The geometry of the Quantum State Space, described by Bloch vectors, is a very intricate one. A deeper understanding of this geometry could lead to the solution of some difficult problems in Quantum Foundations and Quantum Information such as the existence of SIC-POVMs and the cardinality of the maximal set of MUBs. In this paper we show that the geometry of quantum states can be described by the probability distributions that quantum states induce over the outcomes of symmetric POVMs not necessarily of arbitrary rank or informationally complete. We also describe the properties of these symmetric POVMs.The geometry of the Quantum State Space, described by Bloch vectors, is a very intricate one. A deeper understanding of this geometry could lead to the solution of some difficult problems in Quantum Foundations and Quantum Information such as the existence of SIC-POVMs and the cardinality of the maximal set of MUBs. In this paper we show that the geometry of quantum states can be described by the probability distributions that quantum states induce over the outcomes of symmetric POVMs not necessarily of arbitrary rank or informationally complete. We also describe the properties of these symmetric POVMs.

arXiv:1309.7309Sep 27, 20131 citation

Algebraic geometry tools for the study of entanglement: an application to spin squeezed states

Alessandra Bernardi, Iacopo Carusotto

A short review of algebraic geometry tools for the decomposition of tensors and polynomials is given from the point of view of applications to quantum and atomic physics.Examples of application to assemblies of indistinguishable twolevel bosonic atoms are discussed using modern formulations of the classical Sylvester algorithm for the decomposition of homogeneous polynomials in two variables.In particular, the symmetric rank and symmetric border rank of spin squeezed states are calculated as well as their Schrödinger-cat-like decomposition as the sum of macroscopically different coherent spin states; Fock states provide an example of states for which the symmetric rank and the symmetric border rank are different.A short review of algebraic geometry tools for the decomposition of tensors and polynomials is given from the point of view of applications to quantum and atomic physics.Examples of application to assemblies of indistinguishable twolevel bosonic atoms are discussed using modern formulations of the classical Sylvester algorithm for the decomposition of homogeneous polynomials in two variables.In particular, the symmetric rank and symmetric border rank of spin squeezed states are calculated as well as their Schrödinger-cat-like decomposition as the sum of macroscopically different coherent spin states; Fock states provide an example of states for which the symmetric rank and the symmetric border rank are different.

arXiv:1109.0221Journal of Physics A Mathematical and TheoreticalFeb 23, 201224 citations

Geometry of the set of mixed states

Guillaume Aubrun, Stanisław Szarek

Mathematical surveys and monographsAug 30, 20170 citations

Projective construction of the Zk Read-Rezayi fractional quantum Hall states and their excitations on the torus geometry

Cécile Repellin, Titus Neupert, Bogdan Andrei Bernevig +1 more

Multilayer fractional quantum Hall wave functions can be used to construct the non-Abelian states of the ${\mathbb{Z}}_{k}$ Read-Rezayi series upon symmetrization over the layer index. Unfortunately, this construction does not yield the complete set of ${\mathbb{Z}}_{k}$ ground states on the torus. We develop an alternative projective construction of ${\mathbb{Z}}_{k}$ Read-Rezayi states that complements the existing one. On the multilayer torus geometry, our construction consists of introducing twisted boundary conditions connecting the layers before performing the symmetrization. We give a comprehensive account of this construction for bosonic states, and numerically show that the full ground state and quasihole manifolds are recovered for all computationally accessible system sizes. Furthermore, we analyze the neutral excitation modes above the Moore-Read on the torus through an extensive exact diagonalization study. We show numerically that our construction can be used to obtain excellent approximations to these modes. Finally, we extend our symmetrization scheme to the plane and sphere geometries.Multilayer fractional quantum Hall wave functions can be used to construct the non-Abelian states of the ${\mathbb{Z}}_{k}$ Read-Rezayi series upon symmetrization over the layer index. Unfortunately, this construction does not yield the complete set of ${\mathbb{Z}}_{k}$ ground states on the torus. We develop an alternative projective construction of ${\mathbb{Z}}_{k}$ Read-Rezayi states that complements the existing one. On the multilayer torus geometry, our construction consists of introducing twisted boundary conditions connecting the layers before performing the symmetrization. We give a comprehensive account of this construction for bosonic states, and numerically show that the full ground state and quasihole manifolds are recovered for all computationally accessible system sizes. Furthermore, we analyze the neutral excitation modes above the Moore-Read on the torus through an extensive exact diagonalization study. We show numerically that our construction can be used to obtain excellent approximations to these modes. Finally, we extend our symmetrization scheme to the plane and sphere geometries.

arXiv:1504.00367Physical Review BSep 14, 201520 citations

Geometry of State Spaces

Armin Uhlmann, Bernd Crell

Lecture notes in physicsNov 7, 200820 citations

Multilayer fractional quantum Hall wave functions can be used to construct the non-Abelian states of the

{\mathbb{Z}}_{k}

Read-Rezayi series upon symmetrization over the layer index. Unfortunately, this construction does not yield the complete set of

{\mathbb{Z}}_{k}

ground states on the torus. We develop an alternative projective construction of

{\mathbb{Z}}_{k}

Read-Rezayi states that complements the existing one. On the multilayer torus geometry, our construction consists of introducing twisted boundary conditions connecting the layers before performing the symmetrization. We give a comprehensive account of this construction for bosonic states, and numerically show that the full ground state and quasihole manifolds are recovered for all computationally accessible system sizes. Furthermore, we analyze the neutral excitation modes above the Moore-Read on the torus through an extensive exact diagonalization study. We show numerically that our construction can be used to obtain excellent approximations to these modes. Finally, we extend our symmetrization scheme to the plane and sphere geometries.