arXiv:quant-ph/0210007 · Zeitschrift für Naturforschung A

Solution to the King’s Problem in Prime Power Dimensions

P. K. Aravind

Published Mar 1, 2003• 31 citations

Abstract

It is shown how to ascertain the values of a complete set of mutually complementary observables of a prime power degree of freedom by generalizing the solution in prime dimensions given by Englert and Aharonov [Phys. Lett. A284, 1 - 5 (2001)].

DOI: 10.1515/zna-2003-2-302

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arXiv:quant-ph/0210007 · Zeitschrift für Naturforschung A

Solution to the King’s Problem in Prime Power Dimensions

P. K. Aravind

Published Mar 1, 2003• 31 citations

arXiv HTML PDF

Abstract

DOI: 10.1515/zna-2003-2-302

Related papers

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Berthold‐Georg Englert, Yakir Aharonov

We show how one can ascertain the values of a complete set of mutually complementary observables of a prime degree of freedom.

Jan 29, 200172 citations

The mean king's problem: Spin 1

Yakir Aharonov, Berthold‐Georg Englert

We show how one can ascertain the values of four mutually complementary observables of a spin-1 degree of freedom.

Jan 16, 200145 citations

The Mean King's Problem: Spin

Yakir Aharonov, Berthold‐Georg Englert

Abstract We show how one can ascertain the values of four mutually complementary observables of a spin-1 degree of freedom. -Pacs: 03.65.Bz

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Solution to the king’s problem with observables that are not mutually complementary

M. Horibe, A. Hayashi, T. Hashimoto

We investigate the king's problem of the measurement of operators ${\stackrel{P\vec}{n}}_{k}∙\stackrel{P\vec}{\ensuremath{\sigma}}$ ( $k=1$ , 2, 3) instead of the three Cartesian components ${\ensuremath{\sigma}}_{x},{\ensuremath{\sigma}}_{y}$, and ${\ensuremath{\sigma}}_{z}$ of the spin operator $\stackrel{P\vec}{\ensuremath{\sigma}}$. Here, ${\stackrel{P\vec}{n}}_{k}$ are three-dimensional real unit vectors. We show the condition over three vectors ${\stackrel{P\vec}{n}}_{k}$ to ascertain the result for measurement of any one of these operators.We investigate the king's problem of the measurement of operators ${\stackrel{P\vec}{n}}_{k}∙\stackrel{P\vec}{\ensuremath{\sigma}}$ ($k=1$, 2, 3) instead of the three Cartesian components ${\ensuremath{\sigma}}_{x},{\ensuremath{\sigma}}_{y}$, and ${\ensuremath{\sigma}}_{z}$ of the spin operator $\stackrel{P\vec}{\ensuremath{\sigma}}$. Here, ${\stackrel{P\vec}{n}}_{k}$ are three-dimensional real unit vectors. We show the condition over three vectors ${\stackrel{P\vec}{n}}_{k}$ to ascertain the result for measurement of any one of these operators.

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Generalized Numerical Construction of MUBs: A Group Theoretical Investigation

B. Gültekin, S. B. Samuel, Z. Gedik

Mutually Unbiased Bases (MUBs) constitute a fundamental geometric structure in quantum theory, known for providing an optimal measurement scheme for quantum state tomography. In prime and prime-power dimensions, analytical constructions of maximal sets of MUBs are well-known and standard construction relies on the Weyl-Heisenberg (WH) group and finite fields. In non-prime-power dimensions, on the other hand, the existence of such maximal sets remains an open question. We present a generalized numerical method of constructing MUBs without any reliance on \textit{a priori} group structure or specific algebraic frameworks. Formulating the problem at the level of Gram matrix, we reduce the search for complete sets of d+1 MUBs to a phase space optimisation problem. We use the fact that the MUB Gram matrix is a projection matrix, and the third- and fourth-order trace constraints are necessary and sufficient conditions for a projection matrix. We further develop a classification framework based on third-order Bargmann invariants and automorphism groups, allowing us to probe the underlying algebraic and geometric structure of the resulting configurations. Numerical applications of this method in dimensions $3$ , $4$ , and $5$ demonstrate that all numerically constructed solutions are mutually isomorphic, are isolated points in phase space, and possess automorphism groups that coincide exactly with the Clifford group, the normalizer of the WH group. In the lowest non-prime-power dimension $6$ where the existence of a complete set remains open, our numerical search yields no solution. However, for incomplete sets of solutions, our search finds no set of four MUBs, but it yields several inequivalent triples, which we classify and whose neighbourhood we then investigate.Mutually Unbiased Bases (MUBs) constitute a fundamental geometric structure in quantum theory, known for providing an optimal measurement scheme for quantum state tomography. In prime and prime-power dimensions, analytical constructions of maximal sets of MUBs are well-known and standard construction relies on the Weyl-Heisenberg (WH) group and finite fields. In non-prime-power dimensions, on the other hand, the existence of such maximal sets remains an open question. We present a generalized numerical method of constructing MUBs without any reliance on \textit{a priori} group structure or specific algebraic frameworks. Formulating the problem at the level of Gram matrix, we reduce the search for complete sets of d+1 MUBs to a phase space optimisation problem. We use the fact that the MUB Gram matrix is a projection matrix, and the third- and fourth-order trace constraints are necessary and sufficient conditions for a projection matrix. We further develop a classification framework based on third-order Bargmann invariants and automorphism groups, allowing us to probe the underlying algebraic and geometric structure of the resulting configurations. Numerical applications of this method in dimensions $3$, $4$, and $5$ demonstrate that all numerically constructed solutions are mutually isomorphic, are isolated points in phase space, and possess automorphism groups that coincide exactly with the Clifford group, the normalizer of the WH group. In the lowest non-prime-power dimension $6$ where the existence of a complete set remains open, our numerical search yields no solution. However, for incomplete sets of solutions, our search finds no set of four MUBs, but it yields several inequivalent triples, which we classify and whose neighbourhood we then investigate.

quant-pharXiv:2604.04164Journal of Physics A Mathematical and TheoreticalApr 7, 20260 citations

We investigate the king's problem of the measurement of operators ${\stackrel{P\vec}{n}}_{k}∙\stackrel{P\vec}{\ensuremath{\sigma}}$ (

k=1

, 2, 3) instead of the three Cartesian components ${\ensuremath{\sigma}}_{x},{\ensuremath{\sigma}}_{y}$, and ${\ensuremath{\sigma}}_{z}$ of the spin operator $\stackrel{P\vec}{\ensuremath{\sigma}}$. Here, ${\stackrel{P\vec}{n}}_{k}$ are three-dimensional real unit vectors. We show the condition over three vectors ${\stackrel{P\vec}{n}}_{k}$ to ascertain the result for measurement of any one of these operators.

3

4

, and

5

demonstrate that all numerically constructed solutions are mutually isomorphic, are isolated points in phase space, and possess automorphism groups that coincide exactly with the Clifford group, the normalizer of the WH group. In the lowest non-prime-power dimension

6

where the existence of a complete set remains open, our numerical search yields no solution. However, for incomplete sets of solutions, our search finds no set of four MUBs, but it yields several inequivalent triples, which we classify and whose neighbourhood we then investigate.