Multi-body entanglement and information rearrangement in nuclear many-body systems: a study of the Lipkin–Meshkov–Glick model

Abstract

We examine how effective-model-space (EMS) calculations of nuclear many-body systems rearrange and converge multi-particle entanglement. The generalized Lipkin–Meshkov–Glick (LMG) model is used to motivate and provide insight for future developments of entanglement-driven descriptions of nuclei. The effective approach is based on a truncation of the Hilbert space together with a variational rotation of the qubits (spins), which constitute the relevant elementary degrees of freedom. The non-commutivity of the rotation and truncation allows for an exponential improvement of the energy convergence throughout much of the model space. Our analysis examines measures of correlations and entanglement, and quantifies their convergence with increasing cut-off. We focus on one- and two-spin entanglement entropies, mutual information, and n-tangles for $n=2,4$ to estimate multi-body entanglement. The effective description strongly suppresses entropies and mutual information of the rotated spins, while being able to recover the exact results to a large extent with low cut-offs. Naive truncations of the bare Hamiltonian, on the other hand, artificially underestimate these measures. The n-tangles in the present model provide a basis-independent measures of n-particle entanglement. While these are more difficult to capture with the EMS description, the improvement in convergence, compared to truncations of the bare Hamiltonian, is significantly more dramatic. We conclude that the low-energy EMS techniques, that successfully provide predictive capabilities for low-lying observables in many-body systems, exhibit analogous efficacy for quantum correlations and multi-body entanglement in the LMG model, motivating future studies in nuclear many-body systems and effective field theories relevant to high-energy physics and nuclear physics.

Appendices

A Matrix elements of the bare and effective Hamiltonians

The bare Hamiltonian of the LMG model

$$\begin{aligned} \hat{H}= & {} \varepsilon \hat{J}_z - \frac{V}{2} \left( \hat{J}_+^2 + \hat{J}_-^2 \right) - \frac{W}{2} (\hat{J}_+ \hat{J}_- + \hat{J}_- \hat{J}_+ -\hat{N}) \; , \nonumber \\ \end{aligned}$$

(54)

has matrix elements in the original $\{ \left| N_+\right\rangle \equiv \left| J,M\right\rangle \}$ basis that read

$$\begin{aligned} {\langle N_+' | \hat{H} | N_+\rangle }= & {} \left( \varepsilon \, M -\frac{W}{2} \left[ C_+(M)^2 + C_-(M)^2 - N \right] \right) \delta _{N_+',N_+} \nonumber \\{} & {} - \frac{V}{2} \Bigl [ C_+(M) C_+(M+1) \, \delta _{N_+',N_+ +2} \nonumber \\{} & {} +C_-(M) C_-(M-1) \, \delta _{N_+',N_+-2} \Bigr ] \; , \end{aligned}$$

(55)

where $M = N_+ - J$ and $J=\frac{N}{2}$, and where we have defined

$$\begin{aligned} C_\pm (M) = \sqrt{J(J+1) - M(M\pm 1)} \; . \end{aligned}$$

(56)

The Hamiltonian acting in the effective model space can be obtained by performing a rotation $\hat{U}(\beta )$ around the y axis of the collective (quasi-)spin operators $\hat{J}_\alpha$ given in Eq. (3), by an angle $\beta$. The rotated operators $\hat{J}_\alpha (\beta )$ are then related to the original ones by

$$\begin{aligned} \hat{J}_z= & {} \cos (\beta ) \hat{J}_z(\beta ) + \frac{1}{2} \sin \beta \left( \hat{J}_+(\beta ) + \hat{J}_-(\beta )\right) \; , \end{aligned}$$

(57)

$$\begin{aligned} \hat{J}_+= & {} \frac{1}{2} \Bigl [ - 2 \sin (\beta ) \hat{J}_z(\beta ) \nonumber \\{} & {} + \bigl ( \cos \beta +1 \bigr ) \hat{J}_+(\beta ) + \bigl ( \cos \beta -1 \bigr ) \hat{J}_-(\beta ) \Bigr ] \; , \end{aligned}$$

(58)

$$\begin{aligned} \hat{J}_-= & {} (\hat{J}_+)^\dagger = \frac{1}{2} \Bigl [ - 2\sin \beta \hat{J}_z(\beta ) \nonumber \\{} & {} + \bigl ( \cos \beta +1 \bigr ) \hat{J}_-(\beta ) + \bigl ( \cos \beta -1 \bigr ) \hat{J}_+(\beta ) \Bigr ] \; . \end{aligned}$$

(59)

