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Quantum coherence plays a vital role in quantum mechanics and quantum information processing. It can be regarded as an important quantum resource. Quantification of coherence is the core of coherence theory. Geometric coherence as a measure of coherence can effectively quantify quantum coherence. In this work, we drive complementarity relations for geometric coherence with respect to mutually unbiased bases. Then, based on these complementarity relations, we establish the relationship between geometric coherence and quantum steering. [ABSTRACT FROM AUTHOR]

In this study, we give a synthetic construction of a complete system of mutually unbiased bases in C3{{\mathbb{C}}}^{3}.

Characterizing the 6 × 6 complex Hadamard matrices (CHMs) is an open problem in linear algebra and quantum information. We name the 6 × 6 CHMs except the H 2 -reducible matrices and the Tao matrix as the non- H 2 -reducible matrices. As far as we know, no non- H 2 -reducible matrices with analytic form have been found. In this paper, we find some non- H 2 -reducible matrices with analytic form. We also characterize some special 6 × 6 CHMs. Using our result one can further narrow the range of MUB trio (a set of four MUBs in C 6 consists of an MUB trio and the identity). Our results may lead to the complete classification of 6 × 6 CHMs and the solution of MUB problem in C 6 . [ABSTRACT FROM AUTHOR]

In a perfect quantum key distribution (QKD) protocol, quantum states should be prepared and measured with mutually unbiased bases (MUBs). However, in a practical QKD system, quantum states are generally prepared and measured with imperfect MUBs using imperfect devices, possibly reducing the secret key rate and transmission distance. To analyze the security of a QKD system with imperfect MUBs, we propose virtual MUBs to characterize the quantum channel against collective attack, and analyze the corresponding secret key rate under imperfect state preparation and measurement conditions. More generally, we apply the advantage distillation method for analyzing the security of QKD with imperfect MUBs, where the error tolerance and transmission distance can be sharply improved. Our analysis method can be applied to benchmark and standardize a practical QKD system, elucidating the security analysis of different QKD protocols with imperfect devices. [ABSTRACT FROM AUTHOR]

A quantum (t, n) threshold signature (QTS) based on Logistic Chaotic sequences and mutually unbiased bases is proposed, taking advantage of the pseudo-randomness of chaotic sequences and the cyclic characteristics of d-dimensional mutually unbiased bases. In this scheme, only t or more signatories can produce a valid signature on behalf of a signature group consisting of n members. The scheme uses a Logistic Chaotic mapping to generate a chaotic sequence, for which a particle swapping algorithm is used to obtain a position sequence, combined with a generalized Pauli operator to encrypt the message. Also, mutually unbiased bases in d-dimensional Hilbert space are used to verify the threshold value in the scheme. Security analysis proves that the proposed scheme satisfies signature non-repudiation and is resistant to collusive attacks by disloyal signing members. By comparing to existing QTS schemes, the proposed QTS scheme can reduce the resources consumed in the signing phase. The number of particles in the scheme does not increase as the number of signing members increases, except for local operations. That increases the scalability of the signature scheme. [ABSTRACT FROM AUTHOR]

We observe that the discrete Wigner functions (DWFs) of n-partite systems with odd local dimensions are tomographically universal, as reflected in the delta function form of the DWF for any stabilizer. However, in the n-qubit case, this property does not hold due to the non-factorization of the mapping kernel, the explicit form of which depends on a particular partition of the discrete phase space. Nonetheless, it turns out that the DWF for some specific stabilizers, not included in the set used for the construction of the Wigner map, takes on the form of a delta function. This implies that the possibility of classical simulations of Pauli measurements in a given stabilizer state for qubit systems is closely tied to the experimental setup. [ABSTRACT FROM AUTHOR]

We observe that the discrete Wigner functions (DWFs) of n-partite systems with odd local dimensions are tomographically universal, as reflected in the delta function form of the DWF for any stabilizer. However, in the n-qubit case, this property does not hold due to the non-factorization of the mapping kernel, the explicit form of which depends on a particular partition of the discrete phase space. Nonetheless, it turns out that the DWF for some specific stabilizers, not included in the set used for the construction of the Wigner map, takes on the form of a delta function. This implies that the possibility of classical simulations of Pauli measurements in a given stabilizer state for qubit systems is closely tied to the experimental setup.

