Reconstruction Of Density Matrix Research Articles

Extensivity and the contracted Schrödinger equation

We provide an extensive formulation of the contracted Schrödinger equation and other reduced eigenvalue equations. Nonextensive (unconnected) terms in these equations cancel exactly, leading to completely connected one- and two-electron equations that together are equivalent to the Schrödinger equation. We discuss how these equations can be solved for the one- and two-electron cumulants. These cumulants yield a two-electron reduced density matrix that is necessarily size consistent, even for an approximate solution. A diagram technique, introduced to aid the formal manipulations, clarifies the connection between density matrix reconstruction and solution of the CSE.

Contraction relations for Grassmann products of reduced density matrices and implications for density matrix reconstruction

We consider, for systems of indistinguishable fermions, approximate reconstruction of the three- and four-particle reduced density matrices (RDMs) from the one- and two-particle RDMs, $\mathit{\ensuremath{\gamma}}$ and $\mathit{\ensuremath{\Gamma}}.$ Our ansatz for reconstructing the four-particle RDM is the linear combination $a(\mathit{\ensuremath{\Gamma}}\ensuremath{\wedge}\mathit{\ensuremath{\Gamma}})+b(\mathit{\ensuremath{\gamma}}\ensuremath{\wedge}\mathit{\ensuremath{\gamma}}\ensuremath{\wedge}\mathit{\ensuremath{\Gamma}})+c(\mathit{\ensuremath{\gamma}}\ensuremath{\wedge}\mathit{\ensuremath{\gamma}}\ensuremath{\wedge}\mathit{\ensuremath{\gamma}}\ensuremath{\wedge}\mathit{\ensuremath{\gamma}}),$ where ``\ensuremath{\wedge}'' denotes the antisymmetrized (Grassmann) product. This is a generalization of reconstruction functionals employed recently to perform direct RDM calculations without wave functions via the contracted Schr\"odinger equation. Here we consider relationships between the parameters a, b, and c that are required in order for the reconstruction functionals to respect the hierarchy of contraction relations between RDMs. To this end we establish several general theorems concerning contractions of antisymmetrized tensor products of $\mathit{\ensuremath{\gamma}},$ $\mathit{\ensuremath{\Gamma}},$ and various products thereof. The accuracy of proposed reconstruction functionals is evaluated using accurate density matrices for the ground state of Be.

Analysis of density matrix reconstruction in NMR quantum computing

Reconstruction of density matrices is important in NMR quantum computing. An analysis is made for a 2-qubit system by using the error matrix method. It is found that the state tomography method determines well the parameters that are necessary for reconstructing the density matrix in NMR quantum computations. Analysis is also made for a simplified state tomography procedure that uses fewer read-outs. The result of this analysis with the error matrix method demonstrates that a satisfactory accuracy in density matrix reconstruction can be achieved even in a measurement with the number of read-outs being largely reduced.

Pursuit ofN-representability for the contracted Schrödinger equation through density-matrix reconstruction

The solution of N-representability for the two-particle reduced density matrix (2-RDM) is intimately connected through the contracted Schr\"odinger equation with the problem of reconstructing the 3- and 4-RDM's from the 2-RDM [D. A. Mazziotti, Phys. Rev. A 57, 4219 (1998)]. We derive an explicit formula for building the 3-RDM from the 2-RDM which significantly improves the accuracy of the 3-RDM cumulant expansion. Calculations indicate that the improvement is better than that achieved by the correction of Nakatsuji and Yasuda (NY) [Phys. Rev. Lett. 76, 1039 (1996); Phys. Rev. A 56, 2648 (1997)]. We also present a more efficient formulation of the NY correction. Both the present and NY reconstructions are shown to satisfy the particle-hole duality. Comparison of the reconstruction methods is made through several molecules LiH, ${\mathrm{BeH}}_{2},$ ${\mathrm{BH}}_{3},$ ${\mathrm{H}}_{2}\mathrm{O},$ and ${\mathrm{CH}}_{4},$ as well as systems with more general pairwise correlation.

Homodyne reconstruction of density matrix in Fock-state basis: deterministic versus maximum likelihood approach

Abstract Quantum state reconstruction, consisting of mixing an unknown signal with the sequence of probe fields on the beam-splitter and the subsequent counting of the statistics of the output state, is considered here. Besides the standard method based on deterministic inversion of counted data the maximum likelihood estimation is applied. The comparison demonstrates the possible failure of the standard reconstruction procedure in some cases.

Homodyne reconstruction of density matrix in fock-state basis: Deterministic versus maximum likelihood approach

Quantum state reconstruction, consisting of mixing an unknown signal with the sequence of probe fields on the beam-splitter and the subsequent counting of the statistics of the output state, is considered here. Besides the standard method based on deterministic inversion of counted data the maximum likelihood estimation is applied. The comparison demonstrates the possible failure of the standard reconstruction procedure in some cases.

Density-matrix reconstruction by unbalanced homodyning

Recently a method for measuring quasidistributions of single-mode optical fields in unbalanced homo- dyne detection has been proposed [S. Wallentowitz and W. Vogel, Phys. Rev. A 53, 4528 (1996); K. Bana- szek and K. W\'odkiewicz, Phys. Rev. Lett. 76, 4344 (1996)]. We show that the scheme can be used for direct sampling of the signal-mode density matrix in the Fock basis. In this case it is sufficient to vary only the phase of the local oscillator, keeping its amplitude constant.

On density matrix reconstruction from measured distributions

A method to recover the density matrix of radiation states expressed by a finite superpositions of number states is presented. It starts from realistic, not fully efficient, heterodyne or double homodyne detection. Examples are given by means of numerical simulations.