From single-particle physical distributions to probabilistic measures of two-particle entanglement

TL;DR

This paper develops an inversion method to infer two-particle entanglement information from measurable one-particle distributions in a one-dimensional quantum system, advancing the analysis of quantum correlations.

Contribution

It introduces a novel inversion technique based on a shell-like expansion to extract entanglement data from one-particle distributions, tailored for a harmonically confined Heisenberg model.

Findings

The method successfully relates one-particle distributions to entanglement measures.

An approximate optimization scheme is proposed for various inter-particle interactions.

The approach is grounded in exact mathematical constraints derived from the model.

Abstract

An inversion method is formulated for extracting entanglement-related information on two-particle interactions in a one-dimensional system from measurable one-particle position- and momentum-distribution functions. The method is based on a shell-like expansion of these norm-1 measured quantities in terms of product states taken from a parametric orthonormal complete set. The mathematical constraints deduced from these point-wise expansions are restricted by the underlying physics of our harmonically confined and interacting Heisenberg model. Based on these exact results, we introduce an approximate optimization scheme for different inter-particle interactions and discuss it from the point of view of entropic correlation measures.

Equations

$${\hat{H}}_0(x)=-\frac{1}{2}\frac{d^2}{d{x}^2}+\frac{1}{2}{\omega }_0^2{x}^2,$$

$$n(k)=\frac{1}{\sqrt{2\pi }}{e}^{-k^2/(4{\omega }_0)}.$$

$$f(k)=[{\varphi }_0(k,{\omega }_0){]}^2={\left[{\left(\frac{1}{\pi {\omega }_0}\right)}^{1/4}{e}^{-k^2/(2{\omega }_0)}\right]}^2.$$

$$<K_1>=\frac{1}{2}{\int }_{-\infty }^{\infty }dx{\left|\frac{d{\varphi }_o(x,\omega )}{dx}\right|}^2=\frac{1}{4}\omega$$

$$<K_2>={\int }_{-\infty }^{\infty }\frac{1}{2}{k}^{2}f(k,\Omega )dk=\frac{1}{4}\Omega$$

$${\varphi }_1(x,{\omega }_s)={\left(\frac{{\omega }_s}{\pi }\right)}^{1/4}{e}^{-\frac{1}{2}{\omega }_s{x}^2}$$

$${\varphi }_2(k,{\Omega }_s)={\left(\frac{1}{\pi {\Omega }_s}\right)}^{1/4}{e}^{-\frac{1}{2}\frac{k^2}{{\Omega }_s}}.$$

$$N(x,{\omega }_s)=\sum _{m=0}^{\infty }(1-Z)Z^m[{\varphi }_m(\sqrt{\bar{\omega }}x){]}^2$$

$$F(k,{\Omega }_s)=\sum _{m=0}^{\infty }(1-Z)Z^m[{\varphi }_m(k/\sqrt{\bar{\omega }}){]}^2$$

$${\varphi }_m(\sqrt{\alpha }u)={\left(\frac{\alpha }{\pi }\right)}^{1/4}\frac{1}{\sqrt{2^{m}m!}}{e}^{-\frac{1}{2}\alpha u^2}{H}_m(\sqrt{\alpha }u).$$

$${\omega }_s=\bar{\omega }\frac{1-Z}{1+Z}$$

$$\frac{1}{{\Omega }_s}=\frac{1}{\bar{\omega }}\frac{1-Z}{1+Z},$$

$$\Pi =\sum _{m=0}^{\infty }(P_m{)}^2=\frac{1-Z}{1+Z}=\sqrt{\frac{{\omega }_s}{{\Omega }_s}}\le 1,$$

$$S_R(q)=\frac{1}{1-q}\ln \frac{(1-Z{)}^q}{1-Z^q}$$

$$S_N=-{\left[q^2\frac{d}{dq}\left(\frac{1-q}{q}{S}_R(q)\right)\right]}_{q=1}=-\ln (1-Z)-\frac{Z}{1-Z}\ln Z.$$

$$\hat{H}(x_1,x_2)=-\frac{1}{2}\left(\frac{d^2}{d{x}_1^2}+\frac{d^2}{d{x}_2^2}\right)+\frac{1}{2}{\omega }_0^2(x_1^2+x_2^2)+v_H(|x_1-x_2|),$$

$${\Gamma }_1(x_1,x_2)={\varphi }_s(x_1){\varphi }_s(x_2)\times e^{-D[(x_1-x_2)/\sqrt{2}{]}^2}$$

