arXiv:1803.02360v1 [math-ph] 6 Mar 2018

Submitted to Bernoulli

Gaussian optimizers for entropic inequalities
in quantum information
GIACOMO DE PALMA1 DARIO TREVISAN2 VITTORIO
GIOVANNETTI3 and LUIGI AMBROSIO4
1

QMATH, Department of Mathematical Sciences, University of Copenhagen,
Universitetsparken 5, 2100 Copenhagen, Denmark. E-mail: giacomo.depalma@math.ku.dk

2

Università degli Studi di Pisa, 56126 Pisa, Italy.

3

NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, 56126 Pisa, Italy.

4

Scuola Normale Superiore, 56126 Pisa, Italy.

We survey the state of the art for the proof of the quantum Gaussian optimizer conjectures of
quantum information theory. These fundamental conjectures state that quantum Gaussian input
states are the solution to several optimization problems involving quantum Gaussian channels.
These problems are the quantum counterpart of three fundamental results of functional analysis
and probability: the Entropy Power Inequality, the sharp Young’s inequality for convolutions and
the theorem “Gaussian kernels have only Gaussian maximizers”. Quantum Gaussian channels
play a key role in quantum communication theory: they are the quantum counterpart of Gaussian
integral kernels and provide the mathematical model for the propagation of electromagnetic
waves in the quantum regime. The quantum Gaussian optimizer conjectures are needed to
determine the maximum communication rates over optical fibers and free space. The restriction
of the quantum-limited Gaussian attenuator to input states diagonal in the Fock basis coincides
with the thinning, the analogue of the rescaling for positive integer random variables. Quantum
Gaussian channels provide then a bridge between functional analysis and discrete probability.
Keywords: quantum information theory, quantum Gaussian channels, entropic inequalities, thinning, Entropy Power Inequality, sharp Young’s inequality for convolutions.

Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . .
2 Gaussian optimizers in functional analysis . . . . . . .
2.1 Gaussian kernels have only Gaussian maximizers
2.2 The sharp Young’s inequality for convolutions . .
2.3 The Entropy Power Inequality . . . . . . . . . .
3 Quantum Gaussian systems . . . . . . . . . . . . . . .
3.1 Quantum systems . . . . . . . . . . . . . . . . . .
3.2 Quantum channels . . . . . . . . . . . . . . . . .
3.3 Quantum Gaussian systems . . . . . . . . . . . .
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G. De Palma, D. Trevisan, V. Giovannetti, and L. Ambrosio

3.4 Quantum Gaussian channels . . . . . . . . . . . . . . . . . . . . . . . . . .
4 The minimum output entropy conjecture . . . . . . . . . . . . . . . . . . . . . .
5 Gaussian optimizers for entropic inequalities in quantum information . . . . . .
5.1 Quantum Gaussian channels have Gaussian maximizers . . . . . . . . . .
5.1.1 The proof of Conjecture 5.4 for one-mode quantum-limited Gaussian channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.2 The proof of Conjecture 5.1 for all the one-mode attenuators and
ampliﬁers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.3 The proof of Conjecture 5.4 for p=q . . . . . . . . . . . . . . . . .
5.2 The Entropy Photon-number Inequality . . . . . . . . . . . . . . . . . . .
5.3 The sharp Young’s inequality for the beam-splitter . . . . . . . . . . . . .
6 The thinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7 Quantum conditioning and the quantum Entropy Power Inequality . . . . . . .
7.1 The quantum Entropy Power Inequality . . . . . . . . . . . . . . . . . . .
7.2 Quantum conditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3 The quantum conditional Entropy Power Inequality . . . . . . . . . . . .
8 Conclusions and perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Author Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

13
14
16
18
19
20
21
22
22
24
26
26
27
28
31
32
32
32

1. Introduction
Gaussian functions play a key role in both functional analysis and probability, and are the
solution to several optimization problems involving Gaussian kernels. The most prominent among these problems is determining the norms of the Gaussian integral kernels G
that send a function f ∈ Lp (Rm ) to the function Gf ∈ Lq (Rn ) with p, q ≥ 1. In the
seminal paper “Gaussian kernels have only Gaussian maximizers” [86], Lieb proved that
these norms are achieved by Gaussian functions. A closely related fundamental result
is the sharp Young’s inequality for convolutions [7, 11, 41, 18, 8], stating that for any
p, q, r ≥ 1, the ratio kf ∗ gkr / kf kp kgkq with f ∈ Lp (Rn ) and g ∈ Lq (Rn ) is maximized
by Gaussian functions, where f ∗ g denotes the convolution of f with g. This inequality
has several fundamental applications, such as a proof of the Entropy Power Inequality
[85, 36, 19], of the Brunn-Minkowski inequality [45, 19] and Lieb’s solution [85, 87] of
Wehrl’s conjecture [105, 1], stating that coherent states minimize the Wehrl entropy. The
theorem “Gaussian kernels have only Gaussian maximizers” and the sharp Young’s inequality for convolutions are among the most important inequalities of functional analysis
(see e.g. the book [84]).
The Entropy Power Inequality [36, 100, 98] states that the Shannon diﬀerential entropy
of the sum of two independent random variables with values in Rn and given Shannon
diﬀerential entropies is minimum when the two random variables are Gaussian, and
is a fundamental tool of information theory [19]. The Entropy Power Inequality was

Gaussian optimizers for entropic inequalities in quantum information

3

introduced by Shannon to provide an upper bound to the information capacity of nonGaussian channels [98], and was later used to bound the information capacity region of
the Gaussian broadcast channel [9] and the secret information capacity of the Gaussian
wiretap channel [83]. The Entropy Power Inequality was also employed to prove the
convergence in relative entropy for the central limit theorem [6].
Quantum information theory [90, 64, 108, 70] is the theory of the transmission and the
processing of the information stored in quantum systems. Most of nowadays communications are made with electromagnetic signals traveling through optical ﬁbers or free space.
Quantum Gaussian channels [17, 13, 104, 71] are the quantum counterpart of Gaussian
integral kernels, and an n-mode quantum Gaussian channel provides the mathematical
model for the propagation of n modes of the electromagnetic radiation along an optical
ﬁbre or free space in the quantum regime. For this reason, quantum Gaussian channels
play a key role in quantum communication theory.
The subject of this review is the generalization of all the above inequalities for the convolution and for Gaussian integral kernels to quantum Gaussian channels. The solutions
to the resulting quantum optimization problems are conjectured to be quantum Gaussian states, the quantum counterpart of Gaussian probability measures. This Gaussian
optimizer problem arose in quantum information theory for the determination of the classical information capacity of phase-covariant quantum Gaussian channels [69, 49, 50, 26].
Indeed, proving that the coherent states constitute an optimal coding requires to prove
a minimum output entropy conjecture, stating that the coherent input states minimize
the output entropy of n-mode phase-covariant quantum Gaussian channels (Theorem
4.1). This conjecture implies that both the minimum output entropy and the classical
capacity of phase-covariant quantum Gaussian channels are additive with respect to the
tensor product, i.e., that entanglement does not increase the communication rate. While
the minimum output entropy of any classical channel is trivially additive, this property
does not hold in general for quantum channels [62]. The proof of the minimum output
entropy conjecture has then been a fundamental result, which required more than ten
years [51, 43, 44, 48, 88, 46] (see the review [71]; see also [72] for the capacity of non
phase-covariant quantum Gaussian channels).
Proving that the coherent states constitute an optimal coding for the Gaussian broadcast channel requires a constrained version of the minimum output entropy conjecture.
This constrained version states that quantum Gaussian input states minimize the output
entropy of n-mode quantum Gaussian channels among all the input states with a given
entropy [58, 56, 54, 57, 92] (Conjecture 5.1). The constrained minimum output entropy
conjecture also implies the converse theorems for the triple trade-oﬀ coding with the
quantum-limited attenuator and ampliﬁer [106, 107, 92]. The conjecture has been generalized to the Entropy Photon-number Inequality [55, 54], stating that quantum Gaussian
input states minimize the output entropy of the beam-splitter among all the couple of
input states each with a given entropy 5.13. Moreover, it has been realized [33] that the
constrained minimum output entropy conjecture would follow from the generalization of
the theorem “Gaussian kernels have Gaussian maximizers” to n-mode quantum Gaussian channels (Conjecture 5.4). Since the beam-splitter is the quantum counterpart of
the convolution, the Entropy Photon-number Inequality is the quantum counterpart of

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G. De Palma, D. Trevisan, V. Giovannetti, and L. Ambrosio

the Entropy Power Inequality. Based on this relation, we conjecture for the ﬁrst time in
this review the validity of a sharp Young’s inequality for the beam-splitter (Conjecture
5.14).
The proof of all the above quantum inequalities has been completed only in some
particular cases, and is currently an active ﬁeld of research. The constrained minimum
output entropy conjecture has been proven only for one-mode quantum Gaussian channels [32, 34, 21, 33, 93] or for input states diagonal in some joint product basis [35].
These results are based on a new majorization theorem for one-mode quantum Gaussian
channels [32] (Theorem 5.2). The majorization result has been extended to single-jump
lossy quantum channels [29], but unfortunately it fails for multi-mode quantum Gaussian
channels [29]. The proof of the constrained minimum output entropy conjecture for onemode quantum Gaussian channels made possible the proof of the fundamental relation
between the von Neumann and the Wehrl entropy, stating that for any n, n-mode quantum Gaussian states have the minimum Wehrl entropy among all the n-mode quantum
states with a given von Neumann entropy [23]. For generic p, q ≥ 1, the theorem “Gaussian kernels have Gaussian maximizers” has been proven only for one-mode quantum
Gaussian channels [31], while for n-mode channels the theorem has been proven only for
p = 1 [47, 71] and p = q [42, 73]. A proof of the Entropy Photon-number Inequality has
been attempted with the quantum analogue of the heat semigroup technique of the proof
of the Entropy Power Inequality by Blachman and Stam. This technique led instead to
the proof of the quantum Entropy Power Inequality [79, 80, 78, 27, 28, 21] (Theorem 7.1),
which provides a lower bound to the output entropy of the beam-splitter in terms of the
entropies of the two inputs. This bound is strictly lower than the output entropy achieved
by Gaussian input states, hence the quantum Entropy Power Inequality is strictly weaker
than the Entropy Photon-number Inequality, that is still an open conjecture. The same
heat semigroup technique led to the proof of the quantum conditional Entropy Power
Inequality [77, 30] and of the quantum Entropy Power Inequality for the quantum additive noise channels both in the unconditioned [74] and conditional [25] versions. The
quantum conditional Entropy Power Inequality (Theorem 7.2) determines the minimum
quantum conditional von Neumann entropy of the output of the beam-splitter or of the
squeezing among all the input states where the two inputs are conditionally independent given the memory and have given quantum conditional entropies. This inequality
has been exploited to prove an uncertainty relation for the conditional Wehrl entropy
[22]. These Entropy Power Inequalities have stimulated the proof of similar inequalities
in diﬀerent contexts, such as the qubit swap channel [3, 14] and information combining
[66]. The implications among the main results and conjectures for quantum Gaussian
channels are summarized in section 1.
As a possible approach towards the proof of the unsolved entropic inequalities for
quantum Gaussian channels, we mention that sharp functional inequalities in the commutative setting have been recently studied using the theory of optimal transport [103].
These methods led to e.g. quantitative stability results for isoperimetric [39], Sobolev
and log-Sobolev [40, 37] inequalities. Ideas from optimal transport are also implicit in
the solution of Shannon’s problem on the monotonicity of entropy [2]. Recently, transportation distances have been proposed in the quantum fermionic setting [15, 16] and

