Abstract
We consider the quantum multiple hypothesis testing problem, focusing on the case of hypothesis represented by pure states. A sequential adaptive algorithm is derived and analyzed first. This strategy exhibits a decay rate in the error probability with respect to the expected value of measurements greater than the optimal decay rate of the fixed-length methods. A more elaborated scheme is developed next, by serially concatenating multiple implementations of the first scheme. In this case each stage considers as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a priori</i> hypothesis probability the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a posteriori</i> probability of the previous stage. We show that, by means of a fixed number of concatenations, the expected value of measurements to be performed decreases considerably. We also analyze one strategy based on an asymptotically large concatenation of the initial scheme, demonstrating that the expected number of measurements in this case is upper bounded by a constant, even in the case of zero average error probability. A lower bound for the expected number of measurements in the zero error probability setting is also derived.
Highlights
The task of discriminating between hypothesis is essential in a wide variety of scientific fields
The results presented in [5], [6] show that the decay rate associated with this error probability, i.e., the exponent of the error probability as L goes to infinity, is CQ(σ1, σ2) = − log min0≤s≤1 Tr(σ1sσ21−s), which defines the quantum Chernoff distance between two quantum states
In this paper we focus on sequential methods for the multiple hypothesis testing problem and present an adaptive sequential scheme, named Sequential Discarding Method (SDM), that attains an asymptotic decay rate with respect to the expected value of measurements higher than mini,j:i=j CQ(σi, σj)
Summary
The task of discriminating between hypothesis is essential in a wide variety of scientific fields. J. Pérez-Guijarro et al.: Quantum Multiple Hypothesis Testing Based on Sequential Discarding Scheme crimination that measures each copy individually using an adaptive strategy, and attains the minimum error probability achievable with collective measurements. Pérez-Guijarro et al.: Quantum Multiple Hypothesis Testing Based on Sequential Discarding Scheme crimination that measures each copy individually using an adaptive strategy, and attains the minimum error probability achievable with collective measurements This result does not hold for the case of two non-pure states, where the maximum decay rate is strictly lower than the quantum Chernoff distance [10]. In this paper we focus on sequential methods for the multiple hypothesis testing problem and present an adaptive sequential scheme, named Sequential Discarding Method (SDM), that attains an asymptotic decay rate with respect to the expected value of measurements higher than mini,j:i=j CQ(σi, σj).
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
