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Several constructions of Mutually Unbiased Bases (MUBs) borrow tools from combinatorial objects. In this paper we focus on how one can construct Approximate Real MUBs (ARMUBs) with improved parameters using results from the domain of Resolvable Block Designs (RBDs). We first explain the generic idea of our strategy in relating the RBDs with MUBs/ARMUBs, which are sparse (the basis vectors have small number of non-zero co-ordinates). Then specific parameters are presented, for which we can obtain new classes and improve the existing results. To be specific, we present an infinite family of $\lceil \sqrt {d}\rceil $ many ARMUBs for dimension d = q(q + 1), where q ≡ 3 mod 4 and it is a prime power, such that for any two vectors v1,v2 belonging to different bases, $|\langle {v_{1}|v_{2}}\rangle | < \frac {2}{\sqrt {d}}$ . We also demonstrate certain cases, such as d = sq2, where q is a prime power and sq ≡ 0 mod 4. These findings subsume and improve our earlier results in [Cryptogr. Commun. 13, 321-329, January 2021]. This present construction idea provides several infinite families of such objects, not known in the literature, which can find efficient applications in quantum information processing for the sparsity, apart from suggesting that parallel classes of RBDs are intimately linked with MUBs/ARMUBs.

We study mutually unbiased bases formed by special entangled basis with fixed Schmidt number 2 (MUSEB2s) in $\mathbb {C}^{3}\otimes \mathbb {C}^{4p} (p\in \mathbb {Z}^{+})$ . Through analyzing the conditions MUSEB2s satisfy, a systematic way of the concrete construction of MUSEB2s in $\mathbb {C}^{3}\otimes \mathbb {C}^{4p}$ is established. A general approach to constructing MUSMEB2s in $\mathbb {C}^{3}\otimes \mathbb {C}^{4p}(p\in \mathbb {Z}^{+})$ from MUSMEB2s in $\mathbb {C}^{3}\otimes \mathbb {C}^{4}$ is also presented. Detailed examples in $\mathbb {C}^{3}\otimes \mathbb {C}^{4}$ , $\mathbb {C}^{3}\otimes \mathbb {C}^{8}$ and $\mathbb {C}^{3}\otimes \mathbb {C}^{12}$ are given. Especially, by choosing special entangled basis with fixed Schmidt number 2 (SEB2) from [J. Phys. A. Math. Theor. 48245301(2015)], the limitation of $3\nmid p$ in [Quantum Inf. Process. 17:58(2018)] is successfully deleted.

Mutually unbiased bases (MUBs), a kind of important best measurement base, is widely applied to quantum information processing. In this paper, we obtain the different unextendible maximally entangled bases (UMEBs) by constructing unitary matrices in bipartite systems HCode $C^{d}\otimes C^{d^{\prime }}(\frac {d^{\prime }}{2}< d< d^{\prime })$ . Then, we show that the sufficient and necessary conditions for UMEBs extend to MUBs in this bipartite systems. Finally, these results are generalized in bipartite systems $C^{d}\otimes C^{d^{\prime }}(d^{\prime }=qd+r,\ 0

An equiangular tight frame (ETF) yields a type of optimal packing of lines in a Euclidean space. ETFs seem to be rare, and all known infinite families of them arise from some type of combinatorial design. In this paper, we introduce a new method for constructing ETFs. We begin by showing that it is sometimes possible to construct multiple ETFs for the same space that are "mutually unbiased" in a way that is analogous to the quantum-information-theoretic concept of mutually unbiased bases. We then show that taking certain tensor products of these mutually unbiased ETFs with other ETFs sometimes yields infinite families of new complex ETFs.

Construction of Mutually Unbiased Bases (MUBs) is a very challenging combinatorial problem in quantum information theory with several long standing open questions in this domain. With certain relaxations, the object Approximate Mutually Unbiased Bases (AMUBs) is defined in this context. In this paper we provide a method to construct upto $(\sqrt {d} + 1)$ many AMUBs in dimension d = q2, where q is a positive integer. The result is particularly important when q ≡ 0 mod 4, as we obtain Approximate Real MUBs (ARMUBs) assuming the cases where a Hadamard matrix of order q exists. In this construction, we also characterize the inner product values between the elements of two different bases. In particular, when d is of the form (4x)2 where x is a prime, we obtain $(\frac {\sqrt {d}}{4}+1)$ ARMUBs such that for any two vectors v1,v2 belonging to different bases, $|{\left \langle \left . v_{1} \right | v_{2} \right \rangle }| \leq \frac {4}{\sqrt {d}}$ .

