dorsal/arxiv
View SchemaQuamtumness, Generalized Spherical 2-Design and Symmetric Informationally Complete POVM
| Authors | Isaac H. Kim |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0608024 |
| URL | https://arxiv.org/abs/quant-ph/0608024 |
| Journal | Quant. Inf. Comp Vol. 7, No. 8 (2007) 730--737 |
Abstract
C. A. Fuchs and M. Sasaki defined the quantumness of a set of quantum states in \cite{Quantumness}, which is closely related to the fidelity loss in transmission of the quantum states through a classical channel. In \cite{Fuchs}, Fuchs showed that in $d$-dimensional Hilbert space, minimum quantumness is $\frac{2}{d+1}$, and this can be achieved by all rays in the space. He left an open problem, asking whether fewer than $d^2$ states can achieve this bound. Recently, in a different context, A. J. Scott introduced a concept of generalized $t$-design in \cite{GenSphet}, which is a natural generalization of spherical $t$-design. In this paper, we show that the lower bound on the quantumness can be achieved if and only if the states form a generalized 2-design. As a corollary, we show that this bound can be only achieved if the number of states are larger or equal to $d^2$, answering the open problem. Furthermore, we also show that the minimal set of such ensemble is Symmetric Informationally Complete POVM(SIC-POVM). This leads to an equivalence relation between SIC-POVM and minimal set of ensemble achieving minimal quantumness.
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"abstract": "C. A. Fuchs and M. Sasaki defined the quantumness of a set of quantum states\nin \\cite{Quantumness}, which is closely related to the fidelity loss in\ntransmission of the quantum states through a classical channel. In\n\\cite{Fuchs}, Fuchs showed that in $d$-dimensional Hilbert space, minimum\nquantumness is $\\frac{2}{d+1}$, and this can be achieved by all rays in the\nspace. He left an open problem, asking whether fewer than $d^2$ states can\nachieve this bound. Recently, in a different context, A. J. Scott introduced a\nconcept of generalized $t$-design in \\cite{GenSphet}, which is a natural\ngeneralization of spherical $t$-design. In this paper, we show that the lower\nbound on the quantumness can be achieved if and only if the states form a\ngeneralized 2-design. As a corollary, we show that this bound can be only\nachieved if the number of states are larger or equal to $d^2$, answering the\nopen problem. Furthermore, we also show that the minimal set of such ensemble\nis Symmetric Informationally Complete POVM(SIC-POVM). This leads to an\nequivalence relation between SIC-POVM and minimal set of ensemble achieving\nminimal quantumness.",
"arxiv_id": "quant-ph/0608024",
"authors": [
"Isaac H. Kim"
],
"categories": [
"quant-ph"
],
"journal_ref": "Quant. Inf. Comp Vol. 7, No. 8 (2007) 730--737",
"title": "Quamtumness, Generalized Spherical 2-Design and Symmetric Informationally Complete POVM",
"url": "https://arxiv.org/abs/quant-ph/0608024"
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