Information-disturbance theorem for mutually unbiased observables (4 pages)

William K. Wootters, Brian D. Fields

Takayuki Miyadera, Hideki Imai

We derive a version of information-disturbance theorems for mutually unbiased observables. We show that the information gain by Eve inevitably makes the outcomes by Bob in the conjugate basis not only erroneous but random.

K. Urbanik

Howard Barnum

I consider the tradeoff between the information gained about an initially unknown quantum state, and the disturbance caused to that state by the measurement process. I show that for any distribution of initial states, the information-disturbance frontier is convex, and disturbance is nondecreasing with information gain. I consider the most general model of quantum measurements, and all post-measurement dynamics compatible with a given measurement. For the uniform initial distribution over states, I show that the least-disturbing way of making any measurement is with conditional dynamics satisfying a generalization of the projection postulate, the ``square-root dynamics.'' Thus, procedures for achieving a point on the information-disturbance frontier may be assumed to involve such conditional dynamics. Also, the information-disturbance frontier for the uniform ensemble may be achieved with ``isotropic'' (unitarily covariant) dynamics. This results in a significant simplification of the optimization problem for calculating the tradeoff in this case, giving hope for a closed-form solution. I also show that the discrete ensembles uniform on the d(d+1) vectors of a certain set of d+1 ``mutually unbiased'' or conjugate bases in d dimensions form spherical 2-designs in CP_{d-1} when d is a power of an odd prime. This implies that many of the results of the paper apply also to these discrete ensembles.

Nur Rahimah Sakinah Abdul Salam, Jesni Shamsul Shaari, Stefano Mancini

Abstract Making use of the Quantum Network formalism of Phys. Rev. A, 82 (2010) 062 305, we present the case for quantum networks with finite outcomes, more specifically one which could distinguish only between specific unitary operators in a given basis for operators. Despite its simplicity, we proceed to build a network derived from the optimal strategy in Phys. Rev. A, 82 (2010) 062 305 and show that the information-disturbance tradeoff in distinguishing between two operators acting on qubits, selected from mutually unbiased unitary bases is equal to the case of estimating an operator selected randomly from the set of SU(2) based on the Haar measure. This suggests that such strategies in distinguishing between mutually unbiased operators is not any easier than estimating an operator derived from an infinite set. We then show how this network can be used as a natural attack strategy against a bidirectional quantum cryptographic protocol.

Beloslav Riečan