The study of memory effects in quantum channels helps in developing characterization methods for open quantum systems and strategies for quantum error correction. Two main sets of channels exist, corresponding to system dynamics with no memory (Markovian) and with memory (non-Markovian). Interestingly, these sets have a non-convex geometry, allowing one to form a channel with memory from the addition of memoryless channels and vice-versa. Here, we experimentally investigate this non-convexity in a photonic setup by subjecting a single qubit to a convex combination of Markovian and non-Markovian channels. We use both divisibility and distinguishability as criteria for the classification of memory effects, with associated measures. Our results highlight some practical considerations that may need to be taken into account when using memory criteria to study system dynamics given by the addition of Markovian and non-Markovian channels in experiments.
We make a detailed analysis of quantumness for various quantum noise channels, both Markovian and non-Markovian. The noise channels considered include dephasing channels like random telegraph noise, non-Markovian dephasing and phase damping, as well as the non-dephasing channels such as generalized amplitude damping and Unruh channels. We make use of a recently introduced witness for quantumness based on the square $l_1$ norm of coherence. It is found that the increase in the degree of non-Markovianity increases the quantumness of the channel.
The archetypal one-qubit noisy channels ---depolarizing, phase-damping and amplitude-damping channels--- describe both Markovian and non-Markovian evolution. Simple microscopic models for the depolarizing channel, both classical and quantum, are considered. Microscopic models which describe phase damping and amplitude damping channels are briefly reviewed.
A set of schemes for secure quantum communication are analyzed under the influence of non-Markovian channels. By comparing with the corresponding Markovian cases, it is seen that the average fidelity in all these schemes can be maintained for relatively longer periods of time. The effects of non-Markovian noise on a number of facets of quantum cryptography, such as quantum secure direct communication, deterministic secure quantum communication and their controlled counterparts, quantum dialogue, quantum key distribution, quantum key agreement, etc., have been extensively investigated. Specific…
We address continuous variable quantum key distribution (QKD) in non-Markovian lossy channels and show how the non-Markovian features may be exploited to enhance security and/or to detect the presence and the position of an eavesdropper along the transmission line. In particular, we suggest a coherent states QKD protocol which is secure against individual attacks for arbitrarily low values of the overall transmission line. Our scheme relies on specific non-Markovian properties, and cannot be implemented in ordinary Markovian channels characterized by uniform losses.
The Markovian dynamics of a qubit is investigated in the scheme of random unitary dynamics, where Kraus operators are changed by an extra noise. The behavior of Markovianity is explored in the perturbed scenario. We provide a new algorithm for checking CP-divisibility (Markovianity) of a dynamical map.
We study the performance of two measures of non-Markovianity in detecting memory effects in two-qubit dephasing channels. By combining independent Markovian and non-Markovian noise on the qubits, our results show that the trace distance measure is able to detect the memory effects when at least one of the local channels displays non-Markovianity. A measure based on channel capacity, in turn, becomes non-zero when the global two-qubit dynamics shows memory effects. We apply these schemes to a well-known superdense coding protocol and demonstrate an optimal noise configuration to maximize the in…
We study a class of qubit non-Markovian general Pauli dynamical maps with multiple singularities in the generator. We discuss a few easy examples involving trigonometric or other non-monotonic time dependence of the map, and discuss in detail the structure of channels which don't have any trigonometric functional dependence. We demystify the concept of a singularity here, showing that it corresponds to a point where the dynamics can be regular but the map is momentarily non-invertible, and this gives a basic guideline to construct such non-invertible non-Markovian channels. Most members of the…
We study the dynamics of coherence-induced state ordering under incoherent channels, particularly four specific Markovian channels: $-$ amplitude damping channel, phase damping channel, depolarizing channel and bit flit channel for single-qubit states. We show that the amplitude damping channel, phase damping channel, and depolarizing channel do not change the coherence-induced state ordering by $l_1$ norm of coherence, relative entropy of coherence, geometric measure of coherence, and Tsallis relative $α$-entropies, while the bit flit channel does change for some special cases.
