混沌CFT中的量子信息普适性与 ETH 密度矩阵

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"这篇研究论文探讨了混沌共形场理论（CFT）中量子信息的普遍性，重点关注本征态热化假设（ETH）在任意维度的应用。文章指出，使用局部ETH，可以在无限体积限制下计算有限大小球形子系统的密度矩阵。 ETH密度矩阵与系统对称性相关的能量和电荷有关，而与本征态的其他细节无关。在二维情况下，ETH密度矩阵对具有相同中心电荷值的所有理论都是通用的。作者进一步提出，ETH密度矩阵在迹线距离上与微规范集合的简化密度矩阵接近。通过比较ETH密度矩阵的冯·诺依曼熵与低温度下全息系统中黑洞的熵，他们在高维情况下也提供了支持。此外，研究还扩展到能量密度在空间中缓慢变化的相干态，并证明ETH密度矩阵能够有效地描述这些态。文章由Nima Lashkari、Anatoly Dymarsky和Hong Liu合作完成，发表在JHEP03(2018)070上，属于开放获取资源。" 本研究的核心是混沌共形场理论中的量子信息处理，其中混沌特性是指系统行为的高度复杂性和不可预测性。ETH是一个关键的概念，它表明即使单个本征态也能展现出统计热力学性质。在本文中，研究者假设ETH适用于局部，从而允许他们分析整个系统处于能量本征态时，子系统的行为。他们计算的降低密度矩阵揭示了本征态的热化特性，且其近似于一个独立于本征态具体信息的矩阵，仅依赖于能量和全局对称性相关的电荷。在二维CFT中观察到的普遍性表明，对于具有相同中心电荷的理论，ETH密度矩阵是通用的。这可能与二维系统中的特殊对称性和拓扑性质有关。在更高的维度，尽管没有明确的普遍性，但通过与黑洞熵的比较，作者提出了ETH密度矩阵可能与微规范集合的简化密度矩阵相吻合，这在全息对偶理论中是一个重要的概念，全息对偶理论将高维引力问题与低维量子场论联系起来。此外，研究还考虑了能量密度在空间中非均匀分布的相干态，发现ETH密度矩阵仍能有效描述这些态，这扩展了ETH的适用范围，显示了其在处理非平衡量子态时的潜力。这项工作对理解和模拟复杂的量子系统，特别是在量子热力学和量子信息处理领域，有着重要的理论意义。

JHEP03(2018)070

2.1 Diﬀerent ensembles

Consider a QFT with a number of global symmetries living on a sphere. The microcanonical

ensemble ρ

micro

; {α}) is deﬁned as an equal-weight average over all energy eigenstates

lying within a narrow band around E

with a given set of quantum numbers {α} under

various global symmetries,

micro

; {α}) =

E∈(E

−∆,E

+∆),given {α}

|E, {α}ihE, {α}|. (2.1)

As always, we choose the energy band width ∆ to be much larger than the average level

spacing that scales like exp(−O(L

)), but much smaller than the typical energy scales of

interest. Here, N is the total number of states in the band. The density matrix of the

canonical ensemble is

can

(β, {α}) =

{α}

−βH

{α}

, Z

{α}

= Tr P

{α}

−βH

(2.2)

where P

{α}

denotes projection into the subspace of the Hilbert space with given {α}. The

grand canonical density matrix is deﬁned as

grand

(β, {µ}) =

{µ}

−βH−

, Z

{µ}

= Tr e

−βH−

(2.3)

where Q

denote the complete set of commuting charges and {µ} denotes the collection of

the corresponding chemical potentials.

For a general quantum ﬁeld theory, in the thermodynamical limit, for a local operator

O whose quantum numbers we keep ﬁxed as the volume goes to inﬁnity, the microcanonical,

canonical, and grand canonical averages are all equivalent by the standard arguments, pro-

vided that one chooses β and {µ} to give the average energy E

and the average charges

{α}. For example, the micro-canonical and the canonical ensemble which average over

rotationally-invariant states (i.e. with J

= 0 where J

denotes the Casimir operator of

the rotation group) are equivalent to the grand canonical ensemble with the correspond-

ing µ

= 0.

The equivalence of ensembles in conformal ﬁeld theory is more intricate since the rep-

resentations of a conformal group are inﬁnite dimensional. Furthermore, the states which

lie in the same representation of the conformal group in general do not have the same

energy. Let us ﬁrst consider a CFT in d > 2. In this case, the conformal group is the

higher dimensional Mobius transformations, and there are no new conserved charges be-

yond the generators of the conformal transformations. For convenience, let us introduce

ˆρ

(0)

micro

; {J

= 0}) as the (un-normalized) microcanonical density matrix of scalar pri-

maries with energies in a narrow band around E

, where one sums over only the energy

eigenstates which are scalar primaries. Similarly we can deﬁne ˆρ

(n)

micro

; {J

= 0}) to be

the ensemble of states that descend at level n from primary states of energy E

. A state in

the subspace deﬁned by ˆρ

(n)

micro

; {J

= 0}) has energy approximately equal to E

+O(

The standard microcanonical ensemble can then be expressed as

micro

; {J

= 0}) =

(n)

micro



− O





; {J

= 0}



, (2.4)

– 6 –

JHEP03(2018)070

where N is the total number of states at energy E

including both primaries and descen-

dants.

Now, we consider the thermodynamic limit that is L → ∞ with E

ﬁxed. In this

limit, from (1.1) we have that for any n which does not scale with L

|O|E

i =

− O





|O|E

− O



E

+ O(L

−1

), =

(n)

|O|E

(n)

+ O(L

−1

) (2.5)

where |E

i denotes a primary state while |E

(n)

i denotes an n-th level descendant state of

a primary state of approximate energy E

− O(

); see [1]. The density of states grows

exponentially with energy

log Ω(E) ∼ O(E

) 0 < α < 1.

The contribution of states in (2.4) with n scaling as L or larger, is exponentially suppressed

compared to the contribution of those with n = 0; hence we neglect such states. We

conclude that in the thermodynamic limit for any local operator

|O|E

i = Tr



Oρ

micro

; {J

= 0})



+ O(L

−1

) (2.6)

and will also be the same as in the canonical and grand canonical ensembles.

A CFT in d = 2 has an inﬁnite number of conserved charges that commute with both

and

. This is the KdV hierarchy of charges {Q

2k+1

, k = 1, 2, ···}. Here, the

corresponding microcanonical and canonical ensembles are denoted as

micro

; {Q

2k+1

}), ρ

canonical

(β; {Q

2k+1

}) (2.7)

and the corresponding grand canonical ensemble is the so-called Generalized Gibbs En-

semble (GGE)

GGE

(β; {µ

2k+1

, ¯µ

2k+1

}) =

−β(L

)−

2k+1

−

¯µ

2k+1

. (2.8)

Again, ρ

micro

; {Q

2k+1

}) contains descendant states. By descendants we are now

referring to Virasoro descendants. Following the same arguments as above we conclude that



, {Q

2k+1

}|O|E

, {Q

2k+1

}



= Tr



Oρ

micro

; {Q

2k+1

})



. (2.9)

The same holds also for the canonical ensemble and the GGE, provided we assume an

appropriate growth of the density of states Ω as a function of Q.

2.2 Equivalence of reduced density matrices

We now present a general argument showing that given (2.6), the reduced density matrix

for a region B of |E

ihE

|, and the ETH density matrix ψ

ETH

are close in trace distance

to the reduced state ρ of the subsystem B of a thermal state (the two-dimensional case is

diﬀerent and will be discussed in more depth in section 4). The argument works for any of

the three ensembles mentioned earlier.

– 7 –

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