Scientific Series

We study the problem of reconstructing an unknown matrix M, of rank r and dimension d, using O(rd poly log d) Pauli measurements. This has applications to compressed sensing methods for quantum state tomography.  We give a solution to this problem based on the restricted isometry property (RIP), which improves on previous results using dual certificates. In particular, we show that almost all sets of O(rd log^6 d) Pauli measurements satisfy the rank-r RIP. This implies that M can be recovered from a fixed ("universal") set of Pauli measurements, using nuclear-norm minimization (e.g., the matrix Lasso), with nearly-optimal bounds on the error. Our proof uses Dudley's inequality for Gaussian processes, together with bounds on covering numbers obtained via entropy duality.

The covariant formulation of loop quantum gravity has developed strongly during the last few years.  I summarize the current definition of the theory and the results that have been proven. I discuss what is missing towards of the goal of defining a consistent quantum theory whose classical limit is general relativity.

I will discuss some of the most basic questions in fermionic and bosonic transport, such as the conditions for the existence of a steady-state current, its uniqueness, the role of interactions and spin statistics, its entanglement entropy, etc. This will lead me to introduce an alternative viewpoint to conduction - the micro-canonical formalism of transport - which is ideal to study the above issues [1]. I will point out the similarities and differences with the widely used Landauer formalism, and advance a series of predictions that can be verified by loading ultra-cold atoms into artificial optical lattices.

[1] M. Di Ventra, Electrical Transport in Nanoscale Systems (Cambridge University Press, 2008).

In the de Broglie-Bohm pilot-wave theory, an ensemble of fermions is not only described by a spinor,  but also by a distribution of position beables. If the distribution of positions is different from the one predicted by the Born rule, the ensemble is said to be in quantum non-equilibrium. Such ensembles, which can lead to an experimental discrimination between the pilot-wave theory and standard quantum mechanics, are thought to quickly relax to quantum equilibrium in most cases.   In this talk, I will look at the Majorana equation from the point of view of the pilot-wave theory and I will show that it predicts peculiar trajectories for the beables; they have to move luminally at all times and they usually undergo complex helical trajectories to give the illusion that their motion is subluminal. The nature of the Majorana trajectory suggests that relaxation to quantum equilibrium could only be partial and that quantum non-equilibrium could still survive at length scales below the Compton wavelength.  I investigate this claim, thanks to some numerical simulations of the temporal evolution of non-equilibrium distributions,  for three-dimensional confined systems governed by the Dirac and Majorana equations.

Models with right-handed currents have recently attracted attention because of their potential ability to solve the discrepancy in various determinations of |V_{ub}|. We consider a minimal setup with an SU(2)_L x SU(2)_R x U(1)_{B-L} gauge symmetry, for which we perform a simultaneous analysis of the most important constraints from electroweak precision observables, particle anti-particle mixing and the B -> X_{s,d} gamma decays. The main goals of our analysis are the determination of allowed parameter space for the new right-handed mixing matrix and the possible solution of various anomalies in the current flavor data.

One of the biggest challenges in physics is to develop accurate and efficient methods that can solve many currently intractable problems in correlated quantum or statistical systems. Tensor-network model/state is drawing more and more attention since it captures the feature of the area law and is absent from the sign problem. The evaluation of the expectation value of the observables can be reduced to the contraction of a tensor-network, which can be done by means of renormalization group method, and this is exactly what tensor renormalization group (TRG) method has done. In the light of comparison between numerical renormalization group (NRG) and density matrix renormalization group(DMRG), having considered the renormalization effect of the environment, second renormalization group (SRG) method has been proposed to improve the performance of TRG. Although TRG and SRG have achieved great success in 2D lattice, application to 3D is not easy. The talk will give a review of TRG and SRG, and talk about the HOTRG, i.e., TRG based on the higher-order singular value decomposition, and its SRG version, HOSRG. They can be easily applied to 3D lattice with relatively low memory and computation cost. The well-known 3D Ising model on simple cubic lattice has been tested. As far as our group know, HOSRG have achieved by far the most accurate renormalization group results for the 3D Ising model. Some analysis and possible applications will be also discussed.

Instantons and W-bosons in 5d N=2 Yang-Mills theory arise from a circle compactification of the 6d (2,0) theory as Kaluza-Klein modes and winding self-dual strings, respectively. We study an index which counts BPS instantons with electric charges in Coulomb and symmetric phases. We first prove the existence of unique threshold bound state of U(1) instantons for any instanton number. By studying SU(N) self-dual strings in the Coulomb phase, we find novel momentum-carrying degrees on the worldsheet. The total number of these degrees equals the anomaly coefficient of SU(N) (2,0) theory. We finally propose that our index can be used to study the symmetric phase of this theory, and provide an interpretation as the superconformal index of the sigma model on instanton moduli space.