# Quantum Rényi Entropy Functionals for Bosonic Gaussian Systems

Junseo Lee  School of Electrical and Electronic Engineering, Yonsei
University, Seoul 03722, Korea  Quantum Computing R&D, Norma Inc., Seoul
04799, Korea  Kabgyun Jeong<EMAIL_ADDRESS>Research Institute of
Mathematics, Seoul National University, Seoul 08826, Korea  School of
Computational Sciences, Korea Institute for Advanced Study, Seoul 02455, Korea

###### Abstract

In this study, the quantum Rényi entropy power inequality of order $p>1$ and
power $\kappa$ is introduced as a quantum analog of the classical Rényi-$p$
entropy power inequality. To derive this inequality, we first exploit the
Wehrl-$p$ entropy power inequality on bosonic Gaussian systems via the mixing
operation of quantum convolution, which is a generalized beam-splitter
operation. This observation directly provides a quantum Rényi-$p$ entropy
power inequality over a quasi-probability distribution for $D$-mode bosonic
Gaussian regimes. The proposed inequality is expected to be useful for the
nontrivial computing of quantum channel capacities, particularly universal
upper bounds on bosonic Gaussian quantum channels, and a Gaussian entanglement
witness in the case of Gaussian amplifier via squeezing operations.

##  I Introduction

A fundamental open problem addressed in quantum communication theory is
understanding the amount of information that can be reliably transmitted
through a noisy quantum channel BS04 ; H06 . As a measure of information flow,
this quantity is formally characterized by the concept of quantum channel
capacity, which is the maximal rate achievable by a quantum communication
channel with a vanishing error W17 ; W18 . While the classical capacity of all
classical channels is additive, it is known that most quantum channel
capacities are surprisingly “non-additive,” which indicates that determining
the quantum channel capacity is an extremely difficult problem, also referred
to as the _superadditivity_ of quantum channel capacity SY08 ; H09 ; LWZG09 ;
LLSSS23 . It is believed that this effect originates from intrinsic quantum
phenomena, such as quantum entanglement HLW06 ; HHHH09 .

Although determining the exact channel capacity has not been possible to date,
alternative methods have recently been introduced by exploiting a method known
as quantum entropy power inequality (EPI) KS14 ; PMG14 , which is a type of
entropic functional, providing nontrivial upper bounds on quantum channel
capacities, particularly for bosonic Gaussian quantum systems KS13 ; KS13+ ;
JLL19 ; LLKJ19 ; J20 . In general, quantum EPI estimates the output entropy of
a given quantum channel for independent input signals, and the inequality
satisfies a concave property for the functional. In bosonic Gaussian systems,
the quantum channel is described by a symplectic unitary transformation, and
it is formally given by a beam-splitter or amplifier, which mixes two
independent input Gaussian signals into an output signal HW01 ; FOP05 ; W+12 ;
S17 .

The classical EPI, first introduced by Shannon in 1948 S48  along with its
proofs including extensions S59 ; B65 ; B75 ; BL76 ; L78 ; DCT91 ; CS91 ; VG06
; R11 ; WM14 ; R17 , can be applied to the context of additive noise channels.
Thus, it is still an attractive topic studied in standard information theory
to explore the inequality at theoretical levels, such as Rényi entropy and the
associated functionals BC15 ; BM17 . The basic form of EPI is as follows: For
independent random variables $X$ and $Y$ in ${\mathbb{R}}^{d}$ with the
corresponding probability densities $p_{X}$ and $p_{Y}$ under a convolution
operation $\circledcirc$ such that $(p_{X},p_{Y})\mapsto p_{X\circledcirc Y}$,
the classical EPI is expressed as follows:

$\mathbf{V}(X\circledcirc Y)\geq\mathbf{V}(X)+\mathbf{V}(Y),$

where $\mathbf{V}(X):=\frac{1}{2\pi e}e^{\frac{2}{d}H(X)}$ is the “entropy
power” or energy, and $H(X):=-\int_{{\mathbb{R}}^{d}}p_{X}(x)\log
p_{X}(x){\textnormal{d}}x$ is the differential Shannon entropy. The
convolution operation on random variables $X$ and $Y$ is defined as
$p_{X\circledcirc
Y}(z)=\int_{{\mathbb{R}}^{d}}p_{X}(x)p_{Y}(z-x){\textnormal{d}}x$ for any
$z\in{\mathbb{R}}^{d}$. There are three types of mathematical proofs for EPIs:
(i) Gaussian perturbation with de Bruijn’s identity and convexity of Fisher
information S59 ; B65 ; CS91 ; VG06 ; R11 , (ii) sharp Young’s inequality B75
; BL76 ; L78 ; WM14 , and (iii) changes in variables on convex bodies R17 . It
is known that the exponential form of EPI is essentially equivalent to the
linear form, that is, $H(X\circledcirc Y)\geq H(X)+H(Y)$ DCT91 ; VG06 ; R11 .
While the classical EPIs are useful for determining the channel capacity such
as an additive white Gaussian noise (AWGN) channel, the quantum EPIs are more
powerful to quantum Shannon theory when quantum entanglement is involved.
Here, the history of quantum EPI, which is a quantum analog of classical EPI,
is briefly summarized. The quantum EPI on bosonic Gaussian systems was first
derived by König and Smith KS14 , and it was generalized to bosonic Gaussian
PMG14  and discrete systems ADO16  as well as conditional cases K15 ; JLJ18 ;
PT18 ; P19 . While the classical capacity is additive, quantum channel
capacities are generally nonadditive; hence, quantum EPIs have the potential
power of finding (tight upper bounds) realistic capacities on quantum
channels. As an another potential application, it can be devised to
entanglement witness for a continuous-variable (bipartite) quantum system
DGCZ00 ; R00 ; CMYY23  along with a squeezed-type quantum EPIs.