Many deep, mysterious connections have been observed between collections of mutually unbiased bases (MUBs) and combinatorial designs called $k$-nets (and in particular, between complete collections of MUBs and finite affine - or equivalently: finite projective - planes). Here we introduce the notion of a $k$-net over an algebra $\mathfrak{A}$ and thus provide a common framework for both objects. In the commutative case, we recover (classical) $k$-nets, while choosing $\mathfrak{A} := M_d(\mathbb C)$ leads to collections of MUBs. A common framework allows one to find shared properties and proofs that "inherently work" for both objects. As a first example, we derive a certain rigidity property which was previously shown to hold for $k$-nets that can be completed to affine planes using a completely different, combinatorial argument. For $k$-nets that cannot be completed and for MUBs, this result is new, and, in particular, it implies that the only vectors unbiased to all but $k \leq \sqrt{d}$ bases of a complete collection of MUBs in $\mathbb C^d$ are the elements of the remaining $k$ bases (up to phase factors). In general, this is false when $k$ is just the next integer after $\sqrt{d}$; we present an example of this in every prime-square dimension, demonstrating that the derived bound is tight. As an application of the rigidity result, we prove that if a large enough collection of MUBs constructed from a certain type of group representation (e.g. a construction relying on discrete Weyl operators or generalized Pauli matrices) can be extended to a complete system, then in fact every basis of the completion must come from the same representation. In turn, we use this to show that certain large systems of MUBs cannot be completed.

A collection $$\beta = \left\{ {{B_1},{B_2}, \cdots \,,{B_N}} \right\}$$ of orthonormal bases for ℂL is called approximately mutually unbiased bases if $$\left| {\left\langle {u,u} \right\rangle } \right| \leqslant \tfrac{1}{{\sqrt L }}\left( {1 + o\left( 1 \right)} \right)$$ for all u ∈ Bi, v ∈ Bj and 1 ≤ i ≤ j ≤ N. In this paper, the authors construct approximately mutually unbiased bases by using Gauss sums over Frobenius rings.

Using character of mutually orthogonal Latin squares, we first prove that two special squares are orthogonal to a complete set of mutually orthogonal Latin squares of order d. Then using the complete set of mutually orthogonal Latin squares of order d and two special squares, we construct d + 1 mutually unbiased bases in $\mathbb {C}^{d} \otimes \mathbb {C}^{d}$, which include d − 1 mutually unbiased maximally entangled bases and two mutually unbiased product bases. We also present the corresponding construction in $\mathbb {C}^{3} \otimes \mathbb {C}^{3}$, $\mathbb {C}^{4} \otimes \mathbb {C}^{4}$ and $\mathbb {C}^{5} \otimes \mathbb {C}^{5}$.

We study mutually unbiased bases (MUBs) in which all the bases are unextendible maximally entangled ones. We first present a necessary and sufficient condition of constructing a pair of MUBs in $C^2 \otimes C^4$. Based on this condition, an analytical and necessary condition for constructing MUBs is given. Moreover we illustrate our approach by some detailed examples in $C^2 \otimes C^4$. The results are generalized to $C^2 \otimes C^d$ $(d\geq 3)$ and a concrete example in $C^2 \otimes C^8$ is given., 14 pages

We investigate the number of real entries of an n×n complex Hadamard matrix. We analytically derive the numbers when n = 2, 3, 4, 6. In particular, the number can be any one of 0–22, 24, 25, 26, 30...

We expand the quantum variant of the popular game Sudoku by introducing the notion of cardinality of a quantum Sudoku (SudoQ), equal to the number of distinct vectors appearing in the pattern. Our considerations are focused on the genuinely quantum solutions---the solutions of size ${N}^{2}$ that have cardinality greater than ${N}^{2}$, and therefore cannot be reduced to classical counterparts by a unitary transformation. We find the complete parametrization of the genuinely quantum solutions of a $4\ifmmode\times\else\texttimes\fi{}4$ SudoQ game and establish that in this case the admissible cardinalities are 4, 6, 8, and 16. In particular, a solution with the maximal cardinality equal to 16 is presented. Furthermore, the parametrization enabled us to prove a recent conjecture of Nechita and Pillet [I. Nechita and J. Pillet, Quantum Inf. Comput. 21, 781 (2021)] for this special dimension. In general, we proved that for any $N$ it is possible to find an ${N}^{2}\ifmmode\times\else\texttimes\fi{}{N}^{2}$ SudoQ solution of cardinality ${N}^{4}$, which for a prime $N$ is related to a set of $N$ mutually unbiased bases of size ${N}^{2}$. Such a construction of ${N}^{4}$ different vectors of size $N$ yields a set of ${N}^{3}$ orthogonal measurements.

