The cumulant two-particle reduced density matrix as a measure of electron correlation and entanglement

Abstract

Several measures of electron correlation are compared based on two criteria: (i) the presence of a unique mapping between the reduced variables in the measure and the many-electron wave function and (ii) the linear scaling of the measure and its variables with system size. We propose the squared Frobenius norm of the cumulant part of the two-particle reduced density matrix (2-RDM) as a measure of electron correlation that satisfies these criteria. An advantage of this cumulant-based norm is its ability to measure the correlation from spin entanglement, which is not contained in the correlation energy. Alternative measures based on the 2-RDM, such as the von Neumann entropy, do not scale linearly with system size. Properties of the measures are demonstrated with Be, F(2), HF, N(2), and a hydrogen chain.

On the measure of electron correlation and entanglement in quantum chemistry based on the cumulant of the second-order reduced density matrix

“…It has been reported that correlation measures only depending on the 1-RDM exhibit the same degree of correlation for systems states differing significantly, 22 which is obviously unphysical. However, in quantum systems with twoparticle interactions, there is a one to one mapping between the 2-RDM and the wave function for nondegenerate ground states 48 or for degenerate states which can be distinguished by at least one operator containing just two-particle interactions.…”

“…33 Hence, Ref. 22 concluded from this mapping that the 2-RDM is needed for a correlation measure of atomic and molecular systems in order to avoid the above mentioned drawback. For these reasons and according to our previous studies involving the quantities defined by Eqs.…”

“…26 Antecedents in this sense may be found in Ref. 22, where a Frobenius norm of the spin-dependent cumulant matrix has been proposed to provide a quantitative measure of correlation from spin entanglement of the state; up to what we know it is the first definition in the scenario of the two-particle reduced density matrix. Another interesting approach based on entropic concepts to define entanglement in molecular systems has been proposed in Ref.…”

“…25,37,38 The matrix ⌫ possesses the fundamental property that is a size-consistent quantity. 22,39,40 This matrix may be contracted in two different ways as has been published elsewhere 14,30,41 and it therefore shows its connection with the elements of the density matrix of effectively unpaired electrons, u j i , defined as [16][17][18]31,42 …”

“…3,10,11 Nevertheless, electron densities are not enough for a complete description of the electron distribution of molecular systems, especially those undergoing chemical reactions and electron rearrangements ͑nonequilibrium geometrical conformations͒ such as formation and breaking of bonds. In these cases, higher-order density matrices are relevant [20][21][22][23] and particularly the second-order reduced density matrices ͑2-RDM͒ are essentially needed. 3,10 These latter types of system states permit to introduce a natural linkage between the quantum chemistry phenomena of correlation and entanglement 23 ͑the correlation that has not a classical counterpart͒.…”