Algorithmic Aspects of Optimal Channel Coding

The main contributions of this paper are two-fold.Firstly, we present a novel direction in the cryptanalysis of the cryptographic hash function SHA-1. Our work builds on previous cryptanalytic efforts on SHA-1 based on combinations of local collisions. Due to dependencies, previous approaches used heuristic corrections when combining the success probabilities and message conditions of the individual local collisions. Although this leads to success probabilities that are seemingly sufficient for feasible collision attacks, this approach most often does not lead to the maximum success probability possible as desired. We introduce novel techniques that enable us to determine the theoretical maximum success probability for a given set of (dependent) local collisions, as well as the smallest set of message conditions that attains this probability. We apply our new techniques and present an implemented open-source near-collision attack on SHA-1 with a complexity equivalent to 257.5 SHA-1 compressions.Secondly, we present an identical-prefix collision attack and a chosen-prefix collision attack on SHA-1 with complexities equivalent to approximately 261 and 277.1 SHA-1 compressions, respectively.

By using the theory of cavity QED, we study the system in which a two-level atom interacts with a cavity in the case of large detuning. Through the selective detecting of atomic state, Schrödinger cat states and entangled coherent states are easily generated. When the atom is driven by a weak classical field and the cavity field is in the Schrödinger cat state, we study the conditions of generating the Fock states and the maximal success probability. The maximal success probability in our scheme is larger than the previous one.

Channel simulation using coherence, which refers to realizing a target channel with coherent states and free operations, is a fundamental problem in the quantum resource theory of coherence. The limitations of the accuracy of deterministic channel simulation motivate us to consider the more general probabilistic framework. In this paper, we develop the framework for probabilistic channel simulation using coherence with free operations. When the chosen set of free operations is the maximally incoherent operations, we provide an efficiently computable semidefinite program (SDP) to calculate the maximal success probability and derive the analytic expression of success probability for some special cases. When the chosen set of free operations is the dephasing-covariant incoherent operations (DIO), we show that if the target channel is not a resource nonactivating channel then one cannot simulate it exactly both deterministically and probabilistically. The SDP for maximal success probability of simulating a channel by DIO is also given correspondingly. Published by the American Physical Society 2024

Assume that the probability of success is unimodal as a function of dose. Take the response to be binary and the possible treatment space to be a lattice. The Optimizing Up-and-Down Design allocates treatments to pairs of subjects in a way that causes the treatment distribution to cluster around the treatment with maximum success probability. This procedure is constructed to use accruing information to limit the number of patients that are exposed to doses with high probabilities of failure. The Optimizing Up-and-Down Design is motivated by Kiefer and Wolfowitz’s stochastic approximation procedure. In this paper, we compare its performance to stochastic approximation. As an estimator of the best dose, simulation studies demonstrate that the mode of the empirical treatment distribution using the Optimizing Up-and-Down Design converges faster than does the usual estimator using stochastic approximation.Keywordsconstant gain stochastic algorithmMarkov chainsphase I/II clinical trialrandom walk designsstationary treatment distributionstochastic approximation

The ability to distill quantum coherence is pivotal for optimizing the performance of quantum technologies; however, such a task cannot always be accomplished with certainty. Here we develop a general framework of probabilistic distillation of quantum coherence in a one-shot setting, establishing fundamental limitations for different classes of free operations. We first provide a geometric interpretation for the maximal success probability, showing that under maximally incoherent operations (MIO) and dephasing-covariant incoherent operations (DIO) the problem can be simplified into efficiently computable semidefinite programs. Exploiting these results, we find that DIO and its subset of strictly incoherent operations have equal power in the probabilistic distillation of coherence from pure input states, while MIO are strictly stronger. We then prove a fundamental no-go result: Distilling coherence from any full-rank state is impossible even probabilistically. We further find that in some conditions the maximal success probability can vanish suddenly beyond a certain threshold in the distillation fidelity. Finally, we consider probabilistic coherence distillation assisted by a catalyst and demonstrate, with specific examples, its superiority to the unassisted and deterministic cases.

We present some different hyperentanglement concentration protocols (hyper-ECPs) for nonlocal N-photon systems in partially polarization-spatial hyperentangled states with known parameters, resorting to linear optical elements only, including those for hyperentangled Greenberger---Horne---Zeilinger-class states and the ones for hyperentangled cluster-class states. Our hyper-ECPs have some interesting features. First, they require only one copy of nonlocal N-photon systems and do not resort to ancillary photons. Second, they work with linear optical elements, neither Bell-state measurement nor two-qubit entangling gates. Third, they have the maximal success probability with only a round of entanglement concentration, not repeating the concentration process some times. Fourth, they resort to some polarizing beam splitters and wave plates, not unbalanced beam splitters, which make them more convenient in experiment.

