One of the essential features of quantum mechanics is that most pairs of observables cannot be measured simultaneously. This phenomenon is most strongly manifested when observables are related to mutually unbiased bases. In this paper, we shed some light on the connection between mutually unbiased bases and another essential feature of quantum mechanics, quantum entanglement. It is shown that a complete set of mutually unbiased bases of a bipartite system contains a fixed amount of entanglement, independently of the choice of the set. This has implications for entanglement distribution among the states of a complete set. In prime-squared dimensions we present an explicit experiment-friendly construction of a complete set with a particularly simple entanglement distribution. Finally, we describe basic properties of mutually unbiased bases composed only of product states. The constructions are illustrated with explicit examples in low dimensions. We believe that properties of entanglement in mutually unbiased bases might be one of the ingredients to be taken into account to settle the question of the existence of complete sets. We also expect that they will be relevant to applications of bases in the experimental realization of quantum protocols in higher-dimensional Hilbert spaces.
We investigate the interplay between mutual unbiasedness and product bases for multiple qudits of possibly different dimensions. A product state of such a system is shown to be mutually unbiased to a product basis only if each of its factors is mutually unbiased to all the states which occur in the corresponding factors of the product basis. This result implies both a tight limit on the number of mutually unbiased product bases which the system can support and a complete classification of mutually unbiased product bases for multiple qubits or qutrits. In addition, only maximally entangled stat…
We show that maximal families of mutually unbiased bases are characterized in all dimensions by partitioned unitary error bases, up to a choice of a family of Hadamards. Furthermore, we give a new construction of partitioned unitary error bases, and thus maximal families of mutually unbiased bases, from a finite field, which is simpler and more direct than previous proposals. We introduce new tensor diagrammatic characterizations of maximal families of mutually unbiased bases, partitioned unitary error bases, and finite fields as algebraic structures defined over Hilbert spaces.
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 formulate some properties of a set of several mutually unbiased measurements. These properties are used for deriving entropic uncertainty relations. Applications of mutually unbiased measurements in entanglement detection are also revisited. First, we estimate from above the sum of the indices of coincidence for several mutually unbiased measurements. Further, we derive entropic uncertainty relations in terms of the Rényi and Tsallis entropies. Both the state-dependent and state-independent formulations are obtained. Using the two sets of local mutually unbiased measurements, a method of en…
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 in dimension N>2 admits the introduction of (m-1)N/2 free parameters which 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 construct the maximal set of triplets of mutually unbiased bases for two-qubits systems and triplets, quadruplets and quintuplets of mutually unbiased bases with free parame…
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 coher…
Semi-quantum key distribution is traditionally based on two-level quantum systems. In this paper, an unconditional security of a semi quantum key distribution protocol based on higher-dimensional systems using various mutually unbiased bases is presented. We first consider the three dimensional case using three and four mutually unbiased bases and derive a lower bound for the key rate as a function of the quantum channel's noise. We then generalize the result to a semi-quantum key distribution protocol that employs different number of mutually unbiased bases for four-dimensional states. It is …
The thesis is mainly about the construction and implementation of cyclic mutually unbiased bases, dealing with different entanglement structures by discussing the related group structures. A recursive construction for Fermat number dimensions is given and related to Wiedemann's conjecture for systems with more than 2048 qubits. The second part of the thesis analyses a quantum public-key encryption scheme.
Unextendible sets of Mutually Unbiased Bases (MUBs) are examined from the point of view of complementary subalgebras. We show, that the linear span of less than $d+1$ factors of $M_d \otimes M_d$ does not contain pure states, and therefore some complementary decompositions give rise to undextendable sets of MUBs. We provide some new complementary decompositions, and thus prove strong unextendibility of some set of MUBs.
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 $\mathbb{C}^{d} \otimes \mathbb{C}^{d} \otimes \mathbb{C}^{d}$ based on mutually orthogonal Latin squares. Then we generalize the approach to the case of $\mathbb{C}^{d_{1}} \otimes \mathbb{C}^{d_{2}} \otimes \mathbb{C}^{d_{1}d_{2}}$ by mutually weak orthogonal Latin squares. The concise direct constructions of mutually unbiased tripartite absolutely maximally entangled bases are remarkably…
We study entanglement witness and present a construction of entanglement witnesses in terms of the mutually unbiased measurements (MUMs). These witnesses include the entanglement witnesses constructed from mutually unbiased bases (MUBs) as a special case. Comparing with the dimension dependence of MUBs, the witnesses can be always constructed from a complete set of $d+1$ MUMs for any dimension $d$. We show that our witness can detect entanglement better than previous separability criterion given also by MUMs. And the approach can be experimentally implemented.