We present an adaptation of the NPA hierarchy to the setting of synchronous correlation matrices. Our adaptation improves upon the original NPA hierarchy by using smaller certificates and fewer constraints, although it can only be applied to certify synchronous correlations. We recover characterizations for the sets of synchronous quantum commuting and synchronous quantum correlations. For applications, we show that the existence of symmetric informationally complete positive operator-valued measures and maximal sets of mutually unbiased bases can be verified or invalidated with only two certificates of our adapted NPA hierarchy. [ABSTRACT FROM AUTHOR]

We study mutually unbiased maximally entangled bases (MUMEBs) in bipartite system C d ⊗ C d with d ≥ 3 a power of an odd prime number. By using the theory of finite fields, we provide a new and intuitive method to construct MUMEBs in C d ⊗ C d . And we construct d (d - 1) MUMEBs in bipartite system C d ⊗ C d explicitly. [ABSTRACT FROM AUTHOR]

Properties of coherent states for the Heisenberg group over a field of -adic numbers are investigated. It turns out that coherent states form an orthonormal basis. The family of all such bases is parametrized by a set of self-dual lattices in the phase space. The Wehrl entropy is considered and its properties are investigated. In particular, it is proved that and vanishes only on coherent states. For a pair of different bases of coherent states, an entropic uncertainty relation is obtained. It is shown that the lower bound for the sum of the corresponding Wehrl entropies is given by the distance between the lattices. The proof uses the fact that the bases of coherent states corresponding to a pair of lattices are mutually unbiased. [ABSTRACT FROM AUTHOR]

The performance of quantum coherence under different bases is an important subject in quantum theory and quantum information science. This paper mainly focuses on the Tsallis relative 2-entropy of coherence of quantum states under mutually unbiased bases(MUBs) in two, three and four dimensional systems. For single-qubit mixed states, the sum under complete MUBs is less than 6 ; for Gisin states and Bell-diagonal states in four dimensional system, the sum under a new set of “autotensor of mutually unbiased basis” (AMUBs) is less than nine. For three classes of X states in three-dimensional system, each Tsallis relative 2-entropy of coherence under nontrivial unbiased bases is found equal. Also the surfaces of the sum of the Tsallis relative 2-entropy of coherence under MUBs and AMUBs are described, respectively. Among them, a surface of a special class of X states in AMUBs is an ellipsoid. [ABSTRACT FROM AUTHOR]

Quantum secret sharing is an important technique of quantum cryptography. A (t, n) threshold quantum secret sharing scheme with fairness is proposed. Firstly, the proposed scheme only requires the distributor to provide a share for each participant to achieve fairness. Secondly, the distributor interacts with participants only during the secret distribution phase. Another important point is that our scheme combines the privacy features of secure multi-party computing to ensure the reuse of participants’ secret shares. Finally, in our scheme, the correctness, security and fairness are discussed in detail. [ABSTRACT FROM AUTHOR]

This paper considers coherent states for the representation of Weyl commutation relations over a field of p-adic numbers. A geometric object, a lattice in vector space over a field of p-adic numbers, corresponds to the family of coherent states. It is proven that the bases of coherent states corresponding to different lattices are mutually unbiased, and that the operators defining the quantization of symplectic dynamics are Hadamard operators. [ABSTRACT FROM AUTHOR]

Beamforming training or cell discovery (CD) is a fundamental procedure in millimeter wave communication systems in which the user detects the presence of a base station, acquires the identity of that base station, and finds suitable beamforming directions to establish the wireless link for the subsequent data transmission phase. We consider uniform planar array-based beamforming under both separable and non-separable configurations as it allows confined antenna placements and 3-D steering of beams. Our focus is to design the beamforming training phase and develop low-complexity receiver algorithms for the cell discovery procedure. By exploiting the sparse nature of the mm-wave channels and using an equivalent compressive sensing measurement model, we obtain analytical bounds for the cell detection probability in terms of the mutual coherence parameter of the resulting sensing matrix. Using mutually unbiased bases (MUB) from quantum information theory, we design CD schemes with low mutual coherence. Our design includes parameters that control the training phase duration, and we analytically characterize the relationship between the training duration and the corresponding mutual coherence value. We present detailed simulation studies using experimentally driven mm-wave channel simulators and show that our MUB based scheme gives superior detection performance when compared to the conventional beam sweeping based schemes and random beamforming schemes.

