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Masahito Hayashi

Publications and source records attributed to Masahito Hayashi.

At least 253 records · Page 14Linked to original sources
arXiv2011

Changepoint Problem in Quantumn Setting

In the changepoint problem, we determine when the distribution observed has changed to another one. We expand this problem to the quantum case where copies of an unknown pure state are being distributed. We study the fundamental case, which has only two candidates to choose. This problem is equal to identifying a given state with one of the two unknown states when multiple copies of the states are provided. In this paper, we assume that two candidate states are distributed independently and uniformly in the space of the whole pure states. The minimum of the averaged error probability is given and the optimal POVM is defined as to obtain it. Using this POVM, we also compute the error probability which depends on the inner product. These analytical results allow us to calculate the value in the asymptotic case, where this problem approaches to the usual discrimination problem.

quant-ph↗
arXiv2011

Phase estimation with photon number constraint

Many researches proposed the use of the noon state as the input state for phase estimation, which is one topic of quantum metrology. This is because the input noon state provides the maximum Fisher information at the specific point. However, the Fisher information does not necessarily give the attainable bound for estimation error. In this paper, we adopt the local asymptotic mini-max criterion as well as the mini-max criterion, and show that the maximum Fisher information does not give the attainable bound for estimation error under these criteria in the phase estimation. We also propose the optimal input state under the constraints for photon number of the input state instead of the noon state.

quant-ph↗
arXiv2010

Multi-copy and stochastic transformation of multipartite pure states

Characterizing the transformation and classification of multipartite entangled states is a basic problem in quantum information. We study the problem under two most common environments, local operations and classical communications (LOCC), stochastic LOCC and two more general environments, multi-copy LOCC (MCLOCC) and multi-copy SLOCC (MCSLOCC). We show that two transformable multipartite states under LOCC or SLOCC are also transformable under MCLOCC and MCSLOCC. What's more, these two environments are equivalent in the sense that two transformable states under MCLOCC are also transformable under MCSLOCC, and vice versa. Based on these environments we classify the multipartite pure states into a few inequivalent sets and orbits, between which we build the partial order to decide their transformation. In particular, we investigate the structure of SLOCC-equivalent states in terms of tensor rank, which is known as the generalized Schmidt rank. Given the tensor rank, we show that GHZ states can be used to generate all states with a smaller or equivalent tensor rank under SLOCC, and all reduced separable states with a cardinality smaller or equivalent than the tensor rank under LOCC. Using these concepts, we extended the concept of "maximally entangled state" in the multi-partite system.

quant-ph↗
arXiv2010

Comparison between the Cramer-Rao and the mini-max approaches in quantum channel estimation

In a unified viewpoint in quantum channel estimation, we compare the Cramer-Rao and the mini-max approaches, which gives the Bayesian bound in the group covariant model. For this purpose, we introduce the local asymptotic mini-max bound, whose maximum is shown to be equal to the asymptotic limit of the mini-max bound. It is shown that the local asymptotic mini-max bound is strictly larger than the Cramer-Rao bound in the phase estimation case while the both bounds coincide when the minimum mean square error decreases with the order O(1/n). We also derive a sufficient condition for that the minimum mean square error decreases with the order O(1/n).

quant-ph↗
arXiv2010

Additivity and non-additivity of multipartite entanglement measures

We study the additivity property of three multipartite entanglement measures, i.e. the geometric measure of entanglement (GM), the relative entropy of entanglement and the logarithmic global robustness. First, we show the additivity of GM of multipartite states with real and non-negative entries in the computational basis. Many states of experimental and theoretical interests have this property, e.g. Bell diagonal states, maximally correlated generalized Bell diagonal states, generalized Dicke states, the Smolin state, and the generalization of Dür's multipartite bound entangled states. We also prove the additivity of other two measures for some of these examples. Second, we show the non-additivity of GM of all antisymmetric states of three or more parties, and provide a unified explanation of the non-additivity of the three measures of the antisymmetric projector states. In particular, we derive analytical formulae of the three measures of one copy and two copies of the antisymmetric projector states respectively. Third, we show, with a statistical approach, that almost all multipartite pure states with sufficiently large number of parties are nearly maximally entangled with respect to GM and relative entropy of entanglement. However, their GM is not strong additive; what's more surprising, for generic pure states with real entries in the computational basis, GM of one copy and two copies, respectively, are almost equal. Hence, more states may be suitable for universal quantum computation, if measurements can be performed on two copies of the resource states. We also show that almost all multipartite pure states cannot be produced reversibly with the combination multipartite GHZ states under asymptotic LOCC, unless relative entropy of entanglement is non-additive for generic multipartite pure states.