Finding four six-dimensional mutually unbiased bases (MUBs) containing the identity matrix is a long-standing open problem in quantum information. We show that if they exist, then the $H_2$-reducible matrix in the four MUBs has exactly nine $2\times2$ Hadamard submatrices. We apply our result to exclude from the four MUBs some known CHMs, such as symmetric $H_2$-reducible matrix, the Hermitian matrix, Dita family, Bjorck's circulant matrix, and Szollosi family. Our results represent the latest progress on the existence of six-dimensional MUBs., 15 pages, be published on quantum information processing. arXiv admin note: text overlap with arXiv:1904.10181

Can vectors with low Schmidt rank form mutually unbiased bases? Can vectors with high Schmidt rank form positive under partial transpose states? In this work, we address these questions by presenting several new results related to Schmidt rank constraints and their compatibility with other properties. We provide an upper bound on the number of mutually unbiased bases of $\mathbb{C}^m\otimes\mathbb{C}^n$ $(m\leq n)$ formed by vectors with low Schmidt rank. In particular, the number of mutually unbiased product bases of $\mathbb{C}^m\otimes\mathbb{C}^n$ cannot exceed $m+1$, which solves a conjecture proposed by McNulty et al. Then we show how to create a positive under partial transpose entangled state from any state supported on the antisymmetric space and how their Schmidt numbers are exactly related. Finally, we show that the Schmidt number of operator Schmidt rank 3 states of $\mathcal{M}_m\otimes \mathcal{M}_n\ (m\leq n)$ that are invariant under left partial transpose cannot exceed $m-2$.

It is known that a real symmetric circulant matrix with diagonal entries $d\geq0$, off-diagonal entries $\pm1$ 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 $d\geq0$ 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 $\mathbb{Z}_m$. As an application of our results, we show a close connection to mutually unbiased bases, an important open problem in quantum information theory., Comment: 16 pages, revised version: text partly rewritten, several new results added

We resolve a longstanding open problem by reformulating the Grassmannian fusion frames to the case of mixed dimensions and show that this satisfies the proper properties for the problem. In order to compare elements of mixed dimension, we use a classical embedding to send all fusion frame elements to points on a higher dimensional Euclidean sphere, where they are given “equal footing.” Over the embedded images—a compact subset in the higher dimensional embedded sphere—we define optimality in terms of the corresponding restricted coding problem. We then construct infinite families of solutions to the problem by using maximal sets of mutually unbiased bases and block designs. Finally, we show that using Hadamard 3-designs in this construction leads to infinitely many examples of maximal orthoplectic fusion frames of constant-rank. Moreover, any such fusion frames constructed by this method must come from Hadamard 3-designs.

We study the construction of mutually unbiased bases such that all the bases are unextendible maximally entangled ones. By using some results from the theory of finite fields, we construct mutually unbiased unextendible maximally entangled bases in some bipartite systems of higher dimension: $${\mathbb {C}}^{4} \otimes {\mathbb {C}}^{5}$$ , $${\mathbb {C}}^{6} \otimes {\mathbb {C}}^{7}$$ , $${\mathbb {C}}^{10} \otimes {\mathbb {C}}^{11}$$ and $${\mathbb {C}}^{12} \otimes {\mathbb {C}}^{13}$$ , which extend the known result of $${\mathbb {C}}^{2} \otimes {\mathbb {C}}^{3}$$ . We also generalize these results to more bipartie systems of specific dimension.

Grassmannian frames in composite dimensions D are constructed as a collection of orthogonal bases where each is the element-wise product of a mask sequence with a generalized Hadamard matrix. The set of mask sequences is obtained by exponentiation of a q-root of unity by different quadratic forms with m variables, where q and m are the product of the unique primes and total number of primes, respectively, in the prime decomposition of D. This method is a generalization of a well-known construction of mutually unbiased bases, as well as second-order Reed-Muller Grassmannian frames for power-of-two dimension D = 2m, and allows to derive highly symmetric nested families of frames with finite alphabet. Explicit sets of symmetric matrices defining quadratic forms leading to constructions in non-prime-power dimension with good distance properties are identified.