We study the curves of coherence for the Bell-diagonal states including $l_{1}$-norm of coherence and relative entropy of coherence under the Markovian channels in the first subsystem once. For a special Bell-diagonal state under bit-phase flip channel, we find frozen coherence under $l_{1}$ norm occurs, but relative entropy of coherence decrease. It illustrates that the occurrence of frozen coherence depends on the type of the measure of coherence. We study the coherence evolution of Bell-diagonal states under Markovian channels in the first subsystem $n$ times and find coherence under depola…
Quantum non-Markovianity of channels can be produced by mixing Markovian channels, as observed recently by various authors. We consider an analogous question of whether singularities of the channel can be produced by mixing non-singular channels, i.e., ones that lack them. Here we answer the question in the negative in the context of qubit Pauli channels. On the other hand, mixing channels with a singularity can lead to the elimination of singularities in the resultant channel. We distinguish between two types of singular channels, which lead under mixing to broadly quite different properties …
The classical capacity of a quantum channel with arbitrary Markovian correlated noise is evaluated. For the general case of a channel with long-term memory, which corresponds to a Markov chain which does not converge to equilibrium, the capacity is expressed in terms of the communicating classes of the Markov chain. For an irreducible and aperiodic Markov chain, the channel is forgetful, and one retrieves the known expression for the capacity.
We analyze the quantum evolution represented by a time-dependent family of generalized Pauli channels. This evolution is provided by the random decoherence channels with respect to the maximal number of mutually unbiased bases. We derive the necessary and sufficient conditions for the vanishing back-flow of information.
The notion of forgetfulness, used in discrete quantum memory channels, is slightly weakened in order to be applied to the case of continuous channels. This is done in the context of quantum memory channels with Markovian noise. As a case study, we apply the notion of weak-forgetfulness to a bosonic memory channel with additive noise. A suitable encoding and decoding unitary transformation allows us to unravel the effects of the memory, hence the channel capacities can be computed using known results from the memoryless setting.
The quantum channels with memory, known as non-Markovian channels, are of crucial importance for a realistic description of a variety of physical systems, and pave ways for new methods of decoherence control by manipulating the properties of environment such as its frequency spectrum. In this work, the reduced dynamics of coin in a discrete-time quantum walk is characterized as a non-Markovian quantum channel. A general formalism is sketched to extract the Kraus operators for a $t$-step quantum walk. Non-Markovianity, in the sense of P-indivisibility of the reduced coin dynamics, is inferred f…
Non-Markovian Noise Suppression Simplified through Channel Representation
Tsinghua University（清华大学）；Institute for Interdisciplinary Information Sciences, Tsinghua University（清华大学交叉信息研究院）；Center for Quantum Information, Institute for Interdisciplinary Information Sciences, Tsinghua University（清华大学交叉信息研究院量子信息中心）；Technology Innovation Institute（技术创新研究所）；A*STAR Quantum Innovation Centre (Q.InC)（A*STAR量子创新中心）；Agency for Science, Technology and Research (A*STAR)（新加坡科技研究局）；Institute of High Performance Computing (IHPC)（高性能计算研究所）；University of Oxford（牛津大学）；Imperial College London（帝国理工学院）；Quantum Motion
In this paper, we analyze the evolution of the generalized Pauli channels governed by the memory kernel master equation. We provide necessary and sufficient conditions for the memory kernel to give rise to the legitimate (completely positive and trace-preserving) quantum evolution. In particular, we analyze a class of kernels generating the quantum semi-Markov evolution, which is a natural generalization of the Markovian semigroup. Interestingly, the convex combination of Markovian semigroups goes beyond the semi-Markov case. Our analysis is illustrated with several examples.
Establishing the strong converse theorem for a communication channel confirms that the capacity of that channel, that is, the maximum achievable rate of reliable information communication, is the ultimate limit of communication over that channel. Indeed, the strong converse theorem for a channel states that coding at a rate above the capacity of the channel results in the convergence of the error to its maximum value 1 and that there is no trade-off between communication rate and decoding error. Here we prove that the strong converse theorem holds for the product-state capacity of quantum chan…
We investigate the problem of reversing quantum dynamics, specifically via optimal Petz recovery maps. We focus on typical decoherence channels, such as dephasing, depolarizing and amplitude damping. We illustrate how well a physically implementable recovery map simulates an inverse evolution. We extend this idea to explore the use of recovery maps as an approximation of inverse maps, and apply it in the context of non-Markovian dynamics. We show how this strategy attenuates non-Markovian effects, such as the backflow of information.
The time reversal of a completely-positive, nonequilibrium discrete-time quantum Markov evolution is derived via a suitable adjointness relation. Space-time harmonic processes are introduced for the forward and reverse-time transition mechanisms, and their role for relative entropy dynamics is discussed.