For the classical capacity of a (bosonic) Gaussian quantum channel
$\Lambda_{\tau,N_{E}}$, the nontrivial upper bound is determined by quantum
EPI as in KS13 ; KS13+ ; JLL19 , where $C(\Lambda_{\tau,N_{E}},N)$ is the
classical capacity of the channel, and $\tau$, $N$, and $N_{E}$ denote the
mixing parameter, input, and environment mean photon number, respectively.
Then, to use it in an $n$-copy channel, it is known that

$\displaystyle C(\Lambda_{\tau,N_{E}},N)$ $\displaystyle\leq$ $\displaystyle
g(\tau N+(1-\tau)N_{E})$
$\displaystyle-\lim_{n\to\infty}\min_{\langle\rho_{n}^{G}\rangle\leq
nN}\frac{1}{n}S\left(\Lambda_{\tau,N_{E}}^{\otimes n}(\rho_{n}^{G})\right),$

where $\rho_{n}^{G}$ denotes the input (entangled) Gaussian quantum state,
$g(x):=(x+1)\log(x+1)-x\log x$, and
$S(\rho)=-\mbox{$\textnormal{Tr}$}\rho\log\rho$ the von Neumann entropy. Also,
the minimization is taken over all $n$-mode bosonic Gaussian input states such
that its expectation value is less than the total input mean photon number
$nN$. By employing the power of quantum EPI, that is,
$S\left(\Lambda_{\tau,N_{E}}^{\otimes n}(\rho_{n}^{G})\right)\geq\tau
S(\rho_{n}^{G})+(1-\tau)S({\rho_{E}^{G}}^{\otimes n})$, the convex
optimization term can be derived as $(1-\tau)g(N_{E})$. Note that the lower
bound is given by the famous (Gaussian) Holevo capacity formula, and it was
reported some additive properties GGCH14 ; LW22  in the Gaussian regime.

The second case is a Gaussian and/or non-Gaussian entanglement witness. In
general, a quantum EPI for two-mode Gaussian amplifying channel
$\Lambda_{\zeta}$ can provides an inequality known as

$S\left(\Lambda_{\zeta}(\rho^{G})\right)\geq\log(2\zeta-1)$ (1)

with a squeezing parameter $\zeta\in(0,1)$, and it can be used to detecting a
Gaussian quantum entanglement. (We notice that Eq. (1) is violated, then there
exists a Gaussian quantum entanglement.) Because a squeezing-type quantum EPI
can be _trivially_ recasted as
$e^{S\left(\Lambda_{\zeta}(\rho^{G}_{A})\right)}\geq\max\left\\{\zeta
e^{S(\rho_{A}^{G})},(\zeta-1)e^{S(\rho_{B}^{G})}\right\\}$ from the well-known
quantum EPI in the form of
$e^{S\left(\Lambda_{\zeta}(\rho^{G}_{A})\right)}\geq\zeta
e^{S(\rho_{A}^{G})}+(\zeta-1)e^{S(\rho_{B}^{G})}$ PMG14 ; LLKJ19 . Thus, if we
find an entropic measure (or quantity) such as
$\mu\left(\Lambda_{\zeta}(\rho^{G})\right)\leq\frac{1}{2\zeta-1}$ in the help
of the recasted relation above, we can obtain the (non-)Gaussian entanglement
witness in Eq. (1). As an example,
$\mu(\varrho):=\mbox{$\textnormal{Tr}$}(\varrho^{2})$ is a purity of a
Gaussian quantum state $\varrho$ PISS03 .

In this paper, first, a type of quantum Rényi EPI on bosonic Gaussian systems
is derived through a modification of classical Rényi EPIs BM17 , with order
$p$, power $\kappa$, and weight factor $\tau$. As a concrete example of the
bosonic system, first, a (quantum) Wehrl entropy and its associated functional
(i.e., Wehrl entropy power) are introduced, including a quantum convolution
operation (i.e., $\tau$-beam-splitter), prior to stating the Wehrl EPI.
Finally, our main result—the quantum Rényi EPI on the bosonic Gaussian
system—is formulated. Our results offer a novel perspective on fundamental
problems in quantum communication theory in terms of bounding channel
capacities. The arguments employed here are expected to be extendable from
bosonic systems to arbitrary quantum cases.

The remainder of this paper is organized as follows. In Section II, standard
information-theoretic results are provided for the differential Rényi EPI,
which is further generalized. In Section II.1, a quantum addition rule (in
particular, the $\tau$-beam-splitter) is introduced in the language of a
quantum channel. In Section III, a quantum EPI is presented in the framework
of Wehrl entropy along with its functional, which are useful in providing our
main results. In Section IV, the concept of quantum Rényi EPI on bosonic
Gaussian systems is conceived. Finally, in Section V, a discussion and remarks
are presented on this topic, raising some open questions for future research.

