Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1809.06762 · Physica Scripta

Mutually disjoint, maximally commuting set of physical observables for optimum state determination

H S Smitha Rao, Swarnamala Sirsi, Karthik Bharath

Published Jun 18, 2019• 1 citation
arXivHTMLPDF

Abstract

Abstract We consider the state determination problem using mutually unbiased bases (MUBs). For spin-1, spin-3/2 and spin-2 systems, analogous to Pauli operators of spin-1/2 system, which are experimentally implementable and correspond to the optimum measurement in characterizing the density matrix, we describe a procedure to construct an orthonormal set of operators from MUBs. The constructed operators are maximally commuting, can be physically realized, and correspond to physical observables. The method of construction is general enough to allow for extensions to higher-dimensional spin systems and arbitrary density matrices in finite dimensions for which MUBs are known to exist.

DOI
10.1088/1402-4896/ab2a85

Related papers

Unextendible Mutually Unbiased Bases from Pauli Classes

Prabha Mandayam, Somshubhro Bandyopadhyay, Markus Grassl +1 more

We provide a construction of sets of (d/2+1) mutually unbiased bases (MUBs) in dimensions d=4,8 using maximal commuting classes of Pauli operators. We show that these incomplete sets cannot be extended further using the operators of the Pauli group. However, specific examples of sets of MUBs obtained using our construction are shown to be strongly unextendible; that is, there does not exist another vector that is unbiased with respect to the elements in the set. We conjecture the existence of such unextendible sets in higher dimensions (d=2^{n}, n>3) as well. Furthermore, we note an interesting connection between these unextendible sets and state-independent proofs of the Kochen-Specker Theorem for two-qubit systems. Our construction also leads to a proof of the tightness of a H_{2} entropic uncertainty relation for any set of three MUBs constructed from Pauli classes in d=4.

arXiv:1302.3709•Feb 15, 2013•8 citations

On mutually unbiased bases

Rahim Taghikhani

Two orthonormal bases in the complex space of dimension d, are said to be mutually unbiased if the square of the magnitude of the inner product of any vector from one basis with any vector in other basis is equal to the reciprocal of the dimension d. Mutually unbiased bases are used for optimal state determination of mixed quantum states. It is known that in any dimension d, the number of mutually unbiased bases is at most d+1. Ivanovic found a complete set of mutually unbiased bases for prime dimensions. His construction was generalized by Wootters and Fields for prime power dimensions. There is a strong connection between maximally commuting bases of orthogonal unitary matrices and mutually unbiased bases. Based on this connection, there exits a constructive proof of the existence of a complete set of mutually unbiased bases for prime power dimensions. This thesis is an exploration on construction of mutually unbiased bases.

•Aug 26, 2013•1 citation

Unextendible mutually unbiased bases from Pauli C\classes

Prabha Mandayam, Somshubhro Bandyopadhyay, Markus Grassl +1 more

We provide a construction of sets of $d/2+1$ mutually unbiased bases (MUBs) in dimensions $d=4,8$ using maximal commuting classes of Pauli operators. We show that these incomplete sets cannot be extended further using the operators of the Pauli group. Moreover, specific examples of sets of MUBs obtained using our construction are shown to be {\it strongly unextendible}; that is, there does not exist another vector that is unbiased with respect to the elements in the set. We conjecture the existence of such unextendible sets in higher dimensions $d=2^{n} (n>3) $ as well.} {Furthermore, we note an interesting connection between these unextendible sets and state-independent proofs of the Kochen-Specker Theorem for two-qubit systems. Our construction also leads to a proof of the tightness of a $H_{2}$ entropic uncertainty relation for any set of three MUBs constructed from Pauli classes in $d=4$.

Quantum Information and Computation•Jul 1, 2014•9 citations

Mutually unbiased bases and complementary spin 1 observables

Paweł Kurzyński, Wawrzyniec Kaszub, Mikołaj Czechlewski

The two observables are incompatible if they cannot be measured simultaneously; however, they become maximally incompatible (complementary) if their eigenstates are mutually unbiased. Only then does the measurement of one observable give no information about the other observable. The spin projection operators onto three mutually orthogonal directions are complementary only for spin 1/2. For higher spin numbers the corresponding eigenstates are no longer unbiased. In this work we examine the properties of spin 1 mutually unbiased bases (MUB) and look for the physical meaning of the corresponding operators. We show that if the computational basis is chosen to be the eigenbasis of the spin projection operator along some direction z , then all the states, which are unbiased to this basis, have to be squeezed. Next, we study the generation and the measurement of MUB states by introducing the Fourier-like transform through spin squeezing. Finally, we try to ascribe some classical interpretation to the operators corresponding to MUB and study what information the observer gains while measuring them. Higher spin numbers are also considered.

