Truncated geometry on the circle

Hekkelman, Evert-Jan

doi:10.1007/s11005-022-01514-5

“…The operator systems C(S 1 ) (n) and C(S 1 ) (n) are of classical importance. Yet, only recently have they been linked through duality, and through this linkage these operator systems are receiving recent renewed attention in areas such as spectral truncation in noncommutative geometry [5,13,19] and operator system tensor products [6]. The purpose of the present paper is to shed light on how this linkage relates to the factorisation and separability of positive Toeplitz matrices of Hilbert space operators.…”

Section: Introductionmentioning

confidence: 98%

“…Following [5,6,13,19], the operator system of n × n Toeplitz matrices over the complex numbers is denoted by C(S 1 ) (n) . The canonical linear basis for C(S 1 ) (n) is the one given by {r −n+1 , .…”

Section: Introductionmentioning

confidence: 99%

Toeplitz separability, entanglement, and complete positivity using operator system duality

Farenick

¹

,

McBurney

²

2023

Proc. Amer. Math. Soc. Ser. B

0

View full text Add to dashboard Cite

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 ( S 1 ) ( n ) C(S^1)^{(n)} . Our approach also has the advantage of providing some new information concerning positive Toeplitz matrices whose entries are from B ( H ) \mathcal {B}(\mathcal {H}) when H \mathcal {H} has infinite dimension. In particular, we prove that normal positive linear maps ψ \psi on B ( H ) \mathcal {B}(\mathcal {H}) are partially completely positive in the sense that ψ ( n ) ( x ) \psi ^{(n)}(x) is positive whenever x x is a positive n × n n\times n Toeplitz matrix with entries from B ( H ) \mathcal {B}(\mathcal {H}) . We also establish a certain factorisation theorem for positive Toeplitz matrices (of operators), showing an equivalence between the Gurvits approach to separation and an earlier approach of T. Ando [Acta Sci. Math. (Szeged) 31 (1970), pp. 319–334] to universality.

show abstract

“…Following [5,6,13,19], the operator system of n × n Toeplitz matrices over the complex numbers is denoted by C(S 1 ) (n) . The canonical linear basis for C(S 1 ) (n) is the one given by {r −n+1 , .…”

Section: Introductionmentioning

confidence: 99%

Untitled

2023

Proc. Amer. Math. Soc. Ser. B

0

View full text Add to dashboard Cite

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) of n × 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) and C(S 1 ) (n) ⊗ min B(H), where H is an arbitrary Hilbert space and C(S 1 ) (n) is the operator system dual of C(S 1 ) (n) . Our approach also has the advantage of providing some new information concerning positive Toeplitz matrices whose entries are from B(H) when H has infinite dimension. In particular, we prove that normal positive linear maps ψ on B(H) are partially completely positive in the sense that ψ (n) (x) is positive whenever x is a positive n × n Toeplitz matrix with entries from B(H). We also establish a certain factorisation theorem for positive Toeplitz matrices (of operators), showing an equivalence between the Gurvits approach to separation and an earlier approach of T. Ando [Acta Sci. Math. (Szeged) 31 (1970), pp. 319-334] to universality.

show abstract

“…Following [5,6,10,15], the operator system of n × n Toeplitz matrices over the complex numbers is denoted by C(S 1 ) (n) . The canonical linear basis for C(S 1 ) (n) 2020 Mathematics Subject Classification.…”

Section: Introductionmentioning

confidence: 99%

“…The operator system C(S 1 ) (n) and its operator system dual, C(S 1 ) (n) , are receiving recent attention because of their importance to spectral truncation in noncommutative geometry [5,10,15]. It is, therefore, interesting to determine whether there is an analogue of Theorem 1.1 when C(S 1 ) (n) is replaced by its operator system dual C(S 1 ) (n) .…”

Section: Introductionmentioning

confidence: 99%

Toeplitz separability, entanglement, and complete positivity using operator system duality

Farenick¹,

McBurney²

2022

Preprint

0

View full text Add to dashboard Cite

A new proof is presented of a theorem of L. Gurvits, which states that the cone of positive block-Toeplitz matrices with matrix entries has no entangled elements. We also show that in the cone of positive Toeplitz matrices with Toeplitz entries, entangled elements exist in all dimensions. 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) of n × 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) and C(S 1 ) (n) ⊗ min B(H), where H is an arbitrary Hilbert space and C(S 1 ) (n) is the operator system dual of C(S 1 ) (n) . Our approach also has the advantage of providing some new information concerning positive Toeplitz matrices whose entries are from the type I factor B(H). In particular, we prove that normal positive linear maps ψ on B(H) are partially completely positive in the sense that ψ (n) (x) is positive whenever x is a positive n × n Toeplitz matrix with entries from B(H). We also establish a certain factorisation theorem for positive Toeplitz matrices (of operators), and explain how the factorisation theorem can be used to show an equivalence between the Gurvits approach to separation and an earlier approach of T. Ando to universality.

show abstract

Truncated geometry on the circle

Abstract: In this letter, we prove that the pure state space on the $$n \times n$$ n × n complex Toeplitz matrices converges in the Gromov–Hausdorff sense to the state space on $$C(S^1)$$ C ( S 1 ) as n grows … Show more

Cited by 6 publications

References 10 publications

Toeplitz separability, entanglement, and complete positivity using operator system duality

Toeplitz separability, entanglement, and complete positivity using operator system duality

Untitled

Toeplitz separability, entanglement, and complete positivity using operator system duality

Contact Info

Product

Resources

About