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 /tr τ 2 , where τ can be an arbitrary Hermitian operator. We elaborate the 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. *
A variational upper bound on the ground state energy Egs of a quantum system, Egs Ψ|H|Ψ , is well-known (here H is the Hamiltonian of the system and Ψ is an arbitrary wave function). Much less known are variational lower bounds on the ground state. We consider one such bound which is valid for a many-body translation-invariant lattice system. Such a lattice can be divided into clusters which are identical up to translations. The Hamiltonian of such a system can be written as H = M i=1 Hi, where a term Hi is supported on the i'th cluster. The bound reads Egs M inf ρ cl ∈S G cl tr cl ρ cl H cl , where S G cl is some wisely chosen set of reduced density matrices of a single cluster. The implementation of this latter variational principle can be hampered by the difficulty of parameterizing the set M, which is a necessary prerequisite for a variational procedure. The root cause of this difficulty is the nonlinear positivity constraint ρ > 0 which is to be satisfied by a density matrix. The squaring parametrization of the density matrix, ρ = τ 2 / tr τ 2 , where τ is an arbitrary (not necessarily positive) Hermitian operator, accounts for positivity automatically. We discuss how the squaring parametrization can be utilized to find variational lower bounds on ground states of translation-invariant many-body systems. As an example, we consider a one-dimensional Heisenberg antiferromagnet.arXiv:1902.09246v1 [quant-ph]
The stationary Schrödinger equation can be cast in the form Hρ = Eρ, 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 SSEρ, applied to many-body systems with symmetries. For a nondegenerate energy level, the solution ρ of the SSEρ is merely a projection on the corresponding eigenvector. However, in the case of degeneracy ρ in nonunique 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 SSEρ can prove helpful by easing the notations and providing an unobscured insight into the structure of the eigenstates. We work out the SSEρ for the system of spins 1/2 with Heisenberg interactions. Eigenvalue problem for quantum observables other than Hamiltonian can also be formulated in terms of density matrices. We provide an analytical solution to one of them, S 2 ρ = S(S + 1)ρ, where 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 S 2 . Finally, we note that the anti-Hermitian part of the SSEρ can be used to construct sum rules for temperature correlation functions, and provide an example of such sum rule. 03J 12 3J 23 3J 13 J 12 −2J 12 J 23 J 13 J 23 J 12 −2J 23 J 13 J 13 J 12 J 23 −2J 13
В физике большую роль играют системы квантовых спинов $1/2$ с изотропным гейзенберговским взаимодействием. При изучении таких систем может быть полезно иметь полный и притом непереполненный базис операторов, каждый из которых обладает симметрией гамильтониана, т.е. инвариантен относительно вращений (глобальных $\mathrm {SU}(2)$-преобразований матриц Паули). В настоящей статье сформулирован алгоритм построения такого базиса. Алгоритм реализован в программе Wolfram Mathematica.
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