$${\Gamma }_1(k_1,k_2)=\frac{1}{\sqrt{\pi ({\omega }_s+2D)}}{e}^{-\frac{1}{2}(k_1^2+k_2^2)\frac{{\omega }_s+D}{{\omega }_s({\omega }_s+2D)}}{e}^{+\frac{D{k}_1{k}_2}{{\omega }_s({\omega }_s+2D)}}$$

$${\varphi }_s(x)={\left[\frac{{\omega }_s}{\pi }\right]}^{1/4}{e}^{-\frac{1}{2}{\omega }_s{x}^2}$$

$$D=\frac{1}{4}\frac{({\omega }_1-{\omega }_2{)}^2}{{\omega }_1+{\omega }_2}\ge 0.$$

$$v_C(|x_1-x_2|)=\frac{\Lambda }{(x_1-x_2{)}^2}$$

Full text

I Motivation

Let us take the Hamiltonian, in atomic units, of a single harmonic oscillator

[TABLE]

the ground-state normalized solution to the Schrödinger equation is $\phi_{0}(x,\omega_{0})=(\omega_{0}/\pi)^{1/4}\exp(-\omega_{0}x^{2}/2)$. Its square gives the normalized density distribution function, $n(x)=[\phi_{0}(x)]^{2}$, whose Fourier transform is

[TABLE]

This function can be sampled by X-ray scattering, and by Fourier inversion, one may say that $n(x)$ is accessible experimentally. The normalized one-particle momentum density distribution, $f(k)$, is, on the other hand, connected to Compton scattering. The relation between these position and momentum-space distributions, i.e., between the two sets of experimental data, is one-to-one because $\phi_{0}(x)=\sqrt{n(x)}$ and $f(k)$ is given by

[TABLE]

From $\phi_{0}(x,\omega)$ and $\phi_{0}(k,\Omega)$ we can determine the corresponding kinetic energies

[TABLE]

[TABLE]

In the noninteracting case, $\omega=\omega_{0}$ and $\Omega=\omega_{0}$, these are equal, but in the interacting case below, ($\omega_{0}\Rightarrow{\Omega_{s}}$) only Eq.(5) gives the exact result. The definition in Eq.(4), even with the exact density ($\omega_{0}\Rightarrow{\omega_{s}}$) Dreizler90 gives $[<K_{1}>/<K_{2}>]<1$.

When the Gaussian function $f(k)$ is characterized by $\Omega_{s}\neq{\omega_{0}}$, instead of $\omega_{s}$ used for the density distribution, the one-to-one mapping is lost. In case of harmonic external confinement, one should conclude from this observation that we have at least two similar particles in the ground-state under confinement and, moreover, these particles are dynamically correlated. Is there, in the physically most important interacting case, still something useful which one can extract from the above two observables? How could a proper mathematical recipe be formulated? These are the questions motivating this note on the application of reduced information. We feel that examining the weakly interacting two-particle case is a conservative first step having general information-theoretic relevance.

II Decompositions, constraints, and optimization

Four our interacting system, there is no simple connection between the normalized position-space density $n(x)$ and the normalized momentum-space density $f(k)$, despite the fact that they are related by the Fourier transformation between the wave functions, and associated density matrices, in these spaces Dreizler90 . Furthermore, the reduced one-particle density matrices, the sources of $N(x)$ and $F(k)$ are two-variable functions. They contain more information than the corresponding one-variable probability densities, i.e., their diagonals. Besides, since correlation is encoded differently Harriman91 in the observables discussed above, a successful recipe for extracting information must rest on both densities. Such chameleon-like behavior in the observables requires care in their mathematical treatment. One can not simply follow the Duke of Gloucester who, in Shakespeare’s famous play Henry VI, stated: ”I can add colors to the chameleon”.

The details of the one-to-one correspondence outlined above for the one-particle case suggest that decompositions of $N(x)$ and $F(k)$, when properly normalized, into products of one-variable functions belonging to complete sets, and the Fourier-transformation $(\mathcal{F})$ link between these sets, could form the mathematical basis of the recipe. In the extraction of information, one requires spatially-independent decomposition weights. Since these weights are eigenvalues of the underlying one-body matrices they allow a detailed analysis of entanglement entropies. However, the two-variable eigenfunctions, i.e., the natural orbitals, are not directly accessible experimentally. Thus, physically important information, say the energy scales behind extensions of these optimal orbitals in real space, remains intact.