Entanglement and Quantum Coherence in Coupled Double Quantum Dots under Markovian and Non-Markovian Noisy Channels
Mohammed V University in Rabat（拉巴特穆罕默德五世大学）；Ibnou Zohr University（伊本佐赫尔大学）；Princess Nourah bint Abdulrahman University（公主努拉·宾特·阿卜杜勒拉赫曼大学）；University of Bisha（比沙大学）
The protocol of the teleportation of an arbitrary unknown one-qubit state is investigated when the quantum channel is subject to the decoherence from the non-Markovian environments with the memory effects. The quality of the teleportation measured by the average fidelity and minimal fidelity can be enhanced in the non-Markovian channel. The memory effects give rise to the revival of the fidelity which is better than that obtained by the Markovian channel.
We provide a simple and realistic model to study memory effects in a lossy bosonic quantum channel over arbitrary number of uses. The noise correlation among different uses is introduced by contiguous modes interactions which results in an exponential decay of the correlations over channel uses (modes). This model allows us to characterize the asymptotic behavior of the channel for classical information transmission.
Any realistic quantum system is inevitably subject to an external environment. This environment makes the open-system dynamics significant for many quantum tech-nologies, such as entangled-state engineering, quantum simulation, and quantum sensing. The information flow of a system to its environment usually induces a Markovian process, while the backflow of information from the environment exhibits non-Markovianity. The practical environment, usually consisting of a large number of degrees of freedom, is hard to control, despite some attempts on controllable transitions from Markovian to non-M…
We extend the non-Markovian quantum state diffusion (QSD) equation to open quantum systems which exhibit multi-channel coupling to a harmonic oscillator reservoir. Open quantum systems which have multi-channel reservoir coupling are those in which canonical transformation of reservoir modes cannot reduce the number of reservoir operators appearing in the interaction Hamiltonian to one. We show that the non-Markovian QSD equation for multi-channel reservoir coupling can, in some cases, lead to an exact master equation which we derive. We then derive the exact master equation for the three-level…
Quantum technologies rely on the ability to coherently manipulate, process and transfer information, encoded in quantum states, along quantum channels. Decoherence induced by the environment introduces errors, thus setting limits on the efficiency of any quantum-enhanced protocol or device. A fundamental bound on the ability of a noisy quantum channel to transmit quantum (classical) information is given by its quantum (classical) capacity. Generally, the longer is a quantum channel the more are the introduced errors, and hence the worse is its capacity. In this Letter we show that for non-Mark…
The relations between the resource theoretic measures of quantum coherence are rigorously investigated for various Markovian and non-Markovian channels for the two-qubit $X$ states with specific attention to the maximum and minimum attainable coherence and usefulness of these states in performing quantum teleportation in noisy environment. The investigation has revealed that under both dephasing and dissipative type noises the maximally entangled mixed states and Werner states lose their form and usefulness. However, maximally non-local mixed states (MNMSs) lose their identity in dissipative n…
We propose an effective method which can simplify the optimization of the increase of the trace distance over all pairs of initial states in calculating the non-Markovianity $\mathcal{N}$ for single channel open systems. For the amplitude damping channel, we can unify the results of Breuer $et$ $al$. [Phys. Rev. Lett. \bf 103\rm, 210401 (2009)] in the large-detuning case and the results of Xu $et$ $al$. [Phys. Rev. A \bf 81\rm, 044105 (2010)] in the resonant case; furthermore, for the general off-resonant cases we can obtain a very tight lower bound of $\mathcal{N}$. As another application of …
We address the non-Markovian character of quantum maps describing the interaction of a qubit with a random classical field. In particular, we evaluate trace- and capacity-based non-Markovianity measures for two relevant classes of environments showing non-Gaussian fluctuations, described respectively by random telegraph noise and colored noise with spectra of the the form $1/f^α$. We analyze the dynamics of both the trace distance and the quantum capacity, and show that the behavior of non-Markovianity based on both measures is qualitatively similar. Our results show that environments with a s…
The purpose of this paper is to set out the problems of modeling quantum communication and signal processing where the communication between systems via a non-Markovian channel. This is a general feature of quantum transmission lines. Our ultimate objective is to extend the networks rules that have been developed for Markovian models. To this end we recall the Hamiltonian description of such non-Markov models of transmission lines and their quantization. These have occurred in the context of non-quilibrium thermodynamics, but our interest is in the transmission lines as carriers of information…