Gaussian optimizers for entropic inequalities in quantum information

Sharp Young’s inequality
for the beam-splitter
(Conjecture 5.14)

Entropy Photon-number Inequality
(Conjecture 5.13)

5

Quantum Gaussian channels
have Gaussian maximizers
(Conjecture 5.4)

p → p norms of
quantum Gaussian channels
(Theorem 5.11)

1 → p norms of
quantum Gaussian channels
(Corollary 4.4)

Quantum Entropy
Power Inequality
(Theorem 7.1)

Constrained minumum output entropy
of quantum Gaussian channels
(Conjecture 5.1)

Minimum output entropy of
quantum Gaussian channels
(Theorem 4.1)

Majorization for quantum
Gaussian channels
(Theorem 4.3)

Figure 1. Implications among conjectures and results. Green = proven result; Yellow = result proven
in some particular cases; Red = open conjecture.

have then been extended to quantum Gaussian systems [95, 96] (see also [20]).
An interesting particular case of the inequalities for quantum Gaussian channels is
when the input states are diagonal in the Fock basis [59, 75]. This provides a link between
quantum Gaussian channels and classical discrete probability theory. The restriction of
the one-mode quantum-limited attenuator to input states diagonal in the Fock basis
is the linear map acting on discrete classical probability distributions on N known as
thinning [32]. The quantum-limited attenuator is the quantum Gaussian channel that
models the attenuation of electromagnetic signals. The thinning has been introduced by
Rényi [94] as a discrete analogue of the rescaling of a continuous real random variable,
and has been involved with this role in discrete versions of the central limit theorem
[60, 109, 61], of the Entropy Power Inequality [110, 76] and of Young’s inequality [81].
Most of these results require the ad hoc hypothesis of the ultra log-concavity (ULC)
of the input state. In particular, the Restricted Thinned Entropy Power Inequality [76]

6

G. De Palma, D. Trevisan, V. Giovannetti, and L. Ambrosio

states that the Poisson input probability distribution minimizes the output Shannon
entropy of the thinning among all the ULC input probability distributions with a given
Shannon entropy. The results on quantum Gaussian channels presented in this review led
to the proof of new entropic inequalities for the thinning that apply to any probability
distribution, regardless of whether they satisfy the ULC assumption. Quantum Gaussian
states correspond to geometric probability distributions. The inequalities on quantum
Gaussian channels imply that geometric input probability distributions both achieve the
norms of the thinning [31] (Theorem 6.5) and minimize its output entropy among all the
input probability distributions with a given entropy [34] (Theorem 6.4).
The review is structured as follows. In section 2, we present the classical results for
Gaussian optimizers in functional analysis. In section 3, we introduce Gaussian quantum
systems and channels, and the necessary notions of quantum mechanics. In section 4 we
present the minimum output entropy conjecture and its proof. In section 5, we present
all the conjectures on Gaussian optimizers in quantum information including the new
sharp Young’s inequality for the beam-splitter, together with the state of the art in their
proofs. In section 6, we present the thinning and its relation with the results for quantum
Gaussian channel. In section 7, we present the quantum Entropy Power Inequality and
its proof. Moreover, we introduce the quantum conditional entropy, and present the
conditioned version of the quantum Entropy Power Inequality. We conclude in section 8.

2. Gaussian optimizers in functional analysis
2.1. Gaussian kernels have only Gaussian maximizers
For any p ≥ 1, the Lp (Rn ) norm of a function f : Rn → C is
Z
 p1
p
kf kp =
|f (x)| dx
.

(2.1)

Rn

Given p, q ≥ 1, let us consider a Gaussian integral kernel G from Lp (Rm ) to Lq (Rn ):
Z
G(x, y) f (y) dy ,
x ∈ Rn ,
f ∈ Lp (Rm ) ,
(2.2)
(G f )(x) =
Rm

where G(x, y) is a Gaussian function on Rm+n , i.e., the exponential of a quadratic polynomial. The norm of G is
kG f kq
.
(2.3)
kGkp→q =
sup
0<kf kp <∞ kf kp

In the seminal paper “Gaussian kernels have only Gaussian maximizers” [86], Lieb proved
that under certain fairly broad assumptions on G, p and q, this operator is well deﬁned,
and the supremum in (2.3) is attained on a Gaussian function f . If 1 < p < q < ∞,
any function that attains the supremum in (2.3) is a Gaussian function. The proof of
this fundamental result is based on the multiplicativity of the norm of generic integral
kernels with respect to the tensor product.

Gaussian optimizers for entropic inequalities in quantum information

7

Theorem 2.1 ([86, 67]). The norms of integral kernels are multiplicative, i.e., for
any two (not necessarily Gaussian) integral kernels G1 : Lp (Rm1 ) → Lq (Rn1 ) and G2 :
Lp (Rm2 ) → Lq (Rn2 ),
kG1 ⊗ G2 kp→q = kG1 kp→q kG2 kp→q .

(2.4)

Moreover, if the ratios kG1 f1 kq / kf1 kp and kG2 f2 kq / kf2 kp are maximized by the unique
functions f1 = f¯1 ∈ Lp (Rm1 ) and f2 = f¯2 ∈ Lp (Rm2 ), the ratio k(G1 ⊗ G2 )f kq / kf kp is
maximized by the unique function f = f¯1 ⊗ f¯2 ∈ Lp (Rm1 +m2 ).

2.2. The sharp Young’s inequality for convolutions
The convolution operation can be considered as a degenerate Gaussian integral kernel
given by a Dirac delta function centered in the origin. Indeed, the convolution of f ∈
Lp (Rn ) with g ∈ Lq (Rn ) is
Z
f (y) g(z) δ0 (y + z − x) dy dz ,
x ∈ Rn .
(2.5)
(f ∗ g)(x) =
R2n

The sharp Young’s inequality for convolutions states that the supremum
kf ∗ gkr
0<kf kp ,kgkq <∞ kf kp kgkq
sup

(2.6)

is ﬁnite iﬀ

1
1 1
+ =1+ ,
(2.7)
p q
r
and in this case it is achieved by Gaussian functions. This result has been ﬁrst proven by
Beckner [7] and by Brascamp and Lieb [11] using a rearrangement inequality for integrals
[12]. A completely diﬀerent proof based on the heat semigroup has been provided by
Toscani [102].

2.3. The Entropy Power Inequality
Let X be a random variable with values in Rn and whose probability law is absolutely
continuous with respect to the Lebesgue measure, so that it admits a probability density
f (x)dx. The Shannon diﬀerential entropy [19] of X is
Z
f (x) ln f (x) dx ,
(2.8)
S(X) = −
Rn

and quantiﬁes the noise contained in X. Let σ be a symmetric strictly positive n × n real
matrix, and let X be the centered Gaussian random variable with covariance matrix σ
and density
1 T −1
e− 2 x σ x
.
(2.9)
f (x) = p
det(2πσ)

8

G. De Palma, D. Trevisan, V. Giovannetti, and L. Ambrosio

The Shannon diﬀerential entropy of X is proportional to the logarithm of the determinant
of the covariance matrix:
1
(2.10)
S(X) = ln det(2πeσ) .
2
Let us consider the sum of two independent random variables X and Y with values
in Rn . The Entropy Power Inequality [36, 100, 98] states that, if X and Y have Shannon diﬀerential entropy ﬁxed to the values S(X) and S(Y ), respectively, the Shannon
diﬀerential entropy of X + Y is minimum when X and Y have a Gaussian probability
distribution with proportional covariance matrices. The covariance matrix of the sum of
two independent random variables is equal to the sum of their covariance matrices:
σX+Y = σX + σY .

(2.11)

If σY = λ σX for some λ > 0, (2.10) and (2.11) imply
exp

2S(X + Y )
2S(X)
2S(Y )
= exp
+ exp
,
n
n
n

(2.12)

so that the Entropy Power Inequality has the form
exp

2S(X)
2S(Y )
2S(X + Y )
≥ exp
+ exp
.
n
n
n

(2.13)

Two diﬀerent proofs of the Entropy Power Inequality are known. The ﬁrst is due to
Blachman and Stam [100, 10], and is based on perturbing the inputs X and Y with
the heat semigroup. The second is due to Lieb [85], and is based on the sharp Young’s
inequality for convolutions and on the properties of the Rényi entropies. For any p > 1,
the p-Rényi entropy of the random variable X with values in Rn and density f is
Sp (X) =

p
ln kf kp .
1−p

(2.14)

The Rényi entropies are a generalization of the Shannon diﬀerential entropy, which is
recovered in the limit p → 1:
S(X) = lim Sp (X) .
(2.15)
p→1

3. Quantum Gaussian systems
In this Section, we introduce the elements of quantum information and quantum Gaussian
systems that are needed for presenting the entropic inequalities for quantum Gaussian
channels. For a more comprehensive introduction, we refer the reader to the books [70, 97]
and the review [71].