We propose an optimal entanglement concentration protocol (ECP) for nonlocal \(N\)-electron systems in a partially entangled Greenberger–Horne–Zeilinger (GHZ) pure state, resorting to charge detection and the projection measurement on an additional electron. For each nonlocal \(N\)-electron system in a partially entangled GHZ state, one party in quantum communication, say Alice first entangles it with an additional electron, and then, she projects the additional electron into an orthogonal basis for dividing the \(N\)-electron systems into two groups. In the first group, the \(N\) parties obtain a subset of \(N\)-electron systems in a maximally entangled state directly. In the second group, they obtain some less-entangled \(N\)-electron systems which are the resource for the entanglement concentration in the next round. By iterating the entanglement concentration process several times, the present ECP has the maximal success probability, the theoretical limit of an ECP as it just equals to the entanglement of the partially entangled state, far higher than others. Moreover, this ECP for an \(N\)-electron GHZ-type state requires only one additional electron, not two or more, and it does not resort to a collective unitary evolution, far different from others, which may decrease the difficulty for its implementation in experiment. When it is used for an \(N\)-electron W-type state, \(N-1\) additional electrons are required only.

We propose an optimal nonlocal entanglement concentration protocol (ECP) for\nmulti-photon systems in a partially entangled pure state, resorting to the\nprojection measurement on an additional photon. One party in quantum\ncommunication first performs a parity-check measurement on her photon in an\nN-photon system and an additional photon, and then she projects the additional\nphoton into an orthogonal Hilbert space for dividing the original $N$-photon\nsystems into two groups. In the first group, the N parties will obtain a subset\nof $N$-photon systems in a maximally entangled state. In the second group, they\nwill obtain some less-entangled N-photon systems which are the resource for the\nentanglement concentration in the next round. By iterating the entanglement\nconcentration process several times, the present ECP has the maximal success\nprobability which is just equivalent to the entanglement of the partially\nentangled state. That is, this ECP is an optimal one.\n

Hyperentanglement has attracted much attention due to its fascinating applications in quantum communication. However, it is impossible to purify a pair of photon systems in a mixed hyperentangled state with errors in two degrees of freedom using linear optical elements only, far different from all the existing entanglement purification protocols in a degree of freedom (DOF) for quantum systems. Here, we investigate the possibility of purifying a spatial-polarization mixed hyperentangled Bell state with the errors in both the spatial-mode and polarization DOFs, resorting to the nonlinear optics of a nitrogen-vacancy (NV) center in a diamond embedded in a photonic crystal cavity coupled to a waveguide. We present the first hyperentanglement purification protocol for purifying a pair of two-photon systems in a mixed hyperentangled Bell state with the errors in two DOFs. We also propose an efficient hyperentanglement concentration protocol for a partially hyperentangled Bell pure state, which has the maximal success probability in principle. These two protocols are useful in long-distance quantum communication with hyperentanglement.

A channel scheduling problem for a given graph G(V,E) is to select a subset E′ of edges of E and assign channel to each one in E′ under the restriction that all edges of E′ are interference free. We introduce the notion of two sides approximation for the channel scheduling problem. A pair of parameters (f,g) controls the accuracy of approximation in this paper. A (f,g)-approximation satisfies \(\sum_{e\in E'}W(e)\ge{\sum_{e\in Opt}W(e)\over f}\) and \(\sum_{e\in E-E'}W(e)\le g\sum_{e\in E- Opt^*}W(e)\), where Opt * is the set of edges assigned channels in an optimal solution, and W(e) is the weight of edge e. An f-approximation satisfies \(\sum_{e\in E'}W(e)\ge{\sum_{e\in Opt}W(e)\over f}\). We show that a simple greedy algorithm can obtain an (O(1), O(1))-approximation for the single channel scheduling problem. In many cases, the greedy algorithms gives much more accurate result than the worst ratio. Furthermore, we develop an \(|E|^{O({1\over \epsilon})}\) time (1 − ε, O(1))-approximation algorithm for the single channel scheduling problem. We also show that a simple greedy algorithm can obtain an O(1)-approximation for the multi-channel scheduling problem which satisfies \(\sum_{e\in E'}W(e)\ge{\sum_{e\in Opt}W(e)\over \Omega(1)}\). We also develop a \(|E|^{O({d\over \epsilon})}\) time (1 − ε)-approximation algorithm the multi-channel scheduling problem, where d is the number of channels. This improves the existing approximation scheme for multi-channel scheduling problem with \(|E|^{O({d\over \epsilon^2})}\) time by Cheng et al. We also develop a polynomial time constant factor greedy approximation algorithm for the multi-channel scheduling that allows variate radii of interference among those nodes.

