An algorithm to explore entanglement in small systems

A quantum state's entanglement across a bipartite cut can be quantified with entanglement entropy or, more generally, Schmidt norms. Using only Schmidt decompositions, we present a simple iterative algorithm to maximize Schmidt norms. Depending on the choice of norm, the optimizing states maximize or minimize entanglement, possibly across several bipartite cuts at the same time and possibly only among states in a specified subspace. Recognizing that convergence but not success is certain, we use the algorithm to explore topics ranging from fermionic reduced density matrices and varieties of pure quantum states to absolutely maximally entangled states and minimal output entropy of channels.Comment: Published version, 20 page

Generalized Pauli constraints in large systems: the Pauli principle dominates

Lately, there has been a renewed interest in fermionic 1-body reduced density matrices and their restrictions beyond the Pauli principle. These restrictions are usually quantified using the polytope of allowed, ordered eigenvalues of such matrices. Here, we prove this polytope's volume rapidly approaches the volume predicted by the Pauli principle as the dimension of the 1-body space grows, and that additional corrections, caused by generalized Pauli constraints, are of much lower order unless the number of fermions is small. Indeed, we argue the generalized constraints are most restrictive in (effective) few-fermion settings with low Hilbert space dimension.Comment: Published version; 37 pages, 5 figure

Lower bound on entanglement in subspaces defined by Young diagrams

Eigenvalues of 1-particle reduced density matrices of $N$-fermion states are upper bounded by $1/N$, resulting in a lower bound on entanglement entropy. We generalize these bounds to all other subspaces defined by Young diagrams in the Schur-Weyl decomposition of $\otimes^N\mathbb{C}^d$.Comment: Published version, 21 page

Calculation of the critical temperature of a dilute Bose gas in the Bogoliubov approximation

Following an earlier calculation in 3D, we calculate the 2D critical temperature of a dilute, translation-invariant Bose gas using a variational formulation of the Bogoliubov approximation introduced by Critchley and Solomon in 1976. This provides the first analytical calculation of the Kosterlitz-Thouless transition temperature that includes the constant in the logarithm.Comment: Published version, 7 pages, 2 figure

Entropy and entanglement bounds for reduced density matrices of fermionic states

Unlike bosons, fermions always have a non-trivial entanglement. Intuitively, Slater determinantal states should be the least entangled states. To make this intuition precise we investigate entropy and entanglement of fermionic states and prove some extremal and near extremal properties of reduced density matrices of Slater determinantal states.Comment: 20 pages. This version of the paper is a substantial revision of the previous versio

Ground state energy of a dilute two-dimensional Bose gas from the Bogoliubov free energy functional

We extend the analysis of the Bogoliubov free energy functional to two dimensions at very low temperatures. For sufficiently weak interactions, we prove two term asymptotics for the ground state energy.Comment: revised versio

Pseudobudding: ruptured glands do not represent true tumor buds

Tumor budding (TB) is a strong biomarker of poor prognosis in colorectal cancer and other solid cancers. TB is defined as isolated single cancer cells or clusters of up to four cancer cells at the invasive tumor front. In areas with a large inflammatory response at the invasive front, single cells and cell clusters surrounding fragmented glands are observed appearing like TB. Occurrence of these small groups is referred to as pseudobudding (PsB), which arises due to external influences such as inflammation and glandular disruption. Using a combination of orthogonal approaches, we show that there are clear biological differences between TB and PsB. TB is representative of active invasion by presenting features of epithelial-mesenchymal transition and exhibiting increased deposition of extracellular matrix within the surrounding tumor microenvironment (TME), whereas PsB represents a reactive response to heavy inflammation where increased levels of granulocytes within the surrounding TME are observed. Our study provides evidence that areas with a strong inflammatory reaction should be avoided in the routine diagnostic assessment of TB

A Flea on Schroedinger's Cat

We propose a technical reformulation of the measurement problem of quantum mechanics, which is based on the postulate that the final state of a measurement is classical; this accords with experimental practice as well as with Bohr's views. Unlike the usual formulation (in which the post-measurement state is a a unit vector in Hilbert space, such as a wave-function), our version actually admits a purely technical solution within the confines of conventional quantum theory (as opposed to solutions that either modify this theory, or introduce unusual and controversial interpretative rules and/or ontologies). To that effect, we recall a remarkable phenomenon in the theory of Schroedinger operators (discovered in 1981 by Jona-Lasinio, Martinelli, and Scoppola), according to which the ground state of a symmetric double-well Hamiltonian (which is paradigmatically of Schroedinger's Cat type) becomes exponentially sensitive to tiny perturbations of the potential as h -> 0. We show that this instability emerges also from the textbook WKB approximation, extend it to time-dependent perturbations, and study the dynamical transition from the ground state of the double well to the perturbed ground state (in which the cat is typically either dead or alive, depending on the details of the perturbation). Numerical simulations show that, in an individual experiment, certain (especially adiabatically rising) perturbations may (quite literally) cause the collapse of the wavefunction in the classical limit. Thus we combine the technical and conceptual virtues of dynamical collapse models a la GRW (which do solve the measurement problem) with those of decoherence (in that our perturbations come from the environment) without sharing their drawbacks: although single measurement outcomes are obtained (instead of merely diagonal reduced density matrices), no modification of quantum mechanics is needed