![](https://csdnimg.cn/release/download_crawler_static/12273927/bgb.jpg)
entropies of higher orders can detect this peak and become small. For β > ˜α,
S
(β)
R
(λ) =
1
1 − β
log
1
d
β( ˜α−1)
˜α
+ (d − 1)
1 −
1
d
˜α−1
˜α
d − 1
β
≈
˜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:
S
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)
R
(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:
D
(α)
R
(ρkI/d) =
1
α − 1
log(tr{ρ
α
}d
α−1
) = log d − S
(α)
R
(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
˜
S
(α)
R
(ρ) =
1
α
2
∂
α
(
α−1
α
S
(α)
R
(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 –