Mathematics

Unextendible Sets of Mutually Unbiased Basis Obtained from Complete Subgraphs

Andrés García, Pablo Carlos López Vázquez

Published Jun 15, 2021• 0 citations

Abstract

We propose a method, based on the search and identification of complete subgraphs of a regular graph, to obtain sets of Pauli operators whose eigenstates form unextendible complete sets of mutually unbiased bases of n-qubit systems. With this method we can obtain results for complete and inextensible sets of mubs for 2, 3, 4 and 5 qubits.

DOI: 10.3390/math9121388

Related papers

Unextendible Mutually Unbiased Bases from Pauli Classes

Prabha Mandayam, Somshubhro Bandyopadhyay, Markus Grassl +1 more

arXiv:1302.3709Feb 15, 20138 citations

Unextendible mutually unbiased bases from Pauli C\classes

Prabha Mandayam, Somshubhro Bandyopadhyay, Markus Grassl +1 more

Quantum Information and ComputationJul 1, 20149 citations

Generation of bases with definite factorization for ann-qubit system and mutually unbiased sets construction

Andrés García, José Luis Romero, A. B. Klimov

Journal of Physics A Mathematical and TheoreticalAug 4, 20109 citations

Unextendible mutually unbiased bases in prime-squared dimensions

Vishakh Hegde, Prabha Mandayam

arXiv:1508.05892Aug 24, 20152 citations

Unitarily inequivalent mutually unbiased bases for n qubits

Arun Sehrawat, A. B. Klimov

Physical Review ADec 2, 20145 citations

Graph-state formalism for mutually unbiased bases

Christoph Spengler, Barbara Kraus

arXiv:1309.6557Physical Review ANov 20, 201321 citations

Mathematics

Unextendible Sets of Mutually Unbiased Basis Obtained from Complete Subgraphs

Andrés García, Pablo Carlos López Vázquez

Published Jun 15, 2021• 0 citations

Publisher

Abstract

DOI: 10.3390/math9121388

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.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:1302.3709Feb 15, 20138 citations

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 $d=4$ .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 ComputationJul 1, 20149 citations

Generation of bases with definite factorization for ann-qubit system and mutually unbiased sets construction

Andrés García, José Luis Romero, A. B. Klimov

We propose a systematic procedure to construct all the possible bases with a definite factorization structure (eigenstates of n commuting monomials constructed as products of Pauli operators) for an n -qubit system, as well as the possibility of collecting them into mutually unbiased sets. We also discuss an algorithm for the determination of basis separability and propose a criteria for complementarity between such bases. The results are applied to generate non-isomorphic complete sets of mutually unbiased bases.We propose a systematic procedure to construct all the possible bases with a definite factorization structure (eigenstates of n commuting monomials constructed as products of Pauli operators) for an n -qubit system, as well as the possibility of collecting them into mutually unbiased sets. We also discuss an algorithm for the determination of basis separability and propose a criteria for complementarity between such bases. The results are applied to generate non-isomorphic complete sets of mutually unbiased bases.

Journal of Physics A Mathematical and TheoreticalAug 4, 20109 citations

Unextendible mutually unbiased bases in prime-squared dimensions

Vishakh Hegde, Prabha Mandayam

A set of mutually unbiased bases (MUBs) is said to be unextendible if there does not exist another basis that is unbiased with respect to the given set. Here, we prove the existence of smaller sets of MUBs in prime-squared dimensions ( $d=p^{2}$ ) that cannot be extended to a complete set using the generalized Pauli operators. We further observe an interesting connection between the existence of unextendible sets and the tightness of entropic uncertainty relations (EURs) in these dimensions. In particular, we show that our construction of unextendible sets of MUBs naturally leads to sets of $p+1$ MUBs that saturate both a Shannon ( $H_{1}$ ) and a collision ( $H_{2}$ ) entropic lower bound. Such an identification of smaller sets of MUBs satisfying tight EURs is crucial for cryptographic applications as well as constructing optimal entanglement witnesses for higher dimensional systems.A set of mutually unbiased bases (MUBs) is said to be unextendible if there does not exist another basis that is unbiased with respect to the given set. Here, we prove the existence of smaller sets of MUBs in prime-squared dimensions ($d=p^{2}$) that cannot be extended to a complete set using the generalized Pauli operators. We further observe an interesting connection between the existence of unextendible sets and the tightness of entropic uncertainty relations (EURs) in these dimensions. In particular, we show that our construction of unextendible sets of MUBs naturally leads to sets of $p+1$ MUBs that saturate both a Shannon ($H_{1}$) and a collision ($H_{2}$) entropic lower bound. Such an identification of smaller sets of MUBs satisfying tight EURs is crucial for cryptographic applications as well as constructing optimal entanglement witnesses for higher dimensional systems.

arXiv:1508.05892Aug 24, 20152 citations

Unitarily inequivalent mutually unbiased bases for n qubits

Arun Sehrawat, A. B. Klimov

The standard construction of complete sets of mutually unbiased bases (MUBs) in prime power dimensions is based on the quadratic Gauss sums. We introduce complete MUB sets for three, four, and five qubits that are unitarily inequivalent to all existing MUB sets. These sets are constructed by using certain exponential sums, where the degree of the polynomial appearing in the exponent can be higher than 2. Every basis of these MUBs (except the computational) consists of two disjoint blocks of vectors with different factorization structures and associated with a unique hypergraph (or graph) that represents an interaction between the qubits.The standard construction of complete sets of mutually unbiased bases (MUBs) in prime power dimensions is based on the quadratic Gauss sums. We introduce complete MUB sets for three, four, and five qubits that are unitarily inequivalent to all existing MUB sets. These sets are constructed by using certain exponential sums, where the degree of the polynomial appearing in the exponent can be higher than 2. Every basis of these MUBs (except the computational) consists of two disjoint blocks of vectors with different factorization structures and associated with a unique hypergraph (or graph) that represents an interaction between the qubits.

Physical Review ADec 2, 20145 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.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:1309.6557Physical Review ANov 20, 201321 citations

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

d=p^{2}

) that cannot be extended to a complete set using the generalized Pauli operators. We further observe an interesting connection between the existence of unextendible sets and the tightness of entropic uncertainty relations (EURs) in these dimensions. In particular, we show that our construction of unextendible sets of MUBs naturally leads to sets of

p+1

MUBs that saturate both a Shannon (

H_{1}

) and a collision (

H_{2}

) entropic lower bound. Such an identification of smaller sets of MUBs satisfying tight EURs is crucial for cryptographic applications as well as constructing optimal entanglement witnesses for higher dimensional systems.

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.