Quantum secret sharing (QSS) is an important branch of quantum cryptography. Identity authentication is a significant means to achieve information protection, which can effectively confirm the identity information of both communication parties. Due to the importance of information security, more and more communications require identity authentication. We propose a d-level (t , n) threshold QSS scheme in which both sides of the communication use mutually unbiased bases for mutual identity authentication. In the secret recovery phase, the sharing of secrets that only the participant holds will not be disclosed or transmitted. Therefore, external eavesdroppers will not get any information about secrets at this phase. This protocol is more secure, effective, and practical. Security analysis shows that this scheme can effectively resist intercept–resend attacks, entangle–measure attacks, collusion attacks, and forgery attacks. [ABSTRACT FROM AUTHOR]

Sets of signals that meet Welch bounds with equality or near equality are of value in communications and sensing applications, and the construction of such signal sets has been an active research area. Although Welch derived a family of bounds indexed by positive integers k, only the first Welch bound (i.e., for k = 1) has been considered in these constructions. Earlier, a frame-theoretic perspective was introduced on the higher Welch bounds that is valuable in constructing signals that simultaneously meet multiple Welch bounds with equality or near equality. This perspective is used in this paper to examine the existence of signal sets that meet the kth Welch bound with equality by using second order Reed-Muller codes. Some examples of such signal sets are presented and connections to equiangular lines and t-designs are discussed. [ABSTRACT FROM AUTHOR]

Alltop functions have applications to code-division multiple access (CDMA) systems and mutually unbiased bases (MUBs). Alltop functions construct MUBs and CDMA signal sets, and this motivates the continued search for further Alltop functions. The discovery of further Alltop functions is hindered by the computational complexity of verification. This paper narrows the search space through the introduction of a characteristic cubic polynomial and provides a verification process for Alltop monomials with a reduced computational complexity. Computational results lead to the conjecture that there are no further Alltop monomials. [ABSTRACT FROM AUTHOR]

We study the construction of unextendible maximally entangled bases in bipartite systems ℂ d ⊗ ℂ d ′ ( d ′ = q d + r , q ≥ 2 , 0 < r < d). We provide a class of maximally entangled states in ℂ d ⊗ ℂ d ′ by constructing some unitary matrices in M d ′ (ℂ). In particular, we present two complete unextendible maximally entangled bases in ℂ 2 ⊗ ℂ 5 which are mutually unbiased. [ABSTRACT FROM AUTHOR]

Incompatible measurements are of fundamental importance to revealing the peculiar features of quantum theory, and are also useful resources in various quantum information tasks. In this work, we investigate the quantum incompatibility of mutually unbiased bases (MUBs) within the operational framework of quantum resource theory, and report an experimental validation via the task of state discrimination. In particular, we construct an experimentally friendly witness to detect incompatible MUBs, based on the probability of correctly discriminating quantum states. Furthermore, we prove that the noise robustness of MUBs can be retrieved from violating the above witness. Finally, we experimentally test the incompatibility of MUBs of dimensionality ranging from 2 to 4, and demonstrate that it is more robust to noise, as either the dimensionality of measurements or the number of MUBs increases. Our results may aid the exploration of the essential roles of incompatible measurements in both theoretical and practical applications in quantum information.

