Uncertainty relations in the presence of quantum memory for mutually unbiased measurements

Abstract

In [1], uncertainty relations in the presence of quantum memory was formulated for mutually unbiased bases using conditional collision entropy. In this paper, we generalize their results to the mutually unbiased measurements. Our primary result is an equality between the amount of uncertainty for a set of measurements and the amount of entanglement of the measured state, both of which are quantified by the conditional collision entropy. Implications of this equality relation are discussed. We further show that similar equality relation can be obtained for generalized symmetric informationally complete measurements. We also derive an interesting equality for arbitrary orthogonal basis of the space of Hermitian, traceless operators.

Characterizing the uncertainty relation via a class of measurements

“…( ) . It is surprising to find that equation ( 26) is theorem 1 in [35]. Similarly, suppose N = 1 and M = d 2 , the efficiency parameter a for GSIC-POVMs will be obtained, satisfying [43…”

“…Conveniently, the proof of theorem 2 bears resemblance to theorem 1 of [35], with the exception that equations (11), ( 12), (23), and (24) have been suitably substituted.…”

“…In [35], the uncertainty relations in the presence of quantum memory were formulated for MUMs and GSIC-POVMs, where the authors emphatically presented an equation between the uncertainty and the entanglement for a collection of MUMs to show the relation between uncertainty and entanglement. It is commonly known that (N, M)-POVMs can be viewed as a common generalization of MUMs and GSIC-POVMs.…”

“…Hence it is important from a practical standpoint that we utilize conditional collision entropy to analyze the uncertainty relations of quantum memory based on (N, M)-POVMs, which implies the relation between uncertainty and entanglement. Surprisingly, it is a common extension of [35] and [36].…”

Generalization of Pauli channels through mutually unbiased measurements

“…The last type helped to find new separability conditions for bipartite system [8]. Moreover, it was shown that there is an equality between the amounts of uncertainty for MUMs and entanglement of the measured states quantified by the conditional collision entropy [9]. New separability criteria were given for arbitrary d-dimensional bipartite [10][11][12] and multipartite systems [13,14].…”