##  II Rényi entropy power inequality

As a variant of the differential Shannon entropy, the differential Rényi-$p$
entropy is given by

$H_{p}(X)=\frac{1}{1-p}\log\int_{{\mathbb{R}}^{d}}p_{X}^{~{}p}(x){\textnormal{d}}x,$
(2)

where $X$ is a continuous random variable in ${\mathbb{R}}^{d}$, and
$H_{1}(X)=-\int p_{X}(x)\log p_{X}(x){\textnormal{d}}x$ recovers Shannon’s
variable whenever $p\to 1$. It is natural to define Rényi-$p$ entropy power as
an associated functional of the Rényi-$p$ entropy in the form of
$\mathbf{V}_{p}(X)=\exp\left(\frac{2}{d}H_{p}(X)\right)$, and it is known
that, in the sense of a corollary of BM17 , the weighted Rényi EPI is given by

$\mathbf{V}_{p}^{\kappa}(X\circledcirc_{t}Y)\geq
t^{\kappa}\mathbf{V}_{p}^{\kappa}(X)+(1-t)^{\kappa}\mathbf{V}_{p}^{\kappa}(Y),$
(3)

where $\circledcirc_{t}$ denotes the weighted convolution operation with
$t\in(0,1)$ and $\kappa\geq\frac{p+1}{2}$ for any $p>1$.

By exploiting the inequality in Eq. (3), two types of quantum EPIs, namely
Wehrl-$p$ and quantum Rényi-$p$ EPIs, are investigated with respect to bosonic
Gaussian systems with Husimi and Wigner distributions, respectively. Before
the main results, the known “quantum addition rule” (or quantum convolution
operation) is briefly presented in the context of a quantum channel.

Figure 1: Quantum addition rule via $\tau$-beamsplitter operation (or quantum
convolution operation) represented by
$\rho_{Z}:=\rho_{X}\boxplus_{\tau}\rho_{Y}$ where $\tau\in(0,1)$

###  II.1 Quantum addition rule

In the classical case, two independent random variables are mixed by applying
the convolution operation to the probability densities corresponding to each
random variable. However, the quantum case is subtly different. Next, the
convolution of quantum states, referred to as the quantum addition rule or
_quantum convolution_ , is defined BGJ23 .

Let $\rho_{X}$ and $\rho_{Y}$ be two independent quantum states over
${\mathcal{H}}({\mathbb{C}}^{d})$, $d$-dimensional complex Hilbert space, and
let $\Lambda_{\tau}$ be a completely positive and trace-preserving (CPT) map
satisfying any $\tau\in(0,1)$,

$\Lambda_{\tau}:\rho_{X}\otimes\rho_{Y}\mapsto\rho_{X}\boxplus_{\tau}\rho_{Y}=\mbox{$\textnormal{Tr}$}_{Y}\left[U_{\tau}(\rho_{X}\otimes\rho_{Y})U_{\tau}^{\dagger}\right],$
(4)

where $U_{\tau}$ denotes any unitary operator over the total quantum system
$XY$, $\Lambda_{\tau}(\bullet)\in{\mathcal{H}}({\mathbb{C}}^{d})$ is the
output of the CPT map, and $\boxplus_{\tau}$ is the quantum convolution. As an
example of quantum convolution, it is possible to define a partial swap
operation over $d$-dimensional quantum systems (namely, a qudit-system) with a
weight factor $\tau$ as
$U_{\tau}=\sqrt{\tau}{\mathbb{1}}+i\sqrt{1-\tau}{\mathbb{W}}$ ADO16 , where
${\mathbb{W}}$ is the swap operator such that
${\mathbb{W}}\rho_{AB}{\mathbb{W}}^{\dagger}=\rho_{BA}$. Here, the convolution
operation is generalized to the regime of bosonic Gaussian systems
(particularly, $\tau$-beam-splitter) and the quantum EPIs are studied.

While the $d$-dimensional quantum states are described on a finite dimensional
Hilbert space, bosonic Gaussian states are generally represented in the phase
space ${\textnormal{Sp}}(2D,{\mathbb{R}})$, which is $2D$-dimensional real
symplectic space and and it has an infinite dimensional Hilbert space with
continuous eigenvalues of Gaussian observables, and evolve with symplectic
unitaries ($U_{\tau}^{G}$) equipped with canonical commutation relation
algebra FOP05 ; W+12 . If independent $\rho_{X}^{G}$ and $\rho_{Y}^{G}$ in
symplectic space ${\textnormal{Sp}}(2D,{\mathbb{R}})$ with mode $D$ are
defined as bosonic Gaussian states, the quantum addition rule in Eq. (4) can
be expressed by the state representation in
$\rho_{Z}^{G}:=\rho_{X}^{G}\boxplus_{\tau}\rho_{Y}^{G}=\mbox{$\textnormal{Tr}$}_{Y}[U_{\tau}^{G}(\rho_{X}^{G}\otimes\rho_{Y}^{G})U_{\tau}^{G^{\dagger}}]$
or the field-operator representation in the form of

$\hat{a}_{Z}=\sqrt{\tau}\hat{a}_{X}+\sqrt{1-\tau}\hat{a}_{Y},$ (5)

where $\hat{a}_{\bullet}$ denotes the annihilation operator for the number
operator $\hat{n}_{\bullet}:=\hat{a}_{\bullet}^{\dagger}\hat{a}_{\bullet}$
with a creation operator $\hat{a}_{\bullet}^{\dagger}$ on quantum harmonic
oscillitors, i.e., $\hat{n}|n\rangle=n|n\rangle$, and $\tau\in(0,1)$ is the
mixing parameter (Fig. 1).

