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Register a new userMinimum Decision Cost for Operators
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Authors
Bernhard K. Meister
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State discrimination with the aim to minimize the error probability is a well studied problem. Instead, here the binary decision problem for operators with a given prior is investigated. A black box containing the unknown operator is probed by selected wave functions. The output is analyzed with conventional methods developed for state discrimination. An error probability bound for all binary operator choices is provided, and it is shown how probe entanglement enhances the result.
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The problem of quantum state discrimination between two wave functions on a ring is considered. The optimal minimum-error probability is known to be given by the Helstrom bound. A new strategy is introduced by inserting either adiabatically or instantaneously an impenetrable barrier. The insertion point, independent of the shape of the initial wave function, becomes a node. The resulting modified wave functions can be, if the initial functions are judiciously chosen, distinguished with a smaller error probability, and as a consequence the Helstrom bound can be violated under idealised conditions.
The problem of quantum state discrimination between two wave functions of a particle in a square well potential is considered. The optimal minimum-error probability is known to be given by the Helstrom bound. A new strategy is introduced by inserting an impenetrable barrier in the middle of the square well, which is either a nodal or non-nodal point of the wave function. The energy required to insert the barrier is dependent on the initial state. This enables the experimenter to gain additional information beyond the standard probing of the state envisaged by Helstrom and to improve the distinguishability of the states. It is shown that under some conditions the Helstrom bound can be violated, i.e. the state discrimination can be realized with a smaller error probability.
The problem of quantum state discrimination between two wave functions of a particle in a square well potential is considered. The optimal minimum-error probability for the state discrimination is known to be given by the Helstrom bound. A new strategy is introduced here whereby the square well is compressed isoenergetically, modifying the wave-functions. The new contracted chamber is then probed using the conventional optimal strategy, and the error probability is calculated. It is shown that in some cases the Helstrom bound can be violated, i.e. the state discrimination can be realized with a smaller error probability.
In a minimum cost submodular cover problem (MinSMC), given a monotone non-decreasing submodular function $fcolon 2^V rightarrow mathbb{Z}^+$, a cost function $c: Vrightarrow mathbb R^{+}$, an integer $kleq f(V)$, the goal is to find a subset $Asubseteq V$ with the minimum cost such that $f(A)geq k$. MinSMC has a lot of applications in machine learning and data mining. In this paper, we design a parallel algorithm for MinSMC which obtains a solution with approximation ratio at most $frac{H(min{Delta,k})}{1-5varepsilon}$ with probability $1-3varepsilon$ in $O(frac{log mlog nlog^2 mn}{varepsilon^4})$ rounds, where $Delta=max_{vin V}f(v)$, $H(cdot)$ is the Hamornic number, $n=f(V)$, $m=|V|$ and $varepsilon$ is a constant in $(0,frac{1}{5})$. This is the first paper obtaining a parallel algorithm for the weighted version of the MinSMC problem with an approximation ratio arbitrarily close to $H(min{Delta,k})$.
In this paper we study a cybersecurity problem of protecting systems secrets with multiple protections and a required security level, while minimizing the associated cost due to implementation/maintenance of these protections as well as the affected system usability. The target system is modeled as a discrete-event system (DES) in which there are a subset of marker states denoting the services/functions provided to regular users, a subset of secret states, and multiple subsets of protectable events with different security levels. We first introduce usability-aware cost levels for the protectable events, and then formulate the security problem as to ensure that every system trajectory that reaches a secret state contains a specified number of protectable events with at least a certain security level, and the highest usability-aware cost level of these events is minimum. We first provide a necessary and sufficient condition under which this security problem is solvable, and when this condition holds we propose an algorithm to solve the problem based on the supervisory control theory of DES. Moreover, we extend the problem to the case of heterogeneous secrets with different levels of importance, and develop an algorithm to solve this extended problem. Finally, we demonstrate the effectiveness of our solutions with a network security example.
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