Gaussian optimizers for entropic inequalities in quantum information

9

3.1. Quantum systems
Let H be a separable complex Hilbert space with not necessarily ﬁnite dimension. We
adopt the bra-ket notation, where a vector ψ ∈ H is denoted as |ψi and the scalar product
between the vectors φ and ψ is denoted as hφ|ψi, is linear in ψ and antilinear in φ.
For any p ≥ 1, the p-Schatten norm of a linear compact operator X̂ on H is
 
 p2  p1
†
,
(3.1)
X̂ = Tr X̂ X̂
p

where X̂ † is the adjoint operator of X̂. The p-Schatten norm play the role of the Lp norm
of functional analysis. The operators with ﬁnite 1-Schatten norm are called trace-class
operators. The ∞-Schatten norm kp
X̂k∞ of a continuous linear operator X̂ is deﬁned as

the supremum of the spectrum of X̂ † X̂.
Quantum states are the noncommutative counterpart of probability measures. A quantum state is a positive trace-class operator with unit trace. Any quantum state ρ̂ can be
diagonalized in an orthonormal basis:
ρ̂ =

∞
X

k=0

pk |ψk ihψk | ,

(3.2)

where {|ψk ihψk |}k∈N denote the rank-one projectors onto the orthonormal vectors {ψk }k∈N ,
and {pk }k∈N are the eigenvalues of ρ̂. Since ρ̂ is positive and has unit trace, {pk }k∈N is a
probability measure on N. The quantum state ρ̂ is called pure if it is a rank-one projector,
and mixed otherwise. With a small abuse of nomenclature, we call pure state both the
normalized vector ψ ∈ H and the associated rank-one projector |ψihψ|. From (3.2), any
quantum state can be expressed as a convex combination of orthogonal pure states.
The von Neumann entropy of the quantum state ρ̂ in (3.2) is the Shannon entropy of
its eigenvalues:
∞
X
pk ln pk ,
(3.3)
S(ρ̂) = −Tr [ρ̂ ln ρ̂] = −
k=0

and is the quantum counterpart of the Shannon diﬀerential entropy. As the Shannon
entropy and contrarily to the Shannon diﬀerential entropy, the von Neumann entropy is
always positive, and vanishes iﬀ ρ̂ is pure. If ρ̂ is a quantum state of the quantum system
A with Hilbert space HA , we use indistinctly the notations S(ρ̂) or S(A) for the entropy
of ρ̂.
As in the case of classical probability measures, we can deﬁne for any p > 1 the p-Rényi
entropy of the quantum state ρ̂ as
p
Sp (ρ̂) =
ln kρ̂kp .
(3.4)
1−p

The Rényi entropies are a generalization of the von Neumann entropy, which is recovered
in the limit p → 1:
S(ρ̂) = lim Sp (ρ̂) .
(3.5)
p→1

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G. De Palma, D. Trevisan, V. Giovannetti, and L. Ambrosio

The observables of a quantum system are the self-adjoint operators on the Hilbert
space, and the expectation value of the observable Ô on the state ρ̂ is
h i
D E
(3.6)
Ô = Tr Ô ρ̂ .
ρ̂

If A and B are quantum systems with Hilbert spaces HA and HB , the joint system
AB has Hilbert space HA ⊗ HB . A pure state ψ ∈ HA ⊗ HB is called product state if
ψ = ψA ⊗ ψB for some ψA ∈ HA and ψB ∈ HB . A fundamental diﬀerence with respect
to classical probability is that not all pure states are product states. A pure state that
is not a product state is called entangled state. Let ρ̂AB be a quantum state of the joint
quantum system AB. We deﬁne the reduced or marginal states on A and B as
ρ̂A = TrB ρ̂AB ,

ρ̂B = TrA ρ̂AB ,

(3.7)

where TrA and TrB denote the partial trace over the system A and B, respectively. In
other words, ρ̂A is the quantum state of A such that
i

h
i
h
(3.8)
TrA X̂A ρ̂A = TrAB X̂A ⊗ ÎB ρ̂AB

for any bounded operator X̂A on HA , and analogously for ρ̂B .

3.2. Quantum channels
Quantum channels [90, 64, 108, 70] are the noncommutative counterpart of the Markov
operators of probability theory. A quantum channel Φ from the quantum system A with
Hilbert space HA to the quantum system B with Hilbert space HB is a linear completely
positive trace-preserving map from the trace-class operators on HA to the trace-class
operators on HB . Precisely, a map Φ is said to be
• positive if Φ(X̂) ≥ 0 for any trace-class operator X̂ ≥ 0;
• completely positive if the map Φ ⊗ Id is positive for any d ∈ N, where Id denotes
the identity map on the operators on the Hilbert space Cd ;
• trace-preserving if Tr Φ(X̂) = Tr X̂ for any trace-class operator X̂.
These properties ensure that for any d ∈ N the map Φ ⊗ Id sends the quantum states
on HA ⊗ Cd to quantum states on HB ⊗ Cd . Since any joint probability measure of two
random variables is a convex combination of product probability measures, the complete
positivity of any Markov operator is a trivial consequence of its positivity. On the contrary, the existence of entanglement makes complete positivity a nontrivial requirement
for quantum channels. For example, the transposition is a linear positive trace-preserving
map that is not completely positive.
Any quantum channel Φ from A to B can be realized by an isometry followed by a
partial trace, i.e., there exists an Hilbert space HE and an isometry V̂ : HA → HB ⊗ HE
with V̂ † V̂ = ÎA such that for any trace-class operator X̂ on HA ,
i
h
 
(3.9)
Φ X̂ = TrE V̂ X̂ V̂ † .

Gaussian optimizers for entropic inequalities in quantum information

11

The Hilbert space HE and the isometry V̂ are unique up to isometries on HE . The
expression (3.9) is called the Stinespring dilation of Φ. The quantum channel from A to
E deﬁned on trace-class operators as
i
h
 
(3.10)
Φ̃ X̂ = TrA V̂ X̂ V̂ †

is called the complementary channel of Φ [70]. We mention that the Stinespring dilation
has a classical analogue that has found some applications in the commutative setting,
e.g., in inﬁnite-dimensional stochastic analysis [63].
Let Φ : A → B be a quantum channel. The dual channel of Φ is the linear map Φ†
from bounded operators on HB to bounded operators on HA such that for any trace-class
operator Â on HA and any bounded operator B̂ on HB
h   i
 i
h
(3.11)
Tr B̂ Φ Â = Tr Φ† B̂ Â .
For any 1 ≤ p, q ≤ ∞, the p → q norm of a quantum channel Φ is deﬁned as
 
Φ X̂
q
kΦkp→q =
sup
.
0<kX̂ k <∞
X̂
p

(3.12)

p

A fundamental question is whether the p → q norm of a channel is multiplicative with
respect to the tensor product, i.e., whether
Φ⊗n

p→q

= kΦknp→q

∀n∈N.

(3.13)

This property holds for any classical integral kernel [86, 67], but it is known to fail for
generic quantum channels [67].

3.3. Quantum Gaussian systems
A n-mode Gaussian quantum system is the mathematical model for n harmonic oscillators, or n modes of the electromagnetic radiation. For the sake of simplicity, we present
one-mode Gaussian quantum systems ﬁrst.
The Hilbert space of a one-mode Gaussian quantum system is L2 (R), the irreducible
representation of the canonical commutation relation (see [97] or [70], Chapter 12 for a
more complete presentation)


â, â† = Î .
(3.14)

The operator â is called ladder operator, plays the role of a noncommutative complex
variable and acts on ψ in a suitable dense domain in L2 (R) as
(â ψ)(x) =

x ψ(x) + ψ ′ (x)
√
.
2

(3.15)

12

G. De Palma, D. Trevisan, V. Giovannetti, and L. Ambrosio

The quantum states of a one-mode Gaussian quantum system are the quantum counterparts of the probability measures on R2 . Since each mode is associated to a complex
noncommutative variable, the number of real classical components is twice the number
of quantum modes. We deﬁne the Hamiltonian
N̂ = â† â ,

(3.16)

that counts the number of excitations, or photons. The vector annihilated by â is the
vacuum and is denoted by |0i. From the vacuum we can build the eigenstates of the
Hamiltonian, called Fock states:
n
â†
√
|ni =
|0i , hm|ni = δmn , N̂ |ni = n|ni , m, n ∈ N ,
(3.17)
n!
where hφ|ψi denotes the scalar product in L2 (R). An operator diagonal in the Fock basis
is called Fock-diagonal.
A quantum Gaussian state is a quantum state proportional to the exponential of a
quadratic polynomial in â and â† . The most important Gaussian states are the thermal
Gaussian states, where the polynomial is proportional to the Hamiltonian â† â. They
correspond to a geometric probability distribution for the energy:
n
∞ 
E
1 X
|nihn| ,
ω̂(E) =
E + 1 n=0 E + 1

E≥0.

(3.18)

For E = 0, we recover the vacuum state ω̂(0) = |0ih0|. The average energy of ω̂(E) is
i
h
(3.19)
E = Tr N̂ ω̂(E) ,

and the von Neumann entropy is

g(E) := S(ω̂(E)) = (E + 1) ln (E + 1) − E ln E .

(3.20)

As Gaussian probability measures maximize the Shannon diﬀerential entropy among all
the probability measures with a given covariance matrix, thermal quantum Gaussian
states maximize the von Neumann entropy among all the quantum states with a given
average energy.
The Hilbert space of a n-mode Gaussian quantum system is the tensor product of n
Hilbert spaces of a one-mode Gaussian quantum system, i.e., the irreducible representation of the canonical commutation relations
i
h
i
h
i, j = 1, . . . , n ,
(3.21)
[âi , âj ] = â†i , â†j = 0 ,
âi , â†j = δij Î ,

where each ladder operator âi is associated to one mode. An n-mode thermal quantum
⊗n
Gaussian state is the tensor product ω̂(E)
of n identical one-mode thermal quantum
Gaussian states.

