The quantum state of a $d$ -dimensional system can be represented by the $d^2$ probabilities corresponding to a SIC-POVM, and then this distribution of probability can be represented by a vector of $\R^{d^2-1}$ in a simplex, we will call this set of vectors $\mathcal{Q}$ . Other way of represent a $d$ -dimensional system is by the corresponding Bloch vector also in $\R^{d^2-1}$ , we will call this set of vectors $\mathcal{B}$ . In this paper it is proved that with the adequate scaling $\mathcal{B}=\mathcal{Q}$ . Also we indicate some features of the shape of $\mathcal{Q}$ .The quantum state of a $d$-dimensional system can be represented by the $d^2$ probabilities corresponding to a SIC-POVM, and then this distribution of probability can be represented by a vector of $\R^{d^2-1}$ in a simplex, we will call this set of vectors $\mathcal{Q}$. Other way of represent a $d$-dimensional system is by the corresponding Bloch vector also in $\R^{d^2-1}$, we will call this set of vectors $\mathcal{B}$. In this paper it is proved that with the adequate scaling $\mathcal{B}=\mathcal{Q}$. Also we indicate some features of the shape of $\mathcal{Q}$.

Aug 16, 20167 citations

Geometry of the state spaces in quantum probability

Sh. A. Ayupov, N. J. Yadgorov

De Gruyter eBooksDec 31, 19940 citations

SIC-POVMs and the Knaster's Conjecture

S. B. Samuel, Z. Gedik

Symmetric Informationally Complete Positive Operator-Valued Measures (SIC-POVMs) have been constructed in many dimensions using the Weyl-Heisenberg group. In the quantum information community, it is commonly believed that SCI-POVMs exist in all dimensions; however, the general proof of their existence is still an open problem. The Bloch sphere representation of SIC-POVMs allows for a general geometric description of the set of operators, where they form the vertices of a regular simplex oriented based on a continuous function. We use this perspective of the SIC-POVMs to prove the Knaster's conjecture for the geometry of SIC-POVMs and prove the existence of a continuous family of generalized SIC-POVMs where $(n^2-1)$ of the matrices have the same value of $Tr(ρ^k)$ . Furthermore, by using numerical methods, we show that a regular simplex can be constructed such that all its vertices map to the same value of $Tr(ρ^3)$ on the Bloch sphere of $3$ and $4$ dimensional Hilbert spaces. In the $3$ -dimensional Hilbert space, we generate $10^4$ generalized SIC-POVMs for randomly chosen $Tr(ρ^3)$ values such that all the elements are equivalent up to unitary transformations.Symmetric Informationally Complete Positive Operator-Valued Measures (SIC-POVMs) have been constructed in many dimensions using the Weyl-Heisenberg group. In the quantum information community, it is commonly believed that SCI-POVMs exist in all dimensions; however, the general proof of their existence is still an open problem. The Bloch sphere representation of SIC-POVMs allows for a general geometric description of the set of operators, where they form the vertices of a regular simplex oriented based on a continuous function. We use this perspective of the SIC-POVMs to prove the Knaster's conjecture for the geometry of SIC-POVMs and prove the existence of a continuous family of generalized SIC-POVMs where $(n^2-1)$ of the matrices have the same value of $Tr(ρ^k)$. Furthermore, by using numerical methods, we show that a regular simplex can be constructed such that all its vertices map to the same value of $Tr(ρ^3)$ on the Bloch sphere of $3$ and $4$ dimensional Hilbert spaces. In the $3$-dimensional Hilbert space, we generate $10^4$ generalized SIC-POVMs for randomly chosen $Tr(ρ^3)$ values such that all the elements are equivalent up to unitary transformations.

arXiv:2405.16733May 27, 20240 citations

The quantum state of a

d

-dimensional system can be represented by the

d^2

probabilities corresponding to a SIC-POVM, and then this distribution of probability can be represented by a vector of

\R^{d^2-1}

in a simplex, we will call this set of vectors

\mathcal{Q}

. Other way of represent a

d

-dimensional system is by the corresponding Bloch vector also in

\R^{d^2-1}

, we will call this set of vectors

\mathcal{B}

. In this paper it is proved that with the adequate scaling

\mathcal{B}=\mathcal{Q}

. Also we indicate some features of the shape of

\mathcal{Q}

(n^2-1)

of the matrices have the same value of

Tr(ρ^k)

. Furthermore, by using numerical methods, we show that a regular simplex can be constructed such that all its vertices map to the same value of

Tr(ρ^3)

on the Bloch sphere of

3

and

4

dimensional Hilbert spaces. In the

3

-dimensional Hilbert space, we generate

10^4

generalized SIC-POVMs for randomly chosen

Tr(ρ^3)

values such that all the elements are equivalent up to unitary transformations.