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arXiv:1412.7338

Asymptotic analysis of the one-dimensional quantum walks by the Tsallis and Rényi entropies

Ide, Yusuke, Norio Konno, Junji Shikata

Published Dec 23, 2014• 0 citations
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Abstract

The Tsallis and Rényi entropies are important quantities in the information theory, statistics and related fields because the Tsallis entropy is an one parameter generalization of the Shannon entropy and the Rényi entropy includes several useful entropy measures such as the Shannon entropy, Min-entropy and so on, as special choices of its parameter. On the other hand, the discrete-time quantum walk plays important roles in various applications, for example, quantum speed-up algorithm and universal computation. In this paper, we show limiting behaviors of the Tsallis and Rényi entropies for discrete-time quantum walks on the line which are starting from the origin and defined by arbitrary coin and initial state. The results show that the Tsallis entropy behaves in polynomial order of time with the parameter dependent exponent while the Rényi entropy tends to infinity in logarithmic order of time independent of the choice of the parameter. Moreover, we show the difference between the Rényi entropy and the logarithmic function characterizes by the Rényi entropy of the limit distribution of the quantum walk. In addition, we show an example of asymptotic behavior of the conditional Rényi entropies of the quantum walk.

DOI
10.48550/arxiv.1412.7338

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The Tsallis and Rényi entropies are important quantities in the information theory, statistics and related fields because the Tsallis entropy is an one parameter generalization of the Shannon entropy and the Rényi entropy includes several useful entropy measures such as the Shannon entropy, Min-entropy and so on, as special choices of its parameter. On the other hand, the discrete-time quantum walk plays important roles in various applications, for example, quantum speed-up algorithm and universal computation. In this paper, we show limiting behaviors of the Tsallis and Rényi entropies for discrete-time quantum walks on the line which are starting from the origin and defined by arbitrary coin and initial state. The results show that the Tsallis entropy behaves in polynomial order of time with the parameter dependent exponent while the Rényi entropy tends to infinity in logarithmic order of time independent of the choice of the parameter. Moreover, we show the difference between the Rényi entropy and the logarithmic function characterizes by the Rényi entropy of the limit distribution of the quantum walk. In addition, we show an example of asymptotic behavior of the conditional Rényi entropies of the quantum walk.
The discrete-time quantum walk is a quantum counterpart of the random walk. It is expected that the model plays important roles in the quantum field. In the quantum information theory, entanglement is a key resource. We use the von Neumann entropy to measure the entanglement between the coin and the particle's position of the quantum walks. Also we deal with the Shannon entropy which is an important quantity in the information theory. In this paper, we show limits of the von Neumann entropy and the Shannon entropy of the quantum walks on the one dimensional lattice starting from the origin defined by arbitrary coin and initial state. In order to derive these limits, we use the path counting method which is a combinatorial method for computing probability amplitude.
Abstract A comparative study of 1D quantum structures which allows analytic expressions for the position and momentum Rényi R ( α ) and Tsallis T ( α ) entropies, focuses on extracting the most characteristic physical features of these one-parameter functionals. Consideration of the harmonic oscillator reconfirms the special status of the Gaussian distribution: at any parameter α it converts into the equality both Rényi and Tsallis uncertainty relations removing for the latter an additional requirement 1/2 ≤ α ≤ 1 that is a necessary condition for all other geometries. It is shown that the lowest limit of the semi-infinite range of the dimensionless parameter α where momentum components exist strongly depends on the position potential and/or boundary condition for the position wave function. Asymptotic limits reveal that in either space the entropies R ( α ) and T ( α ) approach their Shannon counterpart, α = 1, along different paths. Similarities and differences between the two entropies and their uncertainty relations are exemplified. Some unsolved problems are also indicated.