Mutually unbiased bases plays a central role in quantum mechanics and quantum information processing. As an important class of mutually unbiased bases, mutually unbiased maximally entangled bases (MUMEBs) in bipartite systems have attracted much attention in recent years. In the paper, we try to construct MUMEBs in $${\mathbb {C}}^{2^s}\otimes {\mathbb {C}}^{2^s}$$ by using Galois rings, which is different from the work in [17], where finite fields are used. As applications, we obtain several new types of MUMEBs in $${\mathbb {C}}^{2^s}\otimes {\mathbb {C}}^{2^s}$$ and prove that $$M(2^s,2^s)\ge 3(2^s-1)$$, which raises the lower bound of $$M(2^s,2^s)$$ given in [16].

Analogous to the notion of mutually unbiased bases for Hilbert spaces, we consider mutually unbiased unitary bases (MUUB) for the space of operators, $M(d, \mathbb{C})$, acting on such Hilbert spaces. The notion of MUUB reflects the equiprobable guesses of unitary in one bases of $M(d, \mathbb{C})$ when estimating a unitary operator in another. Though, for prime dimension $d$, the maximal number of MUUBs is known to be $d^{2}-1$, there is no known recipe for constructing them, assuming they exist. However, one can always construct a minimum of three MUUBs, and the maximal number is approached for very large values of $d$. MUUBs can also exists for some $d$-dimensional subspace of $M(d, \mathbb{C})$ with the maximal number being $d$., 11 pages, 1 figure

We study mutually unbiased unextendible maximally entangled bases (MUUMEBs) in bipartite stystem $\mathbb {C}^{d}\otimes \mathbb {C}^{d + 1}$ . By deriving the sufficient and necessary conditions that two MUUMEBs in $\mathbb {C}^{3}\otimes \mathbb {C}^{4}$ need to satisfy, we first establish two pairs of MUUMEBs in $\mathbb {C}^{3}\otimes \mathbb {C}^{4}$ . Then we present the sufficient and necessary conditions that two MUUMEBs in bipartite system $\mathbb {C}^{d}\otimes \mathbb {C}^{d + 1}$ need to satisfy, thus generalize the main results of Halqem et al. (Int. J. Theor. Phys. 54(1), 326, 2015).

We investigate mutually unbiased property between maximally entangled bases and product bases in bipartite systems $\mathbb {C}^{d} \otimes \mathbb {C}^{d}$ . We first visualize the description of $p_{1}^{a_{1}}-1$ -member mutually unbiased maximally entangled bases(MUMEBs) in $\mathbb {C}^{d} \otimes \mathbb {C}^{d}\ $ , while $d=p_{1}^{a_{1}}p_{2}^{a_{2}}...p_{s}^{a_{s}}$ , $3\leq p_{1}^{a_{1}}\leq p_{2}^{a_{2}}\leq ...\leq p_{s}^{a_{s}}$ , $p_{1}^{a_{1}},...,p_{s}^{a_{s}}$ are distinct primes, which was proposed by Liu et al. (Quantum Inf. Process. 16(6), 159, 2017). We then establish two more mutually unbiased product bases which are also mutually unbiased to the above $p_{1}^{a_{1}}-1$ MUMEBs, thus we present $p_{1}^{a_{1}}+ 1$ mutually unbiased bases(MUBs) in $\mathbb {C}^{d} \otimes \mathbb {C}^{d}\ $ . We also show the concrete construction of those MUBs in bipartite systems $\mathbb {C}^{3} \otimes \mathbb {C}^{3}\ $ , $\mathbb {C}^{4} \otimes \mathbb {C}^{4}\ $ , $\mathbb {C}^{5} \otimes \mathbb {C}^{5}\ $ and $\mathbb {C}^{12} \otimes \mathbb {C}^{12}$ .

We propose a method to define quasiprobability distributions for general spin-j systems of dimension n = 2 j + 1 , where n is a prime or power of prime. The method is based on a complete set of orthonormal commuting operators related to Mutually Unbiased Bases which enable (i) a parameterisation of the density matrix and (ii) construction of measurement operators that can be physically realised. As a result we geometrically characterise the set of states for which the quasiprobability distribution is non-negative, and can be viewed as a joint distribution of classical random variables assuming values in a finite set of outcomes. The set is an ( n 2 − 1 ) -dimensional convex polytope with n + 1 vertices as the only pure states, n n + 1 number of higher dimensional faces, and n 3 ( n + 1 ) / 2 edges.