As one of the primitive operations of quantum cryptography, quantum secure multiparty multiplication plays an important role in practical applications. In this paper, we propose a (k, n)-threshold dynamic quantum secure multiparty multiplication protocol. (i) Based on Shamir's threshold scheme, the threshold (k, n) can be implemented; (ii) In the multiparty multiplication phase, the cheating behavior of participants can be detected by the one-to-one correspondence of hash values; (iii) Any m participants can dynamically join or exit during the execution of the protocol. Moreover, the method of this paper can also achieve dynamic quantum secure multiparty summation. Further, the security analysis shows that our protocol is resistant to intercept-resend attack, entangle-measure attack, Trojan horse attack, and participant attack. [ABSTRACT FROM AUTHOR]

Based on maximally entangled states, we explore the constructions of mutually unbiased bases in bipartite quantum systems. We present a new way to construct mutually unbiased bases by difference matrices in the theory of combinatorial designs. In particular, we establish q mutually unbiased bases with q âˆ' 1 maximally entangled bases and one product basis in C q ⊗ C q for arbitrary prime power q. In addition, we construct maximally entangled bases for dimension of composite numbers of non-prime power, such as five maximally entangled bases in C 12 ⊗ C 12 and C 21 ⊗ C 21 , which improve the known lower bounds for d = 3 m, with (3, m) = 1 in C d ⊗ C d . Furthermore, we construct p + 1 mutually unbiased bases with p maximally entangled bases and one product basis in C p ⊗ C p 2 for arbitrary prime number p. [ABSTRACT FROM AUTHOR]

We develop a new technique to construct mutually unbiased tripartite absolutely maximally entangled bases. We first explore the tripartite absolutely maximally entangled bases and mutually unbiased bases in C d ⊗ C d ⊗ C d based on mutually orthogonal Latin squares. Then we generalize the approach to the case of C d 1 ⊗ C d 2 ⊗ C d 1 d 2 by mutually weak orthogonal Latin squares. The concise direct constructions of mutually unbiased tripartite absolutely maximally entangled bases are remarkably presented with generality. Detailed examples in C 3 ⊗ C 3 ⊗ C 3 , C 2 ⊗ C 2 ⊗ C 4 and C 2 ⊗ C 5 ⊗ C 10 are provided to illustrate the advantages of our approach. [ABSTRACT FROM AUTHOR]

This paper considers coherent states for the representation of Weyl commutation relations over a field of p-adic numbers. A geometric object, a lattice in vector space over a field of p-adic numbers, corresponds to the family of coherent states. It is proven that the bases of coherent states corresponding to different lattices are mutually unbiased, and that the operators defining the quantization of symplectic dynamics are Hadamard operators.

As the best measurement bases, mutually unbiased bases play an important role in quantum information processing and they are intimately related to Bohr’s complementarity principle. Whereas, the construction of mutually unbiased bases for arbitrary high-dimensional quantum systems is difficult. In this paper, we derive a useful approach to construct mutually unbiased bases via maximally entangled bases based on an effective division of the three-qubit quantum systems. Furthermore, an explicit quantum teleportation protocol is proposed on account of the constructed mutually unbiased bases. We provides a new direction for the choice of measurement bases in quantum tasks. [ABSTRACT FROM AUTHOR]

A collection B = { B 1 , B 2 , ⋯ , B N } of orthonormal bases of C K is called mutually unbiased bases if | ⟨ v i | v j ⟩ | = 1 K for all v i ∈ B i , v j ∈ B j and 1 ≤ i < j ≤ N . In this paper, we present several new series of mutually unbiased bases constructed by utilizing p-ary weakly regular bent functions, permutation polynomials and PN functions over finite fields. Specifically, we are the first to use weakly regular bent functions to construct MUBs. In addition, we obtain a complete set of MUBs by employing linearized permutation polynomials over finite fields. [ABSTRACT FROM AUTHOR]

Quantum secure multi-party geometry computation is a specific primitive of classical secure multi-party computation. Compared with classical secure multi-party geometry computation based on mathematical difficulty problems which have been potentially threatened due to the development of quantum computer technology, the quantum protocol can provide unconditional security for the geometry computation. A novel quantum protocol based on the mutually biased bases of d-level quantum system is constructed to perform secure two-party Euclidean distance computation. With the aid of a semi-honest third party, the proposed protocol can calculate the Euclidean distance between two secret n-dimensional coordinates held by the participants who do not trust each other. Not only can the protocol resist the attacks from both outside eavesdroppers and participants, but also from semi-honest third party who does not collude with any participant. [ABSTRACT FROM AUTHOR]

