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| Format: | Preprint |
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2019
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| Online Access: | https://arxiv.org/abs/1907.10604 |
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| _version_ | 1866911034106707968 |
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| author | Matsumoto, Keiji |
| author_facet | Matsumoto, Keiji |
| contents | Consider a classical system, which is in the state described by probability distribution $p$ or $q$, and embed these classical informations into quantum system by a physical map $Γ$, $ρ=Γ(p)$ and $σ=Γ(q)$. Intuitively, the pair $\{p_ρ^{M},p_σ^{M}\}$ of the distributions of the data of the measurement $M$ on the pair $\{ρ,σ\}$ should contain strictly less information than the pair $\{p,q\}$ provided the pair $\{ρ,σ\}$ is non-commutative. Indeed, this statement had been shown if the information is measured by $f$-divergence such that $f$ is operator convex. In the paper, the statement is extended to the case where $f$ is strictly convex. Also, we disprove the assertion for the total variation distance $\Vert p-q\Vert_{1}$, the $f$-divergence with $f(r)=|1-r|$: if $\{ρ,σ\}$ satisfies some not very restrictive conditions, $\Vert p_ρ^{M}-p_σ^{M}\Vert_{1}$ equals $\Vert p-q\Vert_{1}$. Here we present sufficient condition for general case, and necessary and sufficient condition for qubit states. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1907_10604 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Reversibility of distance measures of states with some focus on total variation distance Matsumoto, Keiji Quantum Physics Mathematical Physics Consider a classical system, which is in the state described by probability distribution $p$ or $q$, and embed these classical informations into quantum system by a physical map $Γ$, $ρ=Γ(p)$ and $σ=Γ(q)$. Intuitively, the pair $\{p_ρ^{M},p_σ^{M}\}$ of the distributions of the data of the measurement $M$ on the pair $\{ρ,σ\}$ should contain strictly less information than the pair $\{p,q\}$ provided the pair $\{ρ,σ\}$ is non-commutative. Indeed, this statement had been shown if the information is measured by $f$-divergence such that $f$ is operator convex. In the paper, the statement is extended to the case where $f$ is strictly convex. Also, we disprove the assertion for the total variation distance $\Vert p-q\Vert_{1}$, the $f$-divergence with $f(r)=|1-r|$: if $\{ρ,σ\}$ satisfies some not very restrictive conditions, $\Vert p_ρ^{M}-p_σ^{M}\Vert_{1}$ equals $\Vert p-q\Vert_{1}$. Here we present sufficient condition for general case, and necessary and sufficient condition for qubit states. |
| title | Reversibility of distance measures of states with some focus on total variation distance |
| topic | Quantum Physics Mathematical Physics |
| url | https://arxiv.org/abs/1907.10604 |