Gaussian optimizers for entropic inequalities in quantum information

13

3.4. Quantum Gaussian channels
Quantum Gaussian channels are the quantum channels that preserve the set of quantum Gaussian states. The most important families of quantum Gaussian channels are
the beam-splitter, the squeezing, the quantum Gaussian attenuators and the quantum
Gaussian ampliﬁers. The beam-splitter and the squeezing are the quantum counterparts
of the classical linear mixing of random variables, and are the main transformations in
quantum optics. Let A and B be one-mode quantum Gaussian systems with ladder operators â and b̂, respectively. The beam-splitter of transmissivity 0 ≤ λ ≤ 1 is implemented
by the unitary operator


√ 
(3.22)
Ûλ = exp â† b̂ − b̂† â arccos λ ,

and performs a linear rotation of the ladder operators (see e.g. [38], Section 1.4.2):
√
√
Ûλ† â Ûλ = λ â + 1 − λ b̂ ,
√
√
(3.23)
Ûλ† b̂ Ûλ = − 1 − λ â + λ b̂ .
The physical beam-splitter is a passive element, and does not require energy for functioning. Indeed, the mixing unitary operator preserves the Hamiltonian (3.16):


(3.24)
Ûλ† â† â + b̂† b̂ Ûλ = â† â + b̂† b̂ .

The two-mode squeezing [5] of parameter κ ≥ 1 is implemented by the unitary operator


√ 
(3.25)
Ûκ = exp â† b̂† − â b̂ arccosh κ ,

and acts on the ladder operators as

√
√
κ â + κ − 1 b̂† ,
√
√
Ûκ† b̂ Ûκ = κ − 1 â† + κ b̂ .

Ûκ† â Ûκ =

(3.26)

The squeezing is an active operation that requires energy. Indeed, the squeezing unitary
operator does not preserve the Hamiltonian (3.16).
We deﬁne for any joint quantum state ρ̂AB on AB and any λ ≥ 0 the quantum channel
from AB to A
h
i
(3.27)
Bλ (ρ̂AB ) = TrB Ûλ ρ̂AB Ûλ† ,

where TrB denotes the partial trace over the system B. Bλ implements the beam-splitter
for 0 ≤ λ ≤ 1 and the squeezing for λ ≥ 1.
The quantum Gaussian attenuators model the attenuation and the noise aﬀecting electromagnetic signals traveling through optical ﬁbers or free space. The quantum Gaussian
attenuator Eλ,E can be implemented mixing the input state ρ̂ with the thermal Gaussian
state with average energy E ≥ 0 through a beam-splitter of transmissivity 0 ≤ λ ≤ 1:
Eλ,E (ρ̂) = Bλ (ρ̂ ⊗ ω̂(E)) .

(3.28)

14

G. De Palma, D. Trevisan, V. Giovannetti, and L. Ambrosio

The quantum Gaussian attenuators constitute a multiplicative semigroup with composition law
E1,E = I ,

Eλ,E ◦ Eλ′ ,E = Eλλ′ ,E

∀ E ≥ 0, 0 ≤ λ, λ′ ≤ 1 .

(3.29)

The quantum Gaussian ampliﬁers model the ampliﬁcation of electromagnetic signals.
The quantum Gaussian amplifier Aκ,E can be implemented performing a two-mode
squeezing of parameter κ ≥ 1 on the input state ρ̂ and the thermal Gaussian state
with average energy E ≥ 0:
Aκ,E (ρ̂) = Bκ (ρ̂ ⊗ ω̂(E)) .

(3.30)

Also the quantum Gaussian ampliﬁers constitute a semigroup with composition law
A1,E = I ,

Aκ,E ◦ Aκ′ ,E = Aκκ′ ,E

∀ E ≥ 0, κ, κ′ ≥ 1 .

(3.31)

The attenuator Eλ,E and the ampliﬁer Aκ,E are called quantum-limited if E = 0, i.e., if
they mix the input state with the vacuum. Indeed, the vacuum as state of the environment
adds the least possible noise to the input state. In this case, since ω̂(0) = |0ih0| is a pure
state, the expressions (3.28) and (3.30) are the Stinespring dilations of the corresponding
channels.

4. The minimum output entropy conjecture
The minimum output entropy of a quantum channel plays a key role in the determination
of its classical information capacity. The following theorem has been a fundamental result
in quantum communication theory.
Theorem 4.1. For any n ∈ N, the vacuum state minimizes the output entropy of the
n-mode quantum Gaussian attenuators and of the n-mode quantum Gaussian amplifiers,
i.e., for any n-mode quantum state ρ̂, any E ≥ 0, 0 ≤ λ ≤ 1 and κ ≥ 1





⊗n
⊗n
⊗n
= n S(Eλ,E (|0ih0|)) ,
S Eλ,E
(ρ̂) ≥ S Eλ,E
|0ih0|





⊗n
⊗n
= n S(Aκ,E (|0ih0|)) .
(4.1)
S A⊗n
κ,E (ρ̂) ≥ S Aκ,E |0ih0|
Therefore, the minimum output entropy of the quantum attenuators and amplifiers is
additive.

We stress that Theorem 4.1 is trivial for classical Gaussian channels, i.e., for Gaussian
integral kernels that send probability measures on Rm to probability measures on Rn .
Indeed, for the concavity of the entropy it is suﬃcient to prove Theorem 4.1 for pure
input states. In the classical case, the only pure probability measures are the Dirac delta
functions, and they all achieve the same output entropy. As we will see, the proof of

Gaussian optimizers for entropic inequalities in quantum information

15

Theorem 4.1 exploits tools of quantum information theory that do not have a classical
counterpart: the complementary channel and the decomposition of any Gaussian channel
as a quantum-limited attenuator followed by a quantum-limited ampliﬁer.
The proof of Theorem 4.1 is based on majorization theory [89].
Definition 1 (majorization). We say that the quantum state ρ̂ majorizes the quantum
state σ̂, and write ρ̂ ≻ σ̂, iff σ̂ can be obtained applying to ρ̂ a convex combination
of unitary operators, i.e., iff there exists a probability measure µ on the set of unitary
operators such that
Z
 
(4.2)
σ̂ = Û ρ̂ Û † dµ Û .
The link between majorization and the entropy is provided by the following property.
Proposition 4.2. Let ρ̂ and σ̂ be quantum states such that ρ̂ ≻ σ̂. Then, f (ρ̂) ≥ f (σ̂)
for any unitarily invariant convex functional f on the set of quantum states. In particular,
• kρ̂kp ≥ kσ̂kp for any p ≥ 1;
• S(ρ̂) ≤ S(σ̂).
Theorem 4.1 is a consequence of this more fundamental result.
Theorem 4.3 (majorization for quantum Gaussian channels). For any n ∈ N and for
all the n-mode quantum Gaussian attenuators and amplifiers, the output generated by
the vacuum input state majorizes the output generated by any other input state, i.e., for
any 0 ≤ λ ≤ 1, κ ≥ 1 and E ≥ 0 and for any n-mode quantum state ρ̂,




⊗n
⊗n
⊗n
⊗n
⊗n
⊗n
≻ Aκ,E
(ρ̂) .
(4.3)
≻ Eλ,E
(ρ̂) ,
Aκ,E
|0ih0|
Eλ,E
|0ih0|
Besides Theorem 4.1, a fundamental consequence of Theorem 4.3 is the following.
Corollary 4.4 (1 → p norms of quantum Gaussian channels). For any p ≥ 1 and any
n ∈ N, the vacuum input state achieves the 1 → p norm of the n-mode quantum Gaussian
attenuators and amplifiers, i.e., for any 0 ≤ λ ≤ 1, κ ≥ 1 and E ≥ 0


⊗n
n
⊗n
⊗n
Eκ,E
= Eκ,E
|0ih0|
= kEκ,E (|0ih0|)kp ,
1→p
p


⊗n
⊗n
Aκ,E
= Aκ,E
|0ih0|⊗n
= kAκ,E (|0ih0|)knp .
(4.4)
1→p

p

Therefore, the 1 → p norms of the quantum Gaussian attenuators and amplifiers are
multiplicative.
Theorem 4.1 was ﬁrst proven by Giovannetti, Holevo and Garcı́a-Patrón [48]. Shortly
later, Mari, Giovannetti and Holevo realized that the same proof implies the more general

16

G. De Palma, D. Trevisan, V. Giovannetti, and L. Ambrosio

Theorem 4.3, ﬁrst for one-mode quantum Gaussian channels [88], and then for multi-mode
quantum Gaussian channels [47, 68]. We present here a sketch of the proof. For more
details, the reader can also consult the review [71].
The ﬁrst step to prove Theorem 4.3 is the following observation.
Proposition 4.5. For any n, any n-mode quantum Gaussian attenuator or amplifier can be decomposed as an n-mode quantum-limited attenuator followed by a n-mode
quantum-limited amplifier.
Theorem 4.3 is trivial for the quantum-limited attenuator, since the vacuum is a ﬁxed
point. Thanks to Proposition 4.5, it is suﬃcient to prove Theorem 4.3 for the quantumlimited ampliﬁer. It is easy to see that it is suﬃcient to prove Theorem 4.3 for pure input
states. The next step exploits the following properties:
Proposition 4.6. Let Φ be a quantum channel and Φ̃ be its complementary channel.
Then, for any pure input state ψ, the quantum states Φ(|ψihψ|) and Φ̃(|ψihψ|) have the
same spectrum.
From Proposition 4.6, the optimal input states for Φ and Φ̃ must coincide.
Proposition 4.7. The complementary channel of the quantum-limited amplifier is a
quantum-limited attenuator followed by the same quantum-limited amplifier followed by
the transposition, i.e., for any κ ≥ 1,
Ãκ,0 = T ◦ Aκ,0 ◦ E1−1/κ,0 ,

(4.5)

where T is the transposition operation.
From Propositions 4.6 and 4.7, the optimal input states for the quantum-limited ampliﬁer must coincide with the optimal input states for a suitable quantum-limited attenuator composed with the same quantum-limited ampliﬁer. Since the optimal input states
must be pure, they must be left pure by the quantum-limited attenuator. The claim then
follows from the following property.
Proposition 4.8. For any n ∈ N and any 0 < λ < 1, the vacuum is the only n-mode
⊗n
quantum state ρ̂ such that Eλ,0
(ρ̂) is pure.

5. Gaussian optimizers for entropic inequalities in
quantum information
The problem of determining the information capacity region of the quantum Gaussian
degraded broadcast channel has led to a constrained minimum output entropy conjecture
[58], which is a generalization of Theorem 4.1 with a constrained input entropy.