The Welch bounds for a finite set of unit vectors are a family of inequalities indexed by $t=1,2,\ldots $ , which describe how “evenly spread” the vectors are. They have important applications in signal analysis, where sequences giving equality in the first Welch bound are known as Welch bound equality sequences or as unit norm tight frames. Here, we consider sequences of vectors giving equality in the higher order Welch bounds. These are seen to correspond to tight frames for the complex symmetric $t$ –tensors (which we prove always exist). We show that for $t>1$ , the Welch bounds can be sharpened for real vectors, and again, vectors giving equality always exist. We give a unified treatment of various conditions for equality in both the real and complex cases. In particular, we give an explicit description of the corresponding cubature rules ( $t$ –designs). Our results set up a framework for the construction and classification several configurations of vectors of recent interest. These include mutually unbiased bases, complex equiangular lines, spherical half–designs, projective $t$ –designs, and minimisers of the higher order frame potential. One interesting consequence is a construction of sets of complex equiangular lines, which were previously unknown.

Equiangular tight frames are examples of Grassmannian line packings for a Hilbert space. More specifically, according to a bound by Welch, they are minimizers for the maximal magnitude occurring among the inner products of all pairs of vectors in a unit-norm frame. This paper is dedicated to packings in the regime in which the number of frame vectors precludes the existence of equiangular frames. The orthoplex bound then serves as an alternative to infer a geometric structure of optimal designs. We construct frames of unit-norm vectors in K-dimensional complex Hilbert spaces that achieve the orthoplex bound. When K − 1 is a prime power, we obtain a tight frame with K 2 + 1 vectors and when K is a prime power, with K 2 + K − 1 vectors. In addition, we show that these frames form weighted complex projective 2-designs that are useful additions to maximal equiangular tight frames and maximal sets of mutually unbiased bases in quantum state tomography. Our construction is based on Singer's family of difference sets and the related concept of relative difference sets.

Constructing four six-dimensional mutually unbiased bases (MUBs) is an open problem in quantum physics and measurement. We investigate the existence of four MUBs including the identity, and a complex Hadamard matrix (CHM) of Schmidt rank three. The CHM is equivalent to a controlled unitary operation on the qubit-qutrit system via local unitary transformation I 2 ⊗ V and I 2 ⊗ W . We show that V and W have no zero entry, and apply it to exclude constructed examples as members of MUBs. We further show that the maximum of entangling power of controlled unitary operation is log 2 3 ebits. We derive the condition under which the maximum is achieved, and construct concrete examples. Our results describe the phenomenon that if a CHM of Schmidt rank three belongs to an MUB then its entangling power may not reach the maximum.

Finding four six-dimensional mutually unbiased bases (MUBs) is a long-standing open problem in quantum information. By assuming that they exist and contain the identity matrix, we investigate whether the remaining three MUBs have an $$H_2$$ -reducible matrix, namely a $$6\times 6$$ complex Hadamard matrix (CHM) containing a $$2\times 2$$ subunitary matrix. We show that every $$6\times 6$$ CHM containing at least 23 real entries is an $$H_2$$ -reducible matrix. It relies on the fact that the CHM is complex equivalent to one of the two constant $$H_2$$ -reducible matrices. They, respectively, have exactly 24 and 30 real entries, and both have more than eighteen $$2\times 2$$ subunitary matrices. It turns out that such $$H_2$$ - reducible matrices do not belong to the remaining three MUBs. This is the corollary of a stronger claim; namely, any $$H_2$$ -reducible matrix belonging to the remaining three MUBs has exactly nine or eighteen $$2\times 2$$ subunitary matrices.

