| Russian Mathematical Surveys |
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This article is cited in 3 scientific papers (total in 3 papers) Brief Communications Logarithmic Sobolev inequality and Hypothesis of Quantum Gaussian Maximizers A. S. Holevo Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
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     A long-standing problem in quantum Shannon theory is the classical capacity of bosonic Gaussian channels of various kinds [6]. The Hypothesis of Gaussian Maximizers (HGM) states that the full capacity of such channels is attained at Gaussian encodings. A breakthrough was made in the papers [1] and [2], where HGM was proved for the important class of multimode gauge co- or contra-variant Gaussian channels [4]. In [8] and [5] the solution was extended to a much broader class of channels satisfying a certain ‘threshold condition’. At the same time the HGM remains open for a large variety of Gaussian channels lying beyond the scope of the threshold condition [7]. In this note we sketch a novel approach to such problems, based on principles of convex programming, and illustrate it using the characteristic case of an approximate position measurement with energy constraint, underlying noisy Gaussian homodyning in quantum optics. Rather remarkably, for this particular model the method reduces the solution of the optimization problem to a generalization of the celebrated log-Sobolev inequality. Let $\mathfrak{S}$ be the convex set of all density operators in a separable Hilbert space $\mathcal{H}$ of the quantum system and $\mathcal{X}$ and $ \mathcal{Y}$ be standard measurable spaces. Consider a measurement channel $ M\colon \rho \to p_{\rho }(y)=\operatorname{Tr}\rho m(y),$ where $\rho \in \mathfrak{ S}$ and $m(y)$ is a uniformly bounded positive operator-valued function of $ y\in \mathcal{Y}$ such that $\displaystyle\int m(y)\,\mu (dy)=I$ ($I$ is the identity operator on $\mathcal{H}$ and $\mu$ is a measure). The encoding $\mathcal{E}=\{ \pi (dx),\rho (x)\} $ is a probability measure $\pi (dx)$ with a measurable family of states $\rho (x)$, $x\in \mathcal{X}$. The average state $\bar{\rho}_{\mathcal{E}}=\displaystyle\int\rho(x)\,\pi(dx)$. Let $H$ be a Hamiltonian on $\mathcal{H}$ and $E$ be a positive number. Then the energy-constrained classical capacity of the measurement channel $M$ is
$$
\begin{equation}
C(M,H,E)=\sup_{\mathcal{E}:\operatorname{Tr}\! \bar{\rho}_{\mathcal{E}}H\leqslant E}I(\mathcal{E},M),
\end{equation}
\tag{1}
$$
where $I(\mathcal{E},M)$ is the mutual information between $\mathcal{E}$ and $M$. Introducing the well-defined output differential entropy $h_{M}(\rho)= -\displaystyle\int p_{\rho}(y)\ln p_{\rho}(y)\,\mu(dy)$, one has
$$
\begin{equation}
I(\mathcal{E},M)=h_{M}(\bar{\rho}_{\mathcal{E}})- \int h_{M}(\rho(x))\,\pi(dx).
\end{equation}
\tag{2}
$$
Therefore,
$$
\begin{equation}
C(\mathcal{M},H,E)=\sup_{\rho:\operatorname{Tr}\!\rho H\leqslant E} [h_{\mathcal{M}}(\rho)-e_{\mathcal{M}}(\rho)],
\end{equation}
\tag{3}
$$
where an analogue of the convex closure of the output differential entropy for a quantum channel [10] is introduced:
$$
\begin{equation}
e_{M}(\rho)=\inf_{\mathcal{E}:\bar{\rho}_{\mathcal{E}}=\rho} \int h_{M}(\rho(x))\,\pi(dx).
\end{equation}
\tag{4}
$$
The minimization problem (4) is formally analogous to the quantum Bayes problem studied in [3]. Introducing $K(\rho)=-\displaystyle\int m(y)\ln p_{\rho}(y)\,\mu(dy)$, the optimality condition for an ensemble $\mathcal{E}=\{\pi_{0}(dx),\rho_{0}(x)\}$ becomes: There exists a selfadjoint operator $\Lambda_{0}$ such that (i) $\Lambda_{0}\leqslant K(\rho)$ for $\rho \in \mathfrak{S}$; (ii) $[K(\rho_{0}(x))-\Lambda_{0}]\rho_{0}(x)=0\pmod{\pi_{0}}$. Integrating (ii), an equation for determining $\Lambda_{0}$ is obtained:
$$
\begin{equation}
\int_{\mathfrak{S}}K(\rho_{0}(x))\rho_{0}(x)\,\pi_{0}(dx)= \Lambda_{0}\bar{\rho}.
