Yangian Construction of the Virasoro Algebra

We show that a Yangian construction based on the algebra of an infinite number of harmonic oscillators (i.e. a vibrating string) terminates after one step, yielding the Virasoro algebra.Comment: 5 pages, AMS-Latex 2

Non-local properties of multi-particle density matrices

As far as entanglement is concerned, two density matrices of $n$ particles are equivalent if they are on the same orbit of the group of local unitary transformations, $U(d_1)\times...\times U(d_n)$ (where the Hilbert space of particle $r$ has dimension $d_r$). We show that for $n$ greater than or equal to two, the number of independent parameters needed to specify an $n$-particle density matrix up to equivalence is $\Pi_r d_r^2 - \sum_r d_r^2 + n - 1$. For $n$ spin-${1\over 2}$ particles we also show how to characterise generic orbits, both by giving an explicit parametrisation of the orbits and by finding a finite set of polynomial invariants which separate the orbits.Comment: 13 pages RevTe

How entangled can two couples get?

We describe a pure state of four qubits whose single-qubit density matrices are all maximally mixed and whose average entanglement as a system of two pairs of qubits appears to be maximal.Comment: 9 pages. Note added about the robustness of the entanglement in the four-qubit state described in the paper. Version to be published in Phys. Lett.

Single-world theory of the extended Wigner's friend experiment

Frauchiger and Renner have recently claimed to prove that ``Single-world interpretations of quantum theory cannot be self-consistent". This is contradicted by a construction due to Bell, inspired by Bohmian mechanics, which shows that any quantum system can be modelled in such a way that there is only one ``world" at any time, but the predictions of quantum theory are reproduced. This Bell-Bohmian theory is applied to the experiment proposed by Frauchiger and Renner, and their argument is critically examined. It is concluded that it is their version of ``standard quantum theory", incorporating state vector collapse upon measurement, that is not self-consistent

Representations of the Generalized Lie Algebra sl(2)_q

We construct finite-dimensional irreducible representations of two quantum algebras related to the generalized Lie algebra \ssll (2)_q introduced by Lyubashenko and the second named author. We consider separately the cases of $q$ generic and $q$ at roots of unity. Some of the representations have no classical analog even for generic $q$. Some of the representations have no analog to the finite-dimensional representations of the quantised enveloping algebra $U_q(sl(2))$, while in those that do there are different matrix elements.Comment: 14 pages, plain-TEX file using input files harvmac.tex, amssym.de

UMI-tools: Modelling sequencing errors in Unique Molecular Identifiers to improve quantification accuracy.

Unique Molecular Identifiers (UMIs) are random oligonucleotide barcodes that are increasingly used in high-throughput sequencing experiments. Through a UMI, identical copies arising from distinct molecules can be distinguished from those arising through PCR-amplification of the same molecule. However, bioinformatic methods to leverage the information from UMIs have yet to be formalised. In particular, sequencing errors in the UMI sequence are often ignored, or else resolved in an ad-hoc manner. We show that errors in the UMI sequence are common and introduce network based methods to account for these errors when identifying PCR duplicates. Using these methods, we demonstrate improved quantification accuracy both under simulated conditions and real iCLIP and single cell RNA-Seq datasets. Reproducibility between iCLIP replicates and single cell RNA Seq clustering are both improved using our proposed network-based method, demonstrating the value of properly accounting for errors in UMIs. These methods are implemented in the open source UMI-tools software package