We consider the inverse eigenvalue problem for entanglement witnesses, which asks for a characterization of their possible spectra (or equivalently, of the possible spectra resulting from positive linear maps of matrices). We completely solve this problem in the two-qubit case and we derive a large family of new necessary conditions on the spectra in arbitrary dimensions. We also establish a natural duality relationship with the set of absolutely separable states, and we completely characterize witnesses (i.e., separating hyperplanes) of that set when one of the local dimensions is 2.From a mathematical perspective, quantum information theory (and more specifically, quantum entanglement theory) is largely concerned with properties of (Hermitian) positive semidefinite matrices and the tensor product. A pure quantum state |v ∈ C n is a unit (column) vector and a mixed quantum state ρ ∈ M n (C) is a (Hermitian) positive semidefinite matrix with Tr(ρ) = 1 (we use Tr(·) to denote the trace and M n (R) or M n (C) to denote the set of n × n real or complex matrices, respectively). Whenever we use "ket" notation like |v or |w , or lowercase Greek letters like ρ or σ, we are implicitly assuming that they represent pure or mixed quantum states, respectively.where "⊗" is the tensor (Kronecker) product, v| is the dual (row) vector of |v , so |v v| is the rank-1 projection onto |v , and p 1 , p 2 , . . . form a probability distribution (i.e., they are non-negative and add up to 1). Equivalently, ρ is separable if and only if it can be written in the formwhere each X j ∈ M m (C) and Y j ∈ M n (C) is a (Hermitian) positive semidefinite matrix. If ρ is not separable then it is called entangled.Determining whether or not a mixed state is separable is a hard problem [10,11], so in practice numerous one-sided tests are used to demonstrate separability or entanglement (see [12,13] and the references therein for a more thorough introduction to this problem). The most well-known such test says that if we define the partial transpose of a matrix A = ∑ j B j ⊗ C j ∈ M m (C) ⊗ M n (C) viathen ρ being separable implies that ρ Γ is positive semidefinite (so we write ρ Γ O) [14]. This test follows simply from the fact that if ρ is separable then ρ = ∑ j X j ⊗ Y j with each X j , Y j O, sowhich is still positive semidefinite since each Y T j is positive semidefinite, and tensoring and adding positive semidefinite matrices preserves positive semidefiniteness. If a mixed state ρ is such that ρ Γ O then we say that it has positive partial transpose (PPT), and the previous discussion shows that the set of separable states is a subset of the set of PPT states. Entanglement Witnesses and Positive MapsA straightforward generalization of the partial transpose test for entanglement is based on positive linear matrix-valued maps. A linear map Φ : M m (C) → M n (C) is called positive if X O implies Φ(X) O, and the transpose is an example of one such map. Based on positive maps, we can define block-positive matrices, which are matrices of the form W := ...