Minimum error discrimination for an ensemble of linearly independent pure states

Abstract

Inspired by the work done by Belavkin [BelavkinV.P.,Stochastics,1,315(1975)], and independently by Mochon, [Phys.Rev.A73,032328,(2006)], we formulate the problem of minimum error discrimination of any ensemble of n linearly independent pure states by stripping the problem of its rotational covariance and retaining only the rotationally invariant aspect of the problem. This is done by embedding the optimal conditions in a matrix equality as well as matrix inequality. Employing the implicit function theorem in these conditions we get a set of first-order coupled ordinary non-linear differential equations which can be used to drag the solution from an initial point (where solution is known) to another point (whose solution is sought). This way of obtaining the solution can be done through a simple Taylor series expansion and analytic continuation when required. Thus, we complete the work done by Belavkin and Mochon by ultimately leading their theory to a solution for the minimum error discrimination problem of linearly independent pure state ensembles. We also compare the computational complexity of our technique with a barrier-type interior point method of SDP and show that our technique is computationally as efficient as (actually, a bit more than) the SDP algorithm, with the added advantage of being much simpler to implement.

Quantum Annealing: a journey through Digitalization, Control, and hybrid Quantum Variational schemes

“…This is especially severe in cases where an exponentially small spectral gap must be crossed to go from the initial state to the final ground state one is searching for [12,13]. Different strategies have been proposed to cope with such a problem, including heuristic guesses for the initial state [14], strategies for increasing the minimum gap [15,16] or avoiding first-order lines [17], and the quantum adiabatic brachistochrone formulation [18].…”

Structure of minimum error discrimination for linearly independent states

“…In [2], Mochon rediscovered the structure for the case of pure state ensembles, and proved the existence of the map R for LI pure states ensembles. This map was later employed in [18] to obtain the optimal POVM. Equations (13) tells us that to solve the MED problem it suffices to know the map R. However the construction of R requires the optimal POVM.…”

“…In fact it is a difficult problem to get an exact form of R. On the other hand, we have constructed R −1 and thus if one can invert R −1 , then one solves the MED problem. This was done for the case of LI pure state ensembles in [18], where the authors used the implicit function theorem to do so. We would like to see if this can be generalized to the case of LI mixed state ensembles as well.…”

Optimal working point in digitized quantum annealing

“…In this framework, the hardness of a problem is associated with the intrinsic difficulty in following the adiabatic ground state when a (possibly exponentially) small spectral gap must be crossed to go from the initial state to the final target ground state [14,15]. Different strategies have been proposed to cope with such a problem [16][17][18][19]. Among them, the choice of the driving protocol is crucial for obtaining a quantum speed-up, see e.g.…”