Multipartite entanglement ∗

Moyal-star monomials of arbitrary powers. However, the star-logarithm of the Gaussian density W_Gauss can be worked out explicitly, see appendix A,

log^∗exp(−¹₂zΣ⁻¹z) =−¹₂zV z−logC (3.30) where the matrices Σ andV are related by Σ⁻¹=−2J|JV|tan ¹₂JV

, andC⁻¹ =p

det cosJV /2.

The precise form of the right hand side is not important. The crucial thing is that log^∗W_Gauss is a quadratic form in the phase space coordinatez. Because log^∗W_Gauss is integrated against Wρ in the second term of (3.29b), and the (co)variance ofW_ρ andW_Gauss agree by assumption, one may replace the second integral in (3.29b) by

S_vN(ρ_Gauss) =− Z

R²ⁿ

dz

(2π)ⁿW_Gausslog^∗W_Gauss. This yieldsS_vN(ρ)≤S_vN(ρ_Gauss), and is equivalent to the claim (3.28).

3.6 Multipartite entanglement ∗

This section sketches how the computation of the bipartite entanglement entropy in the fully cf. Sec. 5 in [1]

connected spin model (3.1) is generalized to quadripartitions. An example for a quadripartite entanglement measure is the tripartite informationI₃. The tripartite information is also used to measure how information delocalizes in unitary quantum channels [145, 146]

The main result of this section is Eq. (3.35). It shows that the dynamics of the tripartite information after a quench in the fully connected spin model does not yield more information than the dynamics of the bipartite entanglement entropy.

Quadripartite entanglement for time evolved pure states. Let |Ψ_ti = e^−iHt|Ψ₀i be the time evolved state after a quantum quench in Γ of (3.1). The dynamics takes place in the Dicke subspace DN of N spins. The bipartite entanglement dynamics of |Ψti was investigated in [1]. How does this calculation translate to multipartite entanglement measures?

One quadripartite entanglement measure of pure states is the tripartite information I₃(A:C:D) =I(A:C) +I(A:D)−I(A:CD)

=S(A) +S(C) +S(D)−S(AC)−S(AD)−S(CD) +S(ACD) (3.31) For the quadripartition HN = HA⊗ HB⊗ HC ⊗ HD of N spins into four disjoint subsets of A, B, C and D spins with A+B+C+D =N, any state |Ψi = P

N+Ψ(N₊)|N₊i in DN can be

expanded⁷ inDABCD:=DA⊗ DB⊗ DC⊗ DD ⊃ DN as⁸ takes into account the fact that there are more ways to permuteN+up-spins amongN spins than to independently permute the up-spins within the disjoint subsets of spins without inter-subset permutations.

Let ft(n+) be the rate function of |Ψti ∈ DN, i.e. Ψt(n+N) exp [−N ft(n+)]. The mini-mum of<f_t(n₊) at n₊ =n_t evolves according to Hamilton’s equation of motion for the effective HamiltonianH_eff, cf. Sec. 3.2. We denote the curvature off_t at the minimum by f_t⁰⁰(n_t) =:f₂ = entropic rate function of the combinatorial factor (3.33). S_αβγδ(a, b, c, d) depends on the relative subsystem sizes α = A/N, β = B/N, γ = C/N, δ = D/N, and is a function of the fraction of up spins a=A₊/A,b =B₊/B, c=C₊/C, d=D₊/D in the respective subsystems. Convexity (and α+β+γ +δ = 1) of the binary Shannon entropy guarantees that S_αβγδ is non-negative and vanishes if, and only if, a=b=c =d. Hence, the minimum of <f_t atn_t is inherited to the minimum of<f_t⁽⁴⁾ at (a, b, c, d) = (n_t, n_t, n_t, n_t) =:x^∗.

Expandingf_t⁽⁴⁾(a, b, c, d) to second order around its minimum atx^∗, yields a Gaussian approxi-mation Ψ^ABCD_t (x)∝exp[−^N₂(x−x^∗)Γ⁽⁴⁾(x−x^∗)] with four by four inverse covariance

being the Hessian matrix of the quadripartite rate function. Real and imaginary part of Γ⁽⁴⁾ are

7The fact thatDABCDis larger than the Dicke spaceDN can be seen by counting dimensions. Intuitively,DABCD

contains states that are permutation invariant among their elementary spins within each subgroupA, B, C andD.

However, these states do not need to be invariant under permutations of spins between the groups. The state (3.32) is of course invariant under permutations of allN spins.