Gaussian optimizers for entropic inequalities in quantum information

17

Conjecture 5.1 (constrained minimum output entropy conjecture). For any n ∈ N,
quantum Gaussian input states minimize the output entropy of the n-mode Gaussian
quantum attenuators and amplifiers among all the input states with a given entropy. In
other words, let ρ̂ be a generic n-mode quantum state, and let ω̂ be the one-mode thermal
Gaussian state with entropy S(ρ̂)/n, so that ω̂ ⊗n is the n-mode thermal Gaussian state
with the same entropy as ρ̂. Then, for any 0 ≤ λ ≤ 1, κ ≥ 1 and E ≥ 0,








S(ρ̂)
⊗n
⊗n
+ (1 − λ) E ,
S Eλ,E
(ρ̂) ≥ S Eλ,E
ω̂ ⊗n = n S(Eλ,E (ω̂)) = n g λ g −1
n




⊗n
⊗n
⊗n
= n S(Aκ,E (ω̂))
S Aκ,E (ρ̂) ≥ S Aκ,E ω̂




S(ρ̂)
−1
= ng κg
+ (κ − 1) (E + 1) ,
(5.1)
n
where the function g has been defined in (3.20).
Conjecture 5.1 has been proven only in the one-mode case (n = 1) by De Palma, Trevisan and Giovannetti [34, 33], and has been extended to one-mode gauge-contravariant
quantum Gaussian channels by Qi, Wilde and Guha [93]. The proof by De Palma et al. is
based on the following fundamental majorization result for one-mode quantum Gaussian
channels [32], which extends Theorem 4.3.
Theorem 5.2.
ρ̂,
↓

For any 0 ≤ λ ≤ 1, κ ≥ 1 and E ≥ 0 and any one-mode quantum state

Eλ,E (ρ̂) ≺ Eλ,E ρ̂↓ ,


Aκ,E (ρ̂) ≺ Aκ,E ρ̂↓ ,

(5.2)

where ρ̂ is the passive rearrangement of ρ̂, i.e., the passive state with the same spectrum
as ρ̂.
We recall that a passive state is a quantum state that minimizes the average energy
among all the quantum states with the same spectrum [91, 82, 52]. If ρ̂ is diagonalized
in the orthonormal eigenbasis {ψn }n∈N as
ρ̂ =

∞
X

n=0

ρ̂↓ is given by

pn |ψn ihψn | ,

ρ̂↓ =

∞
X

n=0

p0 ≥ p1 ≥ . . . ≥ 0 ,

pn |nihn| ,

(5.3)

(5.4)

where {|ni}n∈N is the Fock basis.
From Theorem 5.2, in the case of one mode the constrained minimization of the output
entropy of Conjecture 5.1 can be restricted to passive input states. Unfortunately, an
analogue majorization theorem does not hold for more than one mode [29].
Conjecture 5.1 has ﬁrst been proven for the one-mode quantum-limited attenuator
[34]. The proof is based on the following isoperimetric inequality, that constitutes the
inﬁnitesimal version of the conjecture.

18

G. De Palma, D. Trevisan, V. Giovannetti, and L. Ambrosio

Theorem 5.3 (isoperimetric inequality for the one-mode quantum-limited attenuator).
Among all the input states with a given entropy, quantum Gaussian input states maximize
the derivative of the output entropy of the one-mode quantum-limited attenuator with
respect to the attenuation parameter. In other words, let ρ̂ be a one-mode quantum state,
and ω̂ the one-mode thermal Gaussian state with the same entropy as ρ̂. Then,
d
S (Eλ,0 (ρ̂))
dλ

λ=1

≤

d
S (Eλ,0 (ω̂))
dλ

λ=1


= g −1 (S(ρ̂)) g ′ g −1 (S(ρ̂)) .

(5.5)

The adjective “isoperimetric” is due to the formal analogy between entropy and volume
[36]. Up to a change of signs, the left hand side in (5.5) plays the role of a perimeter and
the function g −1 (s)g ′ (g −1 (s)) that of an isoperimetric proﬁle.
Thanks to Theorem 5.2, it is suﬃcient to prove Theorem 5.3 for passive states. The
proof is then performed through the Lagrange multipliers. Since the Hilbert space of a
one-mode Gaussian quantum system has inﬁnite dimension, a generic passive state has
inﬁnite parameters. This issue is solved restricting to a ﬁnite dimensional subspace with
bounded maximum energy, and then proving that the maximum of the left-hand side of
(5.5) for passive input states supported in the subspace tends to the right-hand side in
the limit of inﬁnite maximum energy.
Conjecture 5.1 for the one-mode quantum-limited attenuator then follows integrating the isoperimetric inequality (5.5) thanks to the semigroup property (3.29) of the
quantum-limited attenuator.
The generalization of Theorem 5.3 to all the one-mode quantum Gaussian attenuators and ampliﬁers would have implied Conjecture 5.1 for n = 1. However, for any
one-mode quantum Gaussian channel other than the quantum-limited attenuator, the
inﬁnite dimension of the Hilbert space is really an issue. Indeed, for any quantum state
d
S (Eλ,E (ρ̂)) λ=1 is inﬁnite for any E > 0 and
ρ̂ with a support of ﬁnite dimension dλ
d
dκ S (Aλ,E (ρ̂)) κ=1 is inﬁnite for any E ≥ 0, and nothing can be proven restricting to a
ﬁnite dimensional subspace. If one tries to use the Lagrange multipliers directly for the
inﬁnite dimensional problem, the Gaussian state is not the only solution [93], so that a
new approach is needed. This approach is based on the p → q norms and is presented in
subsection 5.1 below.

5.1. Quantum Gaussian channels have Gaussian maximizers
The theorem “Gaussian kernels have Gaussian maximizers” has been conjectured to
apply also to quantum Gaussian channels.
Conjecture 5.4 (quantum Gaussian channels have Gaussian maximizers). For any
n ∈ N and any p, q ≥ 1, quantum Gaussian input states achieve the p → q norm of the
n-mode Gaussian quantum attenuators and amplifiers. In other words, for any 0 ≤ λ ≤ 1,

Gaussian optimizers for entropic inequalities in quantum information

19

κ ≥ 1 and E ≥ 0,
⊗n
Eλ,E

A⊗n
κ,E

p→q

p→q

= sup
E ′ ≥0

= sup
E ′ ≥0



⊗n
⊗n
Eλ,E
ω̂(E ′ )

q

⊗n

ω̂(E ′ )
p


⊗n
⊗n
Aκ,E ω̂(E ′ )
⊗n

ω̂(E ′ )

=

sup

kEλ,E (ω̂(E ′ ))kq
kω̂(E ′ )kp

E ′ ≥0

q

=

sup

!n

kAκ,E (ω̂(E ′ ))kq
kω̂(E ′ )kp

E ′ ≥0

p

,

!n

,

(5.6)

′

where ω̂(E ) is the one-mode thermal Gaussian state with average energy E ′ as in (3.18).
Therefore, the p → q norms of the quantum Gaussian attenuators and amplifiers are
multiplicative.
Remark 5.1. The suprema in (5.6) are
• ﬁnite and achieved for a ﬁnite E ′ ≥ 0 if 1 ≤ p < q;
• ﬁnite and asymptotically achieved in the limit E ′ → ∞ if 1 < p = q;
• inﬁnite and asymptotically achieved in the limit E ′ → ∞ if 1 ≤ q < p.
Remark 5.2. Conjecture 5.4 can be extended to any linear and completely positive
map that preserves the set of unnormalized quantum Gaussian states, i.e., the operators
proportional to a quantum Gaussian state. These maps include all quantum Gaussian
channels and all the probabilistic maps resulting from the conditioning on the outcome of
a Gaussian measurement performed on a subsystem [70, 97]. The generalized conjecture
states that quantum Gaussian input states achieve the p → q norms of all such maps. In
this more general setup, the analogue of the optimization in the right-hand side of (5.6)
cannot be restricted to the thermal Gaussian states, but has to be performed over all
quantum Gaussian states.
Conjecture 5.4 has been proven only in some particular cases. As we have seen in
Corollary 4.4, the majorization result Theorem 4.3 implies Conjecture 5.4 for any n in
the case p = 1. De Palma, Trevisan and Giovannetti proved Conjecture 5.4 in the case of
one-mode quantum-limited channels, i.e., n = 1 and E = 0 [31]. Frank and Lieb proved
Conjecture 5.4 for any n in the case p = q [42], and Holevo extended the result to any
n-mode quantum Gaussian channel (still for p = q) [73].
5.1.1. The proof of Conjecture 5.4 for one-mode quantum-limited Gaussian channels
First, De Palma et al. prove Conjecture 5.4 for the one-mode quantum-limited attenuator.
From the following Lemma, it is suﬃcient to prove Conjecture 5.4 for positive input
operators.
Lemma 5.5 ([4]).

For any p ≥ 1, any quantum channel Φ and any operator X̂,

 
p
X̂ † X̂
.
(5.7)
Φ X̂
≤ Φ
p

p

20

G. De Palma, D. Trevisan, V. Giovannetti, and L. Ambrosio

The proof of Conjecture 5.4 is then based on the following new logarithmic Sobolev
inequality, that constitutes the inﬁnitesimal version of Conjecture 5.4 (in the same way
as Gross’ logarithmic Sobolev inequality is the inﬁnitesimal version of Nelson’s Hypercontractive theorem [53]).
Theorem 5.6 (logarithmic Sobolev inequality for the quantum-limited Gaussian attenuator). Let us fix p ≥ 1. Let ρ̂ be a one-mode quantum state, and let ω̂ be the thermal
Gaussian state such that ω̂ p /Trω̂ p has the same entropy as ρ̂p /Trρ̂p . Then,
d
ln kEλ,0 (ρ̂)kp
dλ

λ=1

≥

d
ln kEλ,0 (ω̂)kp
dλ

.

(5.8)

λ=1

Thanks to Theorem 5.2, it is suﬃcient to prove Theorem 5.6 for passive input states. As
in the case of Theorem 5.3, the proof is then performed through the Lagrange multipliers,
restricting to a ﬁnite dimensional subspace with bounded maximum energy. Conjecture
5.4 for the one-mode quantum-limited attenuator follows integrating (5.8) thanks to the
semigroup property of the attenuator (3.29).
Conjecture 5.4 for the one-mode quantum-limited ampliﬁer follows from the following
duality Lemma for the Schatten norms.
Lemma 5.7.

For any p > 1 and any positive operator X̂,

 h
i
Ŷ p : Ŷ ≥ 0, rank Ŷ < ∞ .
X̂ = sup Tr X̂ Ŷ
p

(5.9)

p−1

Lemma 5.7 implies the following duality for the norms of quantum channels.
Lemma 5.8.

