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Reports on Mathematical Physics

Reduced density matrices of a system of N coupled oscillators 3. the eigenstructure of the p-particle matrix for the ground-state

S. Pruski, Jan F. Maćkowiak, O. Missuno

Published Sep 1, 1972• 17 citations
Publisher
DOI
10.1016/0034-4877(72)90009-2

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

John M. Herbert, John E. Harriman

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.

Physical Review A•Jan 16, 2002•27 citations

Considerations on Coupled Oscillator Entanglement Entropy

Francesco R. Ruggeri

In (1) an interesting calculation of the entanglement entropy of a coupled two oscillator system ground state |0><0| is presented using the von Neumann reduced density matrix. In this note we wish to focus on two points. First it is stated that the reduced density matrix (in a spatial basis) has an angular frequency w of sqrt(w+ w-) (where w+ and w- are defined later). We find this frequency is linked to wavefunctions before a trace over a second spatial variable is performed and that the final angular frequency is different. Secondly, it is stated that the reduced density matrix “happens” (1) to represent that of a single oscillator. We try to show that this follows from the mathematical form of the spatial density matrix written in terms of a product of two single oscillator spatial density matrices. The orthonormality of the basis functions Hn(xb) and Hj(xb) of the second spatial variable force the two basis functions Hn(xa)Hj(xa) to become Hn(xa)Hn(xa) and leads to a spatial density matrix of a single oscillator, so such a result is anticipated. We write a product of ground state oscillator wavefunctions in variables xa+xb and xa-xb in terms of the density of a single oscillator i.e. Sum over n exp(-En/T) Hn(xa)Hn(xb). We do a similar thing for the two similar functions in xa’ and xb’. Setting xb=xb’ and integrating leads to a single oscillator density form. The problem is that the w and T for this result do not match the w and T of the final result obtained by integrating the product of four ground states with xb=xb’. Thus there may be an issue with integrating before summing. In other words, summing Sum over nexp(-En/T) Hn(xa)Hn(xb) and separately summing Sum over j exp(-Ej/T) Hj(x’a)Hj(xb) and then multiplying and integrating over dxb yields a single oscillator density form, but for a particular w1,T1. Integrating Integral dxb Hj(xb)Hn(xb)= delta(n,j) before summing leads again to a single oscillator spatial density, but with a different w2 and T2. Only one set w,T for a single oscillator should map into the integrated result.

•Mar 19, 2022•0 citations
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.
In (1) an interesting calculation of the entanglement entropy of a coupled two oscillator system ground state |0><0| is presented using the von Neumann reduced density matrix. In this note we wish to focus on two points. First it is stated that the reduced density matrix (in a spatial basis) has an angular frequency w of sqrt(w+ w-) (where w+ and w- are defined later). We find this frequency is linked to wavefunctions before a trace over a second spatial variable is performed and that the final angular frequency is different. Secondly, it is stated that the reduced density matrix “happens” (1) to represent that of a single oscillator. We try to show that this follows from the mathematical form of the spatial density matrix written in terms of a product of two single oscillator spatial density matrices. The orthonormality of the basis functions Hn(xb) and Hj(xb) of the second spatial variable force the two basis functions Hn(xa)Hj(xa) to become Hn(xa)Hn(xa) and leads to a spatial density matrix of a single oscillator, so such a result is anticipated. We write a product of ground state oscillator wavefunctions in variables xa+xb and xa-xb in terms of the density of a single oscillator i.e. Sum over n exp(-En/T) Hn(xa)Hn(xb). We do a similar thing for the two similar functions in xa’ and xb’. Setting xb=xb’ and integrating leads to a single oscillator density form. The problem is that the w and T for this result do not match the w and T of the final result obtained by integrating the product of four ground states with xb=xb’. Thus there may be an issue with integrating before summing. In other words, summing Sum over nexp(-En/T) Hn(xa)Hn(xb) and separately summing Sum over j exp(-Ej/T) Hj(x’a)Hj(xb) and then multiplying and integrating over dxb yields a single oscillator density form, but for a particular w1,T1. Integrating Integral dxb Hj(xb)Hn(xb)= delta(n,j) before summing leads again to a single oscillator spatial density, but with a different w2 and T2. Only one set w,T for a single oscillator should map into the integrated result.