Also note that any quantum state of a $D$-mode bosonic system is equivalent to
a quasi-probability density function, such as Wigner or Husimi, over the
$2D$-dimensional phase space (i.e., real symplectic space). For example, if we
introduce Weyl operator in the form of
$\hat{D}({\vec{\eta}}):=\exp\left(i\hat{x}^{T}\Gamma\vec{\eta}\right)$ for
$\vec{\eta}\in{\mathbb{R}}^{2D}$, then a density operator $\rho^{G}$ is
represented by the Wigner characteristic function and Wigner function (via
Fourier transform) as follows:

$\displaystyle\chi(\vec{\eta})$
$\displaystyle=\mbox{$\textnormal{Tr}$}\left(\rho^{G}D(\vec{\eta})\right)\;\;{\textnormal{and}}$
$\displaystyle W(\hat{x})$
$\displaystyle=\frac{1}{(2\pi)^{2D}}\int_{{\mathbb{R}}^{2D}}\exp\left(-i\hat{x}^{T}\Gamma\vec{\eta}\right)\chi(\vec{\eta}){\textnormal{d}}^{2D}\vec{\eta},$

respectively. Here,
$\hat{x}=(\hat{q}_{1},\hat{p}_{1},\ldots,\hat{q}_{D},\hat{p}_{D})^{T}$ with a
commutation relation $[\hat{x}_{j},\hat{x}_{k}]=2i\Gamma_{jk}$ denotes the
quadrature field operator, and $\Gamma:=\bigoplus_{j=1}^{D}\gamma$ with

$\gamma=\begin{pmatrix}0&1\\\ 1&0\\\ \end{pmatrix}.$

Thus, any density matrix $\rho^{G}$ can be equivalently described by a quasi-
probability distribution (i.e., it has a normalization factor 1 but generally
non-positive) FOP05 ; W+12 .

In addition, the Rényi entropy of order $p$ of a $d$-dimensional quantum state
$\rho\in{\mathcal{H}}({\mathbb{C}}^{d})$ can be formally defined as

$S_{p}(\rho)=\frac{1}{1-p}\log\mbox{$\textnormal{Tr}$}\rho^{p}$ (6)

for any $p>0~{}(p\neq 1)$, and converges to the von Neumann entropy in the
limit of $p\to 1$. However, a differential version of Rényi entropy is
required and thus, the related entropic functional on the bosonic Gaussian
system is explained in detail in Section IV. First, consider the Wehrl entropy
and its functional.

##  III Wehrl entropy and associated entropy power inequality

The Wehrl entropy and associated entropy power of order $p$ on a bosonic
Gaussian state $\rho_{X}\in{\textnormal{Sp}}(2D,{\mathbb{R}})$ can be defined
as

$\displaystyle S_{p}^{W}(\rho_{X})$ $\displaystyle=$
$\displaystyle\frac{1}{1-p}\log\int_{{\mathbb{C}}^{D}}Q_{X}^{~{}p}(\alpha){\textnormal{d}}\alpha$
(7) $\displaystyle\mathbf{V}_{p}^{W}(\rho_{X})$ $\displaystyle=$
$\displaystyle\exp\left(\frac{1}{D}S_{p}^{W}(\rho_{X})\right),$ (8)

where $Q_{X}$ denotes the Husimi distribution corresponding to quantum state
$\rho_{X}$, which is formally given by
$Q_{X}(\alpha)=\frac{1}{(2\pi)^{D}}\langle\alpha|\rho_{X}|\alpha\rangle$ with
coherent state
$|\alpha\rangle=e^{-|\alpha|^{2}/2}\sum_{n=0}^{\infty}\frac{\alpha^{n}}{\sqrt{n!}}|n\rangle~{}(\forall\alpha\in{\mathbb{C}})$;
moreover, note that
$\int_{{\mathbb{C}}^{D}}Q_{X}(\alpha){\textnormal{d}}\alpha=1$ and
$Q_{X}(\alpha)\geq 0$. For this distribution, the quantum convolution with a
mixing parameter $\tau\in(0,1)$ is given by KI95

$Q_{Z}(\beta)=\frac{1}{\tau}\int_{{\mathbb{C}}^{D}}Q_{X}(\alpha)Q_{Y}\left(\frac{\beta-\sqrt{1-\tau}\alpha}{\sqrt{\tau}}\right){\textnormal{d}}\alpha,$
(9)

which corresponds to $\rho_{Z}^{G}=\rho_{X}^{G}\boxplus_{\tau}\rho_{Y}^{G}$ or
Eq. (5) for a bosonic Gaussian system involving the Husimi distribution $Q$.