For any quantum channel Φ and any p, q ≥ 1,
kΦkp→q = Φ†

q
p
q−1 → p−1

.

(5.10)

The norms of the quantum-limited ampliﬁer can then be determined from the norms
of the quantum-limited attenuator thanks to the following property.
Lemma 5.9. The dual of the quantum-limited Gaussian amplifier is proportional to a
quantum-limited Gaussian attenuator, i.e., for any κ ≥ 1,
A†κ,0 =

1
E1 .
κ κ ,0

(5.11)

5.1.2. The proof of Conjecture 5.1 for all the one-mode attenuators and amplifiers
De Palma, Trevisan and Giovannetti have exploited the proof of Conjecture 5.4 for the
one-mode quantum-limited ampliﬁer to prove Conjecture 5.1 for all the one-mode attenuators and ampliﬁers [33]. First, they prove Conjecture 5.1 for the one-mode quantumlimited ampliﬁer. The ﬁrst step is rephrasing Conjecture 5.4 for the one-mode quantumlimited ampliﬁer in the following way.

Gaussian optimizers for entropic inequalities in quantum information

21

Theorem 5.10. Let us fix κ ≥ 1. Let ρ̂ be a generic one-mode quantum state, and let
ω̂ be the one-mode thermal Gaussian state with the same entropy as ρ̂. Then, for any
q > 1 there exists 1 ≤ p < q such that the p → q norm of Aκ,0 is achieved by ω̂, and
kAκ,0 (ω̂)kq
kAκ,0 (ρ̂)kq
≤ kAκ,0 kp→q =
.
kρ̂kp
kω̂kp

(5.12)

Rewriting (5.12) in terms of the Rényi entropies we get
Sq (Aκ,0 (ρ̂)) ≥ Sq (Aκ,0 (ω̂)) +

p−1q
(Sp (ρ̂) − Sp (ω̂)) .
q−1p

(5.13)

Taking the limit q → 1 and recalling (3.5) we get the claim
S(Aκ,0 (ρ̂)) ≥ S(Aκ,0 (ω̂)) .

(5.14)

This result implies Conjecture 5.1 for all the one-mode attenuators and ampliﬁers since
any of these channels can be decomposed as a one-mode quantum-limited attenuator followed by a one-mode quantum-limited ampliﬁer (Proposition 4.5), for which Conjecture
5.1 holds.
5.1.3. The proof of Conjecture 5.4 for p=q
The proof by Frank and Lieb is completely diﬀerent from the proof by De Palma et al.,
and is based on the following theorem.
Theorem 5.11.

For any p > 1 and any quantum channel Φ,

kΦkp→p ≤ Φ Î

p−1
p

∞


Φ† Î

1
p

∞

.

(5.15)

Conjecture 5.4 follows directly applying Theorem 5.11 to quantum Gaussian channels.
The proof of Theorem 5.11 is based on Hadamard’s three line lemma [99].
Theorem 5.12 (Hadamard’s three line lemma). Let f be analytic in the strip {z : 0 <
ℜz < 1} and continuous and bounded on its closure. Let
Mt (f ) = sup |f (t + iy)|

(5.16)

Mt ≤ M01−t M1t .

(5.17)

y∈R

for 0 ≤ t ≤ 1. Then

Theorem 5.11 follows applying Theorem 5.12 to
i
h 1−z 
,
f (z) = Tr Ŷ p p−1 Φ X̂ pz

(5.18)

where X̂ and Ŷ are positive and Ŷ has ﬁnite rank, and recalling the duality relation for
the Schatten norms (Lemma 5.7).

22

G. De Palma, D. Trevisan, V. Giovannetti, and L. Ambrosio

5.2. The Entropy Photon-number Inequality
The Entropy Photon-number Inequality is the quantum counterpart of the Entropy Power
Inequality for the beam-splitter. Guha, Erkmen and Shapiro conjectured it [55] as a
generalization of Conjecture 5.1, and De Palma, Mari and Giovannetti extended the
conjecture to the squeezing [27].
Conjecture 5.13 (Entropy Photon-number Inequality). For any n, n-mode thermal
quantum Gaussian states minimize the output entropy of the n-mode beam-splitter or
squeezing among all the n-mode input states where the two inputs have given entropies.
In other words, let ρ̂A and ρ̂B be two n-mode quantum states, and let ω̂A and ω̂B be the
⊗n
one-mode thermal Gaussian states with entropies S(ρ̂A )/n and S(ρ̂B )/n, such that ω̂A
⊗n
and ω̂B are the n-mode thermal Gaussian states with the same entropy as ρ̂A and ρ̂B ,
respectively. Then, for any 0 ≤ λ ≤ 1


⊗n
⊗n
S Bλ⊗n (ρ̂A ⊗ ρ̂B ) ≥ S Bλ⊗n ω̂A
⊗ ω̂B





S(ρ̂B )
S(ρ̂A )
+ (1 − λ) g −1
,
(5.19)
= n g λ g −1
n
n
and for any κ ≥ 1



⊗n
⊗n
⊗ ω̂B
S Bκ⊗n (ρ̂A ⊗ ρ̂B ) ≥ S Bκ⊗n ω̂A







S(ρ̂A )
S(ρ̂B )
−1
−1
=ng κg
+ (κ − 1) g
+1
,
n
n

(5.20)

where the function g has been defined in (3.20).
For any n-mode quantum state ρ̂, the n-mode thermal Gaussian state with the same
entropy as ρ̂ has average photon number per mode g −1 (S(ρ̂/n)). This quantity is called
the entropy photon-number of ρ̂, hence the name Entropy Photon-number Inequality.
In the case where the second input ρ̂B of the beam-splitter or of the squeezing is
a thermal Gaussian state, Conjecture 5.13 reduces to Conjecture 5.1. The only other
particular case where the Entropy Photon-number Inequality has been proven is when
the two inputs are (not necessarily thermal) Gaussian states [21].

5.3. The sharp Young’s inequality for the beam-splitter
The similarity between the Entropy Photon-number Inequality and the Entropy Power
Inequality together with the proof of the latter through the sharp Young’s inequality
for convolutions leads to conjecture a quantum version of the Young’s inequality, here
formulated for the ﬁrst time.
Let us deﬁne for any n ∈ N, any p, q, r ≥ 1 and any λ ≥ 0


Bλ⊗n X̂ ⊗ Ŷ
r
Cn (p, q, r, λ) =
sup
.
(5.21)
0<kX̂kp ,kŶ kq <∞
Ŷ
X̂
p

q

Gaussian optimizers for entropic inequalities in quantum information

23

Conjecture 5.14 (Quantum sharp Young’s inequality). For any n ∈ N, any p, q, r ≥ 1
and any λ ≥ 0, the supremum in (5.21) can be restricted to thermal Gaussian states, i.e.,


⊗n
⊗n
Bλ⊗n ω̂(EA ) ⊗ ω̂(EB )
r
Cn (p, q, r, λ) = sup
⊗n
⊗n
EA , EB ≥0
ω̂(EB )
ω̂(EA )
q
p
!n
kBλ (ω̂(EA ) ⊗ ω̂(EB ))kr
n
=
sup
= C1 (p, q, r, λ) .
(5.22)
kω̂(E
)k
kω̂(E
)k
A p
B q
EA , EB ≥0
Therefore, the constants Cn are multiplicative.
Remark 5.3. We conjecture that the supremum in (5.22) is
• ﬁnite and achieved by ﬁnite EA , EB ≥ 0 if p1 + q1 > 1 + 1r ;
• ﬁnite and asymptotically achieved in the limit EA , EB → ∞ if p1 + 1q = 1 + r1 ;
• inﬁnite and asymptotically achieved in the limit EA , EB → ∞ if 1p + q1 < 1 + r1 .
The striking diﬀerence with respect to the classical case is that the supremum in (5.22)
is ﬁnite when p1 + q1 > 1 + 1r . The divergence in the classical Young’s inequality when
1
1
1
p + q > 1 + r is asymptotically achieved by a sequence of Gaussian probability measures
that tends to a Dirac delta, and can be ascribed to the fact that a probability density
can have arbitrarily high L∞ norm. The divergence disappears in the quantum scenario
since kρ̂k∞ ≤ 1 for any quantum state ρ̂.
The quantum sharp Young’s inequality provides a multiplicative upper bound to the
p → q norms of the quantum Gaussian attenuators and ampliﬁers. Indeed, assuming
Conjecture 5.14, we have for any n ∈ N, p, q ≥ 1, 0 ≤ λ ≤ 1 and E ≥ 0
 


⊗n
⊗n
X̂
Eλ,E
Bλ⊗n X̂ ⊗ ω̂(E)
q
q
⊗n
Eλ,E
=
sup
=
sup
p→q
0<kX̂kp <∞
0<kX̂kp <∞
X̂
X̂
p
p

n
≤ inf C1 (p, r, q, λ) kω̂(E)kr
,
(5.23)
r≥1

and the same holds for the Gaussian quantum ampliﬁers. Since the conjectured quantum
sharp Young’s inequality is saturated by quantum Gaussian states, we conjecture that
the upper bound (5.23) is sharp and coincides with (5.6), i.e., that
sup
E ′ ≥0

kEλ,E (ω̂(E ′ ))kq
kω̂(E ′ )kp

= inf C1 (p, r, q, λ) kω̂(E)kr .
r≥1

(5.24)

Moreover, the quantum sharp Young’s inequality provides a lower bound to the output
entropy of the beam-splitter and of the squeezing. Indeed, rewriting (5.22) in terms of

24

G. De Palma, D. Trevisan, V. Giovannetti, and L. Ambrosio

the Rényi entropies we get for any n ∈ N, λ ≥ 0, p, q, r ≥ 1 and any n-mode quantum
states ρ̂A and ρ̂B



p−1
q−1
r
Sp (ρ̂A ) +
Sq (ρ̂B ) − n ln C1 (p, q, r, λ) .
Sr Bλ⊗n (ρ̂A ⊗ ρ̂B ) ≥
r−1
p
q
(5.25)
r
and set
We choose 0 ≤ α, β < r−1
p = p(r, α) =

r
,
r + α− αr

q = q(r, β) =

r
,
r+β−βr

(5.26)

so that (5.25) becomes

nr
ln C1 (p(r, α), q(r, β), r, λ) .
Sr Bλ⊗n (ρ̂A ⊗ ρ̂B ) ≥ α Sp(r,α) (ρ̂A ) + β Sq(r,β) (ρ̂B ) −
r−1
(5.27)
Finally, taking the limit r → 1 and the supremum over α, β ≥ 0 we get



d
S Bλ⊗n (ρ̂A ⊗ ρ̂B ) ≥ sup α S(ρ̂A ) + β S(ρ̂B ) − n
,
C1 (p(r, α), q(r, β), r, λ)
dr
α,β≥0
r=1
(5.28)
where we used that
lim p(r, α) = lim q(r, β) = 1 ,

r→1

r→1

C1 (1, 1, 1, λ) = 1 .

