A study has been carried out on the relationship between formal scattering theory and the density matrix formalism. The density operator is developed in terms of the scattering operator S and the observable square modulus matrix elements of S are shown to be equivalent to elements of the density matrix. The Møller wave operators are similarly treated and subsequently used in obtaining the density matrix expression for the transition matrix. Finally, using the latter, it is shown that the hierarchy of approximations to the density matrix yields equivalent results to those obtained from the Born series.A study has been carried out on the relationship between formal scattering theory and the density matrix formalism. The density operator is developed in terms of the scattering operator S and the observable square modulus matrix elements of S are shown to be equivalent to elements of the density matrix. The Møller wave operators are similarly treated and subsequently used in obtaining the density matrix expression for the transition matrix. Finally, using the latter, it is shown that the hierarchy of approximations to the density matrix yields equivalent results to those obtained from the Born series.

Molecular PhysicsJan 1, 19717 citations

Density Operator

Werner Stulpe

Compendium of Quantum PhysicsJan 1, 20090 citations

Exact renormalization group flow for matrix product density operators

Kohtaro Kato

Matrix product density operator (MPDO) provides an efficient tensor network representation of mixed states on one-dimensional quantum many-body systems. We study a real-space renormalization group transformation of MPDOs represented by a circuit of local quantum channels. We require that the renormalization group flow is exact, in the sense that it exactly preserves the correlation between the coarse-grained sites and is therefore invertible by another circuit of local quantum channels. Unlike matrix product states (MPS), which always have a well-defined isometric renormalization transformation, we show that general MPDOs do not necessarily admit a converging exact renormalization group flow. We then introduce a subclass of MPDOs with a well-defined renormalization group flow, and show the structure of the MPDOs in the subclass as a representation of a pre-bialgebra structure. As a result, such MPDOs obey generalized symmetry represented by matrix product operator algebras associated with the pre-bialgebra. We also discuss implications with the classification of mixed-state quantum phases.Matrix product density operator (MPDO) provides an efficient tensor network representation of mixed states on one-dimensional quantum many-body systems. We study a real-space renormalization group transformation of MPDOs represented by a circuit of local quantum channels. We require that the renormalization group flow is exact, in the sense that it exactly preserves the correlation between the coarse-grained sites and is therefore invertible by another circuit of local quantum channels. Unlike matrix product states (MPS), which always have a well-defined isometric renormalization transformation, we show that general MPDOs do not necessarily admit a converging exact renormalization group flow. We then introduce a subclass of MPDOs with a well-defined renormalization group flow, and show the structure of the MPDOs in the subclass as a representation of a pre-bialgebra structure. As a result, such MPDOs obey generalized symmetry represented by matrix product operator algebras associated with the pre-bialgebra. We also discuss implications with the classification of mixed-state quantum phases.

arXiv:2410.22696Oct 30, 20241 citation

Inverse problem for the time-dependent density matrix

H. L. Frisch, Pál Ruján

The solution of the inverse problem for the time-dependent density matrix means finding the time-dependent Hamiltonian operator $\stackrel{^}{H}$ corresponding to a given time-dependent density matrix $\stackrel{^}{\ensuremath{\rho}}$. We point out that the standard requirements for the time-dependent density operator imply a number of compatibility relations. Accordingly, a given $N\ifmmode\times\else\texttimes\fi{}N$ density matrix might result from a whole family of Hamiltonians, characterized by $N$ free, real parameters. The cases of a 2\ifmmode\times\else\texttimes\fi{}2 and 3\ifmmode\times\else\texttimes\fi{}3 density matrix and of the free-fermion chain are worked out explicitly.The solution of the inverse problem for the time-dependent density matrix means finding the time-dependent Hamiltonian operator $\stackrel{^}{H}$ corresponding to a given time-dependent density matrix $\stackrel{^}{\ensuremath{\rho}}$. We point out that the standard requirements for the time-dependent density operator imply a number of compatibility relations. Accordingly, a given $N\ifmmode\times\else\texttimes\fi{}N$ density matrix might result from a whole family of Hamiltonians, characterized by $N$ free, real parameters. The cases of a 2\ifmmode\times\else\texttimes\fi{}2 and 3\ifmmode\times\else\texttimes\fi{}3 density matrix and of the free-fermion chain are worked out explicitly.

Physical Review AApr 1, 19823 citations

N

free, real parameters. The cases of a 2\ifmmode\times\else\texttimes\fi{}2 and 3\ifmmode\times\else\texttimes\fi{}3 density matrix and of the free-fermion chain are worked out explicitly.