In this note we consider a weakly interacting two-particle system under common harmonic confinement $(1/2)\omega_{0}^{2}(x_{1}^{2}+x_{2}^{2})$. This is the standard condition, e.g., in recent experiments on optically trapped systems with controllable number of constituents Wenz13 . For this system we introduce, assuming $\omega_{s}\neq{\Omega_{s}}$, the two Gaussians

[TABLE]

[TABLE]

in order to model the correlated density distribution function, $N(x)=[\phi_{1}(x)]^{2}$, and the correlated momentum distribution function, $F(k)=[\phi_{2}(k)]^{2}$. The case $\omega_{0}=\omega_{s}=\Omega_{s}$ obviously corresponds to the noninteracting situation where, of course, $\phi_{2}(k)=\mathcal{F}[\phi_{1}(x)]$.

The desired product-representations of the experimental data-functions, $N(x)=[\phi_{1}(x)]^{2}$ and $F(k)=[\phi_{2}(k)]^{2}$, with common spatially-independent weighting coefficients necessary for a linear mapping, such as Fourier transformation, are given by

[TABLE]

[TABLE]

[TABLE]

Here $\bar{\omega}$ is an orbit-parameter. For Gaussian densities these Mehler’s Erdelyi53 representations are point-wise Riesz55 , and $\sum_{m=0}^{\infty}(1-Z)Z^{m}=1$. The constraints on the expansions are

[TABLE]

[TABLE]

from which $\bar{\omega}=\sqrt{\omega_{s}\Omega_{s}}$ and $Z=[1-\sqrt{\omega_{s}/\Omega_{s}}]/[1+\sqrt{\omega_{s}/\Omega_{s}}]\leq{1}$. Since we know from the physics Dreizler90 of kinetic energy that $[<K_{2}(\Omega_{s})>/<K_{1}(\omega_{s})>]\geq{1}$, we have $(\Omega_{s}/\omega_{s})\geq{1}$.

A useful probabilistic measure of correlation is the purity $\Pi$. From the properties of our normalized occupation numbers, $P_{m}\equiv{(1-Z)Z^{m}}$, this measure is given by

[TABLE]

in terms of the ratio of $\omega_{s}$ and $\Omega_{s}$, which characterize the experimentally accessible distributions in position and momentum spaces, respectively. Related, commonly applied information-theoretic quantities are Renyi70 Rényi’s ($R$) and von Neumann’s ($N$) entropies

[TABLE]

[TABLE]

Von Neumann’s $S_{N}$ is the entropy of thermodynamics. But, in agreement with an earlier remark Srednicki93 , the above measures depend solely on a ratio of physical parameters. For entropies, taken at arbitrary $q$ values, the orbit-extension parameter, $\bar{\omega}$, is not needed. Therefore, pure information-theoretic measures alone are not applicable directly to determine scale-dependent physical quantities. Determining the sign of the inter-particle interaction (see, below) could be a nontrivial problem for reverse engineering, due to duality Pipek09 ; Glasser13 ; Schilling14 .

To proceed in our realistic modeling of a confined system Wenz13 , we add to the Schrödinger Hamiltonian the tunable (via $\lambda$, see below) two-particle interaction

[TABLE]

When $v_{H}\neq{0}$, we get $\Omega_{s}\neq{\omega_{0}}$, $\omega_{s}\neq{\omega_{0}}$, $\Omega_{s}\neq{\omega_{s}}$. In order to have a rigorous foundation for understanding information-extraction from observables, we turn to the specific Heisenberg26 interaction $v_{H}(|x_{1}-x_{2}|=\lambda(\omega_{0}^{2}/2)(x_{1}-x_{2})^{2}$. Based on this interaction, we recently derived closed-form expressions Nagy17 for both two-variable one-matrices

[TABLE]

[TABLE]

where, with $\omega_{s}\equiv{2\omega_{1}\omega_{2}/(\omega_{1}+\omega_{2}})$, we introduced the following abbreviations

[TABLE]

[TABLE]

We derive $\omega_{1}=\omega_{0}$ and $\omega_{2}=\omega_{0}\sqrt{1+2\lambda}$ in the underlying Nagy17 normal-mode separation of the Schrödinger equation with Eq.(16). Thus, for the repulsive harmonic inter-particle interaction, the allowed range is $\lambda\in{(-0.5,0]}$. Both $\omega_{s}$ and $D$, and therefore $Z$, show a dual character Pipek09 ; Glasser13 ; Schilling14 . This means that to any allowed repulsive coupling there exists a corresponding attractive one with the same value for $Z$. Clearly, with Heisenberg’s inter-particle interaction model Heisenberg26 the measurable $N(x)=\Gamma_{1}(x,x)$ and $F(k)=\Gamma_{1}(k,k)$ are known theoretically. Therefore, in this case, we get $\Omega_{s}=\omega_{s}+2D=\omega_{1}+\omega_{2}$, by which the kinetic energy becomes $<K_{2}>=(1/4)(\omega_{s}+2D)>(1/4)\omega_{s}=<K_{1}>$. But, and this is crucial to information-extraction, by taking the diagonal of Eq.(17) to get a measurable distribution we have no separate access to $D(\omega_{1},\omega_{2})$ which produces non-idempotency.