(5.29)

Since the conjectured quantum Young inequality is saturated by quantum Gaussian
states, we conjecture that the lower bound (5.28) is sharp and coincides with the bound
provided by the Entropy Photon-number Inequality (5.19).

6. The thinning
The thinning [94] is the map acting on probability distributions on N that is the discrete
analogue of the continuous rescaling operation on R+ .
Definition 2 (Thinning). Let N be a random variable with values in N. The thinning
with parameter 0 ≤ λ ≤ 1 is defined as
Tλ (N ) =

N
X

Bi ,

(6.1)

i=1

where the {Bn }n∈N+ are independent Bernoulli variables with parameter λ (also independent of N ), i.e., each Bi is 1 with probability λ, and 0 with probability 1 − λ.
From a physical point of view, the thinning can be understood as follows. Let p be the
probability distribution of the number N of photons that are sent through a beam-splitter

Gaussian optimizers for entropic inequalities in quantum information

25

with transmissivity λ, such that for any n ∈ N the probability that n photons are sent is
pn . Each photon has probability λ of being transmitted and probability 1 − λ of being
reﬂected. Then, Tλ (N ) is the random variable associated to the number of transmitted
photons, and has probability distribution
∞  
X
k n
[Tλ (p)]n =
λ (1 − λ)k−n pk
∀n∈N.
(6.2)
n
k=n

The thinning coincides with the restriction of the one-mode quantum-limited Gaussian
attenuator to input states diagonal in the Fock basis.
Theorem 6.1 ([32], Theorem 56). For any 0 ≤ λ ≤ 1 and any probability distribution
p on N
!
∞
∞
X
X
[Tλ (p)]n |nihn| .
(6.3)
Eλ,0
pn |nihn| =
n=0

n=0

We recall that for any E ≥ 0, the thermal quantum Gaussian states ω̂(E) corresponds
to the geometric probability distribution ω(E) for the energy given by

n
E
1
.
(6.4)
ω(E)n =
E+1 E+1
We can then extend to the thinning all the results on the quantum-limited attenuator.
Let p and q be two probability distributions on N. We say that p majorizes q, and
write p ≻ q, iﬀ there exists a doubly stochastic inﬁnite matrix A such that [89]
qn =

∞
X

Ank pk

k=0

∀n∈N.

(6.5)

The inﬁnite matrix A is doubly stochastic iﬀ
Amn ≥ 0 ∀ m, n ∈ N ,

∞
X

Ank =

k=0

∞
X

k=0

Akn = 1

∀n∈N.

(6.6)

The link with the majorization for quantum states of Deﬁnition 1 is the following.
Theorem 6.2. The quantum state ρ̂ majorizes the quantum state σ̂ iff the probability
distribution on N associated to the spectrum of ρ̂ majorizes the probability distribution
on N associated to the spectrum of σ̂.
Theorem 5.2 implies then
Theorem 6.3.

For any 0 ≤ λ ≤ 1 and any probability distribution p on N,
T λ p ≺ T λ p↓ ,

(6.7)

where p↓ is the decreasing rearrangement of p, i.e., p↓n = pσ(n) for any n ∈ N, where
σ : N → N is a bijective function such that pσ(0) ≥ pσ(1) ≥ . . . ≥ 0.

26

G. De Palma, D. Trevisan, V. Giovannetti, and L. Ambrosio

The Shannon entropy of the probability measure p on N is the counterpart of the von
Neumann entropy:
∞
X
S(p) = −
pn ln pn .
(6.8)
n=0

The proof of Conjecture 5.1 for the one-mode quantum-limited attenuator [34] implies

Theorem 6.4. Geometric input probability distributions minimize the output Shannon
entropy of the thinning among all the input probability distribution with a given Shannon
entropy. In other words, let p be a generic probability distribution on N, and let ω be the
geometric probability distribution with the same Shannon entropy as p. Then, for any
0≤λ≤1

S(Tλ (p)) ≥ S(Tλ (ω)) = g λ g −1 (S(p)) .
(6.9)
For any p ≥ 1, the lp norm of the sequence of complex numbers {xn }n∈N is
! p1
X
.
kxkp =
|xn |p

(6.10)

n∈N

For any p, q ≥ 1, the p → q norm of the thinning is
kTλ kp→q =

kTλ xkq

sup
0<kxkp <∞

.

kxkp

(6.11)

The proof of Conjecture 5.4 for the one-mode quantum-limited attenuator [31] implies
then
Theorem 6.5. For any p, q ≥ 1, the p → q norm of the thinning is achieved by
geometric probability distributions, i.e., for any 0 ≤ λ ≤ 1,
kTλ kp→q = sup

E≥0

kTλ ω(E)kq
kω(E)kp

.

(6.12)

Remark 6.1. The supremum in (6.12) is
• ﬁnite and achieved for a ﬁnite E ≥ 0 if 1 ≤ p < q;
• ﬁnite and asymptotically achieved in the limit E → ∞ if 1 < p = q;
• inﬁnite and asymptotically achieved in the limit E → ∞ if 1 ≤ q < p.

7. Quantum conditioning and the quantum Entropy
Power Inequality
7.1. The quantum Entropy Power Inequality
The ﬁrst attempt to prove the Entropy Photon-number Inequality was through the quantum counterpart of the heat semigroup technique of the proof of the Entropy Power

Gaussian optimizers for entropic inequalities in quantum information

27

Inequality by Blachman and Stam. However, this technique only leads to the quantum
Entropy Power Inequality [79, 80, 27], which has the same expression as the Entropy
Power Inequality and provides a lower bound to the output entropy of the beam-splitter
or of the squeezing in terms of the entropies of the two inputs. Since this bound is strictly
lower than the output entropy achieved by thermal Gaussian input states, the quantum
Entropy Power Inequality is strictly weaker than the Entropy Photon-number Inequality.
Theorem 7.1 (quantum Entropy Power Inequality). For any λ ≥ 0 and any two nmode quantum states ρ̂A and ρ̂B with a finite average energy,

S Bλ⊗n (ρ̂A ⊗ ρ̂B )
S(ρ̂A )
S(ρ̂B )
exp
≥ λ exp
+ |1 − λ| exp
.
(7.1)
n
n
n
Remark 7.1. The factors of 2 in the exponents in the classical Entropy Power Inequality (2.13) do not appear in (7.1) because an n-mode quantum state is the counterpart
of a random variable on R2n . The coeﬃcients in front of the exponentials in the righthand side of (7.1) come from the coeﬃcients in the transformation rules for the ladder
operators (3.23) and (3.25).
The quantum Entropy Power Inequality was proposed by König and Smith [79], who
proved it in the case λ = 21 [79, 80]. De Palma, Mari and Giovannetti extended the
proof to any λ ≥ 0 [27]. De Palma, Mari, Lloyd and Giovannetti proposed and proved an
Entropy Power Inequality for the most general linear transformation of bosonic modes
[28]. Huber, König and Vershynina proposed and proved an Entropy Power Inequality
for the quantum additive noise channels [74].

7.2. Quantum conditioning
In the classical scenario, the Shannon entropy of the random variable A conditioned on
the “memory” random variable M with law p is deﬁned as the expectation value of the
Shannon entropy of A conditioned on the values assumed by M [19]:
Z
S(A|M ) = S(A|M = m) dp(m) .
(7.2)
Let now A and M be quantum systems, and let us consider a quantum state ρ̂AM on the
joint system AM . The deﬁnition (7.2) cannot be brought to the quantum setting when
A is entangled with M , since conditioning on the values assumed by M is not possible.
However, (7.2) can be rewritten as
S(A|M ) = S(AM ) − S(M ) ,

(7.3)

that is the right deﬁnition for the quantum conditional entropy [90, 64, 108, 70] (see [101]
for a broad discussion). We write S(A|M )ρ̂AM when the joint quantum state to which
the conditional entropy refers is not clear from the context. A striking feature of the

28

G. De Palma, D. Trevisan, V. Giovannetti, and L. Ambrosio

quantum conditional entropy is that it can be negative, while the quantum entropy is
always positive.
The correlation between two random variables or two quantum systems A and B are
quantiﬁed by the (quantum) mutual information [90, 64, 108, 70]
I(A : B) = S(A) + S(B) − S(AB) .

(7.4)

Both the classical and quantum versions of the mutual information are positive as a
consequence of the subadditivity of the entropy [90, 64, 108, 70]. The classical mutual
information vanishes iﬀ A and B are independent random variables. Analogously, the
quantum mutual information vanishes iﬀ ρ̂AB = ρ̂A ⊗ ρ̂B .
The conditional mutual information between A and B conditioned on the memory M
is
I(A : B|M ) = S(A|M ) + S(B|M ) − S(AB|M ) .
(7.5)

The classical conditional mutual information is positive as a consequence of the expression (7.2) for the conditional entropy and of the positivity of the mutual information [19].
Also the quantum conditional mutual information is positive [90, 64, 108, 70]. Since the
quantum conditional entropy cannot be written as in (7.2), this result is highly nontrivial. The classical conditional mutual information vanishes iﬀ A and B are conditionally
independent given the value of M . The quantum conditional mutual information vanishes
for all the joint quantum states of the following form [65]
ρ̂ABM =

∞
M
n=0

pn ρ̂AM (n) ⊗ ρ̂BM (n) ,
A

(7.6)

B

where p is a probability distribution on N and each ρ̂AM (n) or ρ̂BM (n) is a quantum state
A
B
on the Hilbert space HA ⊗ HM (n) or HB ⊗ HM (n) , respectively, and where
A

HM =

B

∞
M
n=0

HM (n) ⊗ HM (n) .
A

(7.7)

B

If A, B and M have ﬁnite dimension, all the quantum states with vanishing conditional
mutual information are of the form (7.6). The same property is believed to hold for
inﬁnite dimension, but this has not been proven yet.
A fundamental consequence of the positivity of the quantum conditional mutual information is the associated data-processing inequality, stating that discarding a subsystem
always decreases the quantum conditional mutual information, i.e., for any quantum state
on a joint quantum system ABCM
I(AC : B|M ) ≤ I(A : B|M ) .