The non-linear binary Kerdock codes are known to be Gray images of certain extended cyclic codes of length $N = 2^m$ over $\mathbb{Z}_4$. We show that exponentiating these $\mathbb{Z}_4$-valued codewords by $\imath \triangleq \sqrt{-1}$ produces stabilizer states, that are quantum states obtained using only Clifford unitaries. These states are also the common eigenvectors of commuting Hermitian matrices forming maximal commutative subgroups (MCS) of the Pauli group. We use this quantum description to simplify the derivation of the classical weight distribution of Kerdock codes. Next, we organize the stabilizer states to form $N+1$ mutually unbiased bases and prove that automorphisms of the Kerdock code permute their corresponding MCS, thereby forming a subgroup of the Clifford group. When represented as symplectic matrices, this subgroup is isomorphic to the projective special linear group PSL($2,N$). We show that this automorphism group acts transitively on the Pauli matrices, which implies that the ensemble is Pauli mixing and hence forms a unitary $2$-design. The Kerdock design described here was originally discovered by Cleve et al. (arXiv:1501.04592), but the connection to classical codes is new which simplifies its description and translation to circuits significantly. Sampling from the design is straightforward, the translation to circuits uses only Clifford gates, and the process does not require ancillary qubits. Finally, we also develop algorithms for optimizing the synthesis of unitary $2$-designs on encoded qubits, i.e., to construct logical unitary $2$-designs. Software implementations are available at https://github.com/nrenga/symplectic-arxiv18a, which we use to provide empirical gate complexities for up to $16$ qubits., Comment: 16 pages double-column, 4 figures, and some circuits. Accepted to 2019 Intl. Symp. Inf. Theory (ISIT), and PDF of the 5-page ISIT version is included in the arXiv package

Mutually unbiased maximally entangled bases (MUMEBs) in bipartite systems have a close relation with unitary 2-design and have attracted much attention in recent years. In the paper, we construct MUMEBs in $$\mathbb {C}^q\otimes \mathbb {C}^q$$ with q a power of an odd prime number. For this purpose, we introduce the notation of trace-2 excluded subset of the special linear group $$SL(2,\mathbb {F}_q)$$ over the finite field $$\mathbb {F}_q$$ and establish a relation between a trace-2 excluded subset and a set of MUMEBs in $$\mathbb {C}^q\otimes \mathbb {C}^q$$ . Under this relation, we prove that $$M(q,q)\ge \dfrac{q^2-1}{2}$$ by constructing trace-2 excluded subsets in $$SL(2,\mathbb {F}_q)$$ , which highly raises the lower bound of M(q, q) given in Liu et al. (Quantum Inf Process 16(6):159, 2017) and Xu (Quantum Inf. Process 16(3):65, 2017).

We investigate the $l_{1}$ norm of coherence of quantum states in mutually unbiased bases. We find that the sum of squared $l_{1}$ norm of coherence of the mixed state single qubit is less than two. We derive the $l_{1}$ norm of coherence of three classes of $X$ states in nontrivial mutually unbiased bases for $4$-dimensional Hilbert space is equal. We proposed "autotensor of mutually unbiased basis(AMUB)" by the tensor of mutually unbiased bases, and depict the level surface of constant the sum of the $l_{1}$ norm of coherence of Bell-diagonal states in AMUB. We find the $l_{1}$ norm of coherence of Werner states and isotropic states in AMUB is equal respectively., Comment: 9 pages,4 figure

We first construct a new maximally entangled basis in bipartite systems $\mathbb {C}^{d} \otimes \mathbb {C}^{kd}\ (k\in Z^{+})$ which is diffrent from the one in Tao et al. (Quantum Inf. Process. 14, 2291 (2015)), then we generalize such maximally entangled basis into arbitrary bipartite systems $\mathbb {C}^{d} \otimes \mathbb {C}^{d^{\prime }}$ . We also study the mutual unbiased property of the two types of maximally entangled bases in bipartite systems $\mathbb {C}^{d} \otimes \mathbb {C}^{kd}$ . In particular, explicit examples in $\mathbb {C}^{2} \otimes \mathbb {C}^{4}$ , $\mathbb {C}^{2} \otimes \mathbb {C}^{8}$ and $\mathbb {C}^{3} \otimes \mathbb {C}^{3}$ are presented.

The construction of maximally entangled bases for the bipartite system $\mathbb {C}^{d}\otimes \mathbb {C}^{d}$ is discussed firstly, and some mutually unbiased bases with maximally entangled bases are given, where 2≤d≤5. Moreover, we study a systematic way of constructing mutually unbiased maximally entangled bases for the bipartite system $\mathbb {C}^{d}\otimes \mathbb {C}^{d^{k}}$ .

