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    Higher rank numerical ranges and low rank perturbations of quantum channels

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    For a positive integer kk, the rank-kk numerical range Λk(A)\Lambda_k(A) of an operator AA acting on a Hilbert space \cH of dimension at least kk is the set of scalars λ\lambda such that PAP=λPPAP = \lambda P for some rank kk orthogonal projection PP. In this paper, a close connection between low rank perturbation of an operator AA and Λk(A)\Lambda_k(A) is established. In particular, for 1r<k1 \le r < k it is shown that Λk(A)Λkr(A+F)\Lambda_k(A) \subseteq \Lambda_{k-r}(A+F) for any operator FF with \rank (F) \le r. In quantum computing, this result implies that a quantum channel with a kk-dimensional error correcting code under a perturbation of rank r\le r will still have a (kr)(k-r)-dimensional error correcting code. Moreover, it is shown that if AA is normal or if the dimension of AA is finite, then Λk(A)\Lambda_k(A) can be obtained as the intersection of Λkr(A+F)\Lambda_{k-r}(A+F) for a collection of rank rr operators FF. Examples are given to show that the result fails if AA is a general operator. The closure and the interior of the convex set Λk(A)\Lambda_k(A) are completely determined. Analogous results are obtained for Λ(A)\Lambda_\infty(A) defined as the set of scalars λ\lambda such that PAP=λPPAP = \lambda P for an infinite rank orthogonal projection PP. It is shown that Λ(A)\Lambda_\infty(A) is the intersection of all Λk(A)\Lambda_k(A) for k=1,2,>...k = 1, 2, >.... If AμIA - \mu I is not compact for any \mu \in \IC, then the closure and the interior of Λ(A)\Lambda_\infty(A) coincide with those of the essential numerical range of AA. The situation for the special case when AμIA-\mu I is compact for some \mu \in \IC is also studied.Comment: 21 page

    It Is Emphatically the Province and Duty of State Courts to Say What Tort Law Is

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    Following the U.S. Supreme Court’s 2011 decision in PLIVA, Inc. v. Mensing, consumers of generic prescription drugs suffering from unwarnedof side effects largely remain without an avenue of legal recourse due to their inability to sue their own manufacturers. But in the pursuit for legal redress, some generic plaintiffs have pursued a narrow window of liability by bringing failure-to-warn claims, sounding in negligence, against the manufacturer responsible for producing the brand-name equivalent of the generic drug. Such claims rest on the rationale that the sui generis federal regulatory scheme governing the prescription drug industry furnishes an inextricable nexus between the brand-name manufacturer and generic-drug user such that it generates a negligence duty of care between them. The case law on this duty question remains fractured. Until late 2017, the majority of courts confronting the duty issue ruled for the brand-name defendant and held no duty as a matter of law. However, beginning in December of 2017, two landmark decisions by the California and Massachusetts supreme courts, in support of duty, have called for a reexamination of settled case law and, accordingly, given new hope to the generic-drug user’s pursuit of legal remedy. In light of these recent developments, this Note seeks to equip future courts confronting the duty question with a functional understanding of the considerations that lie on both sides of the duty inquiry. In addition, this Note proposes a remedial position that incorporates both the policy concerns cutting against duty and the doctrinal considerations undergirding it. At its core, this Note argues that doctrine demands a duty be recognized and, further, that courts have the core institutional competence to craft tort law in ways that will avert ruinous public policy consequences. In making this argument, this Note conveys a fighting message to courts: where tort doctrine says a duty of care exists, courts should endeavor to give effect to that duty