(7.8)

7.3. The quantum conditional Entropy Power Inequality
Let X and Y be random variables with values in Rn , and let M be a random variable such
that X and Y are conditionally independent given M . Then, the expression (7.2), the

Gaussian optimizers for entropic inequalities in quantum information

29

Entropy Power Inequality (2.13) and Jensen’s inequality imply the conditional Entropy
Power Inequality [30]
exp

2S(X|M )
2S(Y |M )
2S(X + Y |M )
≥ exp
+ exp
.
n
n
n

(7.9)

The inequality (7.9) is saturated by any joint probability measure on ABM such that,
conditioning on any value m of M , A and B are independent Gaussian random variables
with proportional covariance matrices, and the proportionality constant does not depend
on m.
Since the quantum conditional entropy cannot be expressed as in (7.2), the above proof
does not go through in the quantum setting. However, the following quantum conditional
Entropy Power Inequality follows adapting the proof of the quantum Entropy Power
Inequality.
Theorem 7.2 (quantum conditional Entropy Power Inequality). Quantum Gaussian
states minimize the output quantum conditional entropy of the beam-splitter and of the
squeezing among all the input states where the two inputs are conditionally independent
given the memory. In other words, let A and B be n-mode Gaussian quantum systems,
and let M be a generic quantum system. Let ρ̂ABM be a joint quantum state with finite
average energy on AB, finite S(ρ̂M ) and with I(A : B|M ) = 0, and let

(7.10)
ρ̂CM = Bλ⊗n ⊗ IM (ρ̂ABM ) ,

where λ ≥ 0 and A and B are the two inputs of the beam-splitter or of the squeezing.
Then,
S(A|M )
S(B|M )
S(C|M )
≥ λ exp
+ |1 − λ| exp
.
(7.11)
exp
n
n
n
Moreover, let M be a 2n-mode Gaussian quantum system of the form M = MA MB ,
where MA and MB are n-mode Gaussian quantum systems. Then, for any a, b ∈ R
(k)
there exists a sequence {ρ̂ABM }k∈N of 4n-mode quantum Gaussian states of the form
(k)
(k)
(k)
ρ̂ABM = ρ̂AMA ⊗ ρ̂BMB such that
S(A|M )ρ̂(k)

=a,

ABM

S(B|M )ρ̂(k)

=b

(7.12)

ABM

for any k ∈ N, and
lim exp

k→∞

S(C|M )ρ̂(k)

CM

n

= λ exp

a
b
+ |1 − λ| exp .
n
n

(7.13)

If M is trivial, (7.11) becomes the quantum Entropy Power Inequality. The quantum
conditional Entropy Power Inequality was ﬁrst conjectured by König, who proved it in
the case 0 ≤ λ ≤ 1 for Gaussian input states [77]. The general case was proven by
De Palma and Trevisan [30]. De Palma and Huber proved a conditional Entropy Power

30

G. De Palma, D. Trevisan, V. Giovannetti, and L. Ambrosio

Inequality for the quantum additive noise channels [25]. The proofs of [30, 25] settled
some regularity issues that aﬀected the previous proofs of [79, 27, 28, 77].
The proof of the quantum conditional Entropy Power inequality of [30] is the quantum
counterpart of the proof of the classical Entropy Power Inequality by Blachman and Stam
based on the evolution with the heat semigroup. Let A be a n-mode Gaussian quantum
system with ladder operators â1 , . . . , ân . The displacement operator D̂(z) with z ∈ Cn
is the unitary operator that displaces the ladder operators:
†

D̂(z) âi D̂(z) = âi + zi Î

i = 1, . . . , n .

(7.14)

The quantum heat semigroup is the quantum Gaussian channel generated by a convex
combination of displacement operators with a Gaussian probability measure:
Z
√ †
√ 
2 dz
t z ρ̂ D̂
t z e−|z| n , N0 = I , Nt ◦Nt′ = Nt+t′ ∀ t, t′ ≥ 0 .
D̂
Nt (ρ̂) =
π
Cn
(7.15)
This is the quantum counterpart of the classical heat semigroup acting on a probability
density function f on Cn :
Z

√ 
2 dz
(Nt f )(w) =
f w − t z e−|z| n ,
w ∈ Cn
(7.16)
π
Cn

Let ρ̂AM be a joint quantum state on AM . The quantum conditional Fisher information
of the state ρ̂AM is the rate of increase of the quantum
√ conditional mutual information
between A and Z when the system A is displaced by t Z according to (7.15). In other
words, for any t > 0, let σ̂AMZ (t) be the probability measure on Cn with values in
quantum states on AM such that
Z
√ 
√ †
2 dz
dσ̂AMZ (z, t) = D̂
t z ρ̂ D̂
t z e−|z| n ,
dσ̂AMZ (z, t) = (Nt ⊗ IM )(ρ̂AM ) .
π
Cn
(7.17)
Then, the quantum conditional Fisher information of ρ̂AM is
J(A|M )ρ̂AM =

d
I(A : Z|M )σ̂AM Z (t)
dt

.

(7.18)

t=0

The quantum de Bruijn identity links the quantum conditional Fisher information to the
time derivative of the conditional entropy along the heat semigroup.
Lemma 7.3 (quantum de Bruijn identity).
J(A|M )ρ̂AM =

d
S(A|M )(Nt ⊗IM )(ρ̂AM )
dt

.

(7.19)

t=0

The ﬁrst part of the proof of the quantum conditional Entropy Power Inequality is
proving the following quantum conditional Stam inequality, which provides an upper
bound to the quantum conditional Fisher information of the output of the beam-splitter
or of the squeezing in terms of the quantum conditional Fisher information of the two
inputs.

Gaussian optimizers for entropic inequalities in quantum information

31

Theorem 7.4 (quantum conditional Stam inequality). Let ρ̂ABM be a quantum state
on ABM with finite average energy, finite S(M ) and I(A : B|M ) = 0, and let ρ̂CM be
as in (7.10). Then, for any λ ≥ 0 the quantum conditional Stam inequality holds:
λ
|1 − λ|
1
≥
+
.
J(C|M )ρ̂CM
J(A|M )ρ̂AM
J(B|M )ρ̂BM

(7.20)

The quantum conditional Stam inequality follows from the data processing inequality
for the quantum conditional mutual information (7.8), and implies that the quantum
conditional Entropy Power Inequality does not improve along the evolution with the
heat semigroup. Then, the proof of the Entropy Power Inequality is concluded if we show
that it becomes asymptotically an equality in the inﬁnite time limit. This is achieved by
proving that the quantum conditional entropy has an universal scaling independent on
the initial state in the inﬁnite time limit under the evolution with the heat semigroup.
Lemma 7.5.
S(M ),

For any joint quantum state ρ̂AM with finite average energy and finite
S(A|M )(Nt ⊗IM )(ρ̂AM ) = n ln t + n + o(1)

for t → ∞ .

(7.21)

The proof of this scaling is based on the following more general result.
Theorem 7.6. Let A, B be quantum Gaussian systems with m and n modes, respectively, and let Φ : A → B a quantum Gaussian channel. Then, for any quantum system
M and any quantum state ρ̂AM on AM with finite average energy and finite S(M ),
S(B|M )(Φ⊗IM )(ρ̂AM ) ≥ lim S(B|A′ )(Φ⊗IA′ )(τ̂AA′ (E)) ,
E→∞

(7.22)

where A′ is a Gaussian quantum system with m modes, and for any E ≥ 0, τ̂AA′ (E) is
a pure state such that its marginal on A is the thermal Gaussian state ω̂(E)⊗m .
We mention that a result similar to Theorem 7.6 has been proven in the scenario with
a constraint on the average energy of the system A [24].

8. Conclusions and perspectives
The optimization problems of functional analysis whose solutions are Gaussian functions
have stimulated to conjecture that quantum Gaussian states are the solution to the quantum counterparts of these optimization problems. These conjectures play a key role in
quantum information theory, since they are necessary to prove the converse theorems
for many communication scenarios with quantum Gaussian channels. We have reviewed
the state of the art in the proof of these conjectures. In the case of one-mode quantum Gaussian channels, they are almost all solved, with the exceptions of the Entropy
Photon-number Inequality and the sharp Young’s inequality for the beam-splitter. On

32

G. De Palma, D. Trevisan, V. Giovannetti, and L. Ambrosio

the contrary, there are only very few results for multi-mode quantum Gaussian channels.
In this scenario, both the constrained minimum output entropy conjecture (Conjecture
5.1) and the multiplicativity of the p → q norms with 1 < p < q (Conjecture 5.4) are still
completely open challenging problems, and we hope that this review will set the ground
for their solution.
Quantum Gaussian channels also constitute a bridge between continuous and discrete
classical probability. Indeed, on the one hand their properties are very similar to the
properties of Gaussian integral kernels, with quantum Gaussian states playing the role of
Gaussian probability measures. On the other hand, the quantum states diagonal in the
Fock basis of a one-mode Gaussian quantum system are in a one-to-one correspondence
with the probability measures on N. This correspondence establishes a bridge between
Gaussian quantum system and discrete probability. The role of quantum Gaussian states
is here played by the geometric probability distributions. These distributions turn out
to be the solution to many optimization problems involving the thinning, which is the
discrete analogue of the rescaling of a real random variable.
We then hope that this review will stimulate even more cross-fertilization among
functional analysis, discrete probability and quantum information.

Acknowledgements
We thank Elliott Lieb and Rupert Frank for a careful reading of the review and useful
comments.
GdP acknowledges ﬁnancial support from the European Research Council (ERC Grant
Agreements Nos. 337603 and 321029), the Danish Council for Independent Research
(Sapere Aude), VILLUM FONDEN via the QMATH Centre of Excellence (Grant No.
10059), and the Marie Sklodowska-Curie Action GENIUS (Grant No. 792557).

Author Contributions
The authors equally contributed to conceive the review and prepare the sketch. The main
text was written by GdP. All the authors helped to shape and review the manuscript.

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