In this work, the authors propose a novel technique to extend the scope of cognitive radio (CR) networks to multiple source–destination pairs. To accomplish this, a simplified feasible path set is formed and then three types of criteria or methods are employed to reach a solution for the best path for each pair. This set is defined by non‐intersecting paths which do not share any relay nodes. This not only solves the problem at hand but also greatly reduces the complexity which leads to a fast solution. The first method uses the maximum probability of success as a means for selecting the best path. In the second method, a fast and simple technique is used by choosing the minimum number of overall hops for path selection. The element with the minimum overall transmission time is selected for data transmission in the third technique. Finally, the proposed methods are compared through simulation to a unicast single user case in terms of throughput and packet delivery rate. The simulation results show that the first method has the best performance while the second method is slightly worse but faster.

Vehicle flight readiness review - A technique for assuring maximum probability of mission success

It is one of the peculiar features of quantum physics that some sets of measurements are incompatible in the sense that they cannot be performed simultaneously. Incompatibility is an important quantum resource as it enables Bell nonlocality and steering. Here we show that incompatibility is crucial also in certain communication tasks; we prove that quantum random access code performs better than its classical counterpart only when incompatible quantum measurements are used in the decoding task. As a consequence, evaluating the average success probability for quantum random access code provides a semi-device-independent test for the detection of quantum incompatibility. We further demonstrate that any incompatible pair of projective measurements gives an advantage over all classical strategies. Finally, we establish a connection between the maximal average success probability for quantum random access code and earlier quantities introduced to assess incompatibility.

We report on theoretical research in photonic cluster-state computing. Finding optimal schemes of generating non-classical photonic states is of critical importance for this field as physically implementable photon–photon entangling operations are currently limited to measurement-assisted stochastic transformations. A critical parameter for assessing the efficiency of such transformations is the success probability of a desired measurement outcome. At present there are several experimental groups that are capable of generating multi-photon cluster states carrying more than eight qubits. Separate photonic qubits or small clusters can be fused into a single cluster state by a probabilistic optical CZ gate conditioned on simultaneous detection of all photons with 1/9 success probability for each gate. This design mechanically follows the original theoretical scheme of cluster state generation proposed more than a decade ago by Raussendorf, Browne and Briegel. The optimality of the destructive CZ gate in application to linear optical cluster state generation has not been analyzed previously. Our results reveal that this method is far from the optimal one. Employing numerical optimization we have identified that the maximal success probability of fusing n unentangled dual-rail optical qubits into a linear cluster state is equal to an m-tuple of photonic Bell pair states, commonly generated via spontaneous parametric down-conversion, can be fused into a single cluster with the maximal success probability of (1/4)m−1.

The FIBA Under-17 Basketball World Cup showcases elite youth talent and shapes international basketball's future. This study identified game-related performance indicators differentiating winning from losing teams in different (Group, Second, and Final) rounds of the Under-17 Men World Cup. Data from 122 games across two consecutive Under-17 Men World Cups (Malaga 2022, Istanbul 2024) were analysed, classifying matches as closed, balanced, or unbalanced based on final score differences. Descriptive statistics quantified performance differences, while linear discriminant analysis identified key predictors of game outcomes. In the group stage, first-quarter scoring consistently predicted victory across all game types. Closed games were primarily influenced by three-point field goals made and assists, while balanced games were determined by defensive rebounds and turnovers. Second-round analysis revealed different predictors: points from turnovers in closed games, bench scoring in balanced games, and points from the paint in unbalanced games. The final round showed that points scored in the first and fourth quarters, alongside fast-break points, were consistent predictors regardless of game classification. These findings provide valuable insights for coaches developing tournament preparation strategies. Understanding how different performance indicators influence game outcomes across tournament rounds, enables tactical adjustments that maximize success probability in international youth basketball competitions.