Probabilistic channel simulation using coherence

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.

A central question in information theory is to determine the maximum success probability that can be achieved in sending a fixed number of messages over a noisy channel. This was first studied in the pioneering work of Shannon who established a simple expression characterizing this quantity in the limit of multiple independent uses of the channel. Here we consider the general setting with only one use of the channel. We observe that the maximum success probability can be expressed as the maximum value of a submodular function. Using this connection, we establish the following results: 1. There is a simple greedy polynomial-time algorithm that computes a code achieving a (1-1/e)-approximation of the maximum success probability. Moreover, for this problem it is NP-hard to obtain an approximation ratio strictly better than (1-1/e). 2. Shared quantum entanglement between the sender and the receiver can increase the success probability by a factor of at most 1/(1-1/e). In addition, this factor is tight if one allows an arbitrary non-signaling box between the sender and the receiver. 3. We give tight bounds on the one-shot performance of the meta-converse of Polyanskiy-Poor-Verdu.

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.

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

Classical data compression focuses on the most accurate reproduction of data. However, modern applications, particularly in artificial intelligence, increasingly impose new statistical or behavioral constraints. This thesis extends compression theory and practice to these new settings. Our main tool is channel simulation—a generalization of data compression for remotely generating samples from a specified distribution. We first establish a framework for universal channel simulation, analogous to universal source coding for when the target distribution is unknown, and provide a provably near-optimal algorithm. We then develop a computationally efficient algorithm for simulating the Gaussian channel, a ubiquitous model for which no such practical method existed. In distributed learning, we formalize remote reinforcement learning under communication constraints and propose a general, highly efficient solution employing channel simulation. Furthermore, we uncover a fundamental connection between speculative decoding—a method to accelerate large language models—and channel simulation, yielding new algorithms and speed-up guarantees. Finally, we introduce a goal-oriented compression framework for agents whose actions and communications are concurrent, for which we derive performance bounds and design practical coding schemes. The contributions in this thesis highlight the fundamental nature of channel simulation as a novel and effective tool to address a variety of the challenges in federated learning, reinforcement learning, realistic image compression, and large language model inference.

Recently, Markham and Vedral [Phys. Rev. A 67, 042113 (2003)] investigated the effect of beam splitting on the spin, or SU(2), coherent states for a single mode field. The spin coherent state is a binomial coherent state related to the Holstein-Primakoff realization of the su(2) Lie algebra given in terms of a set of single mode bose annihilation and creation operators. Upon beam splitting, the ordinary (or Glauber) coherent states merely split into products of ordinary coherent states with reduced amplitudes without becoming entangled, as one would expect for a classical-like field. The above authors expected the spin coherent states to go over to the ordinary coherent states in the limit of high spin, $j\ensuremath{\rightarrow}\ensuremath{\infty}$, and thus to become product states after beam splitting. But this expectation was not confirmed through numerical calculation of the entropy which, instead of going to zero, leveled off with increasing spin. In this paper we find similar behavior for SU(1,1) coherent states of the Perelomov type for large Bargman index $k$, but also find that the Barut-Girardello SU(1,1) coherent states appear to rapidly become product states after beam splitting for increasing $k$. We explain these results by showing that, in reality, neither the spin coherent states nor the Perelomov SU(1,1) coherent states go over to ordinary coherent states in the limits of large $j$ or $k$, and that the Barut-Girardello coherent states merely go over to the vacuum in the large $k$ limit. Finally, we examine the correct limiting procedure for obtaining separable states (i.e., products of coherent states) upon beam splitting by performing contractions of the su(2) and su(1,1) Lie algebras and of their associated coherent states.

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.

As an important quantum resource, quantum coherence play key role in quantum information processing. It is often concerned with manipulation of families of quantum states rather than individual states in isolation. Given two pairs of coherent states $(\rho_1,\rho_2)$ and $(\sigma_1,\sigma_2)$, we are aimed to study how can we determine if there exists a strictly incoherent operation $\Phi$ such that $\Phi(\rho_i) =\sigma_i,i = 1,2$. This is also a classic question in quantum hypothesis testing. In this note, structural characterization of coherent preorder under strongly incoherent operations is provided. Basing on the characterization, we propose an approach to realize coherence distillation from rank-two mixed coherent states to $q$-level maximally coherent states. In addition, one scheme of coherence manipulation between rank-two mixed states is also presented.

As the title indicates, this chapter presents a brief, self-contained introduction to five fundamental problems in Quantum Information Science (QIS) that are especially well-suited to be formulated as Semi-definite Programs (SDP). We have in mind two audiences. The primary audience comprises of Operations Research (and Computer Science) graduate students who have familiarity with SDPs, but have found it daunting to become even minimally conversant with pre-requisites of QIS. The second audience consists of Physicists (and Electrical Engineers) already knowledgeable with modeling of QIS via SDP but interested in computational tools that are applicable more generally. For both audiences, we strive for rapid access to the unfamiliar material. For the first, we provide just enough required background material (from Quantum Mechanics, treated via matrices, and mapping them in Dirac notation) and simultaneously for the second audience we recreate, computationally in Jupyter notebooks, known closed-form solutions. We hope you will enjoy this introduction and gain understanding of the marvelous connection between SDP and QIS by self-study, or as a short seminar course. Ultimately, we hope this disciplinary outreach will fuel advances in QIS through their fruitful study via SDPs.

Efficient dissemination of videos is an important problem for mobile telecom carriers. In this paper, to facilitate massive video dissemination, we study joint caching and multicasting for multi-quality videos encoded using two video encoding techniques, namely, scalable video coding (SVC) and HEVC or H.264 as in dynamic adaptive streaming over HTTP (DASH) respectively, in a large-scale wireless network. First, for each type of video, we propose a random caching and multicasting scheme, carefully reflecting the relationship between layers of an SVC-based video or descriptions of a DASH-based video. Then, for each type of video, we derive tractable expressions for the successful transmission probability in the general and high user density regions, respectively, utilizing tools from stochastic geometry. The analytical results reveal that in the high user density region, the marginal increase of the successful transmission probability with respect to the caching probability of a video with a certain quality reduces when the caching probability increases. Next, for each type of video, we consider the maximization of the successful transmission probability in the high user density region, which is a convex problem with an exceedingly large number of optimization variables. We propose a two-stage optimization method to obtain a low-complexity near-optimal solution by solving a relaxed convex problem and a related packing problem. The optimization results reveal the impact of the caching gain of a layer for an SVC-based video or a description for a DASH-based video on its caching probability. Finally, we show that the proposed solutions for SVC-based and DASH-based videos achieve significant performance gains over baseline schemes in the general and high user density regions, and demonstrate their respective operating regions, using numerical results based on real video sequences.

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.