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

Bounds on the number of mutually unbiased entangled bases

Fei Shi, Yi Shen, Lin Chen +1 more

Quantum Information ProcessingOct 1, 20203 citations

Mutually unbiased bases with free parameters

Dardo Goyeneche, Santiago Gómez

We present a systematic method to introduce free parameters in sets of mutually unbiased bases. In particular, we demonstrate that any set of $m$ real mutually unbiased bases existing in dimension $N>2$ admits the introduction of $(m\ensuremath{-}1)N/2$ free parameters that cannot be absorbed by a global unitary operation. As consequence, there are $m=k+1$ mutually unbiased bases in every dimension $N={k}^{2}$ with ${k}^{3}/2$ free parameters, where $k$ is even. We explicitly construct the maximal set of triplets of mutually unbiased bases for two-qubit systems and triplets, quadruplets, and quintuplets of mutually unbiased bases with free parameters for three-qubit systems. Furthermore, we study the richness of the entanglement structure of such bases and provide the quantum circuits required to implement all these bases with free parameters in the laboratory. We also show that the free parameters introduced can be controlled by a single party of the system. Finally, we find the upper bound for the maximal number of real and complex mutually unbiased bases existing in every dimension. This proof is simple, short, and considers basic matrix algebra.We present a systematic method to introduce free parameters in sets of mutually unbiased bases. In particular, we demonstrate that any set of $m$ real mutually unbiased bases existing in dimension $N>2$ admits the introduction of $(m\ensuremath{-}1)N/2$ free parameters that cannot be absorbed by a global unitary operation. As consequence, there are $m=k+1$ mutually unbiased bases in every dimension $N={k}^{2}$ with ${k}^{3}/2$ free parameters, where $k$ is even. We explicitly construct the maximal set of triplets of mutually unbiased bases for two-qubit systems and triplets, quadruplets, and quintuplets of mutually unbiased bases with free parameters for three-qubit systems. Furthermore, we study the richness of the entanglement structure of such bases and provide the quantum circuits required to implement all these bases with free parameters in the laboratory. We also show that the free parameters introduced can be controlled by a single party of the system. Finally, we find the upper bound for the maximal number of real and complex mutually unbiased bases existing in every dimension. This proof is simple, short, and considers basic matrix algebra.

arXiv:1506.08283Physical Review ADec 14, 20158 citations

Quantum Coherence of Qubit States with respect to Mutually Unbiased Bases

Ming-Yang Shen, Yi-Hao Sheng, Yuan-Hong Tao +2 more

International Journal of Theoretical PhysicsNov 4, 20206 citations

Structure of the sets of mutually unbiased bases for N qubits

José Luis Romero, Gunnar G. Bjork, A. B. Klimov +1 more

For a system of $N$ qubits, living in a Hilbert space of dimension $d={2}^{N}$ , it is known that there exists $d+1$ mutually unbiased bases. Different construction algorithms exist, and it is remarkable that different methods lead to sets of bases with different properties as far as separability is concerned. Here we derive four sets of nine bases for three qubits, and show how they are unitarily related. We also briefly discuss the four-qubit case, give the entanglement structure of 16 sets of bases, and show some of them and their interrelations, as examples. The extension of the method to the general case of $N$ qubits is outlined.For a system of $N$ qubits, living in a Hilbert space of dimension $d={2}^{N}$, it is known that there exists $d+1$ mutually unbiased bases. Different construction algorithms exist, and it is remarkable that different methods lead to sets of bases with different properties as far as separability is concerned. Here we derive four sets of nine bases for three qubits, and show how they are unitarily related. We also briefly discuss the four-qubit case, give the entanglement structure of 16 sets of bases, and show some of them and their interrelations, as examples. The extension of the method to the general case of $N$ qubits is outlined.

arXiv:quant-ph/0508129Physical Review ADec 7, 200595 citations

We present a systematic method to introduce free parameters in sets of mutually unbiased bases. In particular, we demonstrate that any set of

m

real mutually unbiased bases existing in dimension

N>2

admits the introduction of $(m\ensuremath{-}1)N/2$ free parameters that cannot be absorbed by a global unitary operation. As consequence, there are

m=k+1

mutually unbiased bases in every dimension

N={k}^{2}

with

{k}^{3}/2

free parameters, where

k

is even. We explicitly construct the maximal set of triplets of mutually unbiased bases for two-qubit systems and triplets, quadruplets, and quintuplets of mutually unbiased bases with free parameters for three-qubit systems. Furthermore, we study the richness of the entanglement structure of such bases and provide the quantum circuits required to implement all these bases with free parameters in the laboratory. We also show that the free parameters introduced can be controlled by a single party of the system. Finally, we find the upper bound for the maximal number of real and complex mutually unbiased bases existing in every dimension. This proof is simple, short, and considers basic matrix algebra.

For a system of

N

qubits, living in a Hilbert space of dimension

d={2}^{N}

, it is known that there exists

d+1

mutually unbiased bases. Different construction algorithms exist, and it is remarkable that different methods lead to sets of bases with different properties as far as separability is concerned. Here we derive four sets of nine bases for three qubits, and show how they are unitarily related. We also briefly discuss the four-qubit case, give the entanglement structure of 16 sets of bases, and show some of them and their interrelations, as examples. The extension of the method to the general case of

N

qubits is outlined.