The resulting effective Hamiltonian then becomes

$$\begin{aligned} \hat{H}(\beta )= & {} \hat{U}^\dagger (\beta ) \hat{H} \hat{U}(\beta ) \nonumber \\= & {} \hat{H}_\varepsilon (\beta ) + \hat{H}_V(\beta ) + \hat{H}_W(\beta ) \; , \end{aligned}$$

(60)

where

$$\begin{aligned} H_\varepsilon (\beta )= & {} \varepsilon \; \left[ \cos \beta \hat{J}_z(\beta ) + \frac{1}{2} \sin \beta \bigl ( \hat{J}_+(\beta ) + \hat{J}_-(\beta ) \bigr ) \right] \; , \nonumber \\ \end{aligned}$$

(61)

$$\begin{aligned} \hat{H}_V(\beta )= & {} - \frac{V}{4} \Bigl [ \sin ^2\beta \bigl ( 4 \hat{J}_z(\beta )^2 - \{ \hat{J}_+(\beta ), \hat{J}_-(\beta ) \} \bigr ) \nonumber \\{} & {} - 2 \sin \beta \cos \beta \bigl ( \{ \hat{J}_z(\beta ),\hat{J}_+(\beta )\} \!+\! \{ \hat{J}_z(\beta ), \hat{J}_-(\beta ) \} \bigr ) \nonumber \\{} & {} + (1 + \cos ^2\beta ) \bigl (\hat{J}_+(\beta )^2 + \hat{J}_-(\beta )^2 \bigr ) \Bigr ] \; , \end{aligned}$$

(62)

and

$$\begin{aligned} \hat{H}_W(\beta )= & {} - \frac{W}{4} \Bigl [ 4 \sin ^2\beta \hat{J}_z(\beta )^2 \nonumber \\{} & {} + (1 + \cos ^2\beta ) \{ \hat{J}_+(\beta ), \hat{J}_-(\beta ) \}- 2 \hat{N} \nonumber \\{} & {} - 2 \sin \beta \cos \beta \bigl ( \{ \hat{J}_z(\beta ),\hat{J}_+(\beta )\} \!+\! \{ \hat{J}_z(\beta ), \hat{J}_-(\beta ) \} \bigr ) \nonumber \\{} & {} - \sin ^2\beta \bigl (\hat{J}_+(\beta )^2 + \hat{J}_-(\beta )^2 \bigr ) \Bigr ] \; .\nonumber \\ \end{aligned}$$

(63)

Matrix elements of the effective Hamiltonian in Eq. (60) in the basis $\{ N_+, \beta \}$ can now be computed. In contrast to the bare Hamiltonian in Eq. (54), the effective Hamiltonian can clearly couple configurations with values of $N_+$ differing by one unit, in addition to those differing by zero, or two units. Since the rotation preserves the value of J, we can use $M = N_+ - J = N_+ -\frac{N}{2}$, and write the non-zero matrix elements as

$$\begin{aligned}{} & {} {\langle N_+, \beta |\hat{H}(\beta ) | N_+, \beta \rangle }\nonumber \\{} & {} \quad = \varepsilon M \cos \beta -\frac{V}{4} \sin ^2\beta \left[ 4 M^2 - [C_+(M)^2 +C_-(M)^2] \right] \nonumber \\{} & {} \qquad -\frac{W}{4} \Big [ 4 \sin ^2\beta M^2 + (1 + \cos ^2\beta ) [C_+(M)^2 \nonumber \\ {}{} & {} \qquad +C_-(M)^2] - 2N \Big ] , \end{aligned}$$

(64)

$$\begin{aligned}{} & {} {\langle N_+\pm 1, \beta |\hat{H}(\beta ) | N_+, \beta \rangle } = \frac{\varepsilon }{2} \sin \beta \, C_\pm (M) \nonumber \\{} & {} \qquad + \frac{V+W}{4} \left[ 2 \sin \beta \cos \beta (2M \pm 1) C_\pm (M) \right] , \end{aligned}$$

(65)

and

$$\begin{aligned}{} & {} {\langle N_+\pm 2, \beta |\hat{H}(\beta ) | N_+, \beta \rangle }\nonumber \\{} & {} \quad = - \frac{V}{4} \Big [ (1+ \cos ^2 \beta ) \, C_\pm (M) C_\pm (M\pm 1) \Big ] \nonumber \\{} & {} \qquad + \frac{W}{4} \left[ \sin ^2 \beta \, C_\pm (M) \, C_\pm (M\pm 1) \right] \; . \end{aligned}$$

(66)

It is straightforward to observe that the bare Hamiltonian in Eq. (54) and the matrix elements in Eq. (55) are recovered when $\beta =0$.

