Quantum channels that preserve entanglement. (English) Zbl 1167.46042
Let \(M\) and \(N\) be full matrix algebras. A unital completely positive (UCP) map \(\phi:M\to N\) (quantum channel) preserves entanglement if its inflation \(\phi\otimes \mathrm{Id}_N : M\otimes N\to N\otimes N\) maps every maximally entangled pure state \(\rho\) (or every state of maximal Schmidt rank, which are called marginally cyclic in this paper) of \(N\otimes N\), into an entangled state of \(M\otimes N\).
In Section 2, complementing a result of M. Horodecki, P. W. Shor, and M. B. Ruskai [Rev. Math. Phys. 15, No. 6, 629–641 (2003; Zbl 1080.81006)], the author shows that every UCP map that is not entanglement breaking must preserve entanglement. Further, the parametrization of states given by W. Arveson [The probability of entanglement, arXiv:0712.4163 (2007)] can be appropriately adapted to UCP maps so as to make the space \(\Phi^r\) of all UCP maps \(\phi:\mathcal B(K)\to \mathcal B(H)\) of rank \(\leq r\) into a compact probability space that carries a unique invariant probability measure \(P^r\), and it is shown in Section 3 that \(P^r\) is concentrated on the set of maps of rank \(r\). Thus, the probability space \((\Phi^r,P^r)\) represents choosing a UCP map of rank \(r\) at random. A zero-one law for channels is proven which expresses in probabilistic terms the dichotomy that a UCP map either preserves entanglement or is entanglement breaking.
The main results of W. Arveson [loc. cit.] are then applied in Sect. 4 to show that there are plenty of entanglement preserving UCP maps of every possible rank, and that almost surely every UCP map \(\Phi^r:\mathcal B(K)\to\mathcal B(H)\) of rank \(r\leq n/2\) preserves entanglement. Here \(n=\dim H, m=\dim K\), and it is assumed that \(n\leq m\). Moreover, for every \(r=1,2,\dots,mn\), the set of entanglement preserving UCP maps of rank \(r\) is a relatively open subset of \(\Phi^r\) of positive measure and for the maximum rank \(r=mn\) its probability is strictly between 0 and 1.
Sect. 5 concludes with a discussion of extreme points of the convex set of UCP maps that implies: for every integer \(r\) satisfying \(1\leq r\leq n\), the extremals of rank \(r\) constitute a relatively open dense set having probability \(1\). There are no extremal UCP maps \(\phi:\mathcal B(K)\to \mathcal B(H)\) of rank \(>n\). Thus, whenever an extremal UCP map of rank \(r\) exists, then almost surely every UCP map of rank \(r\) is extremal.
In Section 2, complementing a result of M. Horodecki, P. W. Shor, and M. B. Ruskai [Rev. Math. Phys. 15, No. 6, 629–641 (2003; Zbl 1080.81006)], the author shows that every UCP map that is not entanglement breaking must preserve entanglement. Further, the parametrization of states given by W. Arveson [The probability of entanglement, arXiv:0712.4163 (2007)] can be appropriately adapted to UCP maps so as to make the space \(\Phi^r\) of all UCP maps \(\phi:\mathcal B(K)\to \mathcal B(H)\) of rank \(\leq r\) into a compact probability space that carries a unique invariant probability measure \(P^r\), and it is shown in Section 3 that \(P^r\) is concentrated on the set of maps of rank \(r\). Thus, the probability space \((\Phi^r,P^r)\) represents choosing a UCP map of rank \(r\) at random. A zero-one law for channels is proven which expresses in probabilistic terms the dichotomy that a UCP map either preserves entanglement or is entanglement breaking.
The main results of W. Arveson [loc. cit.] are then applied in Sect. 4 to show that there are plenty of entanglement preserving UCP maps of every possible rank, and that almost surely every UCP map \(\Phi^r:\mathcal B(K)\to\mathcal B(H)\) of rank \(r\leq n/2\) preserves entanglement. Here \(n=\dim H, m=\dim K\), and it is assumed that \(n\leq m\). Moreover, for every \(r=1,2,\dots,mn\), the set of entanglement preserving UCP maps of rank \(r\) is a relatively open subset of \(\Phi^r\) of positive measure and for the maximum rank \(r=mn\) its probability is strictly between 0 and 1.
Sect. 5 concludes with a discussion of extreme points of the convex set of UCP maps that implies: for every integer \(r\) satisfying \(1\leq r\leq n\), the extremals of rank \(r\) constitute a relatively open dense set having probability \(1\). There are no extremal UCP maps \(\phi:\mathcal B(K)\to \mathcal B(H)\) of rank \(>n\). Thus, whenever an extremal UCP map of rank \(r\) exists, then almost surely every UCP map of rank \(r\) is extremal.
Reviewer: Alexandr S. Holevo (Moskva)
MSC:
| 46L60 | Applications of selfadjoint operator algebras to physics |
| 81P68 | Quantum computation |
| 94B27 | Geometric methods (including applications of algebraic geometry) applied to coding theory |
Citations:
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