Average R\'{e}nyi Entropy of a Subsystem in Random Pure State

• URL: http://arxiv.org/abs/2301.09074v2
• Date: Tue, 16 Jan 2024 05:01:14 GMT
• Title: Average R\'{e}nyi Entropy of a Subsystem in Random Pure State
• Authors: MuSeong Kim, Mi-Ra Hwang, Eylee Jung, and DaeKil Park
• Abstract summary: We plot the $ln m$-dependence of the quantum information derived from $widetildeS_alpha (m,n)$. It is remarkable to note that the nearly vanishing region of the information becomes shorten with increasing $alpha$, and eventually disappears in the limit of $alpha rightarrow infty$.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In this paper we examine the average R\'{e}nyi entropy $S_{\alpha}$ of a subsystem $A$ when the whole composite system $AB$ is a random pure state. We assume that the Hilbert space dimensions of $A$ and $AB$ are $m$ and $m n$ respectively. First, we compute the average R\'{e}nyi entropy analytically for $m = \alpha = 2$. We compare this analytical result with the approximate average R\'{e}nyi entropy, which is shown to be very close. For general case we compute the average of the approximate R\'{e}nyi entropy $\widetilde{S}_{\alpha} (m,n)$ analytically. When $1 \ll n$, $\widetilde{S}_{\alpha} (m,n)$ reduces to $\ln m - \frac{\alpha}{2 n} (m - m^{-1})$, which is in agreement with the asymptotic expression of the average von Neumann entropy. Based on the analytic result of $\widetilde{S}_{\alpha} (m,n)$ we plot the $\ln m$-dependence of the quantum information derived from $\widetilde{S}_{\alpha} (m,n)$. It is remarkable to note that the nearly vanishing region of the information becomes shorten with increasing $\alpha$, and eventually disappears in the limit of $\alpha \rightarrow \infty$. The physical implication of the result is briefly discussed.

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