The formulas for differential Shannon entropy and Wehrl entropy have similar
forms in the logarithmic components; thus, the proof techniques introduced in
the literature can be applied in a straightforward manner to classical Rényi
EPIs J20 ; BM17 —modified Young’s inequality in the power $\kappa$. Then, the
Wehrl EPI of order $p$ is obtained as follows:

######  Theorem 1 (Wehrl-$p$ EPI).

Let $\rho_{X}^{G}$ and $\rho_{Y}^{G}$ in ${\textnormal{Sp}}(2D,{\mathbb{R}})$
be any independent $D$-mode Gaussian states corresponding to Husimi
distributions $Q_{X}$ and $Q_{Y}$, respectively. Let $\tau\in(0,1)$ be the
mixing parameter. Then,

$\displaystyle\mathbf{V}_{p}^{W,\kappa}(\rho_{Z}^{G})\geq\tau^{\kappa}\mathbf{V}_{p}^{W,\kappa}(\rho_{X}^{G})+(1-\tau)^{\kappa}\mathbf{V}_{p}^{W,\kappa}(\rho_{Y}^{G}),$
(10)

where $\rho_{Z}^{G}=\rho_{X}^{G}\boxplus_{\tau}\rho_{Y}^{G}$ is the output
bosonic Gaussian state with power $\kappa\geq\frac{p+1}{2}$ for any $p>1$.

The explicit proof of Theorem 1 is provided in Appendix A and Appendix B. It
is natural that, as a generalization of Wehrl-$p$ entropy and EPI (i.e., a
probability distribution), a “quantum Rényi entropy” and its associated
entropic functional (i.e., a quasi-probability distribution) be considered on
the bosonic Gaussian systems. To address a general case in Gaussian regime,
the Husimi distribution $Q_{X}$ can be replaced with a quasi-probability
distribution density function $q_{X}$ (e.g., Wigner) corresponding to the
bosonic system $\rho_{X}$. In general, the Husimi distribution function has
only positive values, but Wigner function can be negative in the probability
space. Moreover, it is well-known that the Husimi function is well-behaved and
non-singular for all Gaussian states, whereas generic quasi-probability
functions can potentially exhibit a singularity or ill-defined behavior even
for Gaussian states. In this reason, it is believed that a Rényi-$p$ EPI with
a Wigner function allows us a strong bound instead of the Wehrl-$p$ EPI with a
Husimi function, which imply some potential applications such a Gaussian
entanglement detection or determination of Gaussian channel capacities. Here,
a generalized quantum convolution of two independent quasi-distributions takes
the form
$q_{Z}(z^{\prime})=\frac{1}{\tau}\int_{{\mathbb{C}}^{D}}q_{X}(z)q_{Y}(\frac{z^{\prime}-\sqrt{1-\tau}z}{\sqrt{\tau}}){\textnormal{d}}z$,
where $z,z^{\prime}\in{{\mathbb{C}}}^{D}$, as in Eq. (9). The main result can
now be stated in the following.

Table 1: Summary of Rényi-$p$ entropies and its functionals.

| Classical Rényi | Wehrl (quantum) | Quantum Rényi
---|---|---|---
State rep. | $X,Y\in{\mathbb{R}}^{d}$ | $\rho_{X},\rho_{Y}\in{\textnormal{Sp}}(2D,{\mathbb{R}})$ | $\rho_{X},\rho_{Y}\in{\textnormal{Sp}}(2D,{\mathbb{R}})$
PDFs | $p_{X},p_{Y}$ | $Q_{X},Q_{Y}$ | $q_{X},q_{Y}$
Entropy | $H_{p}(X)=\frac{1}{1-p}\log\int_{{\mathbb{R}}^{d}}p_{X}^{p}(x){\textnormal{d}}x$ | $S_{p}^{W}(\rho_{X})=\frac{1}{1-p}\log\int_{{\mathbb{C}}^{D}}Q_{X}^{p}(\alpha){\textnormal{d}}\alpha$ | $S_{p}(\rho_{X})=\frac{1}{1-p}\log\int_{{\mathbb{C}}^{D}}q_{X}^{p}(z){\textnormal{d}}z$
Parameter | $t\in(0,1)$ | $\tau\in(0,1)$ | $\tau\in(0,1)$
Addition | $Z=X\circledcirc_{t}Y=\sqrt{t}X+\sqrt{\tilde{t}}Y$ | $\rho_{Z}=\rho_{X}\boxplus_{\tau}\rho_{Y}$ | $\rho_{Z}=\rho_{X}\boxplus_{\tau}\rho_{Y}$
$L^{p}$-norm | $\|p_{X}\|_{p}=\left(\int_{{\mathbb{R}}^{d}}p_{X}^{p}(x){\textnormal{d}}x\right)^{1/p}$ | $\|Q_{X}\|_{p}=\left(\int_{{\mathbb{C}}^{D}}Q_{X}^{p}(\alpha){\textnormal{d}}\alpha\right)^{1/p}$ | $\|q_{X}\|_{p}=\left(\int_{{\mathbb{C}}^{D}}q_{X}^{p}(z){\textnormal{d}}z\right)^{1/p}$
Mixing ops. | $p_{Z}(z)=\int_{{\mathbb{R}}^{d}}p_{X}(x)p_{Y}(z-x){\textnormal{d}}x$ | $Q_{Z}(\beta)=\frac{1}{\tau}\int_{{\mathbb{C}}^{D}}Q_{X}(\alpha)Q_{Y}(\frac{\beta-\sqrt{\tilde{\tau}}\alpha}{\sqrt{\tau}}){\textnormal{d}}\alpha$ | $q_{Z}(z^{\prime})=\frac{1}{\tau}\int_{{\mathbb{C}}^{D}}q_{X}(z)q_{Y}(\frac{z^{\prime}-\sqrt{\tilde{\tau}}z}{\sqrt{\tau}}){\textnormal{d}}z$
Power | $\kappa\geq\frac{p+1}{2}~{}(p>1)$ | $\kappa\geq\frac{p+1}{2}~{}(p>1)$ | $\kappa\geq\frac{p+1}{2}~{}(p>1)$
EPIs | $\mathbf{V}_{p}^{\kappa}(Z)\geq t^{\kappa}\mathbf{V}_{p}^{\kappa}(X)+\tilde{t}^{\kappa}\mathbf{V}_{p}^{\kappa}(Y)$ | $\mathbf{V}_{p}^{W,\kappa}(\rho_{Z})\geq\tau^{\kappa}\mathbf{V}_{p}^{W,\kappa}(\rho_{X})+\tilde{\tau}^{\kappa}\mathbf{V}_{p}^{W,\kappa}(\rho_{Y})$ | $\mathbf{V}_{p}^{\kappa}(\rho_{Z})\geq\tau^{\kappa}\mathbf{V}_{p}^{\kappa}(\rho_{X})+\tilde{\tau}^{\kappa}\mathbf{V}_{p}^{\kappa}(\rho_{Y})$