We introduce a method to construct non-Markovian variants of completely positive (CP) dynamical maps, particularly, qubit Pauli channels. We identify non-Markovianity with the breakdown in CP-divisibility of the map, i.e., appearance of a not-completely-positive (NCP) intermediate map. In particular, we consider the case of non-Markovian dephasing in detail. The eigenvalues of the Choi matrix of the intermediate map crossover at a point which corresponds to a singularity in the canonical decoherence rate of the corresponding master equation, and thus to a momentary non-invertibility of the map…
We study the competing effects of simultaneous Markovian and non-Markovian decoherence mechanisms acting on a single spin. We show the existence of a threshold in the relative strength of such mechanisms above which the spin dynamics becomes fully Markovian, as revealed by the use of several non-Markovianity measures. We identify a measure-dependent nested structure of such thresholds, hinting at a causality relationship amongst the various non-Markovianity witnesses used in our analysis. Our considerations are then used to argue the unavoidably non-Markovian evolution of a single-electron qua…
We suggest and demonstrate an all-optical experimental setup to observe and engineer entanglement oscillations of a pair of polarization qubits in a non-Markovian channel. We generate entangled photon pairs by spontaneous parametric downconversion (SPDC), and then insert a programmable spatial light modulator in order to impose a polarization dependent phase-shift on the spatial domain of the SPDC output and to create an effective non-Markovian environment. Modulation of the enviroment spectrum is obtained by inserting a spatial grating on the signal arm. In our experiment, programmable oscill…
We study the memory property of the channels obtained by convex combinations of Markovian channels that are not necessarily quantum dynamical semigroups (QDSs). In particular, we characterize the geometry of the region of (non-)Markovian channels obtained by the convex combination of the three Pauli channels, as a function of deviation from the semigroup form in a family of channels. The regions are highly convex, and interestingly, the measure of the non-Markovian region shrinks with greater deviation from the QDS structure for the considered family, underscoring the counterintuitive nature o…
We investigate the dynamics of quantum and classical correlations in a system of two qubits under local colored-noise dephasing channels. The time evolution of a single qubit interacting with its own environment is described by a memory kernel non-Markovian master equation. The memory effects of the non-Markovian reservoirs introduce new features in the dynamics of quantum and classical correlations compared to the white noise Markovian case. Depending on the geometry of the initial state, the system can exhibit frozen discord and multiple sudden transitions between classical and quantum decoh…
We consider two non-Markovian models: Random Telegraph Noise (RTN) and non-Markovian dephasing (NMD). The memory in these models is studied from the perspective of quantum Fisher information flow. This is found to be consistent with the other well known witnesses of non-Markovianity. The two noise channels are characterized quantum information theoretically by studying their gate and channel fidelities. Further, the quantum coherence and its balance with mixedness is studied. This helps to put in perspective the role that the two noise channels can play in various facets of quantum information…
In this paper, we investigate the cohering and decohering power for the one-qubit Markovian channels with respect to coherence in terms of the $l_{1}$-norm, the R$\acute{e}$nyi $α$-relative entropy and the Tsallis $α$-relative entropy. In the case of $α=2$, the cohering and decohering power of the amplitude damping channel, the phase damping channel, the depolarizing channel, and the flip channels under the three measures of coherence are calculated analytically. The decohering power on the $x, y, z$ basis referring to the amplitude damping channel, the phase damping channel, the flip channel …
We study the classical capacity of a forgetful quantum memory channel that switches between two qubit depolarizing channels according to an ergodic Markov chain. The capacity of this quantum memory channel depends on the parameters of the two depolarizing channels and the memory of the ergodic Markov chain. When the number of input qubit's is two, we show that depending on channel parameters either the maximally entangled input states or product input states achieve the classical capacity. Our conjecture based on numerics is that as the number of input qubits are increased the classical capaci…
We study the non-Markovian dynamics of a two-mode bosonic system interacting with two uncorrelated thermal bosonic reservoirs. We present the solution to the exact microscopic Master equation in terms of the quantum characteristic function and study in details the dynamics of entanglement for bipartite Gaussian states. In particular, we analyze the effects of short-time system-reservoir correlations on the separability thresholds and show that the relevant parameter is the reservoir spectral density. If the frequencies of the involved modes are within the reservoir spectral density entanglemen…
