arXiv subjects

E. Shpagina

Publications and source records attributed to E. Shpagina.

3 recordsLinked to original sources

arXiv2021

Thermodynamics at zero temperature: inequalities for the ground state of a quantum many-body system

We prove that for a single-component many-body system at zero temperature the inequality $E_{\rm int} \leq\, P\,V$ holds, where $E_{\rm int}$ is the interaction energy, $P$ is pressure and $V$ is volume. This inequality is proven under rather general assumptions with the use of Anderson-type bound relating ground state energies of systems with different numbers of particles. We also consider adding impurity particles to the system and derive inequalities on the chemical potential of the impurity and binding energy of the bound state of two impurities.

cond-mat.stat-mech↗

arXiv2020

Squaring parametrization of constrained and unconstrained sets of quantum states

A mixed quantum state is represented by a Hermitian positive semi-definite operator $ρ$ with unit trace. The positivity requirement is responsible for a highly nontrivial geometry of the set of quantum states. A known way to satisfy this requirement automatically is to use the map $ρ=τ^2 / \mathrm {tr} \, τ^2$, where $τ$ can be an arbitrary Hermitian operator. We elaborate a parametrization of the set of quantum states induced by the parametrization of the linear space of Hermitian operators by virtue of this map. In particular, we derive an equation for the boundary of the set. Further, we discuss how this parametrization can be applied to a set of quantum states constrained by some symmetry, or, more generally, some linear condition. As an example, we consider the parametrization of sets of Werner states of qubits.

quant-ph↗

arXiv2020

Merits of using density matrices instead of wave functions in the stationary Schrödinger equation for systems with symmetries

The stationary Schrödinger equation can be cast in the form $H ρ= E ρ$, where $H$ is the system's Hamiltonian and $ρ$ is the system's density matrix. We explore the merits of this form of the stationary Schrödinger equation, which we refer to as~SSE$_ρ$, applied to many-body systems with symmetries. For a nondegenerate energy level, the solution $ρ$ of the SSE$_ρ$ is merely a projection on the corresponding eigenvector. However, in the case of degeneracy $ρ$ is non-unique and not necessarily pure. In fact, it can be an arbitrary mixture of the degenerate pure eigenstates. Importantly, $ρ$ can always be chosen to respect all symmetries of the Hamiltonian, even if each pure eigenstate in the corresponding degenerate multiplet spontaneously breaks the symmetries. This and other features of the solutions of the SSE$_ρ$ can prove helpful by easing the notations and providing an unobscured insight into the structure of the eigenstates. We work out the SSE$_ρ$ for a general system of spins $1/2$ with Heisenberg interactions, and address simple systems of spins $1$. Eigenvalue problem for quantum observables other than Hamiltonian can also be formulated in terms of density matrices. As an illustration, we provide an analytical solution to the eigenproblem ${\bf S}^2 ρ=S(S+1) ρ$, where $\bf S$ is the total spin of $N$ spins $1/2$, and $ρ$ is chosen to be invariant under permutations of spins. This way we find an explicit form of projections to the invariant subspaces of ${\bf S}^2$.

quant-ph↗