Quantum uncertainty relations of Tsallis relative $\alpha$ entropy coherence baed on MUBs

In this paper, we discuss quantum uncertainty relations of Tsallis relative $\alpha$ entropy coherence for a single qubit system based on three mutually unbiased bases. For $\alpha\in[\frac{1}{2},1)\cup(1,2]$, the upper and lower bounds of sums of coherence are obtained. However, the above results cannot be verified directly for any $\alpha\in(0,\frac{1}{2})$. Hence, we only consider the special case of $\alpha=\frac{1}{n+1}$, where $n$ is a positive integer, and we obtain the upper and lower bounds. By comparing the upper and lower bounds, we find that the upper bound is equal to the lower bound for the special $\alpha=\frac{1}{2}$, and the differences between the upper and the lower bounds will increase as $\alpha$ increases. Furthermore, we discuss the tendency of the sum of coherence, and find that it has the same tendency with respect to the different $\theta$ or $\varphi$, which is opposite to the uncertainty relations based on the R\'{e}nyi entropy and Tsallis entropy.

Multiplicative functions arising from the study of mutually unbiased bases

We introduce two families of multiplicative functions, which generalize the somewhat unusual function that was serendipitously discovered in 2010 during a study of mutually unbiased bases in the Hilbert space of quantum physics. In addition, we report yet another multiplicative function, which is also suggested by that example; it can be used to express the squarefree part of an integer in terms of an exponential sum.

Evidence for and against Zauner's MUB conjecture in C^6

The problem of finding provably maximal sets of mutually unbiased bases in $\CC^d$, for composite dimensions $d$ which are not prime powers, remains completely open. In the first interesting case,~$d=6$, Zauner predicted that there can exist no more than three MUBs. We explore possible algebraic solutions in~$d=6$ by looking at their~`shadows' in vector spaces over finite fields. The main result is that if a counter-example to Zauner's conjecture were to exist, then it would leave no such shadow upon reduction modulo several different primes, forcing its algebraic complexity level to be much higher than that of current well-known examples. In the case of prime powers~$q \equiv 5 \bmod 12$, however, we are able to show some curious evidence which --- at least formally --- points in the opposite direction. In $\CC^6$, not even a single vector has ever been found which is mutually unbiased to a set of three MUBs. Yet in these finite fields we find sets of three `generalised MUBs' together with an orthonormal set of four vectors of a putative fourth MUB, all of which lifts naturally to a number field.

Unextendible Sets of Mutually Unbiased Basis Obtained from Complete Subgraphs

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.

Circulant matrices with orthogonal rows and off-diagonal entries of absolute value 1

It is known that a real symmetric circulant matrix with diagonal entries d ≥ 0, off-diagonal entries ±1 and orthogonal rows exists only of order 2d + 2 (and trivially of order 1) [Turek and Goyeneche 2019]. In this paper we consider a complex Hermitian analogy of those matrices. That is, we study the existence and construction of Hermitian circulant matrices having orthogonal rows, diagonal entries d ≥ 0 and any complex entries of absolute value 1 off the diagonal. As a particular case, we consider matrices whose off-diagonal entries are 4th roots of unity; we prove that the order of any such matrix with d different from an odd integer is n = 2d + 2. We also discuss a similar problem for symmetric circulant matrices defined over finite rings ℤ m . As an application of our results, we show a close connection to mutually unbiased bases, an important open problem in quantum information theory.