Searcharxiv⌕ Search

arXiv subjects

M. Horibe

Publications and source records attributed to M. Horibe.

At least 19 recordsLinked to original sources
arXiv2023

Center Preserving Automorphisms of Finite Heisenberg Group over $\mathbb Z_N$

We investigate the group structure of center-preserving automorphisms of the finite Heisenberg group over $\mathbb Z_N$ with $U(1)$ extension, which arises in finite-dimensional quantum mechanics on a discrete phase space. Constructing an explicit splitting, it is shown that, for $N=2(2k+1)$, the group is isomorphic to the semidirect product of $Sp_N$ and $\mathbb Z_N^2$. Moreover, when N is divisible by $2^l (l \ge 2)$, the group has a non-trivial 2-cocycle, and its explicit form is provided. By utilizing the splitting, it is demonstrated that the corresponding projective Weil representation can be lifted to linear representation.

quant-ph↗
arXiv2021

Simple criterion for local distinguishability of generalized Bell states in prime dimension

Local distinguishability of sets of generalized Bell states (GBSs) is investigated. We first clarify the conditions such that a set of GBSs can be locally transformed to a certain type of GBS set that is easily distinguishable within local operations and one-way classical communication. We then show that, if the space dimension $d$ is a prime, these conditions are necessary and sufficient for sets of $d$ GBSs in $\mathbb{C}^d \otimes \mathbb{C}^d$ to be locally distinguishable. Thus we obtain a simple computable criterion for local distinguishability of sets of $d$ GBSs in prime dimension $d$.

quant-ph↗
arXiv2019

Minimum uncertainty states and completeness of non-negative quasi probability of finite-dimensional quantum systems

We construct minimum-uncertainty states and a non-negative quasi probability distribution for quantum systems on a finite-dimensional space. We reexamine the theorem of Massar and Spindel for the uncertainty relationof the two unitary operators related by the discrete Fourier transformation. It is shown that some assumptions in their proof can be justified by the use of the Perron-Frobenius theorem. The minimum-uncertainty states are the ones that saturate this uncertainty inequality. The continuum limit is closely analyzed by introducing a scale factor in the limiting scheme. Using the minimum-uncertainty states, we construct a non-negative quasi probability distribution. Its marginal distributions are smeared out. However, we show that this quasi probability is optimal in the sense that there does not exist a non-negative quasi probability distribution with sharper marginal properties if the translational covariance in the phase space is assumed. Generally, it is desirable that the quasi probability is complete, i.e., it contains full information of the state. We show that the obtained quasi probability is indeed complete if the dimension of the state space is odd, whereas it is not if the dimension is even.

quant-ph↗
arXiv2018

Quantum-state comparison and discrimination

We investigate the performance of discrimination strategy in the comparison task of known quantum states. In the discrimination strategy, one infers whether or not two quantum systems are in the same state on the basis of the outcomes of separate discrimination measurements on each system. In some cases with more than two possible states, the optimal strategy in minimum-error comparison is that one should infer the two systems are in different states without any measurement, implying that the discrimination strategy performs worse than the trivial "no-measurement" strategy. We present a sufficient condition for this phenomenon to happen. For two pure states with equal prior probabilities, we determine the optimal comparison success probability with an error margin, which interpolates the minimum-error and unambiguous comparison. We find that the discrimination strategy is not optimal except for the minimum-error case.

quant-ph↗
arXiv2018

Unitary Representation of Symplectic Group for Phase Point Operators on Discrete Phase Space

The phase point operator $Δ(q,p)$ is the quantum mechanical counterpart of the classical phase point $(q,p)$. The discrete form of $Δ(q,p)$ was formulated for an odd number of lattice points by Cohendet et al. and for an even number of lattice points by Leonhardt. Both versions have symplectic covariance, which is of fundamental importance in quantum mechanics. However, an explicit form of the projective representation of the symplectic group that appears in the covariance relation is not yet known. We show in this paper the existence and uniqueness of the representation, and describe a method to construct it using the Euclidean algorithm.

