arXiv subjects

Alexander Klyachko

Publications and source records attributed to Alexander Klyachko.

9 recordsLinked to original sources

arXiv2012

Rank Reduction for the Local Consistency Problem

We address the problem of how simple a solution can be for a given quantum local consistency instance. More specifically, we investigate how small the rank of the global density operator can be if the local constraints are known to be compatible. We prove that any compatible local density operators can be satisfied by a low rank global density operator. Then we study both fermionic and bosonic versions of the N-representability problem as applications. After applying the channel-state duality, we prove that any compatible local channels can be obtained through a global quantum channel with small Kraus rank.

quant-ph↗

arXiv2008

Dynamic symmetry approach to entanglement

In this lectures I explain a connection between geometric invariant theory and entanglement, and give a number of examples how this approach works.

quant-ph↗

arXiv2005

Measurable entanglement

Amount of entanglement carried by a quantum bipartite state is usually evaluated in terms of concurrence (see Ref. 1). We give a physical interpretation of concurrence that reveals a way of its direct measurement and discuss possible generalizations.

quant-ph↗

arXiv2005

Quantum marginal problem and N-representability

A variant of the quantum marginal problem was known from early sixties as N-representability problem. In 1995 it was designated by National Research Council of USA as one of ten most prominent research challenges in quantum chemistry. In spite of this recognition the progress was very slow, until a couple of years ago the problem came into focus again, now in framework of quantum information theory. In the paper I give an account of the recent development.

quant-ph↗

arXiv2004

Single-particle entanglement

Using the approach to quantum entanglement based on the quantum fluctuations of observables, we show the existence of perfect entangled states of a single "spin-1" particle. We give physical examples related to the photons, condensed matter physics, and particle physics.

quant-ph↗

arXiv2004

Quantum marginal problem and representations of the symmetric group

We discuss existence of mixed state of multicomponent system with given spectrum and given reduced density matrices. We give a complete solution of the problem in terms of linear inequalities on the spectra, accompanied with extensive tables of marginal inequalities, including arrays up to 4 qubits. In the second part of the paper we pursue another approach based on reduction of the problem to representation theory of the symmetric group.

quant-ph↗

arXiv2003

Vector bundles, linear representations, and spectral problems

This paper is based on my talk at ICM on recent progress in a number of classical problems of linear algebra and representation theory, based on new approach, originated from geometry of stable bundles and geometric invariant theory.

math.RT↗

arXiv2002

Robust Entanglement in Atomic Systems via Lambda-Type Processes

It is shown that the system of two three-level atoms in $Λ$ configuration in a cavity can evolve to a long-lived maximum entangled state if the Stokes photons vanish from the cavity by means of either leakage or damping. The difference in evolution picture corresponding to the general model and effective model with two-photon process in two-level system is discussed.

quant-ph↗

arXiv2002

Coherent states, entanglement, and geometric invariant theory

The main objective of the paper is to unveil an adequate mathematics hidden behind entanglement, that is Geometric Invariant Theory. More specifically relation between these two subjects can be described by the following theses. (i) Total variance of completely entangled state is maximal. (ii) This distinguishes the state as a minimal vector in its orbit under action of complexified dynamic group. (iii) An ultimate aim of Geometric Invariant Theory is a description of complex orbits and their minimal vectors. It suggests that noncompletely entangled states are just GIT semistable vectors.

quant-ph↗