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Register a new userMultiplicative functions arising from the study of mutually unbiased bases
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Added by
Berthold-Georg Englert
Publication date
2020
fields
Physics
and research's language is
English
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We embed the somewhat unusual multiplicative function, which was serendipitously discovered in 2010 during a study of mutually unbiased bases in the Hilbert space of quantum physics, into two families of multiplicative functions that we construct as generalizations of that particular example. 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.
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We study the connection between mutually unbiased bases and mutually orthogonal extraordinary supersquares, a wider class of squares which does not contain only the Latin squares. We show that there are four types of complete sets of mutually orthogonal extraordinary supersquares for the dimension $d=8$. We introduce the concept of physical striation and show that this is equivalent to the extraordinary supersquare. The general algorithm for obtaining the mutually unbiased bases and the physical striations is constructed and it is shown that the complete set of mutually unbiased physical striations is equivalent to the complete set of mutually orthogonal extraordinary supersquares. We apply the algorithm to two examples: one for two-qubit systems ($d=4$) and one for three-qubit systems ($d=8$), by using the Type II complete sets of mutually orthogonal extraordinary supersquares of order 8.
In this contribution we relate two different key concepts: mutually unbiased bases (MUBs) and entanglement; in particular we focus on bound entanglement, i.e. highly mixed states which cannot be distilled by local operations and classical communications. For a certain class of states --for which the state-space forms a magic simplex-- we analyze the set of bound entangled states detected by the MUB criterion for different dimensions d and number of particles n. We find that the geometry is similar for different d and n, consequently, the MUB criterion opens possibilities to investigate the typicality of PPT-bound and multipartite bound entanglement deeper and provides a simple experimentally feasible tool to detect bound entanglement.
We derive new inequalities for the probabilities of projective measurements in mutually unbiased bases of a qudit system. These inequalities lead to wider ranges of validity and tighter bounds on entropic uncertainty inequalities previously derived in the literature.
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 states can be mutually unbiased to a maximal set of mutually unbiased product bases.
Two equivalent ways of looking for mutually unbiased bases are discussed in this note. The passage from the search for d+1 mutually unbiased bases in C(d) to the search for d(d+1) vectors in C(d*d) satisfying constraint relations is clarified. Symmetric informationally complete positive-operator-valued measures are briefly discussed in a similar vein.
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