We present a new quantum communication complexity protocol, the promise--Quantum Random Access Code, which allows us to introduce a new measure of unbiasedness for bases of Hilbert spaces. The proposed measure possesses a clear operational meaning and can be used to investigate whether a specific number of mutually unbiased bases exist in a given dimension by employing Semi--Definite Programming techniques.
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.
In order to describe the right setting to handle Zauner's conjecture on mutually unbiased bases (MUBs) (saying that in $\mathbb{C}^d$, a set of MUBs of the theoretical maximal size $d + 1$ exists only if $d$ is a prime power), we pose some fundamental questions which naturally arise. Some of these questions have important consequences for the construction theory of (new) sets of maximal MUBs.
Mutually unbiased measurements (MUMs) are generalized from the concept of mutually unbiased bases (MUBs) and include the complete set of MUBs as a special case, but they are superior to MUBs as they do not need to be rank one projectors. We investigate entanglement detection using sets of MUMs and derived separability criteria for $d$-dimensional multipartite systems, and arbitrary high-dimensional bipartitie and multipartite systems. These criteria provide experimental implementation in detecting entanglement of unknown quantum states.
Mutual unbiasedness of the eigenstates of phase-space operators-such as position and momentum, or their standard coarse grained versions-exists only in the limiting case of infinite squeezing. In [Phys. Rev. Lett. 120, 040403 (2018)] it was shown that mutual unbiasedness can be recovered for periodic coarse grainings of these two operators. Here we investigate mutual unbiasedness of coarse-grained measurements for more than two phase-space variables. We show that mutual unbiasedness can be recovered between periodic coarse graining of any two non-parallel phase-space operators. We illustrate t…
An uncertainty relation is introduced for a symmetric arrangement of three mutually unbiased bases in continuous variable phase space, and then used to derive a bipartite entanglement criterion based on the variance of global operators composed of these three phase space variables. We test this criterion using spatial variables of photon pairs, and show that the entangled photons are correlated in three pairs of bases.
We propose a new method for implementing mutually unbiased generation and measurement of time-bin qudits using a cascade of electro-optic phase modulator -- coded fiber Bragg grating pairs. Our approach requires only a single spatial mode and can switch rapidly between basis choices. We obtain explicit solutions for dimensions $d=2,3,4$ that realize all $d+1$ possible mutually unbiased bases and analyze the performance of our approach in quantum key distribution. Given its practicality and compatibility with current technology, our approach provides a promising springboard for scalable process…
The number of measurements necessary to perform the quantum state reconstruction of a system of qubits grows exponentially with the number of constituents, creating a major obstacle for the design of scalable tomographic schemes. We work out a simple and efficient method based on cyclic generation of mutually unbiased bases. The basic generator requires only Hadamard and controlled-phase gates, which are available in most practical realizations of these systems. We show how complete sets of mutually unbiased bases with different entanglement structures can be realized for three and four qubits…
We provide a class of entanglement witnesses constructed in terms of Mutually Unbiased Bases (MUBs). This construction reproduces many well-known examples like the celebrated reduction map and Choi map together with its generalizations. We illustrated our construction by the detailed analysis of the 3-dimensional case: in this case, one obtains a family of entanglement witnesses parameterized by an L-dimensional torus (L = 2,3,4 being a number of MUBs used in the construction).
We derive a framework for quantifying entanglement in multipartite and high dimensional systems using only correlations in two unbiased bases. We furthermore develop such bounds in cases where the second basis is not characterized beyond being unbiased, thus enabling entanglement quantification with minimal assumptions. Furthermore, we show that it is feasible to experimentally implement our method with readily available equipment and even conservative estimates of physical parameters.
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.
We introduce orthogonal quantum Latin squares, which restrict to traditional orthogonal Latin squares, and investigate their application in quantum information science. We use quantum Latin squares to build maximally entangled bases, and show how a pair of mutually unbiased maximally entangled bases can be constructed in square dimension from orthogonal quantum Latin squares. We also compare our construction to an existing construction due to Beth and Wocjan and show that ours is strictly more general.