We investigate the computational hardness of estimating the quantum ααα-Rényi entropy SαR(ρ)=ln⁡Tr(ρα)1−α{\rm S}^{\tt R}_α(ρ) = \frac{\ln {\rm Tr}(ρ^α)}{1-α}SαR​(ρ)=1−αlnTr(ρα)​ and the quantum qqq-Tsallis entropy SqT(ρ)=1−Tr(ρq)q−1{\rm S}^{\tt T}_q(ρ) = \frac{1-{\rm Tr}(ρ^q)}{q-1}SqT​(ρ)=q−11−Tr(ρq)​, both of which converge to the von Neumann entropy as the order approaches 111. The promise problems Quantum ααα-Rényi Entropy Approximation (RényiQEAα_αα​) and Quantum qqq-Tsallis Entropy Approximation (TsallisQEAq_qq​) ask whether $ {\rm S}^ {\tt R}_α(ρ)$ or SqT(ρ){\rm S}^{\tt T}_q(ρ)SqT​(ρ), is at least τ1τ_1τ1​ or at most τ2τ_2τ2​, where τ1−τ2τ_1 - τ_2τ1​−τ2​ is typically a positive constant. Previous hardness results cover only the von Neumann entropy (order 111) and some cases of the quantum qqq-Tsallis entropy, while existing approaches do not readily extend to other orders. We establish that for all positive real ααα and qqq, and also for α=∞α=\inftyα=∞, the rank-222 variants Rank2RényiQEAα_αα​ and Rank2TsallisQEAq_qq​ are BQP-hard. Combined with prior (rank-dependent) quantum query algorithms in Wang, Guan, Liu, Zhang, and Ying (TIT 2024), Wang, Zhang, and Li (TIT 2024), and Liu and Wang (SODA 2025), as well as the one derived from O'Donnell and Wright (STOC 2016), our results imply: - For all real orders α>0α> 0α>0 or α=∞α=\inftyα=∞, and for all real orders 0<q≤10 < q \leq 10<q≤1, LowRankRényiQEAα_αα​ and LowRankTsallisQEAq_qq​ are BQP-complete, where both are restricted versions of RényiQEAα_αα​ and TsallisQEAq_qq​ with ρρρ of polynomial rank. - For all real order q>1q>1q>1, TsallisQEAq_qq​ is BQP-complete. Our hardness results stem from reductions based on new inequalities relating the ααα-Rényi or qqq-Tsallis binary entropies of different orders. These reductions differ substantially from previous approaches, and the inequalities are of independent interest.
The discrete-time quantum walk is a quantum counterpart of the random walk. It is expected that the model plays important roles in the quantum field. In the quantum information theory, entanglement is a key resource. We use the von Neumann entropy to measure the entanglement between the coin and the particle's position of the quantum walks. Also we deal with the Shannon entropy which is an important quantity in the information theory. In this paper, we show limits of the von Neumann entropy and the Shannon entropy of the quantum walks on the one dimensional lattice starting from the origin defined by arbitrary coin and initial state. In order to derive these limits, we use the path counting method which is a combinatorial method for computing probability amplitude.
We report a systematic study of asymptotic behavior of entanglement between the position and coin degrees of freedom for a one-dimensional discrete quantum walk. Effects of coin bias, initial coin asymmetry and phase difference and non-locality of initial position state on the fluctuations and asymptotic value of entanglement are investigated by using Fourier approach. Fluctuations in entropy are found to die out as power law in time as t-α with α = 1/2,3/2 and 5/2 depending on the initial state parameters of the system.
We consider a discrete-time quantum walk W_{t,\kappa} at time t on a graph with joined half lines J_\kappa, which is composed of \kappa half lines with the same origin. Our analysis is based on a reduction of the walk on a half line. The idea plays an important role to analyze the walks on some class of graphs with symmetric initial states. In this paper, we introduce a quantum walk with an enlarged basis and show that W_{t,\kappa} can be reduced to the walk on a half line even if the initial state is asymmetric. For W_{t,\kappa}, we obtain two types of limit theorems. The first one is an asymptotic behavior of W_{t,\kappa} which corresponds to localization. For some conditions, we find that the asymptotic behavior oscillates. The second one is the weak convergence theorem for W_{t,\kappa}. On each half line, W_{t,\kappa} converges to a density function like the case of the one-dimensional lattice with a scaling order of t. The results contain the cases of quantum walks starting from the general initial state on a half line with the general coin and homogeneous trees with the Grover coin.