B Derivations the n-tangles

In the case where the many-body state is real, i.e., $\left| \Psi ^*\right\rangle = \left| \Psi \right\rangle$, the n-tangles $\tau _n$ are the following,

$$\begin{aligned} \tau _n= & {} |{\langle \Psi | \hat{\sigma }_y^{\otimes n} |\Psi ^*\rangle }|^2 \ =\ |{\langle \Psi | \hat{\sigma }_y^{\otimes n} |\Psi \rangle }|^2 \; . \end{aligned}$$

(67)

Their calculation thus requires the evaluation of the expectation value of the tensor products of single-spin Pauli operators $\hat{\sigma }_y$. The fact that all spins are equivalent in the LMG model simplifies the derivation of such expectation values which can be obtained from averages of expectation values of collective operators $\hat{J_y} = \sum _p \hat{\sigma }_y^p/2$, where $\sigma _y^p$ acts on the single spin p.

For example, the 2-tangle requires the calculation of

$$\begin{aligned} {\langle \hat{\sigma }_y^p \otimes \hat{\sigma }_y^q \rangle }= & {} \frac{1}{N(N-1)} \sum _{p \ne q} {\langle \hat{\sigma }_y^p \otimes \hat{\sigma }_y^q \rangle } \nonumber \\= & {} \frac{1}{N(N-1)} \left[ \sum _{pq} {\langle \hat{\sigma }_y^p \otimes \hat{\sigma }_y^q \rangle } - \sum _p {\langle (\hat{\sigma }_y^p )^2 \rangle } \right] \nonumber \\= & {} \frac{1}{N(N-1)} \Bigl [ 4 {\langle \hat{J}_y^2\rangle } - N \Bigr ] \; , \end{aligned}$$

(68)

where we used the notation ${\langle .\rangle } = {\langle \Psi |.|\Psi \rangle }$, and the fact that $(\sigma _p)^2 =1$. One can further insert

$$\begin{aligned} {\langle \hat{J}_y^2\rangle }{} & {} = - \frac{1}{4} \left( {\langle \hat{J}_+^2 \rangle } +{\langle \hat{J}_-^2\rangle } - {\langle \hat{J}_+ \hat{J}_- + \hat{J}_- \hat{J}_+\rangle } \right) \nonumber \\{} & {} = - \frac{1}{4} \left( {\langle \hat{J}_+^2\rangle } +{\langle \hat{J}_-^2\rangle } + 2 {\langle \hat{J}_z^2\rangle } - N \left( \frac{N}{2} + 1 \right) \right) \; , \nonumber \\ \end{aligned}$$

(69)

which is easily calculated in the $\{\left| J,M\right\rangle \}$ basis.

Similarly the expectation value for the calculation of the 3-tangle is

$$\begin{aligned} {\langle \hat{\sigma }_y^p \otimes \hat{\sigma }_y^q \otimes \hat{\sigma }_y^r \rangle }= & {} \frac{1}{N(N-1)(N-2)}\nonumber \\ \sum _{p \ne q \ne r} {\langle \hat{\sigma }_y^p \otimes \hat{\sigma }_y^q \otimes \hat{\sigma }_y^r\rangle }= & {} \frac{1}{N(N-1)(N-2)} \left[ \sum _{pqr} {\langle \hat{\sigma }_y^p \otimes \hat{\sigma }_y^q \otimes \hat{\sigma }_y^r\rangle } \right. \nonumber \\{} & {} \left. - 3 \sum _{p \ne r} {\langle (\hat{\sigma }_y^p )^2 \otimes \hat{\sigma }_y^r\rangle } - \sum _p {\langle (\hat{\sigma }_y^p )^3 \rangle } \right] \nonumber \\= & {} \frac{1}{N(N-1)(N-2)}\nonumber \\ {}{} & {} \times \Bigl [ 8 {\langle \hat{J}_y^3\rangle } - 2(3N-2) {\langle \hat{J}_y\rangle } \Bigr ] . \end{aligned}$$

(70)

For the 4-tangle the expectation value reads

$$\begin{aligned}{} & {} {\langle \hat{\sigma }_y^p \otimes \hat{\sigma }_y^q \otimes \hat{\sigma }_y^r \otimes \hat{\sigma }_y^s \rangle }\nonumber \\{} & {} \quad = \frac{1}{\widetilde{\Omega }_N} \sum _{p \ne q\ne r \ne s} {\langle \hat{\sigma }_y^p \otimes \hat{\sigma }_y^q \otimes \hat{\sigma }_y^r \otimes \hat{\sigma }_y^s\rangle } \end{aligned}$$

