| Russian Mathematical Surveys |
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Brief communications Tight lower bounds for Shannon entropy from ‘quantum pyramids’ A. S. Holevo Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
Computing accessible information for an ensemble of quantum states is a basic problem in quantum information theory, going back to its origin and important, in particular, for applications in quantum cryptography. The problem is solved in some special cases but, in general, remains open. In this note we show that a recently obtained optimality criterion [1], when applied to specific ensembles of states, leads to non-trivial entropy inequalities that are discrete relatives of the log-Sobolev inequality [2]. In this light, the hypothesis of an information-optimal measurement for an ensemble of equiangular equiprobable states — the quantum pyramid [3] — is reconsidered and the corresponding tight entropy inequalities are proposed. Let $\mathfrak{S}$ be the convex set of quantum states (density operators) on $d$-dimensional Hilbert space $\mathcal{H}$. An ensemble of quantum states is a collection $\mathcal{E}=\{\pi_{j},\rho_{j}\}_{j=1,\dots,m}$, where $\rho_{j}\in \mathfrak{S}$ and $\pi=\{\pi_{j}\}$ is a probability distribution. The average state is $\overline{\rho}=\sum_{j}\pi_{j}\rho_{j}$. An observable is a set of Hermitian operators $\mathcal{M}=\{M_{k}\}_{k=1,\dots,n}$, where $M_{k}\geqslant 0$ and $\sum_{k=1}^{n}M_{k}=I$. The joint probability distribution of an ‘input’ $j$ and an ‘output’ $k$ is $p_{jk}=\pi_{j}\operatorname{Tr}\rho_{j}M_{k}$. The accessible information of the ensemble $\mathcal{E}$ is equal to
$$
\begin{equation}
A(\mathcal{E})=\sup_{\mathcal{M}}I(\mathcal{E},\mathcal{M}),
\end{equation}
\tag{1}
$$
where $I(\mathcal{E},\mathcal{M})=\sum_{j,k}p_{jk}\log(p_{jk}/(\pi_{j}p_{\cdot k}))$ is the Shannon mutual information between $j$ and $k$. It has been shown [4] that the supremum is attained at an observable $ \mathcal{M}$ with linearly independent components $M_{k}=|\varphi_{k}\rangle \langle \varphi_{k}|$, $k=1,\dots,n\leqslant d^{2}$, of rank 1.
Theorem 1. An observable $\mathcal{M}= \{|\varphi_{k}\rangle \langle \varphi_{k}|\}$ is optimal for problem (1) if and only if there exists a Hermitian operator $\Lambda$ such that the entropy inequality The value of the accessible information is $A(\mathcal{E})=H(\pi)-\operatorname{Tr}\overline{\rho}\Lambda$. Consider an ensemble $\mathcal{E}$ of $m$ equiprobable equiangular pure states on $m$-dimensional Hilbert space — a quantum pyramid [3]. Denote by $\xi $ the inner product between the state vectors and introduce the parameter $a=m^{-1}[\sqrt{1+(m-1)\xi}+(m-1)\sqrt{1-\xi}\,]$. Then inequality (2) reduces to tight lower bounds for the Shannon entropy of a discrete probability distribution. Namely, in the case of a moderately acute pyramid we have the following result. Theorem 2. For $m\geqslant 2$ and $p\equiv a^2>(m-1)/m$ For $m\geqslant 3$ the value $p=(m-1)/m$ is allowed, giving the inequality
$$
\begin{equation}
-\sum_{j=1}^{m}t_{j}\log t_{j}\geqslant \log m-\frac{m\log(m-1)}{m-2}[1-B(P;P_{U})^{2}],
\end{equation}
\tag{6}
$$
where $B(P;P_{U})=\sum_{j=1}^{m}\sqrt{t_{j}/m}$ is the Bhattacharyya coefficient between the probability distribution $P=(t_{1},\dots ,t_{m}) $ and the uniform distribution $P_{U}$. It can be compared with the lower bound of Theorem 3.11 in [5] in terms of the total variation distance. For $P$ close to $P_{U}$ the bound (6) is considerably better, while close to degenerate probability distributions it can be negative. This bound also turns out to be the entropy inequality (2) for strongly acute pyramids with $1/m\leqslant p\leqslant (m-1)/m$.
In the case of the ensemble of flat pyramid ($\xi =-1)$ the corresponding tight entropy inequality is given by the following statement. Theorem 3. For all complex $z_{j}$ satisfying $\sum_{j=1}^{m}|z_{j}|^{2}=1$ and $\sum_{j=1}^{m}z_{j}=0$, Corollary. The conjectures concerning globally optimal observables for the accessible information of an ensemble of quantum pyramid [3] hold true in the cases of acute and flat pyramids. The proofs are presented in the preprint [6], which also contains derivations of tight entropy inequalities in the case of obtuse ($-1<\xi<0$) quantum pyramids.
Citation in format AMSBIB
\Bibitem{Hol25} |
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