Finally, we turn to an optimization procedure which may connect two Schrödinger Hamiltonians. One could replace, following a recent proposal Brabec13 , a realistic two-particle Hamiltonian having non-harmonic inter-particle interaction by the Heisenberg ($H$) Hamiltonian. For instance, one could apply total-energy correspondence as a constraint on such replacement. One may argue, of course, that the $\lambda$-coupling in Eq.(16) has been chosen qualitatively and it is this mapping correspondence which would allow it to be determined quantitatively. We expect, based on physical considerations, that such an optimized correspondence between two Hamiltonians can be reasonable only if the harmonically confined Wenz13 particles interact weakly. At that small coupling the non-idempotency driver scales as $D\sim{\lambda^{2}}$, i.e., the deviations of $\omega_{s}$ and $\Omega_{s}>\omega_{s}$ from $\omega_{0}$ are small. The associated entropies are small as well. However, at stronger couplings, one may get a serious problem by applying such an optimization scheme to information-theoretic measures.

We now quantify this problem by considering the attractive ($\lambda>0$) case in Eq. (16). Say we construct a more realistic model by taking for the inter-particle interaction

[TABLE]

with $\Lambda>0$. We focus here on the strong coupling limit Koscik15 ; Anna15 . From the prescribed equivalence of ground-state energies $E_{H}(\lambda\rightarrow{\infty})\propto{\sqrt{1+2\lambda}}$ and $E_{C}(\Lambda\rightarrow{\infty})\propto{\sqrt{1+4\Lambda}}$, with $v_{H}(\lambda)$ and $v_{C}(\Lambda)$ respectively, we get the simple correspondence $\lambda=2\Lambda$. However, with the singular interaction above. one gets, in the associated Wigner-crystal limit at strong coupling, $\Lambda$-independent occupation numbers Koscik15 ; Anna15 . Thus the purity is approximately $\Pi_{C}(\Lambda\rightarrow{\infty})\simeq{0.528}$. In the energetically optimized Heisenberg case, i.e., with a harmonic interaction $v_{H}(\lambda=2\Lambda)$, we obtain the different behavior, $\Pi_{H}(\Lambda)\simeq{\sqrt{2}/\Lambda^{1/4}}$ at $\Lambda\rightarrow{\infty}$.

This, seemingly, moderate numerical difference in an information-theoretic measure is related, physically, to crucially different behaviors of the underlying wave function as a function of the relative coordinate. Our quantitative observation at strong coupling is not in contradiction with the prediction Brabec13 which relies on perturbation theory. Clearly, one can only get a physically reasonable approximation for the linear entropy, $L_{C}=1-\Pi_{C}$, at small (i.e., perturbative) coupling within the proposed optimization framework.

III Summary

An inversion method is formulated for extracting entanglement-related information on two-particle interactions from measurable one-particle distribution functions in position and momentum spaces. The method is based on shell-like expansions of these measurable norm-1 quantities in terms of properly weighted product states taken from a parametric complete orthonormal set. It is found that without further physical details, encoded in the two-variable reduced one-particle density matrices, an unambiguous characterization of the inter-particle interaction is not possible by inverting such information.

We have, therefore, given a concrete answer to Pauli’s general question Ballentine98 of whether the position and momentum probability densities are sufficient to determine the statistical state operator. These distributions are not sufficient. One method for resolving the dual character in the sign of an inter-particle interaction is to make use of the dynamical evolution Nagy17 of the correlated state. In such evolution, one of the normal-mode frequencies, $\omega_{2}=\omega_{0}\sqrt{1+2\lambda}$ in the Heisenberg model, could be measurable via the corresponding breathing mode Brabec13 .

Based on exact results obtained with Heisenberg’s Hamiltonian, a recently suggested optimization procedure for introducing a different inter-particle interaction is formulated and analyzed quantitatively from the point of view of entropic correlation measures. This analysis shows that, as expected on physical grounds, an energy-based optimization scheme could be useful only at weak inter-particle couplings.

Acknowledgements.

The authors thank Professor Ricardo Díez Muino for the warm hospitality at the DIPC. The kind help of Professor Anna Okopinska is gratefully acknowledged. MLG acknowledges the financial support of MINECO (Project MTM2014-57129-C2-1-P) and Junta de Castilla y Leon (UIC 011).