A new proof is presented of a theorem of L. Gurvits [LANL Unclassified Technical Report (2001), LAUR–01–2030], which states that the cone of positive block-Toeplitz matrices with matrix entries has no entangled elements. The proof of the Gurvits separation theorem is achieved by making use of the structure of the operator system dual of the operator system C ( S 1 ) ( n ) C(S^1)^{(n)} of n × n n\times n Toeplitz matrices over the complex field, and by determining precisely the structure of the generators of the extremal rays of the positive cones of the operator systems C ( S 1 ) ( n ) ⊗ min B ( H ) C(S^1)^{(n)}\otimes _{\text {min}}\mathcal {B}(\mathcal {H}) and C ( S 1 ) ( n ) ⊗ min B ( H ) C(S^1)_{(n)}\otimes _{\text {min}}\mathcal {B}(\mathcal {H}) , where H \mathcal {H} is an arbitrary Hilbert space and C ( S 1 ) ( n ) C(S^1)_{(n)} is the operator system dual of C
A new proof is presented of a theorem of L. Gurvits [LANL Unclassified Technical Report (2001), LAUR–01–2030], which states that the cone of positive block-Toeplitz matrices with matrix entries has no entangled elements. The proof of the Gurvits separation theorem is achieved by making use of the structure of the operator system dual of the operator system C ( S 1 ) ( n ) C(S^1)^{(n)} of n × n n\times n Toeplitz matrices over the complex field, and by determining precisely the structure of the generators of the extremal rays of the positive cones of the operator systems C ( S 1 ) ( n ) ⊗ min B ( H ) C(S^1)^{(n)}\otimes {\text {min}}\mathcal {B}(\mathcal {H}) and C ( S 1 ) ( n ) ⊗ min B ( H ) C(S^1){(n)}\otimes {\text {min}}\mathcal {B}(\mathcal {H}) , where H \mathcal {H} is an arbitrary Hilbert space and C ( S 1 ) ( n ) C(S^1){(n)} is the operator system dual of C
The structure of the set of positivity-preserving maps between matrix algebras is notoriously difficult to describe. The notable exceptions are the results by Størmer and Woronowicz from 1960s and 1970s settling the low dimensional cases. By duality, these results are equivalent to the Peres-Horodecki positive partial transpose criterion being able to unambiguously establish whether a state in a 2 x 2 or 2 x 3 quantum system is entangled or separable. However, even in these low dimensional cases, the existing arguments (known to the authors) were based on long and seemingly ad hoc computations. We present a simple proof, based on Brouwer's fixed point theorem, for the 2 x 2 case (Størmer's theorem). For completeness, we also include another argument (following the classical outline, but highly streamlined) based on a characterization of extreme self-maps of the Lorentz cone and on a link - noticed by R. Hildebrand - to the S-lemma, a well-known fact from control theory and quadratic/semi-definite programming.
Drawing on results of Choi, Størmer and Woronowicz, we present a nearly complete characterization of certain important classes of positive maps. In particular, we construct a general class of positive linear maps acting between two matrix algebras ℬ( ) and ℬ( ), where and are finite-dimensional Hilbert spaces. It turns out that elements of this class are characterized by operators from the dual cone of the set of all separable states on ℬ( ⊗ ). Subsequently, the relation between entanglements and positive maps is described. Finally, a new characterization of the cone ℬ( ) + ⊗ℬ( ) + is given.
There are various notions of positivity for matrices and linear matrix-valued maps that play important roles in quantum information theory. The cones of positive semidefinite matrices and completely positive linear maps, which represent quantum states and quantum channels respectively, are the most ubiquitous positive cones. There are also many natural cones that can been regarded as "more" or "less" positive than these standard examples. In particular, entanglement theory deals with the cones of separable operators and entanglement witnesses, which satisfy very strong and weak positivity properties respectively. Rather complementary to the various cones that arise in entanglement theory are norms. The trace and operator norms for operators and the diamond and completely bounded norms for superoperators are the typical norms that are seen throughout quantum information theory. In this work our main goal is to develop a family of norms that play a role analogous to the cone of entanglement witnesses. We investigate the basic mathematical properties of these norms, including their relationships with other well-known norms, their isometry groups, and their dual norms. We also make the place of these norms in entanglement theory rigorous by showing that entanglement witnesses arise from minimal operator systems, and analogously our norms arise from minimal operator spaces. Finally, we connect the various cones and norms considered here to several seemingly unrelated problems from other areas. We characterize the problem of whether or not non-positive partial transpose bound entangled states exist in terms of one of our norms, and provide evidence in favour of their existence. We also characterize the minimum gate fidelity of a quantum channel, the maximum output purity and its completely bounded counterpart, and the geometric measure of entanglement in terms of these norms.
An alternative, geometrical proof of a known theorem concerning the decomposition of positive maps of the matrix algebra has been presented. The premise of the proof is the identification of positive maps with operators preserving the Lorentz cone in four dimensions, and it allows to decompose the positive maps with respect to those preserving the boundary of the cone. In addition, useful conditions implying complete positivity of a map of have been given, together with a sufficient condition for complete positivity of an extremal Schwarz map of . Lastly, following the same geometrical approach, a description in topological terms of maps that are simultaneously completely positive and completely copositive has been presented.
The structure of cones of positive and k-positive maps acting on a finite-dimensional Hilbert space is investigated. Special emphasis is given to their duality relations to the sets of superpositive and k-superpositive maps. We characterize k-positive and k-superpositive maps with regard to their properties under taking compositions. A number of results obtained for maps are also rephrased for the corresponding cones of block positive, k-block positive, separable and k-separable operators, due to the Jamio̷lkowski-Choi isomorphism. Generalizations to a situation where no such simple isomorphism is available are also made, employing the idea of mapping cones. As a side result to our discussion, we show that extreme entanglement witnesses, which are optimal, should be of special interest in entanglement studies.
Let K, K' be convex cones residing in finite-dimensional real vector spaces E, E'. An element in the tensor product E \otimes E' is K \otimes K'-separable if it can be represented as finite sum \sum_l x_l \otimes x'_l with x_l \in K and x_l' \in K' for all l. Let S(n), H(n), Q(n) be the spaces of n x n real symmetric, complex hermitian and quaternionic hermitian matrices, respectively. Let further S_+(n), H_+(n), Q_+(n) be the cones of positive semidefinite matrices in these spaces. If a matrix in H(mn) = H(m) \otimes H(n) is H_+(m) \otimes H_+(n)-separable, then it fulfills also the so-called PPT condition, i.e. it is positive semidefinite and has a positive semidefinite partial transpose. The same implication holds for matrices in the spaces S(m) \otimes S(n), H(m) \otimes S(n), and for m < 3 in the space Q(m) \otimes S(n). We provide a complete enumeration of all pairs (n,m) when the inverse implication is also true for each of the above spaces, i.e. the PPT condition is sufficient for separability. We also show that a matrix in Q(n) \otimes S(2) is Q_+(n) \otimes S_+(2)-separable if and only if it is positive semidefinite.