Efficient description of many-body systems with Matrix Product Density Operators

N. Schuch, Condensed Matter Applications of Entanglement Theory , arXiv:1306.5551

Original

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Note that due to the monotonicity of the α \alpha -Rényi entropy in α \alpha , we can also choose any smaller α \alpha in ( 3

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This is readily checked by working in the two-dimensional space spanned by the two vectors; note that | χ D ⟩ |\chi_{D}\rangle need not be normalized

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Different purifications are related by a unitary on the purifying system A ′ ​ B ′ A^{\prime}B^{\prime} , which however mixes A ′ A^{\prime} and B ′ B^{\prime} and thus changes the entanglement properties

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Specifically, ‖ P 1 ​ ( X ) ‖ 1 \displaystyle\|P_{1}(X)\|_{1} ≤ ∑ ‖ A ^ i ⊗ tr A ​ [ ( A ^ i ′ ⊗ 𝟙 ) † ​ 𝕏 ] ‖ 𝟙 \displaystyle\leq\sum\displaylimits\|\hat{A}_{i}\otimes\mathrm{tr}_{A}[(\hat{A}_{i}^{\prime}\otimes\openone)^{\dagger}X]\|_{1} = ∑ ‖ A ^ i ‖ 1 ​ ‖ tr A ​ [ ( A ^ i ′ ⊗ 𝟙 ) † ​ 𝕏 ] ‖ 𝟙 \displaystyle=\sum\displaylimits\|\hat{A}_{i}\|_{1}\|\mathrm{tr}_{A}[(\hat{A}_{i}^{\prime}\otimes\openone)^{\dagger}X]\|_{1} ≤ ( ∗ ) ∑ ∥ A ^ i ∥ 1 ∥ ( A ^ i ′ ⊗ 𝟙 ) † 𝕏 ] ∥ 𝟙 \displaystyle\stackrel{{\scriptstyle\smash{(*)}}}{{\leq}}\sum\displaylimits\|\hat{A}_{i}\|_{1}\|(\hat{A}_{i}^{\prime}\otimes\openone)^{\dagger}X]\|_{1} ≤ ∑ ‖ A ^ i ‖ 1 ​ ‖ A ^ i ′ ⊗ 𝟙 ‖ ∞ ​ ‖ 𝕏 ‖ 𝟙 \displaystyle\leq\sum\displaylimits\|\hat{A}_{i}\|_{1}\|\hat{A}_{i}^{\prime}\otimes\openone\|_{\infty}\|X\|_{1} ≤ ∑ ‖ A ^ i ‖ 1 ​ ‖ A ^ i ′ ‖ ∞ ​ ‖ X ‖ 1 , \displaystyle\leq\sum\displaylimits\|\hat{A}_{i}\|_{1}\|\hat{A}_{i}^{\prime}\|_{\infty}\|X\|_{1}\ , where ( ∗ ) (*) uses the contractivity of the partial trace

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