A set of $k$ orthonormal bases of $\mathbb C^d$ is called mutually unbiased if $|\langle e,f\rangle |^2 = 1/d$ whenever $e$ and $f$ are basis vectors in distinct bases. A natural question is for which pairs $(d,k)$ there exist~ $k$ mutually unbiased bases in dimension $d$ . The (well-known) upper bound $k \leq d+1$ is attained when~ $d$ is a power of a prime. For all other dimensions it is an open problem whether the bound can be attained. Navascués, Pironio, and Acín showed how to reformulate the existence question in terms of the existence of a certain $C^*$ -algebra. This naturally leads to a noncommutative polynomial optimization problem and an associated hierarchy of semidefinite programs. The problem has a symmetry coming from the wreath product of $S_d$ and $S_k$ . We exploit this symmetry (analytically) to reduce the size of the semidefinite programs making them (numerically) tractable. A key step is a novel explicit decomposition of the $S_d \wr S_k$ -module $\mathbb C^{([d]\times [k])^t}$ into irreducible modules. We present numerical results for small $d,k$ and low levels of the hierarchy. In particular, we obtain sum-of-squares proofs for the (well-known) fact that there do not exist $d+2$ mutually unbiased bases in dimensions~ $d=2,3,4,5,6,7,8$ . Moreover, our numerical results indicate that a sum-of-squares refutation, in the above-mentioned framework, of the existence of more than $3$ MUBs in dimension $6$ requires polynomials of total degree at least~ $12$ .A set of $k$ orthonormal bases of $\mathbb C^d$ is called mutually unbiased if $|\langle e,f\rangle |^2 = 1/d$ whenever $e$ and $f$ are basis vectors in distinct bases. A natural question is for which pairs $(d,k)$ there exist~$k$ mutually unbiased bases in dimension $d$. The (well-known) upper bound $k \leq d+1$ is attained when~$d$ is a power of a prime. For all other dimensions it is an open problem whether the bound can be attained. Navascués, Pironio, and Acín showed how to reformulate the existence question in terms of the existence of a certain $C^*$-algebra. This naturally leads to a noncommutative polynomial optimization problem and an associated hierarchy of semidefinite programs. The problem has a symmetry coming from the wreath product of $S_d$ and $S_k$. We exploit this symmetry (analytically) to reduce the size of the semidefinite programs making them (numerically) tractable. A key step is a novel explicit decomposition of the $S_d \wr S_k$-module $\mathbb C^{([d]\times [k])^t}$ into irreducible modules. We present numerical results for small $d,k$ and low levels of the hierarchy. In particular, we obtain sum-of-squares proofs for the (well-known) fact that there do not exist $d+2$ mutually unbiased bases in dimensions~$d=2,3,4,5,6,7,8$. Moreover, our numerical results indicate that a sum-of-squares refutation, in the above-mentioned framework, of the existence of more than $3$ MUBs in dimension $6$ requires polynomials of total degree at least~$12$.

arXiv:2111.05698Nov 10, 20212 citations

On approximate real mutually unbiased bases in square dimension

Ajeet Kumar, Subhamoy Maitra, Chandra Sekhar Mukherjee

Cryptography and CommunicationsJan 11, 20214 citations

Mutually unbiased bases: polynomial optimization and symmetry

Sander Gribling, Sven C. Polak

A set of k orthonormal bases of Cd is called mutually unbiased if |⟨e,f⟩|2=1/d whenever e and f are basis vectors in distinct bases. A natural question is for which pairs (d,k) there exist k mutually unbiased bases in dimension d . The (well-known) upper bound k≤d+1 is attained when d is a power of a prime. For all other dimensions it is an open problem whether the bound can be attained. Navascués, Pironio, and Acín showed how to reformulate the existence question in terms of the existence of a certain C∗ -algebra. This naturally leads to a noncommutative polynomial optimization problem and an associated hierarchy of semidefinite programs. The problem has a symmetry coming from the wreath product of Sd and Sk .We exploit this symmetry (analytically) to reduce the size of the semidefinite programs making them (numerically) tractable. A key step is a novel explicit decomposition of the Sd≀Sk -module C([d]×[k])t into irreducible modules. We present numerical results for small d,k and low levels of the hierarchy. In particular, we obtain sum-of-squares proofs for the (well-known) fact that there do not exist d+2 mutually unbiased bases in dimensions d=2,3,4,5,6,7,8 . Moreover, our numerical results indicate that a sum-of-squares refutation, in the above-mentioned framework, of the existence of more than 3 MUBs in dimension 6 requires polynomials of total degree at least 12 .A set of k orthonormal bases of Cd is called mutually unbiased if |⟨e,f⟩|2=1/d whenever e and f are basis vectors in distinct bases. A natural question is for which pairs (d,k) there exist k mutually unbiased bases in dimension d . The (well-known) upper bound k≤d+1 is attained when d is a power of a prime. For all other dimensions it is an open problem whether the bound can be attained. Navascués, Pironio, and Acín showed how to reformulate the existence question in terms of the existence of a certain C∗ -algebra. This naturally leads to a noncommutative polynomial optimization problem and an associated hierarchy of semidefinite programs. The problem has a symmetry coming from the wreath product of Sd and Sk .We exploit this symmetry (analytically) to reduce the size of the semidefinite programs making them (numerically) tractable. A key step is a novel explicit decomposition of the Sd≀Sk -module C([d]×[k])t into irreducible modules. We present numerical results for small d,k and low levels of the hierarchy. In particular, we obtain sum-of-squares proofs for the (well-known) fact that there do not exist d+2 mutually unbiased bases in dimensions d=2,3,4,5,6,7,8 . Moreover, our numerical results indicate that a sum-of-squares refutation, in the above-mentioned framework, of the existence of more than 3 MUBs in dimension 6 requires polynomials of total degree at least 12 .