Notice that $\circledcirc$: convolution, $t,\tau\in(0,1)$: mixing parameters,
$\boxplus_{\tau}$: $\tau$-beamsplitter, $D$: $D$-mode, $d$: $d$-dimension,
$\tilde{t}=1-t$, $\tilde{\tau}=1-\tau$.

##  IV Quantum Rényi entropy power inequality

In the framework of arbitrary $D$-mode bosonic Gaussian systems, quantum
(differential) Rényi-$p$ entropy, as an extension of Eq. (7), can be defined
as

$S_{p}(\rho_{X})=\frac{1}{1-p}\log\int_{{\mathbb{C}}^{D}}q_{X}^{p}(z){\textnormal{d}}z$
(11)

for any order $p>0~{}(p\neq 1)$ and for any quasi-probability distribution
$q_{X}$ corresponding to $\rho_{X}\in{\textnormal{Sp}}(2D,{\mathbb{R}})$. From
this, it is also natural to define an associated functional, namely quantum
Rényi-$p$ entropy power, as
$\mathbf{V}_{p}(\rho_{X})=\exp\left(\frac{1}{D}S_{p}(\rho_{X})\right)$ for the
bosonic Gaussian state. Adopting the logarithmic form and without loss of
generality, the sharp Young’s inequality in BM17  can be used again.

In mixing input signals, the beam-splitter operation, $\boxplus_{\tau}$, is
referred to as quantum convolution, which can be obtained by substituting the
Husimi $Q$-function with an arbitrary probability ($q_{\bullet}$) on bosonic
Gaussian states. The field operator representation is described in the form of
Eq. (5), with a weight factor of $\tau\in(0,1)$.

######  Theorem 2 (Quantum Rényi-$p$ EPI).

For any $D$-mode Gaussian states $\rho_{X}^{G}$ and $\rho_{Y}^{G}$ in
${\textnormal{Sp}}(2D,{\mathbb{R}})$, and for any $\tau\in(0,1)$, we have

$\mathbf{V}_{p}^{\kappa}(\rho_{X}^{G}\boxplus_{\tau}\rho_{Y}^{G})\geq\tau^{\kappa}\mathbf{V}_{p}^{\kappa}(\rho_{X}^{G})+(1-\tau)^{\kappa}\mathbf{V}_{p}^{\kappa}(\rho_{Y}^{G}),$
(12)

where the power satisfies $\kappa\geq\frac{p+1}{2}$ for any $p>1$.

More precisely, it can be expressed using the exponential form as

$\displaystyle\exp\left(\frac{\kappa}{D}S_{p}(\rho_{X}\boxplus_{\tau}\rho_{Y})\right)$
$\displaystyle\geq\tau^{\kappa}\exp\left(\frac{\kappa}{D}S_{p}(\rho_{X})\right)+(1-\tau)^{\kappa}\exp\left(\frac{\kappa}{D}S_{p}(\rho_{Y})\right).$

The components of EPIs in both classical and quantum cases are summarized in
Table 1. Furthermore, note that the inequality in Eq. (12) can be easily
derived from the fundamental quantum EPI for $\kappa=1$ KS14 ; PMG14 :
$\mathbf{V}_{p}^{1}(\rho_{Z})\geq\tau\mathbf{V}_{p}^{1}(\rho_{X})+(1-\tau)\mathbf{V}_{p}^{1}(\rho_{Y})$
because the $\kappa$-power of the total equation and removing trivial
(positive) terms on the right-hand side generates our main claim in Theorem 2.
However, there are some drawbacks to $\kappa>1$ in classical as well as
quantum Rényi EPIs when considering Young’s constant argument.

##  V Conclusions

The quantum entropy power inequalities are very powerful tool for determining
quantum channel capacities in quantum Shannon theory when quantum entanglement
is involved. In this paper, an identity called quantum Rényi EPI of order
$p>1$ and power $\kappa$ is obtained via a generalization of Wehrl-$p$ EPI on
bosonic Gaussian regimes based on the classical Rényi-$p$ EPIs, which can be
useful for the nontrivial computing of quantum channel capacities. In the
quantum case, it is conjectured that
$\mathbf{V}_{p}^{1}(\rho_{X}\boxplus_{\tau}\rho_{Y})\geq\tau\mathbf{V}_{p}^{1}(\rho_{X})+(1-\tau)\mathbf{V}_{p}^{1}(\rho_{Y})$
for the order $p>0$; however, this still requires an explicit proof. The main
result (i.e., Theorem 2) is true for any order of $p$ in the Rényi
functionals. In particular, it can help achieve a recent result of the
modified breakthrough of Young’s convolution inequality.

