A new representation for the density matrix

G. Ramachandran, M. V. N. Murthy

E. E. Perepelkin, Б. И. Садовников, Наталья Германовна Иноземцева +1 more

In this paper, a new representation of the Wigner function for a quantum system in the phase space is proposed. The new representation is of the form $W=\operatorname{Sp}\left[ ρ\mathcal{W} \right]$, where $ρ$ is the density matrix, and $\mathcal{W}$ is the universal density matrix. The density matrix $ρ$ for each quantum system is different, and the universal matrix $\mathcal{W}$ is the same for any quantum system. Thus, the matrix $\mathcal{W}$ has a fundamental character. In the work, the elements of the universal matrix $\mathcal{W}$ were found explicitly and their properties were investigated. The diagonal elements of the matrix $\mathcal{W}$ are the Wigner functions of the harmonic oscillator, which do not introduce dissipation into the quantum system. The off-diagonal elements of the matrix $\mathcal{W}$ contain frequency oscillations responsible for dissipations in the quantum systems.

Mary Beth Ruskai, John E. Harriman

The $N$-representability problem for the first- and second-order density matrices is considered when $N$ is an odd integer. Attention is restricted to those density matrices derivable from functions whose natural spin orbitals have certain paired properties. Neither pairing of spatial orbitals nor special spin properties are required in general, but they are discussed as special cases. New sufficient conditions for $N$-representability of the first-order density matrix are given. The second-order density matrix is studied in detail. Special cases in which the $N$-representability problem for the second-order density matrix can be solved are considered.

Ravinder Nath

Hari Prakash, Naresh Chandra, Vachaspati

Some new non-diagonal representations for density operators of radiation in terms of coherent states are given. These involve only two integrations per mode. The weight functions are expressed in terms of coherence functions and of occupation number space matrix elements of the density operators. When a compact form of the weight function in the diagonal representation does not exist, one of these new representations may become preferable. One such example is discussed.

K. Husimi, Yoshihiro Kitano, Toshiyuki Nishiyama