量子复杂性研究：纠缠、随机性和加扰的深度探索

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"纠缠，量子随机性和加扰之外的复杂性" 这篇研究论文深入探讨了量子系统中的复杂性，特别是关注纠缠、随机性和加扰的概念。加扰是一个量子系统中的重要现象，它涉及到全局纠缠如何使得初始的局域量子信息变得难以解析，从而达到随机化的效果。这种随机化在量子引力、多体物理、量子统计力学以及量子信息科学等领域具有核心地位。文中提到，量子混沌和热化是与加扰密切相关的概念。量子混沌是量子系统中的行为类似于经典混沌的现象，而热化则指系统达到热力学平衡时的状态。加扰不等同于完全随机化，虽然完全随机的系统一定是加扰的，但反向推理并不成立，这暗示了两者之间复杂的相互关系。作者们通过分析广义Rényi纠缠熵来研究这一复杂性。Rényi纠缠熵是一种衡量量子系统中信息纠缠程度的工具，其在特定条件下可以模拟单一均匀分布。研究发现，在相同阶数设计上平均的Rényi纠缠熵接近最大值，这揭示了熵与设计顺序之间的联系，并提出Rényi纠缠熵可以作为评估系统随机性的有效指标。文章进一步证明，存在一种（状态）2-设计，它可以限制高阶Rényi纠缠熵与最大值的差距。同时，他们还表明，最小的纠缠熵可以通过对数阶设计来最大化，这意味着对数设计在纠缠特性上已经达到了最优。这项工作对于理解量子信息加扰和Haar随机性之间的层次结构至关重要。Haar随机性是量子力学中描述完全随机态的一种标准方式。通过这些发现，研究者为理解和操纵量子系统中的复杂性提供了新的数学工具和理论框架，这对于未来的量子计算、量子通信以及量子信息处理技术的发展具有深远影响。这篇研究论文不仅深化了我们对量子系统复杂性的理解，而且为量子信息科学的研究提供了新的理论基础和实验指导。通过揭示加扰、随机性和纠缠之间的微妙关系，科学家们能够更好地设计和控制量子系统，以实现更高效、更安全的量子计算和通信任务。

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JHEP07(2018)041

entropies of higher orders can detect this peak and become small. For β > ˜α,

(β)

(λ) =

1 − β

log







β( ˜α−1)

˜α

+ (d − 1)





1 −

˜α−1

˜α

d − 1











≈

˜αβ − β

˜αβ − ˜α

log d, (2.9)

which is Θ(log d) (linear in the number of qubits) smaller than the maximal value log d. In

fact, λ produces Θ(log d) gaps between all higher orders. The extreme case min entropy

only cares about the largest eigenvalue by deﬁnition:

min

(λ) = −log λ

max

˜α − 1

˜α

log d, (2.10)

which is small for all ﬁnite ˜α. That is, the slope of S

(β)

(λ) in log d decreases with β.

It equals one for β = ˜α, and approaches

˜α−1

˜α

in the β → ∞ limit. So there can be an

asymptotic separation between R´enyi entropies of any orders. The intuition is simply that

promoting the power of eigenvalues essentially ampliﬁes the nonuniformity of the spectrum.

We shall construct a similar separation for certain 2-designs, which indicates that R´enyi

entropies can distinguish low-degree pseudorandom states from truly random states.

In our calculations we often assume equal partitions for simplicity. Since the sub-

systems contain half the total degrees of freedom, the equal partitions admit the largest

possible entanglement entropy. Also, the following simple argument ensures that as long

as the (R´enyi) entanglement entropies between all equal partitions are close to the max-

imum, then that between generic partitions must be close to the maximum as well. No-

tice that the quantum R´enyi divergence/relative entropy (either the non-sandwiched or

sandwiched/non-commutative version, see e.g. [37] for deﬁnitions) between ρ and the max-

imally mixed state yields the gap between the R´enyi entropy of ρ and the maximum:

(α)

(ρkI/d) =

α − 1

log(tr{ρ

α−1

) = log d − S

(α)

(ρ). (2.11)

For sandwiched R´enyi divergence with α ≥ 1/2 (which covers the parameter range of

interest in this paper), it is shown in [38, 39] that the data processing inequality holds,

which implies that the divergence is monotonically nonincreasing under partial trace. So

the gap can only be smaller when we look at smaller subsystems.

In the appendix, we derive more properties of R´enyi entropies, including inequalities

relating diﬀerent orders of R´enyi entropies (appendix A), and a weaker form of subadditivity

(appendix B). The above discussions on R´enyi entropies are more or less tailored to our

needs. We refer the interested readers to [37] for a more comprehensive discussion of the

motivations and properties of quantum R´enyi entropies and divergences. We also note that

a close variant of the quantum R´enyi entropy known as the “modular entropy”, given by

(α)

(ρ) =

∂

(

α−1

(α)

(ρ)), is found to be meaningful in the context of holography and

admits a natural thermodynamic interpretation [40, 41].