Akin to the idea of complete sets of Mutually Unbiased Bases for prime dimensional Hilbert spaces, $\mathcal{H}_d$, we study its analogue for a $d$ dimensional subspace of $M (d,\mathbb{C})$, i.e. Mutually Unbiased Unitary Bases (MUUBs) comprising of unitary operators. We note an obvious isomorphism between the vector spaces and beyond that, we define a relevant monoid structure for $\mathcal{H}_d$ isomorphic to one for the subspace of $M (d,\mathbb{C})$. This provides us not only with the maximal number of such MUUBs, but also a recipe for its construction., Comment: Lemma 5 corrected, new appendixes B, D included, typos corrected

The mutually unbiasedness between a maximally entangled basis (MEB) and an unextendible maximally entangled system (UMES) in the bipartite system \(\mathbb {C}^{2}\otimes \mathbb {C}^{2^{k}} (k>1)\) are introduced and discussed first in this paper. Then two mutually unbiased pairs of a maximally entangled basis and an unextendible maximally entangled system are constructed; lastly, explicit constructions are obtained for mutually unbiased MEB and UMES in \(\mathbb {C}^{2}\otimes \mathbb {C}^{4}\) and \(\mathbb {C}^{2}\otimes \mathbb {C}^{8}\), respectively.

We study mutually unbiased bases in $${\mathbb {C}}^d\otimes {\mathbb {C}}^{2^{l}d'}$$Cd?C2ld?. A systematic way of constructing mutually unbiased maximally entangled bases (MUMEBs) in $${\mathbb {C}}^d\otimes {\mathbb {C}}^{2^{l}d'} (l\in {\mathbb {Z}}^{+})$$Cd?C2ld?(l?Z+) from MUMEBs in $${\mathbb {C}}^d \otimes {\mathbb {C}}^{d'}(d'=kd, k\in {\mathbb {Z}}^+)$$Cd?Cd?(d?=kd,k?Z+) and a general approach to construct mutually unbiased unextendible maximally entangled bases (MUUMEBs) in $${\mathbb {C}}^d\otimes {\mathbb {C}}^{2^ld'} (l \in {\mathbb {Z}}^{+})$$Cd?C2ld?(l?Z+) from MUUMEBs in $${\mathbb {C}}^d \otimes {\mathbb {C}}^{d'}(d'=kd+r, 0

We study maximally entangled bases in bipartite systems $$\mathbb {C}^d \otimes \mathbb {C}^{kd}\ (k\in Z^{+})$$Cd?Ckd(k?Z+), which are mutually unbiased. By systematically constructing maximally entangled bases, we present an approach in constructing mutually unbiased maximally entangled bases. In particular, five maximally entangled bases in $$\mathbb {C}^2 \otimes \mathbb {C}^{4}$$C2?C4 and three maximally entangled bases in $$\mathbb {C}^2 \otimes \mathbb {C}^{6}$$C2?C6 that are mutually unbiased are presented.

Excluding the existence of four MUBs in $\bbC^6$ is an open problem in quantum information. We investigate the number of product vectors in the set of four mutually unbiased bases (MUBs) in dimension six, by assuming that the set exists and contains a product-vector basis. We show that in most cases the number of product vectors in each of the remaining three MUBs is at most two. We further construct the exceptional case in which the three MUBs respectively contain at most three, two and two product vectors. We also investigate the number of vectors mutually unbiased to an orthonormal basis., 10 pages, the total number of vectors in the introduction and the conclusion was revised. arXiv admin note: text overlap with arXiv:1610.04875

We study mutually unbiased maximally entangled bases (MUMEB's) in bipartite system $$\mathbb {C}^d\otimes \mathbb {C}^d (d \ge 3)$$Cd?Cd(d?3). We generalize the method to construct MUMEB's given in Tao et al. (Quantum Inf Process 14:2291---2300, 2015), by using any commutative ring R with d elements and generic character of $$(R,+)$$(R,+) instead of $$\mathbb {Z}_d=\mathbb {Z}/d\mathbb {Z}$$Zd=Z/dZ. Particularly, if $$d=p_1^{a_1}p_2^{a_2}\ldots p_s^{a_s}$$d=p1a1p2a2?psas where $$p_1, \ldots , p_s$$p1,?,ps are distinct primes and $$3\le p_1^{a_1}\le \cdots \le p_s^{a_s}$$3≤p1a1≤?≤psas, we present $$p_1^{a_1}-1$$p1a1-1 MUMEB's in $$\mathbb {C}^d\otimes \mathbb {C}^d$$Cd?Cd by taking $$R=\mathbb {F}_{p_1^{a_1}}\oplus \cdots \oplus \mathbb {F}_{p_s^{a_s}}$$R=Fp1a1???Fpsas, direct sum of finite fields (Theorem 3.3).