We consider sparse representations of signals from redundant dictionaries which are unions of several orthonormal bases. The spark introduced by Donoho and Elad plays an important role in sparse representations. However, numerical computations of sparks are generally combinatorial. For unions of several orthonormal bases, two lower bounds on the spark via the mutual coherence were established in previous work. We constructively prove that both of them are tight. Our main results give positive answers to Gribonval and Nielsen’s open problem on sparse representations in unions of orthonormal bases. Constructive proofs rely on a family of mutually unbiased bases which first appears in quantum information theory. [ABSTRACT FROM AUTHOR]

Quantum secret sharing (QSS) is an important branch of quantum cryptography. Identity authentication is a significant means to achieve information protection, which can effectively confirm the identity information of both communication parties. Due to the importance of information security, more and more communications require identity authentication. We propose a d-level (t,n) threshold QSS scheme in which both sides of the communication use mutually unbiased bases for mutual identity authentication. In the secret recovery phase, the sharing of secrets that only the participant holds will not be disclosed or transmitted. Therefore, external eavesdroppers will not get any information about secrets at this phase. This protocol is more secure, effective, and practical. Security analysis shows that this scheme can effectively resist intercept–resend attacks, entangle–measure attacks, collusion attacks, and forgery attacks.

Kirkwood-Dirac (KD) distribution is a representation of quantum states. Recently, KD distribution has been found applications in many areas such as in quantum metrology, quantum chaos and foundations of quantum theory. KD distribution is a quasiprobability distribution, and negative or nonreal elements may signify quantum advantages in certain tasks. A quantum state is called KD classical if its KD distribution is a probability distribution. Since most quantum information processings use pure states as ideal resources, then a key problem is to determine whether a quantum pure state is KD classical. In this paper, we provide some characterizations for the general structure of KD classical pure states. As an application of our results, we prove a conjecture raised by De Bièvre (2021) [56] which finds out all KD classical pure states for discrete Fourier transformation. • We provide some characterizations for the general structure of Kirkwood-Dirac classical pure states. • We provide a necessary and sufficient condition for Kirkwood-Dirac classical pure states under mutually unbiased bases. • We provide a sufficient condition for Kirkwood-Dirac nonclassical pure states. [ABSTRACT FROM AUTHOR]

In this paper, we present a new method for the construction of maximally entangled states in C d ⊗ C d ′ when d ′ ≥ 2 d . A systematic way of constructing a set of maximally entangled bases (MEBs) in C d ⊗ C d ′ was established. Both cases when d ′ is divisible by d and not divisible by d are discussed. We give two examples of maximally entangled bases in C 2 ⊗ C 4 , which are mutually unbiased bases. Finally, we found a new example of an unextendible maximally entangled basis (UMEB) in C 2 ⊗ C 5 . [ABSTRACT FROM AUTHOR]

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 for n = 6. We apply our result to the existence of four mutually unbiased bases in dimension six, which is a long-standing open problem in quantum physics and information. We show that if four MUBs containing the identity matrix exist then the real entries in any one of the remaining three matrices do not exceed 22. [ABSTRACT FROM AUTHOR]

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 × 2 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. [ABSTRACT FROM AUTHOR]

In this article we consider partial abelianization of associative algebra with respect to some fixed subalgebra. This notion is a generalization of usual abelianization of an associative algebra and has some applications in Quantum Mechanics and Quantum Information Theory. Using various methods, we study partial abelianization of free product of finite-dimensional algebras. [ABSTRACT FROM AUTHOR]

We study mutually unbiased bases formed by special entangled basis with fixed Schmidt number 2 (MUSEB2s) in ℂ 3 ⊗ ℂ 4 p (p ∈ ℤ +) . Through analyzing the conditions MUSEB2s satisfy, a systematic way of the concrete construction of MUSEB2s in ℂ 3 ⊗ ℂ 4 p is established. A general approach to constructing MUSMEB2s in ℂ 3 ⊗ ℂ 4 p (p ∈ ℤ +) from MUSMEB2s in ℂ 3 ⊗ ℂ 4 is also presented. Detailed examples in ℂ 3 ⊗ ℂ 4 , ℂ 3 ⊗ ℂ 8 and ℂ 3 ⊗ 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 ∤ p in [Quantum Inf. Process. 17:58(2018)] is successfully deleted. [ABSTRACT FROM AUTHOR]

