Mutually unbiased unitary bases and its context in uncertainty relation for unitary operators

Abstract

Analogous to Mutually Unbiased Bases (MUB) for d-dimensional Hilbert space, H_d capturing the notion of equiprobable transition between states in one basis to another, we consider a similar notion for some subspace of linear operators instead. Working mainly in terms of matrices, the notion of Mutually Unbiased Unitary Bases (MUUB) of M(d,C) can be understood in terms of the equiprobable guess of a unitary operator in one basis for that in another. MUUBs has in fact shown to be useful in specific quantum key distribution (QKD) protocols, namely bidirectional QKD protocols akin to the role of MUBs for prepare and measure QKD schemes like the well-known BB84 protocol. The MUUB structure is strongly related to the notion of MUBs consisting only of maximally entangled states of space H_d ⨂▒H_d or, mutually unbiased maximally entangled bases (MUMEBS). The two are essentially equivalent though much remains to be explored. In fact, for a d^2-dimensional space of M(d,C), while it is known that the maximal numbers that MUUBs can have is d^2-1, there is no known recipe for constructing the maximal number of such bases. It is not even known if such a number may even be achieved for any d Focusing on the case for d being the prime numbers, we show that the minimal number for MUUBs is 3 and approaches its maximal d^2-1 for very large values of d. We further provide a numerical recipe in constructing MUUBs which gives us an explicit construction for the maximal number of MUUBs for subspaces of M(3,C) and M(2,C). Despite the possible use of the numerical search for any dimension, it quickly becomes inefficient as d grows. For a more analytical solution, we turn our focus to the case of some d-dimensional subspace for any prime d and report on the maximal number of MUUBs for such a subspace. By constructing monoids based on the underlying sets of H_d and a subspace of M(d,C), an isomorphism between the monoids lead to an important theorem for constructing d MUUBs, i.e. the maximal possible number for such a subspace. Finally, we show how the notion of MUUBs arise in some setup relevant to the problem of incompatibility/uncertainty between pairs of unitary operators. Departing from some earlier works making use of standard deviations to quantify the uncertainty of pairs of unitary operators (similar to the uncertainties of observables), we formulate a more ‘operational’ notion of uncertainty of pairs of unitary operators in the context of a guessing game and derive an entropic uncertainty relation for such a pair. We show how distinguishable operators are compatible while maximal incompatibility of unitary operators can be connected to bases for some subspace of operators which are mutually unbiased. We conclude the thesis with some suggestions for future works.