Entanglement quasiprobabilities of squeezed light

We demonstrate the feasibility to completely characterize entanglement by negativities of quasiprobabilities. This requires the complete solution of a sophisticated mathematical problem, the so-called separability eigenvalue problem. Its solution is obtained for a non-Gaussian continuous variable quantum state, a two-mode squeezed state undergoing dephasing. This is a standard scenario for experiments with quantum-correlated radiation fields

Covariant 4-dimensional fuzzy spheres, matrix models and higher spin

We study in detail generalized 4-dimensional fuzzy spheres with twisted extra dimensions. These spheres can be viewed as $SO(5)$-equivariant projections of quantized coadjoint orbits of $SO(6)$. We show that they arise as solutions in Yang-Mills matrix models, which naturally leads to higher-spin gauge theories on $S^4$. Several types of embeddings in matrix models are found, including one with self-intersecting fuzzy extra dimensions $S^4 \times \mathcal{K}$, which is expected to entail 2+1 generations.Comment: 41+7 pages, 4 figure

Operational Gaussian Schmidt-Number Witnesses

The general class of Gaussian Schmidt-number witness operators for bipartite systems is studied. It is shown that any member of this class is reducible to a convex combination of two types of Gaussian operators using local operations and classical communications. This gives rise to a simple operational method, which is solely based on measurable covariance matrices of quantum states. Our method bridges the gap between theory and experiment of entanglement quantification. In particular, we certify lower bounds of the Schmidt number of squeezed thermal and phase-randomized squeezed vacuum states, as examples of Gaussian and non-Gaussian quantum states, respectively.Comment: 9 pages, 4 figure

Prospective Overruling and the Revival of ‘Unconstitutional\u27 Statutes

The Supreme Court\u27s decision in Planned Parenthood v. Casey reshaped the law of abortion in this country. The Court overturned two of its previous decisions invalidating state restrictions on abortions, Thornburgh v. American College of Obstetricians and Gynecologists and Akron v. Akron Center for Reproductive Health, and it abandoned the trimester analytic framework established in Roe v. Wade. At the time Casey was handed down, twenty states had restrictive abortion statutes on the books that were in conflict with Akron or Thornburgh and which were unenforced. In six of these states, courts had held the statutes unconstitutional. Almost as soon as the Casey ruling was announced, the campaign to secure enforcement of these restrictions began. Are these statutes good law, despite the fact that they were once in conflict with governing Supreme Court precedent (and in some cases had been judicially determined to violate women\u27s constitutional rights)? Alternatively, will they have to be re-enacted by the legislature to be enforceable? These questions highlight the revival issue. The revival issue arises when a court overrules a prior decision in which it had held a statute unconstitutional. (We will throughout this article refer to the first decision as the invalidating decision, and to the second decision as the overruling decision. ) Should the enforceability of a statute passed prior to the overruling decision be determined by reference to the invalidating decision--in which case the statute would have to be repassed to be in effect--or by reference to the overruling decision--in which case the statute would not have to be repassed? In other words, does the overruling decision automatically revive a previously unenforceable statute? The way in which the revival issue is resolved will thus determine whether, in light of Casey, previously unenforced statutes became enforceable without the need for any post-Casey legislative action. In addition to affecting what kind of abortion regulations are in effect in twenty states in the immediate wake of Casey, this determination has profound consequences for the kind of abortion regulations that will be in effect in these states in the future. Such long-term consequences reflect the fact that our governmental system is not one of pure majoritarianism and that the burden of inertia in our legislative process is heavy: as we will discuss, statutes on the books can stay on the books even if a current majority no longer desires them; in contrast, proposed statutes need supermajoritarian support to secure passage. Therefore, the starting point for future legislative action--such as whether pre-Casey abortion regulations are enforceable--influences the legislative action that in fact develops