Let $L$ be a quantum logic and let $S(L)$ denote the set of all states on $L$ . (By a state we mean a nonnegative bounded $\sigma$ -additive measure, not necessarily normalized.) We ask whether every logic whose state space is a lattice has to be Boolean. We prove that this is so for finite logics and "projection logics." On the other hand, we show that there exist even concrete non-Boolean logics with a lattice state space (in fact, we prove that every countable concrete logic can be enlarged to a logic with a lattice state space). In the appendix we shortly consider the lattice properties of the set of observables and correct the paper [10].Let $L$ be a quantum logic and let $S(L)$ denote the set of all states on $L$. (By a state we mean a nonnegative bounded $\sigma$-additive measure, not necessarily normalized.) We ask whether every logic whose state space is a lattice has to be Boolean. We prove that this is so for finite logics and "projection logics." On the other hand, we show that there exist even concrete non-Boolean logics with a lattice state space (in fact, we prove that every countable concrete logic can be enlarged to a logic with a lattice state space). In the appendix we shortly consider the lattice properties of the set of observables and correct the paper [10].

Proceedings of the American Mathematical SocietyJan 1, 19875 citations

Density of states of the lattice Schwinger model

Irene Papaefstathiou, Daniel Robaina, J. I. Cirac +1 more

Using a recently introduced tensor network method, we study the density of states of the lattice Schwinger model, a standard testbench for lattice gauge theory numerical techniques, but also the object of recent experimental quantum simulations. We identify regimes of parameters where the spectrum appears to be symmetric and displays the expected continuum properties even for finite lattice spacing and number of sites. However, we find that for moderate system sizes and lattice spacing of $ga\ensuremath{\sim}O(1)$, the spectral density can exhibit very different properties with a highly asymmetric form. We also explore how the method can be exploited to extract thermodynamic quantities.Using a recently introduced tensor network method, we study the density of states of the lattice Schwinger model, a standard testbench for lattice gauge theory numerical techniques, but also the object of recent experimental quantum simulations. We identify regimes of parameters where the spectrum appears to be symmetric and displays the expected continuum properties even for finite lattice spacing and number of sites. However, we find that for moderate system sizes and lattice spacing of $ga\ensuremath{\sim}O(1)$, the spectral density can exhibit very different properties with a highly asymmetric form. We also explore how the method can be exploited to extract thermodynamic quantities.

arXiv:2104.08170Physical review. D/Physical review. D.Jul 30, 20218 citations

State Property Systems and Orthogonality

Diederik Aerts, Didier Deses

arXiv:quant-ph/0211095International Journal of Theoretical PhysicsJul 1, 20054 citations

Finite-Dimensional State Spaces

‎Louis Marchildon

Springer eBooksJan 1, 20020 citations

Real-space renormalization-group method for the ground-state properties of quantum lattice models

Sanjoy Kumar Sarker

A real-space renormalization-group transformation is constructed between blocks of linear dimensions $\mathrm{bL}$ and $L$ to study the ground-state properties of quantum lattice models. In this scheme it is possible (i) to estimate the errors due to the irrelevant variables, including boundary effects, and (ii) to test scaling explicitly, so that an accurate calculation can be performed over the entire range of the parameter space. Application to the one-dimensional Ising model in a tranverse field yields excellent results.A real-space renormalization-group transformation is constructed between blocks of linear dimensions $\mathrm{bL}$ and $L$ to study the ground-state properties of quantum lattice models. In this scheme it is possible (i) to estimate the errors due to the irrelevant variables, including boundary effects, and (ii) to test scaling explicitly, so that an accurate calculation can be performed over the entire range of the parameter space. Application to the one-dimensional Ising model in a tranverse field yields excellent results.

Physical review. B, Condensed matterSep 1, 19848 citations

Let

L

be a quantum logic and let

S(L)

denote the set of all states on

L

. (By a state we mean a nonnegative bounded

\sigma

-additive measure, not necessarily normalized.) We ask whether every logic whose state space is a lattice has to be Boolean. We prove that this is so for finite logics and "projection logics." On the other hand, we show that there exist even concrete non-Boolean logics with a lattice state space (in fact, we prove that every countable concrete logic can be enlarged to a logic with a lattice state space). In the appendix we shortly consider the lattice properties of the set of observables and correct the paper [10].

A real-space renormalization-group transformation is constructed between blocks of linear dimensions

\mathrm{bL}

and

L

to study the ground-state properties of quantum lattice models. In this scheme it is possible (i) to estimate the errors due to the irrelevant variables, including boundary effects, and (ii) to test scaling explicitly, so that an accurate calculation can be performed over the entire range of the parameter space. Application to the one-dimensional Ising model in a tranverse field yields excellent results.