quant-ph↗
arXiv2010

Complete solution for unambiguous discrimination of three pure states with real inner products

Complete solutions are given in a closed analytic form for unambiguous discrimination of three general pure states with real mutual inner products. For this purpose, we first establish some general results on unambiguous discrimination of n linearly independent pure states. The uniqueness of solution is proved. The condition under which the problem is reduced to an (n-1)-state problem is clarified. After giving the solution for three pure states with real mutual inner products, we examine some difficulties in extending our method to the case of complex inner products. There is a class of set of three pure states with complex inner products for which we obtain an analytical solution.

quant-ph↗
arXiv2010

Unitary-process discrimination with error margin

We investigate a discrimination scheme between unitary processes. By introducing a margin for the probability of erroneous guess, this scheme interpolates the two standard discrimination schemes: minimum-error and unambiguous discrimination. We present solutions for two cases. One is the case of two unitary processes with general prior probabilities. The other is the case with a group symmetry: the processes comprise a projective representation of a finite group. In the latter case, we found that unambiguous discrimination is a kind of "all or nothing": the maximum success probability is either 0 or 1. We also closely analyze how entanglement with an auxiliary system improves discrimination performance.

quant-ph↗
arXiv2009

Discrimination with error margin between two states - Case of general occurrence probabilities -

We investigate a state discrimination problem which interpolates minimum-error and unambiguous discrimination by introducing a margin for the probability of error. We closely analyze discrimination of two pure states with general occurrence probabilities. The optimal measurements are classified into three types. One of the three types of measurement is optimal depending on parameters (occurrence probabilities and error margin). We determine the three domains in the parameter space and the optimal discrimination success probability in each domain in a fully analytic form. It is also shown that when the states to be discriminated are multipartite, the optimal success probability can be attained by local operations and classical communication. For discrimination of two mixed states, an upper bound of the optimal success probability is obtained.

quant-ph↗
arXiv2008

State discrimination with error margin and its locality

There are two common settings in a quantum-state discrimination problem. One is minimum-error discrimination where a wrong guess (error) is allowed and the discrimination success probability is maximized. The other is unambiguous discrimination where errors are not allowed but the inconclusive result "I don't know" is possible. We investigate discrimination problem with a finite margin imposed on the error probability. The two common settings correspond to the error margins 1 and 0. For arbitrary error margin, we determine the optimal discrimination probability for two pure states with equal occurrence probabilities. We also consider the case where the states to be discriminated are multipartite, and show that the optimal discrimination probability can be achieved by local operations and classical communication.

quant-ph↗
arXiv2008

Locality and nonlocality in quantum pure-state identification problems

Suppose we want to identify an input state with one of two unknown reference states, where the input state is guaranteed to be equal to one of the reference states. We assume that no classical knowledge of the reference states is given, but a certain number of copies of them are available instead. Two reference states are independently and randomly chosen from the state space in a unitary invariant way. This is called the quantum state identification problem, and the task is to optimize the mean identification success probability. In this paper, we consider the case where each reference state is pure and bipartite, and generally entangled. The question is whether the maximum mean identification success probability can be attained by means of a local operations and classical communication (LOCC) measurement scheme. Two types of identification problems are considered when a single copy of each reference state is available. We show that a LOCC scheme attains the globally achievable identification probability in the minimum-error identification problem. In the unambiguous identification problem, however, the maximal success probability by means of LOCC is shown to be less than the globally achievable identification probability.

quant-ph↗
arXiv2008

Local Quantum Pure-state Identification without Classical Knowledge

Suppose we want to distinguish two quantum pure states. We consider the case in which no classical knowledge on the two states is given and only a pair of samples of the two states is available. This problem is called quantum pure-state identification problem. Our task is to optimize the mean identification success probability, which is averaged over an independent unitary invariant distribution of the two reference states. In this paper, the two states are assumed bipartite states which are generally entangled. The question is whether the maximum mean identification success probability can be attained by means of an LOCC (Local Operations and Classical Communication) measurement scheme. We will show that this is possible by constructing a POVM which respects the conditions of LOCC.

