Quantum non-Markovianity of a quantum noisy channel is typically identified with information backflow, or, more generally, with departure of the intermediate map from complete positivity. But here, we also indicate certain non-Markovian channels that can't be witnessed by the CP-divisibility criterion. In complex systems, non-Markovianity becomes more involved on account of subsystem dynamics. Here we study various facets of non-Markovian evolution, in the context of coined quantum walks, with particular stress on disambiguating the internal vs. environmental contributions to non-Markovian backflow. For the above problem of disambiguation, we present a general power-spectral technique based on a distinguishability measure such as trace-distance or correlation measure such as mutual information. We also study various facets of quantum correlations in the transition from quantum to classical random walks, under the considered non-Markovian noise models. The potential for the application of this analysis to the quantum statistical dynamics of complex systems is indicated.
We study the dynamics of discrete-time quantum walk using quantum coin operations,Ĉ(θ1) and C(θ2) in time-dependent periodic sequence. For the two-period quantum walk with the parameters θ1 and θ2 in the coin operations we show that the standard deviation [σ θ 1 ,θ 2 (t)] is the same as the minimum of standard deviation obtained from one of the one-period quantum walks with coin operations θ1 or θ2, σ θ 1 ,θ 2 (t) = min{σ θ 1 (t), σ θ 2 (t)}. Our numerical result is analytically corroborated using the dispersion relation obtained from the continuum limit of the dynamics. Using the dispersion relation for one-and two-period quantum walks, we present the bounds on the dynamics of three-and higher period quantum walks. We also show that the bounds for the two-period quantum walk will hold good for the split-step quantum walk which is also defined using two coin operators using θ1 and θ2. Unlike the previous known connection of discrete-time quantum walks with the massless Dirac equation where coin parameter θ = 0, here we show the recovery of the massless Dirac equation with non-zero θ parameters contributing to the intriguing interference in the dynamics in a totally non-relativistic situation. We also present the effect of periodic sequence on the entanglement between coin and position space.
Non-Markovian quantum effects are typically observed in systems interacting with structured reservoirs. Discrete-time quantum walks are prime example of such systems in which, quantum memory arises due to the controlled interaction between the coin and position degrees of freedom. Here we show that the information backflow that quantifies memory effects can be enhanced when the particle is subjected to uncorrelated static or dynamic disorder. The presence of disorder in the system leads to localization effects in 1-dimensional quantum walks. We shown that it is possible to infer about the nature of localization in position space by monitoring the information backflow in the reduced system. Further, we study other useful properties of quantum walk such as entanglement, interference and its connection to quantum non-Markovianity.
This paper questions one of the fundamental assumptions made in options pricing: that the daily returns of a stock are independent and identically distributed (IID). We apply an estimation procedure to years of daily return data for all stocks in the French CAC-40 index. We find six stocks whose log returns are best modeled by a first-order Markov chain, not an IID sequence. We further propose the Markov tree (MT) model, a modification of the standard binomial options pricing model, that takes into account this first-order Markov behavior. Empirical tests reveal that, for the six stocks found earlier, the MT model's option prices agree very closely with market prices. IntroductionIn the Black-Scholes model for the price of a European option, one of the main assumptions is that the price of the underlying asset follows a geometric Brownian motion [8]. If S t is the underlying asset price at time t, one assumes dS t = μS t dt + σS t dW t , where μ and σ are constants and W t is a Brownian motion. For fixed t > 0, define X n = log(S (n+1)t /S nt ). Then
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