QuantumApr 30, 20246 citations

Mutually unbiased bases and bound entanglement

Beatrix C. Hiesmayr, W. Löffler

In this contribution we relate two different key concepts: mutually unbiased bases (MUBs) and entanglement. We provide a general toolbox for analyzing and comparing entanglement of quantum states for different dimensions and numbers of particles. In particular we focus on bound entanglement, i.e. highly mixed states which cannot be distilled by local operations and classical communications. For a certain class of states—for which the state-space forms a ‘ magic ’ simplex—we analyze the set of bound entangled states detected by the MUB criterion for different dimensions d and number of particles n . We find that the geometry is similar for different d and n , consequently the MUB criterion opens possibilities to investigate the typicality of positivity under partial transposition (PPT)-bound and multipartite bound entanglement more deeply and provides a simple experimentally feasible tool to detect bound entanglement.In this contribution we relate two different key concepts: mutually unbiased bases (MUBs) and entanglement. We provide a general toolbox for analyzing and comparing entanglement of quantum states for different dimensions and numbers of particles. In particular we focus on bound entanglement, i.e. highly mixed states which cannot be distilled by local operations and classical communications. For a certain class of states—for which the state-space forms a ‘ magic ’ simplex—we analyze the set of bound entangled states detected by the MUB criterion for different dimensions d and number of particles n . We find that the geometry is similar for different d and n , consequently the MUB criterion opens possibilities to investigate the typicality of positivity under partial transposition (PPT)-bound and multipartite bound entanglement more deeply and provides a simple experimentally feasible tool to detect bound entanglement.

arXiv:1309.2271Physica ScriptaApr 1, 20149 citations

Dual bounds for the positive definite functions approach to mutually unbiased bases

Afonso S. Bandeira, Nikolaus Doppelbauer, Dmitriy Kunisky

arXiv:2202.13259Sampling Theory Signal Processing and Data AnalysisSep 22, 20220 citations

A set of

k

orthonormal bases of

\mathbb C^d

is called mutually unbiased if

|\langle e,f\rangle |^2 = 1/d

whenever

e

and

f

are basis vectors in distinct bases. A natural question is for which pairs

(d,k)

there exist~

k

mutually unbiased bases in dimension

d

. The (well-known) upper bound

k \leq d+1

is attained when~

d

is a power of a prime. For all other dimensions it is an open problem whether the bound can be attained. Navascués, Pironio, and Acín showed how to reformulate the existence question in terms of the existence of a certain

C^*

-algebra. This naturally leads to a noncommutative polynomial optimization problem and an associated hierarchy of semidefinite programs. The problem has a symmetry coming from the wreath product of

S_d

and

S_k

. We exploit this symmetry (analytically) to reduce the size of the semidefinite programs making them (numerically) tractable. A key step is a novel explicit decomposition of the

S_d \wr S_k

-module

\mathbb C^{([d]\times [k])^t}

into irreducible modules. We present numerical results for small

d,k

and low levels of the hierarchy. In particular, we obtain sum-of-squares proofs for the (well-known) fact that there do not exist

d+2

mutually unbiased bases in dimensions~

d=2,3,4,5,6,7,8

. Moreover, our numerical results indicate that a sum-of-squares refutation, in the above-mentioned framework, of the existence of more than

3

MUBs in dimension

6

requires polynomials of total degree at least~

12