arXiv:0905.1723•Journal of Physics A Mathematical and Theoretical•Jun 4, 2010•9 citations

Mutually unbiased bases, spherical designs, and frames

Andreas Klappenecker, Martin Roetteler

The principle of complementarity lies at the heart of quantum mechanics. In finite dimensional quantum systems this principle is captured by pairs of observables which are given by mutually unbiased bases (MUBs). Two orthonormal bases B and C of Cd are mutually unbiased if |<b|c>|2 = 1/d holds for all vectors b ∈ B and c ∈ C. This implies that whenever we are given a vector from one of these bases and perform a measurement with respect to any other of the bases, then there is no information gained from this measurement. A basic question about MUBs is how many of them can be found in a given dimension d. While constructions of maximal sets of d+1 such bases are known for system of prime power dimension d, it is unknown whether this bound can be achieved for any non-prime power dimension. We review the known constructions of MUBs and demonstrate that maximal sets of MUBs come with a rich combinatorial structure by showing that they actually are the same objects as the complex projective 2-designs with angle set {0,1/d}. Furthermore, we address the problem of constructing positive operator-valued measures (POVMs) in finite dimension d consisting of d2 operators of rank one which have an inner product equal to uniform or very close to uniform. This is motivated by the related question of constructing symmetric informationally complete POVMs (SIC-POVMs) for which the inner products are perfectly uniform. We also give a simple proof of the fact that symmetric informationally complete POVMs are complex projective 2-designs with angle set {1/(d+1)}. Moreover, we show that MUBs and SIC-POVMs form uniform tight frames in Cd.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Aug 18, 2005•24 citations

Graph-state formalism for mutually unbiased bases

Christoph Spengler, Barbara Kraus

A pair of orthonormal bases is called mutually unbiased if all mutual overlaps between any element of one basis and an arbitrary element of the other basis coincide. In case the dimension, $d$, of the considered Hilbert space is a power of a prime number, complete sets of $d+1$ mutually unbiased bases (MUBs) exist. Here we present a method based on the graph-state formalism to construct such sets of MUBs. We show that for $n$ $p$-level systems, with $p$ being prime, one particular graph suffices to easily construct a set of ${p}^{n}+1$ MUBs. In fact, we show that a single $n$-dimensional vector, which is associated with this graph, can be used to generate a complete set of MUBs and demonstrate that this vector can be easily determined. Finally, we discuss some advantages of our formalism regarding the analysis of entanglement structures in MUBs, as well as experimental realizations.