(71)

where $\widetilde{\Omega }_N = N (N-1)(N-2)(N-3)$, and,

$$\begin{aligned}{} & {} \sum _{p \ne q\ne r \ne s} {\langle \hat{\sigma }_y^p \otimes \hat{\sigma }_y^q \otimes \hat{\sigma }_y^r \otimes \hat{\sigma }_y^s\rangle } \nonumber \\{} & {} \quad = \sum _{pqrs} {\langle \hat{\sigma }_y^p \otimes \hat{\sigma }_y^q \otimes \hat{\sigma }_y^r \otimes \hat{\sigma }_y^s\rangle }\nonumber \\{} & {} \qquad - 6 \sum _{p \ne r \ne s}\! {\langle (\hat{\sigma }_y^p)^2 \!\otimes \hat{\sigma }_y^r \otimes \hat{\sigma }_y^s\rangle } \nonumber \\{} & {} \qquad - 3 \sum _{p \ne s} {\langle (\hat{\sigma }_y^p)^2 \otimes (\hat{\sigma }_y^s )^2\rangle } - 4 \sum _{p \ne s} {\langle (\hat{\sigma }_y^p)^3 \otimes \hat{\sigma }_y^s\rangle }\nonumber \\{} & {} \qquad - \sum _p {\langle (\hat{\sigma }_y^p)^4 \rangle } \nonumber \\{} & {} \quad = \sum _{pqrs} {\langle \hat{\sigma }_y^p \otimes \hat{\sigma }_y^q \otimes \hat{\sigma }_y^r \otimes \hat{\sigma }_y^s\rangle } \nonumber \\{} & {} \qquad - 2( 3N -4) \sum _{p \ne s} {\langle \hat{\sigma }_y^p \otimes \hat{\sigma }_y^s\rangle } - 3N^2 +2N \end{aligned}$$

(72)

where

$$\begin{aligned} \sum _{pqrs} {\langle \hat{\sigma }_y^p \otimes \hat{\sigma }_y^q \otimes \hat{\sigma }_y^r \otimes \hat{\sigma }_y^s\rangle } = 16 {\langle \hat{J}_y^4\rangle } \; , \end{aligned}$$

(73)

and

$$\begin{aligned} \sum _{p\ne s} {\langle \hat{\sigma }_y^p \otimes \hat{\sigma }_y^s\rangle } = 4 {\langle \hat{J}_y^2\rangle } - N \; . \end{aligned}$$

(74)

Thus we obtain

$$\begin{aligned} {\langle \hat{\sigma }_y^p \otimes \hat{\sigma }_y^q \otimes \hat{\sigma }_y^r \otimes \hat{\sigma }_y^s \rangle }= & {} \frac{1}{\widetilde{\Omega }_N} \Bigl [ 16 {\langle \hat{J}_y^4\rangle } - 8 (3N-4) {\langle \hat{J}_y^2\rangle } \nonumber \\{} & {} \quad + 3N(N-2) \Bigr ] . \end{aligned}$$

(75)

The explicit expression for ${\langle \hat{J}_y^2\rangle }$ given in Eq. (69) can be used to finally obtain

$$\begin{aligned} {\langle \hat{\sigma }_y^p \otimes \hat{\sigma }_y^q \otimes \hat{\sigma }_y^r \otimes \hat{\sigma }_y^s \rangle }= & {} \frac{1}{\widetilde{\Omega }_N} \left[ {\langle (\hat{J}_+-\hat{J}_-)^4\rangle } + 2(3N-4) \right. \nonumber \\{} & {} \left. \left( {\langle \hat{J}_+^2\rangle } + {\langle \hat{J}_-^2\rangle } + 2 {\langle \hat{J}_z^2\rangle }\right) \right. \nonumber \\{} & {} \left. - 3N^3 +N^2 +2N \right] \; , \end{aligned}$$

(76)

The expectation values of products of $\hat{J}_\pm$ and $\hat{J}_z^2$ operators can again be easily calculated in the $\{\left| J,M\right\rangle \}$ basis.