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 ⊗ C d ′ ( d ′ 2 < d < d ′) . 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 ⊗ C d ′ (d ′ = q d + r , 0 < r < d , q , r ∈ Z +) . [ABSTRACT FROM AUTHOR]

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 C m ⊗ C n (m ≤ n) formed by vectors with low Schmidt rank. In particular, the number of mutually unbiased product bases of C m ⊗ 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 M m ⊗ M n (m ≤ n) that are invariant under left partial transpose cannot exceed m - 2 . [ABSTRACT FROM AUTHOR]

In this article, we investigate the problem of state reconstruction of four-level quantum systems. A realistic scenario is considered with measurement results distorted by random unitary operators. Two frames that define injective measurements are applied and compared. By introducing arbitrary rotations, we can test the performance of the framework versus the amount of experimental noise. The results of numerical simulations are depicted in graphs and discussed. In particular, a class of entangled states is reconstructed. The concurrence is used as a figure of merit to quantify how well entanglement is preserved through noisy measurements. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we present a new method for the construction of maximally entangled states in Cd⊗Cd′ when d′≥2d. A systematic way of constructing a set of maximally entangled bases (MEBs) in Cd⊗Cd′ was established. Both cases when d′ is divisible by d and not divisible by d are discussed. We give two examples of maximally entangled bases in C2⊗C4, which are mutually unbiased bases. Finally, we found a new example of an unextendible maximally entangled basis (UMEB) in C2⊗C5.

The complementarity (upper bound) and uncertainty relations (lower bound) of the coherence via skew information under mutually unbiased bases (MUBs) are studied. The complementarity relation for the geometric measure of coherence is also obtained based on the relation between the coherence via skew information and the geometric measure of coherence. As applications, two tighter upper bounds are presented on the minimum error probabilities that discriminate a set of pure states with the least square measurement, which improve the results of Xiong et al. (Phys Rev A 98:032324, 2018). [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We study entanglement witnesses (EWs) and construct a linear map by taking two groups of mutually unbiased bases (MUBs) of two Hilbert spaces. We provide two new relations about matrices (may not be square matrices) with properties similar to unitary matrix. Using these two new relations, we obtain a positive map between two different linear spaces of complex matrices. Finally, we calculate a special case by applying our map to get an EW on ℂ 3 ⊗ ℂ 4 . We also find three entangled states detected by this EW. [ABSTRACT FROM AUTHOR]

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: C 4 ⊗ C 5 , C 6 ⊗ C 7 , C 10 ⊗ C 11 and C 12 ⊗ C 13 , which extend the known result of C 2 ⊗ C 3 . We also generalize these results to more bipartie systems of specific dimension. [ABSTRACT FROM AUTHOR]

Many deep connections have been observed between collections of mutually unbiased bases (MUBs) and combinatorial designs called k‐nets (and in particular, between collections of MUBs and finite affine planes). Here we introduce the notion of a k‐net over a C*‐algebra, providing a common framework for both objects. In the commutative case, we recover (classical) k‐nets, while the choice of Md(C) leads to collections of MUBs. In this framework, we derive a rigidity property which hence automatically applies to both objects. For k‐nets that can be completed to affine planes, this was already known by a completely different, combinatorial argument. For k‐nets that cannot be completed and for MUBs, this result is new, and it implies that the only vectors unbiased to all but k≤d bases of a complete collection of MUBs in Cd are the elements of the remaining k bases (up to phase factors). Further, we show that this bound is tight with counterexamples for k>d in every prime‐square dimension. Applying our rigidity result, we prove that if a large enough collection of MUBs constructed from a certain unitary error basis (like, the generalized Pauli operators) can be extended to a complete system, then every basis of the completion must come from the same error basis. In turn, we use this to show that certain large systems of MUBs cannot be completed. [ABSTRACT FROM AUTHOR]