In the sense of explicit computing of the quantum channel capacities, this
type of research is the only way to highlight open problems, but it is still
far from reaching the exact solution. As an example, it is expected that the
following nontrivial upper bound of the channel capacity on
$\Lambda_{\tau,N_{E}}$ can be computed. That is, for all $p>0$ and all
$\rho^{G}\in{\textnormal{Sp}}(2D,{\mathbb{R}})$,

$\displaystyle C(\Lambda_{\tau,N_{E}},N)$
$\displaystyle\leq\max_{\rho^{G}}S_{p}\left(\Lambda_{\tau,N_{E}}(\rho^{G})\right)-\lim_{n\to\infty}\min_{\langle\rho_{n}^{G}\rangle\leq
nN}\frac{1}{n}S_{p}\left(\Lambda_{\tau,N_{E}}^{\otimes
n}(\rho_{n}^{G})\right),$

where parameters $\tau$, $\kappa$, and $p$ encompass all ranges in terms of an
extended notion of quantum EPIs. Furthermore, we remark that, for any Gaussian
amplifying channel with a squeezing parameter $\zeta$, it might be possible to
construct an efficient Gaussian entanglement measure such that
$S_{p}\left(\Lambda_{\zeta}(\rho^{G})\right)<\log(2\zeta-1)$ under the
ordering of $S(\rho^{G})\geq\cdots\geq S_{p}(\rho^{G})$ for a symmetric
Gaussian state as well as non-Gaussian cases. In addition, neither the
nonadditive property of qudit systems nor the subtle major problem of the
entropy photon number inequality (EPNI) in the Gaussian regime are known. EPNI
was conjectured by Guha _et al._ GES08  and is given by
$\mathbf{N}(\rho_{X}\boxplus_{\tau}\rho_{Y})\geq\tau\mathbf{N}(\rho_{X})+(1-\tau)\mathbf{N}(\rho_{Y})$,
where $\mathbf{N}(\rho)=g^{-1}(\frac{S(\rho)}{D})$, which is the “photon
number” of $D$-mode $\rho$. If this conjecture is true, the classical
capacities of various bosonic Gaussian quantum channels can be evaluated GSE07
.

Because most advances in quantum Shannon theory emerge from information
theory, there are still reservations regarding the EPI itself—ranging from the
classical, Tsallis, and unified EPIs to the unknown quantum perspective.
Furthermore, there are several intriguing open questions on the quantum EPI.
First, it is not known whether other types of nontrivial quantum convolutions
on qudit systems beyond the usual beam-splitter or amplifier on bosonic
quantum systems exist. Second, it is not known how far the resulting quantum
EPIs are from known lower bounds, such as the Holevo capacity, data-
processing-inequalities, or coherent and private information. This kind of
research may extend our knowledge of original quantum channel capacity
problems while providing novel insights into the theory and applications of
emerging quantum networks or quantum internet.

## Acknowledgments

The authors thank to anonymous reviewer’s valuable comments. This work was
supported by the National Research Foundation of Korea (NRF) through a grant
funded by the Ministry of Science and ICT (NRF-2020M3E4A1077861,
NRF-2022M3H3A1098237) and the Ministry of Education (NRF-2021R1I1A1A01042199).
This research was partly supported by Korea Institute of Science and
Technology Information (KISTI, P23031).

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## Appendix

Quantum Rényi Entropy Functionals for Bosonic Gaussian Systems

Kabgyun Jeong & Junseo Lee

###  V.1 Sharp Young’s inequality

Some mathematical background is required for the proof of Theorem 1, which is
essentially based on the modification of Bobkov and Marsiglietti’s derivation
BM17  for the classical Rényi entropy power inequality. Notably, the Husimi
distribution of a Gaussian quantum state is almost equivalent to that of a
classical density function.

First, the $L^{p}$-norm for the positive Husimi function $Q_{X}$ in the
Gaussian regime is defined as follows:

$\|Q_{X}\|_{p}=\left(\int_{{\mathbb{C}}^{D}}Q_{X}^{p}(\alpha){\textnormal{d}}\alpha\right)^{1/p}.$
(13)

Then, let $\rho_{X}^{G}$ and
$\rho_{Y}^{G}\in{\textnormal{Sp}}(2D,{\mathbb{R}})$ be independent Gaussian
quantum states associated with Husimi functions $Q_{X}$ and $Q_{Y}$,
respectively. The sharp Young’s inequality implies that, for any $r$, $s$, and
$t\geq 1$ such that $\frac{1}{r}+\frac{1}{s}-\frac{1}{t}=1$,

$\|Q_{X}\circledcirc_{\tau}Q_{Y}\|_{t}\leq K^{D/2}\|Q_{X}\|_{r}\|Q_{Y}\|_{s},$
(14)

where $K=\frac{k_{r}k_{s}}{k_{t}}\leq 1$ with
$k_{p}=\left(1-\frac{1}{p}\right)\sqrt[p]{\frac{p^{2}}{p-1}}$. Note that
$\circledcirc_{\tau}$ denotes the quantum convolution (or $\tau$-beam-
splitter), as in Eq. (9). In addition, any contribution of mixing parameter
$\tau$ can be resolved at the end of the proof of Theorem 1 as a natural
generalization of Eq. (3).

