Probabilistic Distillation of Quantum Coherence.

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

We introduce and study the task of assisted coherence distillation. This task arises naturally in bipartite systems where both parties work together to generate the maximal possible coherence on one of the subsystems. Only incoherent operations are allowed on the target system, while general local quantum operations are permitted on the other; this is an operational paradigm that we call local quantum-incoherent operations and classical communication. We show that the asymptotic rate of assisted coherence distillation for pure states is equal to the coherence of assistance, an analog of the entanglement of assistance, whose properties we characterize. Our findings imply a novel interpretation of the von Neumann entropy: it quantifies the maximum amount of extra quantum coherence a system can gain when receiving assistance from a collaborative party. Our results are generalized to coherence localization in a multipartite setting and possible applications are discussed.

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.

Quantifying quantum coherence is a key task in the resource theory of\ncoherence. Here we establish a good coherence monotone in terms of a state\nconversion process, which automatically endows the coherence monotone with an\noperational meaning. We show that any state can be produced from some input\npure states via the corresponding incoherent channels. It is especially found\nthat the coherence of a given state can be well characterized by the least\ncoherence of the input pure states, so a coherence monotone is established by\nonly effectively quantifying the input pure states. In particular, we show that\nour proposed coherence monotone is the supremum of all the coherence monotones\nthat give the same coherence for any given pure state. Considering the\nconvexity, we prove that our proposed coherence measure is a subset of the\ncoherence measure based on the convex roof construction. As an application, we\ngive a concrete expression of our coherence measure by employing the geometric\ncoherence of a pure state. We also give a thorough analysis on the states of\nqubit and finally obtain series of analytic coherence measures.\n

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.

We define the coherence non-generating channel as the completely positive trace-preserving map which does not generate quantum coherence from an incoherent state. The incoherent operations are the strict subset of the coherence non-generating channels. We prove that the relative entropy of coherence is monotonically decreasing under the coherence non-generating channels, while the coherence of formation may increase under such channels. Interestingly, by combining mathematically the coherence of formation to the entanglement of formation, we show that the coherence of formation of a single-qubit is never increased by a coherence non-generating channel. This leads to the superadditivity property for the coherence increasing power of quantum channels, namely, while two channels can not increase coherence individually, they may increase the quantum coherence of a composed system. Further, we derive the general form of the rank-2 coherence non-generating qubit channels. Our results contribute to the resource theory of quantum coherence.

Quantum coherence as an important physical resource plays the key role in implementing various quantum tasks, whereas quantum coherence is generally nonincreasing under incoherent operations. In this paper, we analyse under which dynamical conditions the l 1-norm or the relative entropy of coherence can remain unchanged under strictly incoherent operations (freezing coherence). We provide a detailed analysis of their structure together with exact geometric conditions of freezing coherence. It reveals a remarkable feature: any strictly incoherent operation freezing coherence can be decomposed as a convex combination of unitary operations. This partially answers an open question named unitary decomposition of doubly-stochastic quantum operations [M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, (Cambridge University Press, Cambridge, 2000)]. Based on this analysis, we also give a complete classification of coherent states from operational coherence theory. This builds the counterpart of entanglement classification under LOCC.

Catalytic coherence transformations allow the otherwise impossible state transformations using only incoherent operations with the aid of an auxiliary system with finite coherence which is not being consumed in anyway. Here we find the necessary and sufficient conditions for the deterministic and stochastic catalytic coherence transformations between pair of pure quantum states. In particular, we show that the simultaneous decrease of a family of R\'enyi entropies of the diagonal parts of the states under consideration are necessary and sufficient conditions for the deterministic catalytic coherence transformations. Similarly, for stochastic catalytic coherence transformations we find the necessary and sufficient conditions for achieving higher optimal probability of conversion. We, thus, completely characterize the coherence transformations amongst pure quantum states under incoherent operations. We give numerous examples to elaborate our results. We also explore the possibility of the same system acting as a catalyst for itself and find that indeed {\it self catalysis} is possible. Further, for the cases where no catalytic coherence transformation is possible we provide entanglement assisted coherence transformations and find the necessary and sufficient conditions for such transformations.

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 investigate assisted enhancement of quantum coherence in a bipartite setting with control and target systems, which converts the coherence of the control qubit into the enhanced coherence of the target qubit. We assume that only incoherent operations and measurements can be applied locally and classical information can be exchanged. In addition, the two subsystems are also coupled by a fixed Hamiltonian whose interaction strength can be controlled. This coupling does not generate any local coherence from incoherent input states. We show that in this setting a measurement and feed-forward based protocol can deterministically enhance the coherence of the target system while fully preserving its purity. The protocol can be iterated and several copies of the control state can be consumed to drive the target system arbitrarily close to a maximally coherent state. We experimentally demonstrate this protocol with a photonic setup and observe the enhancement of coherence for up to five iterations of the protocol.

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.

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.

In this paper, by minimizing the coherence quantifiers over all states in an\n$\\epsilon$ ball around a given state, we define a generalized smooth\nquantifier, called the $\\epsilon$-smooth measure of coherence. We use it to\nestimate the difference between the expected state and the actually prepared\nstate and quantify quantum coherence contained in an actually prepared state,\nand it can been interpreted as the minimal coherence guaranteed to present in\nan $\\epsilon$ ball around given quantum state. We find that the\n$\\epsilon$-smooth measure of any coherence monotone is still a coherence\nmonotone, but it does not satisfy monotonicity on average under incoherent\noperations. We show the $\\epsilon$-smooth measure of coherence is continuous\neven if the original coherence quantifier is not. We also study the\n$\\epsilon$-smooth measure of distance-based coherence quantifiers, and some\ninteresting properties are given. Moreover, we discuss the dual form of the\n$\\epsilon$-smooth measure of coherence by maximizing over all states in an\n$\\epsilon$ ball around the given state and show that the dual $\\epsilon$-smooth\nmeasure of coherence provides an upper bound of one-shot coherence\ndistillation.\n

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.