A new method of construction of all sets of mutually unbiased bases for two-qubit systems

Abstract

Mutually unbiased bases are an important tool in many applications of quantum information theory. We present a new algorithm for finding the mutually unbiased bases for two-qubit systems. We derive a system of four equations in the Galois field GF(4) and show that the solutions of this system are sufficient for obtaining the most general set of mutually unbiased bases. Further, our algorithm is applied to an example and we show that there are three possible solutions of the system of four equations, each solution leading to a different set of mutually unbiased bases.

Mutually Unbiased Maximally Entangled Bases for the Bipartite System ℂ d ⊗ ℂ d k $\mathbb {C}^{d}\otimes \mathbb {C}^{d^{k}}$

“…Theorem If (1) and (3) are MUMEBs for C d ⊗ C d , then the maximally entangled bases (11) and (12) are MUMEBs for C d ⊗ C d k .…”

“…Different constructions of MUBs, especially for prime power and partial qubits systems, have been presented in [12,13]. In [14] the authors first considered the unextendible maximally entangled bases and mutually unbiased bases.…”

Construction of the mutually orthogonal extraordinary supersquares

“…A 1 is an extraordinary subgroup and A j = a j + A 1 with j = 2, 3, ..., d. The importance of the extraordinary subgroup is explained briefly in the Introduction, where we emphasized that the condition tr(x 1 y 2 − x 2 y 1 ) = 0 is equivalent to the commutation of two operators, whose associated elements are (x 1 , y 1 ) and (x 2 , y 2 ) in F d × F d [13], [5].…”

Mutually unbiased maximally entangled bases in $$\mathbb {C}^d\otimes \mathbb {C}^{kd}$$ C d ⊗ C k d

“…One main concern is about the maximal number of MUBs for given dimension d. It has been shown that the maximum number N (d) of MUBs in C d is no more than d + 1 [21] and N (d) = d + 1 if d is a prime power. Different constructions of MUBs, especially for prime power and qubits systems, have been presented in [33][34][35][36][37][38][39][40]…”