By the way, the relationship between $L^{p}$-norm and Rényi-$p$ entropy is
easily induced as follows:

$\displaystyle\|Q_{X}\|_{p}$ $\displaystyle=$
$\displaystyle\left(\int_{{\mathbb{C}}^{D}}Q_{X}^{p}(\alpha){\textnormal{d}}\alpha\right)^{1/p}=\exp\left(\frac{1}{p}\log\int_{{\mathbb{C}}^{D}}Q_{X}^{~{}p}(\alpha){\textnormal{d}}\alpha\right)$
$\displaystyle=$
$\displaystyle\exp\left(\frac{1-p}{p}S_{p}(\rho_{X})\right)=\mathbf{V}_{p}(\rho_{X}^{G})^{-\frac{D}{2}\left(1-\frac{1}{p}\right)}.$

By modifying Equation (A.2) through this property, the following can be
obtained.

$\mathbf{V}_{t}(\rho_{X}^{G}\boxplus_{\tau}\rho_{Y}^{G})^{1-\frac{1}{t}}\geq\frac{1}{K}\mathbf{V}_{r}(\rho_{X}^{G})^{1-\frac{1}{r}}\mathbf{V}_{s}(\rho_{Y}^{G})^{1-\frac{1}{s}}.$
(15)

As a quantum analog, this result corresponds to Beckner’s information-
theoretic formulation B75 .

By exploiting Hölder’s inequality, Eq. (15) can be substituted into one
functional $\mathbf{V}_{t}(\cdot)$ as follows: For any $r,s,t\geq 1$
satisfying the constant $K$,

$\mathbf{V}_{t}(\rho_{X}^{G}\boxplus_{\tau}\rho_{Y}^{G})^{1-\frac{1}{t}}\geq\frac{1}{K}\mathbf{V}_{t}(\rho_{X}^{G})^{1-\frac{1}{r}}\mathbf{V}_{t}(\rho_{Y}^{G})^{1-\frac{1}{s}}.$
(16)

To prove the main result, two technical lemmas (Sec. IV and its derivations in
BM17 ) are introduced.

Lemma 1 (Homogeneity property). For any $t>1$, suppose that any positive $a$
and $b$ satisfy $a+b=1-\frac{1}{t}$. Then, $r,s\geq 1$ exists such that

$K^{-\frac{\kappa
t}{t-1}}a^{\frac{t(r-1)}{r(t-1)}}b^{\frac{t(s-1)}{s(t-1)}}\geq 1-\frac{1}{t},$
(17)

where the constant $K=\frac{k_{r}k_{s}}{k_{t}}$.

Lemma 2 (Calculus lemma). For a given $0<c<1$ and $d\geq\frac{2}{c}-1$, the
algebraic function is

$F(x)=\frac{(1-x)^{d(1-x)}(1-y)^{d(1-y)}}{x^{x}y^{y}}\;\;\;(y=c-x)$ (18)

attaining a minimum in the region of $0\leq x\leq c$ either at the points
$x=0$, $x=c$, or at the center $x=\frac{c}{2}$. Furthermore, in the case of
$d=\frac{2}{c}-1$, the function reaches its minimum at the endpoints.

###  V.2 Proof of Theorem 1

Proof. First, recall the inequality given in Eq. (16). Now, let us rewrite the
inequality by using $\kappa=\frac{t+1}{2}$ in Lemma 1 such that

$\mathbf{V}_{t}^{\kappa}(\rho_{X}^{G}\boxplus_{\tau}\rho_{Y}^{G})^{1-\frac{1}{t}}\geq\frac{1}{K^{\frac{\kappa
t}{t-1}}}\mathbf{V}_{t}^{\kappa}(\rho_{X}^{G})^{\frac{t(r-1)}{r(t-1)}}\mathbf{V}_{t}^{\kappa}(\rho_{Y}^{G})^{\frac{t(s-1)}{s(t-1)}}.$
(19)

Finally, by substituting $a=\mathbf{V}_{t}^{\kappa}(\rho_{X}^{G})$ and
$b=\mathbf{V}_{t}^{\kappa}(\rho_{Y}^{G})$ and exploiting the assumption of
$a+b=1-\frac{1}{t}$ in Lemma 1, we have

$K^{-\frac{\kappa
t}{t-1}}a^{\frac{t(r-1)}{r(t-1)}}b^{\frac{t(s-1)}{s(t-1)}}\geq 1-\frac{1}{t}$
(20)

for some admissible $r,s\geq 1$ such that
$\frac{1}{r}+\frac{1}{s}-\frac{1}{t}=1$. This completes the proof.
$\blacksquare$

Note that Eq. (20) implies that

$\displaystyle\mathbf{V}_{t}^{W,\kappa}(\rho_{X}^{G}\boxplus_{\tau}\rho_{Y}^{G})\geq\tau^{\kappa}\mathbf{V}_{t}^{W,\kappa}(\rho_{X}^{G})+(1-\tau)^{\kappa}\mathbf{V}_{t}^{W,\kappa}(\rho_{Y}^{G})$

is true, and the quantum generalization of the classical case given in Eq. (3)
can be utilized.