Gaussian quantum marginal problem

The quantum marginal problem asks what local spectra are consistent with a given spectrum of a joint state of a composite quantum system. This setting, also referred to as the question of the compatibility of local spectra, has several applications in quantum information theory. Here, we introduce the analogue of this statement for Gaussian states for any number of modes, and solve it in generality, for pure and mixed states, both concerning necessary and sufficient conditions. Formally, our result can be viewed as an analogue of the Sing-Thompson Theorem (respectively Horn's Lemma), characterizing the relationship between main diagonal elements and singular values of a complex matrix: We find necessary and sufficient conditions for vectors (d1, ..., dn) and (c1, ..., cn) to be the symplectic eigenvalues and symplectic main diagonal elements of a strictly positive real matrix, respectively. More physically speaking, this result determines what local temperatures or entropies are consistent with a pure or mixed Gaussian state of several modes. We find that this result implies a solution to the problem of sharing of entanglement in pure Gaussian states and allows for estimating the global entropy of non-Gaussian states based on local measurements. Implications to the actual preparation of multi-mode continuous-variable entangled states are discussed. We compare the findings with the marginal problem for qubits, the solution of which for pure states has a strikingly similar and in fact simple form.Comment: 18 pages, 1 figure, material added, references updated, except from figure identical with version to appear in Commun. Math. Phy

RACK1 Is a Ribosome Scaffold Protein for β-actin mRNA/ZBP1 Complex

In neurons, specific mRNAs are transported in a translationally repressed manner along dendrites or axons by transport ribonucleic-protein complexes called RNA granules. ZBP1 is one RNA binding protein present in transport RNPs, where it transports and represses the translation of cotransported mRNAs, including β-actin mRNA. The release of β-actin mRNA from ZBP1 and its subsequent translation depends on the phosphorylation of ZBP1 by Src kinase, but little is known about how this process is regulated. Here we demonstrate that the ribosomal-associated protein RACK1, another substrate of Src, binds the β-actin mRNA/ZBP1 complex on ribosomes and contributes to the release of β-actin mRNA from ZBP1 and to its translation. We identify the Src binding and phosphorylation site Y246 on RACK1 as the critical site for the binding to the β-actin mRNA/ZBP1 complex. Based on these results we propose RACK1 as a ribosomal scaffold protein for specific mRNA-RBP complexes to tightly regulate the translation of specific mRNAs