2.2 Designs

In quantum information theory, the notion of t-designs characterizes distributions of pure

states or unitary channels that mimic the uniform distribution up to the ﬁrst t moments,

– 8 –

JHEP07(2018)041

and so can be considered as good approximations to Haar randomness. Analogous classical

notions such as t-wise independence and t-universal hash functions are also found to be very

useful in computer science and combinatorics. We shall formally introduce the deﬁnitions

of state and unitary designs relevant to this work in the following.

2.2.1 Complex projective designs

Complex projective t-designs, which we may call “t-designs” for short throughout the paper,

are distributions of vectors on the complex unit sphere that are good approximations to the

uniform distribution, or pseudorandom, in the sense that they reproduce the ﬁrst t moments

of the uniform distribution [42–44]. They are of interest in many research areas, such as

approximation theory, experimental designs, signal processing, and quantum information.

There are many equivalent deﬁnitions of exact designs (see [44] for an introduction). Here

we mention a few that are directly relevant to the current study.

The canonical deﬁnition based on polynomials of vector entries will be directly used

in deriving our results. Deﬁne Hom

(t,t)

) as the space of polynomials homogeneous of

degree t both in the coordinates of vectors in C

and in their complex conjugates.

Deﬁnition 4 (t-designs by polynomials). An ensemble ν of pure state vectors in dimension

d is a (complex projective) t-design if

p(ψ) =

p(ψ)dψ ∀p ∈ Hom

(t,t)

), (2.12)

where the integral is taken with respect to the (normalized) uniform measure on the com-

plex unit sphere in C

The second deﬁnition, based on the frame operator, is also widely used. Let Sym

)

be the t-partite symmetric subspace of (C

)

⊗t

with corresponding projector P

[t]

. The

dimension of Sym

) reads

[t]



d + t − 1



. (2.13)

Deﬁnition 5 (t-designs by frame). The t-th frame operator of ν is deﬁned as

(ν) := D

[t]

(|ψihψ|)

⊗t

, (2.14)

and the t-th frame potential is

(ν) := tr



(ν)



. (2.15)

The ensemble ν is a t-design if and only if F

(ν) = P

[t]

or, equivalently, if Φ

(ν) = D

[t]

[44].

The above deﬁnitions for exact designs are equivalent. However, they lead to slightly

diﬀerent ways to deﬁne approximate designs by directly considering the deviations from

equality, which essentially represent diﬀerent norms. We shall discuss the approximate

designs in more detail later for error analysis.

– 9 –

JHEP07(2018)041

2.2.2 Unitary designs

In analogy to complex projective t-designs, unitary t-designs are distributions on the uni-

tary group that are good approximations to the Haar measure, in the sense that they

reproduce the Haar measure up to the ﬁrst t moments [44–48]. They also play key roles

in many research areas, such as randomized benchmarking, data hiding, and decoupling.

As in the case of state designs there are also many equivalent deﬁnitions of exact unitary

designs (see [44]). Similarly, we formally deﬁne unitary designs by polynomials and frame

operators/potentials.

Let Hom

(t,t)

(U(d)) be the space of polynomials homogeneous of degree t both in the

matrix elements of U ∈ U(d) and in their complex conjugates.

Deﬁnition 6 (Unitary t-designs by polynomials). An ensemble ν of unitary operators in

dimension d is a unitary t-design if

p(U) =

dUp(U ) ∀p ∈ Hom

(t,t)

(U(d)), (2.16)

where the integral is taken over the normalized Haar measure on U(d).

Deﬁnition 7 (Unitary t-designs by frame). The t-th frame operator of ν is deﬁned as

(ν) := E

⊗t

⊗ U

†

⊗t

, (2.17)

and the t-th frame potential is

(ν) := tr



(ν)



(2.18)

The ensemble ν is a unitary t-design if and only if F

(ν) = F

(U(d)), where F

(U(d)) is

the tth frame operator of the unitary group U(d) with Haar measure [44]. In addition,

(ν) ≥ γ(t, d) :=

dU|tr{U }|

, (2.19)

and the lower bound is saturated if and only if ν is a unitary t-design [44, 48, 49]. When

t ≤ d, which is the case we are mostly interested in,

γ(t, d) = t!. (2.20)

Again, the deﬁnitions are equivalent for exact unitary designs, but lead to diﬀerent

ways to deﬁne approximate unitary designs, which we shall look into later.

3 Generalized entanglement entropies and random unitary channels

Unitary channels describe the evolutions of closed quantum systems. Here we study the

entanglement and scrambling properties of random unitary channels, which directly moti-

vates this work. As suggested by [9], we employ the Choi isomorphism to map a unitary

channel to a dual state, and study scrambling by the relevant entanglement properties of

this state. In this section, we ﬁrst brieﬂy introduce the Choi state model, and then present

the explicit calculations of generalized entanglement entropies averaged over unitary de-

signs. The results lead to an entropic notion of scrambling or randomness complexities,

which we shall discuss in depth.

– 10 –

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