We construct mutually unbiased maximally entangled bases (MUMEBs) in bipartite system $${\mathbb {C}}^d\otimes {\mathbb {C}}^d (d\ge 3)$$Cd?Cd(d?3) with d a power of a prime number. Precisely, by means of permutation matrices and Hadamard matrices, we construct $$2(d-1)$$2(d-1) MUMEBs in $${\mathbb {C}}^d\otimes {\mathbb {C}}^d$$Cd?Cd. It follows that $$M(d,d)\ge 2(d-1)$$M(d,d)?2(d-1), which is twice the number given in Liu et al. (2016), where M(d, d) denotes the maximal size of all sets of MUMEBs in $${\mathbb {C}}^d\otimes {\mathbb {C}}^d$$Cd?Cd. In addition, let q be another power of a prime number, we construct MUMEBs in $${\mathbb {C}}^d\otimes {\mathbb {C}}^{qd}$$Cd?Cqd from those in $${\mathbb {C}}^d\otimes {\mathbb {C}}^d$$Cd?Cd by the use of the tensor product of unitary matrices.

We consider two (natural) families of observables ${O}_{k}$ for systems with dimension $d=3,4,5$: the spin observables ${S}_{x},\phantom{\rule{0.16em}{0ex}}{S}_{y}$, and ${S}_{z}$, and the observables that have mutually unbiased bases as eigenstates. We derive tight entropic uncertainty relations for these families, in the form ${\ensuremath{\sum}}_{k}H({O}_{k})\ensuremath{\ge}{\ensuremath{\alpha}}_{d}$, where $H({O}_{k})$ is the Shannon entropy of the measurement outcomes of ${O}_{k}$ and ${\ensuremath{\alpha}}_{d}$ is a constant. We show that most of our bounds are stronger than previously known ones. We also give the form of the states that attain these inequalities.

We generalize the notion of unextendible maximally entangled basis from bipartite systems to multipartite quantum systems. It is proved that there do not exist unextendible maximally entangled bases in three-qubit systems. Moreover, two types of unextendible maximally entangled bases are constructed in tripartite quantum systems and proved to be not mutually unbiased.

We explore the underlying algebraic structure of Mutually Unbiased Bases (MUBs), and their application to code design. Columns in MUBs have inner products with absolute values less or equal to 1/√N. MUBs provide a systematic way of generating optimal codebooks for various coding and precoding applications. A maximal set of MUBs (MaxMUBs) in N = 2m dimensions, with m ∊ Z, can produce codebooks of QPSK lines with good distance properties and alphabets which limit processing complexity. We expand the construction by identifying that in N = 2m dimensions there exists N(m-1)/2 families of MUB, each with N matrices. Inner products of columns of these matrices are less or equal to 1/√2. As an example, we construct Grassmannian line codes from the columns of these matrices. Then decoding or encoding these codebooks can be performed without multiplications, and with a number of additions that scales linearly with the number of codewords, irrespectively of the dimension.

Two (\(i=1,2\)) \(d\)-dimensional (d=prime, \(\ne 2\)) particles are accounted for via universal maximally entangled states (MES) bases. The bases are labeled by the particles’ collective (“center of mass” and “relative”) coordinates in a phase space like formalism. Schmidt’s decomposition represents MES via d-terms of single particle product states that form a ‘line’ in a \(d^2\) array of points each designating a single particles product states. Correspondingly, a ‘line’ in a \(d^2\) array of phase space like points designating a \(d^2\) orthonormal MES, represents a two particles product state.

We systematically study the unextendible maximally entangled basis in arbitrary bipartite spaces $\mathbb {C}^{d} \otimes \mathbb {C}^{d^{\prime }}\ (d

We systematically study the construction of mutually unbiased bases in $\mathbb{C}^{2}\bigotimes\mathbb{C}^{3}$, such that all the bases are unextendible maximally entangled ones. Necessary conditions of constructing a pair of mutually unbiased unextendible maximally entangled bases in $\mathbb{C}^{2}\bigotimes\mathbb{C}^{3}$ are derived. Explicit examples are presented.

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 present a construction for complementary pairs of arrays that exploits a set of mutually-unbiased bases, and enumerate these arrays as well as the corresponding set of complementary sequences obtained from the arrays by projection. We also sketch an algorithm to uniquely generate these sequences. The pairwise squared inner-product of members of the sequence set is shown to be $\frac{1}{2}$. Moreover, a subset of the set can be viewed as a codebook that asymptotically achieves $\sqrt{\frac{3}{2}}$ times the Welch bound., 25 pages, 1 figure