8This follows from a straightforward generalization of the bipartite case, cf. (12) in [1],

|N+i= X

3.6 Multipartite entanglement∗

given by

X⁽⁴⁾=g₂M +1 2

1 (1−nt)nt

[diag(α, β, γ, δ)−M], (3.34a)

Y⁽⁴⁾=−θ2M, (3.34b)

respectively, where the four by four matrixM = (α, β, γ, δ)⊗(α, β, γ, δ) denotes the dyadic square of the relative subsystem sizes, andH₂⁰⁰(x) =−[(1−x)x]⁻¹ is the second derivative of the binary Shannon entropy.

To compute the entanglement entropies on the right hand side of (3.31), one needs the symplectic eigenvalues of the covariance matrix of the reduced Wigner function, cf. Eq. (3.26). LetW_ABCDbe the Wigner function of the quadripartite state Ψ^ABCD. The symplectic spectrum of the covariance Σ ofWABCD is equal to{1/2}, becauseWABCD comes from a pure state. The covariance matrices of the marginals of W_ABCD are denoted by Σ^A,Σ^B,· · ·Σ^AB,· · ·Σ^ABC, etc. It turns out that exactly one symplectic eigenvalue of each marginal covariance is different from one half. This special symplectic eigenvalue is denoted by λA, λB,· · ·λAB,· · ·λABC, etc., respectively. We refer to symplectic eigenvalues equal to one half as trivial symplectic eigenvalues, because they do not contribute to the entanglement entropy. Aided by a computer algebra system (Mathematica), we

find cf. Eq. (14)

in [1]

λ_A= 1 2

r

1−2C_α+C_α varn₊ var_Bnt

+ 4C_αvar_B(n_t) var(p), C_α=α(1−α), (3.35a) λ_AB = 1

2 r

1−2C_αβ+C_αβ varn₊ varBnt

+ 4C_αβvar_B(n_t) var(p), C_αβ = (α+β)[1−(α+β)], (3.35b) λ_ABC = λ_D, and analogously for the other combinations. Here, varn₊ = _2g¹

2, varp = ^g22_2g^+θ22

2 , and var_Bn_t = n_t(1−n_t) is the variance of a Bernoulli random variable with success probability 0< n_t<1.

We elaborate on the observation that only one symplectic eigenvalue is non-trivial. For the sake of concreteness, we focus on the four by four matrix Σ^AB. Since only one of its two symplectic eigenvalues is nontrivial, it must be identical to the symplectic eigenvalue of the reduced covariance w.r.t. the bipartition into A⁰ = AB and B⁰ = CD. Otherwise, this would contradict the result on the bipartite entanglement entropyS(A⁰). Indeed, the symplectic eigenvalues (3.35) are consis-tent with the symplectic eigenvalues in Eq. (24) of [1]. Hypothetically, Σ^AB could also have two non-trivial symplectic eigenvalues, say λ₁ and λ₂, such thatS(AB) would be the sum of the two contributions S(λ₁) and S(λ₂), adding up to the bipartite entanglement S(A⁰) =S(λ₁) +S(λ₂), according to (3.26). If this was the case, then the way howS(A⁰) is distributed onto the summands S(λ₁) andS(λ₂) could be a characterization of entanglement beyond bipartite entanglement. How-ever, the fact that only a single non-trivial symplectic eigenvalue in (3.35) carries all contribution toS(AB), rules out this possibility.

The upshot is that any quadripartite entanglement measure that depends solely on the symplec-tic eigenvalues of the reduced covariance does not yield more information than already contained in bipartite entanglement measures for permutation invariant states. In this sense, the quadripar-tition is not more general than a biparquadripar-tition for permutation invariant states.

We comment on the fact that the variance varn₊ only occurs relative to var_Bn_t in Eq. (3.35).

In other words, the magnetization variance never occurs in absolute units, but only in units of the variance of a Bernoulli random variable with success probability n_t. This rather technical

detail is important and has a natural geometric interpretation on the Bloch sphere. Imagine the Bloch sphere embedded in three dimensional Euclidean space such that the south pole is at the origin and the north pole at (0,0,1). Consider two (quasi) probability distributions on the Bloch sphere that differ only by a translation on the sphere, and are otherwise identical. For the sake of concreteness, assume, one is localized close to the north pole, the other is centered around a point close to the equator. The variance along the z-axis of the former is much smaller than the variance in z-direction of the latter. Not surprisingly, the variance in z direction is not invariant under rotations of the Bloch sphere. However, as the entanglement is independent of the choice of the spin quantization axis, i.e. invariant under rotations of the Bloch sphere, the entanglement entropy cannot depend on the magnetization variance inzdirection directly. It turns out that the ratio between varn₊ and var_BEn₊ = En₊(1−En₊) is invariant under rotations of the sphere.

The variance of the Bernoulli random variable vanishes at the poles and attains its maximum at the equator. Locally, this counteracts the transformation of the distances between two points on a meridian under projection onto thez-axis.