quant-ph↗
arXiv2007

Stationary quantum Markov process for the Wigner function

As a stochastic model for quantum mechanics we present a stationary quantum Markov process for the time evolution of the Wigner function on a lattice phase space Z_N x Z_N with N odd. By introducing a phase factor extension to the phase space, each particle can be treated independently. This is an improvement on earlier methods that require the whole distribution function to determine the evolution of a constituent particle. The process has branching and vanishing points, though a finite time interval can be maintained between the branchings. The procedure to perform a simulation using the process is presented.

quant-ph↗
arXiv2006

Optimal estimation of a physical observable's expectation value for pure states

We study the optimal way to estimate the quantum expectation value of a physical observable when a finite number of copies of a quantum pure state are presented. The optimal estimation is determined by minimizing the squared error averaged over all pure states distributed in a unitary invariant way. We find that the optimal estimation is "biased", though the optimal measurement is given by successive projective measurements of the observable. The optimal estimate is not the sample average of observed data, but the arithmetic average of observed and "default nonobserved" data, with the latter consisting of all eigenvalues of the observable.

quant-ph↗
arXiv2006

Unambiguous pure state identification without classical knowledge

We study how to unambiguously identify a given quantum pure state with one of the two reference pure states when no classical knowledge on the reference states is given but a certain number of copies of each reference quantum state are presented. By the unambiguous identification, we mean that we are not allowed to make a mistake but our measurement can produce an inconclusive result. Assuming the two reference states are independently distributed over the whole pure state space in a unitary invariant way, we determine the optimal mean success probability for an arbitrary number of copies of the reference states and a general dimension of the state space. It is explicitly shown that the obtained optimal mean success probability asymptotically approaches that of the unambiguous discrimination as the number of the copies of the reference states increases.

quant-ph↗
arXiv2005

Quantum pure-state identification

We address a problem of identifying a given pure state with one of two reference pure states, when no classical knowledge on the reference states is given, but a certain number of copies of them are available. We assume the input state is guaranteed to be either one of the two reference states. This problem, which we call quantum pure state identification, is a natural generalization of the standard state discrimination problem. The two reference states are assumed to be independently distributed in a unitary invariant way in the whole state space. We give a complete solution for the averaged maximal success probability of this problem for an arbitrary number of copies of the reference states in general dimension. It is explicitly shown that the obtained mean identification probability approaches the mean discrimination probability as the number of the reference copies goes to infinity.

quant-ph↗
arXiv2005

Reexamination of optimal quantum state estimation of pure states

A direct derivation is given for the optimal mean fidelity of quantum state estimation of a d-dimensional unknown pure state with its N copies given as input, which was first obtained by M. Hayashi in terms of an infinite set of covariant positive operator valued measures (POVM's) and by Bruss and Macchiavello establishing a connection to optimal quantum cloning. An explicit condition for POVM measurement operators for optimal estimators is obtained, by which we construct optimal estimators with finite POVM using exact quadratures on a hypersphere. These finite optimal estimators are not generally universal, where universality means the fidelity is independent of input states. However, any optimal estimator with finite POVM for M(>N) copies is universal if it is used for N copies as input.

quant-ph↗
arXiv2005

Mean king's problem with mutually unbiased bases and orthogonal Latin squares

The mean king's problem with maximal mutually unbiased bases (MUB's) in general dimension d is investigated. It is shown that a solution of the problem exists if and only if the maximal number (d+1) of orthogonal Latin squares exists. This implies that there is no solution in d=6 or d=10 dimensions even if the maximal number of MUB's exists in these dimensions.

quant-ph↗