Robust Hadamard matrices, unistochastic rays in Birkhoff polytope and equi-entangled bases in composite spaces
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2018
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| _version_ | 1866914580785004544 |
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| author | Rajchel-Mieldzioć, Grzegorz Gąsiorowski, Adam Życzkowski, Karol |
| author_facet | Rajchel-Mieldzioć, Grzegorz Gąsiorowski, Adam Życzkowski, Karol |
| contents | We study a special class of (real or complex) robust Hadamard matrices, distinguished by the property that their projection onto a $2$-dimensional subspace forms a Hadamard matrix. It is shown that such a matrix of order $n$ exists, if there exists a skew Hadamard matrix of this size. This is the case for any even dimension $n\le 20$, and for these dimensions we demonstrate that a bistochastic matrix $B$ located at any ray of the Birkhoff polytope, (which joins the center of this body with any permutation matrix), is unistochastic. An explicit form of the corresponding unitary matrix $U$, such that $B_{ij}=|U_{ij}|^2$, is determined by a robust Hadamard matrix. These unitary matrices allow us to construct a family of orthogonal bases in the composed Hilbert space of order $n \times n$. Each basis consists of vectors with the same degree of entanglement and the constructed family interpolates between the product basis and the maximally entangled basis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1804_10715 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Robust Hadamard matrices, unistochastic rays in Birkhoff polytope and equi-entangled bases in composite spaces Rajchel-Mieldzioć, Grzegorz Gąsiorowski, Adam Życzkowski, Karol Combinatorics Quantum Physics We study a special class of (real or complex) robust Hadamard matrices, distinguished by the property that their projection onto a $2$-dimensional subspace forms a Hadamard matrix. It is shown that such a matrix of order $n$ exists, if there exists a skew Hadamard matrix of this size. This is the case for any even dimension $n\le 20$, and for these dimensions we demonstrate that a bistochastic matrix $B$ located at any ray of the Birkhoff polytope, (which joins the center of this body with any permutation matrix), is unistochastic. An explicit form of the corresponding unitary matrix $U$, such that $B_{ij}=|U_{ij}|^2$, is determined by a robust Hadamard matrix. These unitary matrices allow us to construct a family of orthogonal bases in the composed Hilbert space of order $n \times n$. Each basis consists of vectors with the same degree of entanglement and the constructed family interpolates between the product basis and the maximally entangled basis. |
| title | Robust Hadamard matrices, unistochastic rays in Birkhoff polytope and equi-entangled bases in composite spaces |
| topic | Combinatorics Quantum Physics |
| url | https://arxiv.org/abs/1804.10715 |