quant-ph↗
arXiv2010

Practical implementation and error bounds of integer-type general algorithm for higher order differential equations

In our preceding paper, we have proposed an algorithm for obtaining finite-norm solutions of higher-order linear ordinary differential equations of the Fuchsian type [\sum_m p_m (x) (d/dx)^m] f(x) = 0 (where p_m is a polynomial with rational-number-valued coefficients), by using only the four arithmetical operations on integers, and we proved its validity. For any nonnegative integer k, it is guaranteed mathematically that this method can produce all the solutions satisfying \int |f(x)|^2 (x^2+1)^k dx < \infty, under some conditions. We materialize this algorithm in practical procedures. An interger-type quasi-orthogonalization used there can suppress the explosion of calculations. Moreover, we give an upper limit of the errors. We also give some results of numerical experiments and compare them with the corresponding exact analytical solutions, which show that the proposed algorithm is successful in yielding solutions with high accuracy (using only arithmetical operations on integers).

math.NA↗
arXiv2010

Local hypothesis testing between a pure bipartite state and the white noise state

In this paper, we treat a local discrimination problem in the framework of asymmetric hypothesis testing. We choose a known bipartite pure state $\ketΨ$ as an alternative hypothesis, and the completely mixed state as a null hypothesis. As a result, we analytically derive an optimal type 2 error and an optimal POVM for one-way LOCC POVM and Separable POVM. For two-way LOCC POVM, we study a family of simple three-step LOCC protocols, and show that the best protocol in this family has strictly better performance than any one-way LOCC protocol in all the cases where there may exist difference between two-way LOCC POVM and one-way LOCC POVM.

quant-ph↗
arXiv2010

General theory for integer-type algorithm for higher order differential equations

Based on functional analysis, we propose an algorithm for finite-norm solutions of higher-order linear Fuchsian-type ordinary differential equations (ODEs) P(x,d/dx)f(x)=0 with P(x,d/dx):=[\sum_m p_m (x) (d/dx)^m] by using only the four arithmetical operations on integers. This algorithm is based on a band-diagonal matrix representation of the differential operator P(x,d/dx), though it is quite different from the usual Galerkin methods. This representation is made for the respective CONSs of the input Hilbert space H and the output Hilbert space H' of P(x,d/dx). This band-diagonal matrix enables the construction of a recursive algorithm for solving the ODE. However, a solution of the simultaneous linear equations represented by this matrix does not necessarily correspond to the true solution of ODE. We show that when this solution is an l^2 sequence, it corresponds to the true solution of ODE. We invent a method based on an integer-type algorithm for extracting only l^2 components. Further, the concrete choice of Hilbert spaces H and H' is also given for our algorithm when p_m is a polynomial or a rational function with rational coefficients. We check how our algorithm works based on several numerical demonstrations related to special functions, where the results show that the accuracy of our method is extremely high.

math.NA↗
arXiv2010

Differentiability of eigenfunctions of the closures of differential operators with rational coefficient functions

In this paper, for an operator defined by the action of an M-th order differential operator with rational-type coefficients on the function space L_k^2(R):={f: measurable | \|f\|_k <\infty} with norm \|f\|_k^2:= \int |f(x)|^2 (x^2+1)^k dx (k \in Z), we prove the regularity (continuity and differentiability up to M times) of the eigenfunctions of its closure (with respect to the graph norm), except at singular points of the corresponding ordinary differential equation without any assumptions for the Sobolev space, i.e., without any assumptions about the m-th order derivatives of the eigenfunctions with m=1,2,.., M-1. (For the special case of k=0, we prove this regularity for the usual L^2(R).) Especially, we show a one-to-one correspondence between the eigenfunctions of its closure and the solutions in C^M(R)\cap L_k^2(R) of the corresponding differential equation under the condition above when there is no singular point for this differential equation. This one-to-one correspondence is shown in the basic framework of an algorithm proposed in our preceding paper, which can determine all solutions in C^M\cap L_k^2(R) of the ordinary differential equation then.

