Strong Converse Bounds for Compression of Mixed States
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866917990785613824 |
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| author | Khanian, Zahra Baghali |
| author_facet | Khanian, Zahra Baghali |
| contents | In this paper, we study strong converse properties for both visible and blind compression of mixed states. The optimal rate of a visible compression scheme is obtained in terms of the entanglement of purification, whose additivity remains unknown so far. For a variation of extendible states, we prove that the entanglement of purification is additive and apply this to obtain a "pretty strong" converse bound for the blind and visible compression of such states. Namely, when the rate decreases below the optimal rate, the error exhibits a discontinuous jump from 0 to at least $\frac{1}{3\sqrt{2}}$.
To deal with the visible case for general states, we define a new quantity $E_{α,p}(A:R)_ρ$ for a bipartite state $ρ^{AR}$ and $α\in (0,1)\cup (1,\infty)$ as the $α$-Rényi generalization of the entanglement of purification $E_{p}(A:R)_ρ$. For $α=1$, we define $E_{1,p}(A:R)_ρ:=E_{p}(A:R)_ρ$. We show that for any rate below the regularization $\lim_{α\to 1^+}E_{α,p}^{\infty}(A:R)_ρ:=\lim_{α\to 1^+} \lim_{n \to \infty} \frac{E_{α,p}(A^n:R^n)_{ρ^{\otimes n}}}{n}$ the fidelity for the visible compression exponentially converges to zero.
Moreover, we consider blind compression of a general mixed-state source $ρ^{AR}$ shared between an encoder and an inaccessible reference system $R$. We obtain a strong converse bound for the compression of this source by assuming that the decoder is a super-unital channel. This immediately implies a strong converse for the blind compression of ensembles of mixed states, by assuming a super-unital decoder, as this is a special case of the general mixed-state source $ρ^{AR}$ where the reference system $R$ has a classical structure. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2206_09415 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Strong Converse Bounds for Compression of Mixed States Khanian, Zahra Baghali Quantum Physics Information Theory In this paper, we study strong converse properties for both visible and blind compression of mixed states. The optimal rate of a visible compression scheme is obtained in terms of the entanglement of purification, whose additivity remains unknown so far. For a variation of extendible states, we prove that the entanglement of purification is additive and apply this to obtain a "pretty strong" converse bound for the blind and visible compression of such states. Namely, when the rate decreases below the optimal rate, the error exhibits a discontinuous jump from 0 to at least $\frac{1}{3\sqrt{2}}$. To deal with the visible case for general states, we define a new quantity $E_{α,p}(A:R)_ρ$ for a bipartite state $ρ^{AR}$ and $α\in (0,1)\cup (1,\infty)$ as the $α$-Rényi generalization of the entanglement of purification $E_{p}(A:R)_ρ$. For $α=1$, we define $E_{1,p}(A:R)_ρ:=E_{p}(A:R)_ρ$. We show that for any rate below the regularization $\lim_{α\to 1^+}E_{α,p}^{\infty}(A:R)_ρ:=\lim_{α\to 1^+} \lim_{n \to \infty} \frac{E_{α,p}(A^n:R^n)_{ρ^{\otimes n}}}{n}$ the fidelity for the visible compression exponentially converges to zero. Moreover, we consider blind compression of a general mixed-state source $ρ^{AR}$ shared between an encoder and an inaccessible reference system $R$. We obtain a strong converse bound for the compression of this source by assuming that the decoder is a super-unital channel. This immediately implies a strong converse for the blind compression of ensembles of mixed states, by assuming a super-unital decoder, as this is a special case of the general mixed-state source $ρ^{AR}$ where the reference system $R$ has a classical structure. |
| title | Strong Converse Bounds for Compression of Mixed States |
| topic | Quantum Physics Information Theory |
| url | https://arxiv.org/abs/2206.09415 |