arXiv:1309.6557•Physical Review A•Nov 20, 2013•21 citations
Abstract We consider the state determination problem using mutually unbiased bases (MUBs). For spin-1, spin-3/2 and spin-2 systems, analogous to Pauli operators of spin-1/2 system, which are experimentally implementable and correspond to the optimum measurement in characterizing the density matrix, we describe a procedure to construct an orthonormal set of operators from MUBs. The constructed operators are maximally commuting, can be physically realized, and correspond to physical observables. The method of construction is general enough to allow for extensions to higher-dimensional spin systems and arbitrary density matrices in finite dimensions for which MUBs are known to exist.
We provide a construction of sets of (d/2+1) mutually unbiased bases (MUBs) in dimensions d=4,8 using maximal commuting classes of Pauli operators. We show that these incomplete sets cannot be extended further using the operators of the Pauli group. However, specific examples of sets of MUBs obtained using our construction are shown to be strongly unextendible; that is, there does not exist another vector that is unbiased with respect to the elements in the set. We conjecture the existence of such unextendible sets in higher dimensions (d=2^{n}, n>3) as well. Furthermore, we note an interesting connection between these unextendible sets and state-independent proofs of the Kochen-Specker Theorem for two-qubit systems. Our construction also leads to a proof of the tightness of a H_{2} entropic uncertainty relation for any set of three MUBs constructed from Pauli classes in d=4.
Two orthonormal bases in the complex space of dimension d, are said to be mutually unbiased if the square of the magnitude of the inner product of any vector from one basis with any vector in other basis is equal to the reciprocal of the dimension d. Mutually unbiased bases are used for optimal state determination of mixed quantum states. It is known that in any dimension d, the number of mutually unbiased bases is at most d+1. Ivanovic found a complete set of mutually unbiased bases for prime dimensions. His construction was generalized by Wootters and Fields for prime power dimensions. There is a strong connection between maximally commuting bases of orthogonal unitary matrices and mutually unbiased bases. Based on this connection, there exits a constructive proof of the existence of a complete set of mutually unbiased bases for prime power dimensions. This thesis is an exploration on construction of mutually unbiased bases.
We provide a construction of sets of d/2+1d/2+1d/2+1 mutually unbiased bases (MUBs) in dimensions d=4,8d=4,8d=4,8 using maximal commuting classes of Pauli operators. We show that these incomplete sets cannot be extended further using the operators of the Pauli group. Moreover, specific examples of sets of MUBs obtained using our construction are shown to be {\it strongly unextendible}; that is, there does not exist another vector that is unbiased with respect to the elements in the set. We conjecture the existence of such unextendible sets in higher dimensions $d=2^{n} (n>3) $ as well.} {Furthermore, we note an interesting connection between these unextendible sets and state-independent proofs of the Kochen-Specker Theorem for two-qubit systems. Our construction also leads to a proof of the tightness of a H2H_{2}H2​ entropic uncertainty relation for any set of three MUBs constructed from Pauli classes in d=4d=4d=4.
The two observables are incompatible if they cannot be measured simultaneously; however, they become maximally incompatible (complementary) if their eigenstates are mutually unbiased. Only then does the measurement of one observable give no information about the other observable. The spin projection operators onto three mutually orthogonal directions are complementary only for spin 1/2. For higher spin numbers the corresponding eigenstates are no longer unbiased. In this work we examine the properties of spin 1 mutually unbiased bases (MUB) and look for the physical meaning of the corresponding operators. We show that if the computational basis is chosen to be the eigenbasis of the spin projection operator along some direction z , then all the states, which are unbiased to this basis, have to be squeezed. Next, we study the generation and the measurement of MUB states by introducing the Fourier-like transform through spin squeezing. Finally, we try to ascribe some classical interpretation to the operators corresponding to MUB and study what information the observer gains while measuring them. Higher spin numbers are also considered.
The principle of complementarity lies at the heart of quantum mechanics. In finite dimensional quantum systems this principle is captured by pairs of observables which are given by mutually unbiased bases (MUBs). Two orthonormal bases B and C of Cd are mutually unbiased if |<b|c>|2 = 1/d holds for all vectors b ∈ B and c ∈ C. This implies that whenever we are given a vector from one of these bases and perform a measurement with respect to any other of the bases, then there is no information gained from this measurement. A basic question about MUBs is how many of them can be found in a given dimension d. While constructions of maximal sets of d+1 such bases are known for system of prime power dimension d, it is unknown whether this bound can be achieved for any non-prime power dimension. We review the known constructions of MUBs and demonstrate that maximal sets of MUBs come with a rich combinatorial structure by showing that they actually are the same objects as the complex projective 2-designs with angle set {0,1/d}. Furthermore, we address the problem of constructing positive operator-valued measures (POVMs) in finite dimension d consisting of d2 operators of rank one which have an inner product equal to uniform or very close to uniform. This is motivated by the related question of constructing symmetric informationally complete POVMs (SIC-POVMs) for which the inner products are perfectly uniform. We also give a simple proof of the fact that symmetric informationally complete POVMs are complex projective 2-designs with angle set {1/(d+1)}. Moreover, we show that MUBs and SIC-POVMs form uniform tight frames in Cd.
A pair of orthonormal bases is called mutually unbiased if all mutual overlaps between any element of one basis and an arbitrary element of the other basis coincide. In case the dimension, ddd, of the considered Hilbert space is a power of a prime number, complete sets of d+1d+1d+1 mutually unbiased bases (MUBs) exist. Here we present a method based on the graph-state formalism to construct such sets of MUBs. We show that for nnn ppp-level systems, with ppp being prime, one particular graph suffices to easily construct a set of pn+1{p}^{n}+1pn+1 MUBs. In fact, we show that a single nnn-dimensional vector, which is associated with this graph, can be used to generate a complete set of MUBs and demonstrate that this vector can be easily determined. Finally, we discuss some advantages of our formalism regarding the analysis of entanglement structures in MUBs, as well as experimental realizations.