math.CA↗
arXiv2010

Construction of wiretap codes from ordinary channel codes

From an arbitrary given channel code over a discrete or Gaussian memoryless channel, we construct a wiretap code with the strong security. Our construction can achieve the wiretap capacity under mild assumptions. The key tool is the new privacy amplification theorem bounding the eavesdropped information in terms of the Gallager function.

cs.IT↗
arXiv2009

The geometric measure of entanglement for a symmetric pure state with positive amplitudes

In this paper for a class of symmetric multiparty pure states we consider a conjecture related to the geometric measure of entanglement: 'for a symmetric pure state, the closest product state in terms of the fidelity can be chosen as a symmetric product state'. We show that this conjecture is true for symmetric pure states whose amplitudes are all non-negative in a computational basis. The more general conjecture is still open.

quant-ph↗
arXiv2009

Updating Quantum Cryptography Report ver. 1

Quantum cryptographic technology (QCT) is expected to be a fundamental technology for realizing long-term information security even against as-yet-unknown future technologies. More advanced security could be achieved using QCT together with contemporary cryptographic technologies. To develop and spread the use of QCT, it is necessary to standardize devices, protocols, and security requirements and thus enable interoperability in a multi-vendor, multi-network, and multi-service environment. This report is a technical summary of QCT and related topics from the viewpoints of 1) consensual establishment of specifications and requirements of QCT for standardization and commercialization and 2) the promotion of research and design to realize New-Generation Quantum Cryptography.

quant-ph↗
arXiv2009

Quantum hypothesis testing with group symmetry

The asymptotic discrimination problem of two quantum states is studied in the setting where measurements are required to be invariant under some symmetry group of the system. We consider various asymptotic error exponents in connection with the problems of the Chernoff bound, the Hoeffding bound and Stein's lemma, and derive bounds on these quantities in terms of their corresponding statistical distance measures. A special emphasis is put on the comparison of the performances of group-invariant and unrestricted measurements.

quant-ph↗
arXiv2009

Capacity with energy constraint in coherent state channel

We consider two kind of energy constraints when the output state is a coherent state. One is a constraint on the total energy during a fixed period; the other is a constraint on the total energy for a single code. The first setting can be easily dealt with by using the conventional capacity formula. The second setting requires the general capacity formula for a classical-quantum channel.

quant-ph↗
arXiv2008

Fourier Analytic Approach to Phase Estimation

For a unified analysis on the phase estimation, we focus on the limiting distribution. It is shown that the limiting distribution can be given by the absolute square of the Fourier transform of $L^2$ function whose support belongs to $[-1,1]$. Using this relation, we study the relation between the variance of the limiting distribution and its tail probability. As our result, we prove that the protocol minimizing the asymptotic variance does not minimize the tail probability. Depending on the width of interval, we derive the estimation protocol minimizing the tail probability out of a given interval. Such an optimal protocol is given by a prolate spheroidal wave function which often appears in wavelet or time-limited Fourier analysis. Also, the minimum confidence interval is derived with the framework of interval estimation that assures a given confidence coefficient.

quant-ph↗
arXiv2008

Information Spectrum Approach to Second-Order Coding Rate in Channel Coding

Second-order coding rate of channel coding is discussed for general sequence of channels. The optimum second-order transmission rate with a constant error constraint $ε$ is obtained by using the information spectrum method. We apply this result to the discrete memoryless case, the discrete memoryless case with a cost constraint, the additive Markovian case, and the Gaussian channel case with an energy constraint. We also clarify that the Gallager bound does not give the optimum evaluation in the second-order coding rate.

cs.IT↗