\end{equation}
\tag{5}
$$
Passing to bosonic Gaussian systems we denote by $\rho_{\alpha}$ the centred Gaussian state of the canonical commutation relations with covariance matrix $\alpha$ [6], by $\mathfrak{S}(\alpha)$ the set of all states $\rho$ with fixed matrix $\alpha$ of second moments, and set $C(M;\alpha)\equiv \sup_{\mathcal{E}:\bar{\rho}_{\mathcal{E}}\in \mathfrak{S}(\alpha)}I(\mathcal{E},M)$. The following theorem was proved in [7]. Let $M$ be a general Gaussian measurement channel. Then the optimizing density operator $\rho$ in (3) is a (centred) Gaussian operator $\rho_{\alpha}$:
$$
\begin{equation}
C(M;\alpha)=h_{M}(\rho_{\alpha})-e_{M}(\rho_{\alpha}).
\end{equation}
\tag{6}
$$
Hence for a quadratic bosonic Hamiltonian $H=R\epsilon R^{t}$
$$
\begin{equation}
C(M,H,E)=\max_{\alpha:\operatorname{Tr}\!\alpha\epsilon \leqslant E} C(M;\alpha)=\max_{\alpha:\operatorname{Tr}\!\alpha \epsilon \leqslant E} [h_{M}(\rho_{\alpha})-e_{M}(\rho_{\alpha})].
\end{equation}
\tag{7}
$$
The approximate measurement of the position $q$ in one mode $R=(q,p)$ corresponds to $m(y)=g_{\beta}(q-y)$, where $g_{\beta}(y)$ is the probability density of the normal distribution $\mathcal{N}(0,\beta)$. We take the Hamiltonian $H=(q^{2}+p^{2})/2$ and the covariance matrix $\alpha=\operatorname{diag}[\alpha_{q},\alpha_{p}]$. Theorem. The maximum in (1) is attained at the Gaussian encoding $\{\pi_{0}(dx), \rho_{0}(x)\}$, where $\rho_{0}(x)=|x\rangle_{\delta}\langle x|$ are the squeezed states ($|x\rangle_{\delta}=\mathrm{e}^{-ipx}|0\rangle_{\delta}$) such that $\delta=(4\alpha_{p})^{-1}$, The distribution $\pi_{0}(dx)$ is centred Gaussian with variance $\gamma =\alpha_{q}-(4\alpha_{p})^{-1}$. Sketch of proof. The computation of (5) results in This is a selfadjoint operator satisfying condition (ii).
Checking condition (i) requires a generalization of the logarithmic Sobolev inequality (see, for instance, [9]). Let $f(x)=|\psi(x)|^{2}$ be a smooth probability density on $\mathbb{R}$. Then the inequality we have to prove is
$$
\begin{equation}
\int[g_{\beta}\ast f(y)]\ln[g_{\beta}\ast f(y)]\,dy+ \ln\sqrt{2\pi \mathrm{e}(\beta+\delta)}+\frac{\delta}{2(\beta+\delta)} \leqslant \frac{2\delta^{2}}{\beta+\delta}\int|\psi'(x)|^{2}\,dx
\end{equation}
\tag{9}
$$
for $\beta,\delta \geqslant 0$. For $\beta=0$ this is equivalent to the version of the log-Sobolev inequality in [9]. To prove (9) we multiply the difference between the left- and right-hand sides by $\beta+\delta$ and compute the derivative with respect to $\beta$. Its non-positivity is equivalent to the inequality in [9] with properly selected parameters. Thus (9) holds for all $\beta,\delta \geqslant 0$, which yields (i). $\Box$ 
Citation in format AMSBIB
\Bibitem{Hol22} |
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