With continuing improvements on the quality of fabricated quantum devices, it becomes increasingly crucial to analyze noisy quantum process in greater details such as characterizing the non-Markovianity in a quantitative manner. In this work, we propose an experimental protocol, termed Spectral Transfer Tensor Maps (SpecTTM), to accurately predict the RHP non-Markovian measure of any Pauli channels without state-preparation and measurement (SPAM) errors. In fact, for Pauli channels, SpecTTM even allows the reconstruction of highly-precised noise power spectrum for qubits. At last, we also disc…
We investigate the entanglement dynamics of continuous-variable quantum channels in terms of an entangled squeezed state of two cavity fields in a general non-Markovian environment. Using the Feynman-Vernon influence functional theory in the coherent-state representation, we derive an exact master equation with time-dependent coefficients reflecting the non-Markovian influence of the environment. The influence of environments with different spectral densities, e.g., Ohmic, sub-Ohmic, and super-Ohmic, is numerically studied. The non-Markovian process shows its remarkable influences on the entan…
We explore the impact of non-Markovian channels on the quantum correlations (QCs) of Haar uniformly generated random two-qubit input states with different ranks -- either one of the qubits (single-sided) or both the qubits independently (double-sided) are passed through noisy channels. Under dephasing and depolarizing channels with varying non-Markovian strength, entanglement and quantum discord of the output states collapse and revive with the increase of noise. We find that in case of the depolarizing double-sided channel, both the QCs of random states show a higher number of revivals on ave…
We develop a theory of linear witnesses for detecting non-Markovianity, based on the geometric structure of the set of Choi states for all Markovian evolutions having Lindblad type generators. We show that the set of all such Markovian Choi states form a convex and compact set under the small time interval approximation. Invoking geometric Hahn-Banach theorem, we construct linear witnesses to separate a given non-Markovian Choi state from the set of Markovian Choi states. We present examples of such witnesses for dephasing channel and Pauli channel in case of qubits. We further investigate the…
We study spin squeezing under non-Markovian channels, and consider an ensemble of $N$ independent spin-1/2 particles with exchange symmetry. Each spin interacts with its own bath, and the baths are independent and identical. For this kind of open system, the spin squeezing under decoherence can be investigated from the dynamics of the local expectations, and the multi-qubit dynamics can be reduced into the two-qubit one. The reduced dynamics is obtained by the hierarchy equation method, which is a exact without rotating-wave and Born-Markov approximation. The numerical results show that the sp…
A sequence of controlled collisions between a quantum system and its environment (composed of a set of quantum objects) naturally simulates (with arbitrary precision) any Markovian quantum dynamics of the system under consideration. In this paper we propose and study the problem of simulation of an {\it arbitrary} quantum channel via collision models. We show that a correlated environment is capable to simulate {\it non-Markovian} evolutions leading to any indivisible qubit channel. In particular, we derive the corresponding master equation generating a continuous time non-Markovian dynamics i…
We analyze the dynamics of the entanglement in two independent non-Markovian channels. In particular, we focus on the entanglement dynamics as a function of the initial states and the channel parameters like the temperature and the ratio $r$ between $ω_0$ the characteristic frequency of the quantum system of interest, and $ω_c$ the cut-off frequency of Ohmic reservoir. We give a stationary analysis of the concurrence and find that the dynamic of non-markovian entanglement concurrence $\mathcal{C}_ρ(t)$ at temperature $k_BT=0$ is different from the $k_BT>0$ case. We find that "entanglement sudd…
Finite-time Markovian channels, unlike their infinitesimal counterparts, do not form a convex set. As a particular instance of this observation, we consider the problem of mixing the three Pauli channels, conservatively assumed to be quantum dynamical semigroups, and fully characterize the resulting ``Pauli simplex.'' We show that neither the set of non-Markovian (completely positive indivisible) nor Markovian channels is convex in the Pauli simplex, and that the measure of non-Markovian channels is about 0.87. All channels in the Pauli simplex are P divisible. A potential application in the c…
In the present study, time evolution of quantum Cramer-Rao bound of entangled N00N state, as phase sensitivity, is determined by the aid of quantum estimation theory in the presence decoherence channels. Also, the dynamic quantum process as decoherence approach is characterized by quantum fisher information flow and entanglement amount in order to distinguish between Markovian and Non-Markovian process. The comparison between quantum fisher information and quantum fisher information flow assists to comprehend the phase sensitivity evolution corresponding to Non-Markovian and Markovian process.…
