S.-Y. Qi et al., Phys. Rev. A 110, 012406 (2024)], we prove a bipartite partial-trace inequality for every matrix of rank at most two. As applications, we prove the two-copy undistillability of NPT Werner states in arbitrary local dimension, thereby resolving this open problem highlighted in [P. Horodecki et al., PRX Quantum 3, 010101 (2022)]. We further prove a two-parameter extension of the matrix inequality and show that two individually one-copy-undistillable NPT Werner states cannot activate each other's one-copy distillability. We also resolve the singular-value maximization problem associated with the two-ququart case.
Abstract One of the oldest problems in quantum information theory is to study if there exists a state with negative partial transpose which is undistillable [1]. This problem has been open for almost 30 years, and still no one has been able to give a complete answer to it. This work presents a new strategy to try to solve this problem by translating the distillability condition on the family of Werner states into a problem of partial trace inequalities, this is the aim of our first main result. As a consequence, we obtain a new bound for the 2-distillability of Werner states, which does not depend on the dimension of the system. On the other hand, our second main result provides new partial trace inequalities for bipartite systems, connecting some of them also with the separability of Werner states. Throughout this work, we also present numerous partial trace inequalities, which are valid for many families of matrices.
Several classical majorization relations for sums or products of matrices involve a majorizing vector of perfectly aligned eigenvalues or singular values. By relaxing the order of alignment to a partial order, we show that the majorization can be strengthened, provided the change-of-basis matrices admit an LU-approximation with respect to this partial order. In this way, we obtain refined versions of Ky Fan's majorization relations, Horn's log-majorization relation, and von Neumann's trace inequality. As an application, we give a short proof of the separable Ky Fan majorization relation for an arbitrary number of tensor factors and extend it to a sum of tensor products of arbitrary matrices. Further applications concern majorization relations for sums of (anti-)symmetric powers and for products of Kronecker sums.
Abstract. We consider a family of vector and operator norms which we refer to as Schmidt norms. We show that these norms have several uses in quantum information theory – they can be used to help classify k-positive linear maps (and hence entanglement witnesses), they are useful for approaching the problem of finding non-positive partial transpose bound entangled Werner states, they are related to the quantum fidelity and trace distance measures, and they are connected to the recently-defined local numerical range. We show that the vector norms can be explicitly calculated, and we derive several inequalities in order to bound the operator norms and compute them in special cases. We show that one particular entangled Werner state is bound entangled if and only if a certain norm inequality holds on a given family of projections, and we use our inequalities to study that family of projections. We also develop a family of semidefinite programs that can be used to further bound the operator norms. We extend these norms to arbitrary convex mapping cones and explore their implications with positive partial transpose states. 1.
Problem 5 in {\it Five Open Problems in Quantum Information Theory} [PRX Quantum 3, 010101 (2022)], asks whether the two-ququart Werner state
ϱ(4,−21) is two-copy distillable, where
ϱ(d,α)=(I+αF)/(d2+αd). We answer it in the negative. To this end, we show the following stronger statement: for all
C∈Md1d2(C) of rank at most
r≤d1d2,
tr1(C)∥F2+∥tr2(C)∥F2≤r∥C∥F2+r1∣tr(C)∣2. A result by Costa Rico on the equivalence of this inequality with two-copy undistillability at
r=2 then settles Problem 5:
ϱ(4,−21) is not two-copy distillable. Furthermore, we show that
ϱ(d,α) is two-copy undistillable for every
d≥2, if and only if
α≥−21. Thus, the one and two-copy distillability regions of
ϱ(d,α) coincide. These results have been found and written up with AI tools, pointing towards a structural change affecting the field of quantum information and computation.