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Register a new userJoint numerical ranges, quantum maps, and joint numerical shadows
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We associate with k hermitian Ntimes N matrices a probability measure on R^k. It is supported on the joint numerical range of the k-tuple of matrices. We call this measure the joint numerical shadow of these matrices. Let k=2. A pair of hermitian Ntimes N matrices defines a complex Ntimes N matrix. The joint numerical range and the joint numerical shadow of the pair of hermitian matrices coincide with the numerical range and the numerical shadow, respectively, of this complex matrix. We study relationships between the dynamics of quantum maps on the set of quantum states, on one hand, and the numerical ranges, on the other hand. In particular, we show that under the identity resolution assumption on Kraus operators defining the quantum map, the dynamics shrinks numerical ranges.
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In this work, we investigate the joint measurability of quantum effects and connect it to the study of free spectrahedra. Free spectrahedra typically arise as matricial relaxations of linear matrix inequalities. An example of a free spectrahedron is the matrix diamond, which is a matricial relaxation of the $ell_1$-ball. We find that joint measurability of binary POVMs is equivalent to the inclusion of the matrix diamond into the free spectrahedron defined by the effects under study. This connection allows us to use results about inclusion constants from free spectrahedra to quantify the degree of incompatibility of quantum measurements. In particular, we completely characterize the case in which the dimension is exponential in the number of measurements. Conversely, we use techniques from quantum information theory to obtain new results on spectrahedral inclusion for the matrix diamond.
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We give a brief introduction to private quantum codes, a basic notion in quantum cryptography and key distribution. Private code states are characterized by indistinguishability of their output states under the action of a quantum channel, and we show that higher rank numerical ranges can be used to describe them. We also show how this description arises naturally via conjugate channels and the bridge between quantum error correction and cryptography.
We establish two results concerning the Quantum Limits (QLs) of some sub-Laplacians. First, under a commutativity assumption on the vector fields involved in the definition of the sub-Laplacian, we prove that it is possible to split any QL into several pieces which can be studied separately, and which come from well-characterized parts of the associated sequence of eigenfunctions. Secondly, building upon this result, we classify all QLs of a particular family of sub-Laplacians defined on products of compact quotients of Heisenberg groups. We express the QLs through a disintegration of measure result which follows from a natural spectral decomposition of the sub-Laplacian in which harmonic oscillators appear.Both results are based on the construction of an adequate elliptic operator commuting with the sub-Laplacian, and on the associated joint spectral calculus. They illustrate the fact that, because of the possibly high degeneracy of the spectrum, the spectral theory of sub-Laplacians can be very rich.
We have investigated the sliding of droplets made of solutions of Xanthan, a stiff rodlike polysaccharide exhibiting a non-newtonian behavior, notably characterized by a shear-rate dependence of the viscosity. These experimental results are quantitatively compared with those of newtonian fluids (water). The impact of the non-newtonian behavior on the sliding process was shown through the relation between the average dimensionless velocity (i.e. the Capillary number) and the dimensionless volume forces (i.e. the Bond number). To this aim, it is needed to define operative strategies to compute the Capillary number for the shear thinning fluids and compare with the corresponding newtonian case. Results from experiments were complemented with lattice Boltzmann numerical simulations of sliding droplets, aimed to disentangle the influence that non-newtonian flow properties have on the sliding.
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