1.1 B.1 Example of 4-tangles

To reinforce the reader’s intuition about the physical meaning of n-tangles, we provide the example of $\tau _4^{abcd}$ in a system of five spins with a select wavefunction. Consider a system with

$$\begin{aligned} \left| \Psi \right\rangle= & {} {1\over \sqrt{3}}\left[ |11110\rangle + |00000\rangle + |10101\rangle \right] \ \ \ . \end{aligned}$$

(77)

Forming the five possible $\left| \tilde{\Psi }\right\rangle =\hat{\sigma }_y^a\hat{\sigma }_y^b\hat{\sigma }_y^c\hat{\sigma }_y^d \left| \Psi ^*\right\rangle$ gives

$$\begin{aligned} \langle \Psi \vert \hat{\sigma }_y^1\hat{\sigma }_y^2\hat{\sigma }_y^3\hat{\sigma }_y^4 \left| \Psi ^*\right\rangle= & {} {2\over 3} \ \ ,\ \ \tau _4^{(1234)}={4\over 9} \ \ \ , \end{aligned}$$

(78)

with the other four possible $\tau _4^{abcd}$ vanishing.

1.2 B.2 Two- and four-body entanglement in the $N=4$ system

The $N=4$ system is simple enough that it can be investigated analytically in the general case. Considering a (real) wave function expanded in the $\{ \left| N_+\right\rangle = \left| JM\right\rangle \}$ basis, as in Eq. (11):

$$\begin{aligned} \left| \Psi \right\rangle = \sum _{N_+=0}^N A_{N_+} \left| N_+\right\rangle \; , \end{aligned}$$

(79)

we find,

$$\begin{aligned} \tau _2= & {} \Bigl | \frac{1}{3} A_0^2 + \frac{5}{6} A_1^2 + A_2^2 + \frac{5}{6} A_3^2 + \frac{1}{3} A_4^2 \nonumber \\{} & {} -\frac{\sqrt{6}}{3} A_0 A_2 - A_1 A_3 -\frac{\sqrt{6}}{3} A_2 A_4 -\frac{1}{3} \Bigr |^2 \; , \nonumber \\= & {} \Bigl | \frac{1}{2} A_1^2 + \frac{2}{3} A_2^2 + \frac{1}{2} A_3^2 - A_1 A_3\nonumber \\{} & {} \qquad -\sqrt{\frac{2}{3}} A_0 A_2 -\sqrt{\frac{2}{3}} A_2 A_4 \Bigr |^2 \; , \end{aligned}$$

(80)

$$\begin{aligned} \tau _4= & {} \Big | 7 A_0^2 + 7 A_1^2 +8 A_2^2 +7 A_3^2 +7 A_4^2 -2 A_1 A_3 \nonumber \\ {}{} & {} \qquad + 2 A_0 A_4 -7\Big |^2 \; , \nonumber \\= & {} \left| A_2^2 -2 A_1 A_3 + 2 A_0 A_4 \right| ^2 \; , \end{aligned}$$

(81)

where the norm of the wavefunction has been used to simplify the expressions. It is clear that $\tau _2$ can be non-zero only if $\Lambda \ge 1$ is considered, and $\tau _4$ can be non-zero only if $\Lambda \ge 2$. Subsequently, one can determine the “irreducible” 4-spin entanglement, for the $N=4$ system, given by Eq. (51) as

$$\begin{aligned} \eta _{4} \!=\! 1 - \frac{ {\langle \hat{J}_z\rangle }^2}{4} - 3 \tau _2 , \ \ \text{ with } \ \ {\langle \hat{J}_z\rangle } \!=\! \sum _{N_+} A_{N_+} \left( N_+ - \frac{N}{2}\right) .\nonumber \\ \end{aligned}$$

(82)

As a pedagogical case, we consider the limit when the wave function reduces to a single $\left| N_+\right\rangle$ basis state. In that case there is only one non-zero coefficient $A_{N_+} =1$.

• For $N_+=0$ ($M=-2$), the state consists of a single tensor-product state with spins aligned (all particles on the lower level). Such state is unentangled and indeed it is easy to see that all $\tau _2$, $\tau _4$ and $\eta _4$ reduce to zero.

• For $N_+=1$ ($M= -1$), the state a W-like state, i.e. a superposition of configurations with only one of the spins up (one particle on the upper level). Such state has $\tau _2=1/9$, while $\tau _4=\eta _4 =0$. This of course is in agreement with the well known fact that W states have only 2-body entanglement.

• For $N_+=2$ ($M= 0$), the state is a superposition of configurations with equal number of spins up and down (same number of particles on the upper and lower level). Such state has $\tau _2 = 1/9$, $\tau _4 =1$, and $\eta _4=2/3$, suggesting that genuine 4-spin entanglement is larger than the 2-spin entanglement.

A state with $M>0$ has same entanglement properties as that with $M<0$. The general case when the wave function is a superposition of states $\left| N_+\right\rangle$ depends on the values of the coefficients which are governed by the interaction of the LMG model. The numerical results are shown in Sect. 3.2.2.