基于有限域的离散相空间

From the discrete Weyl -- Wigner formalism for symmetric ordering to a number -- phase Wigner function

The general Weyl -- Wigner formalism in finite dimensional phase spaces is investigated. Then this formalism is specified to the case of symmetric ordering of operators in an odd -- dimensional Hilbert space. A respective Wigner function on the discrete phase space is found and the limit, when the dimension of Hilbert space tends to infinity, is considered. It is shown that this limit gives the number -- phase Wigner function in quantum optics. Analogous results for the `almost' symmetric ordering in an even -- dimensional Hilbert space are obtained. Relations between the discrete Wigner funct…

离散相空间和最小不确定状态

Discrete phase space and minimum-uncertainty states

量子比特系统的量子态可以由离散相空间上的维格纳函数表示，相空间的每个轴取有限域中的值。在这个框架内，我们表明，人们可以理解的概念的“旋转不变状态”的任何集合的量子位，任何这样的状态是，在一个定义明确的意义上，最小的不确定性的状态。

Discrete phase space and continuous time relativistic quantum mechanics I: Planck oscillators and closed string-like circular orbits

The discrete phase space continuous time representation of relativistic quantum mechanics involving a characteristic length $l$ is investigated. Fundamental physical constants such as $\hbar$, $c$, and $l$ are retained for most sections of the paper. The energy eigenvalue problem for the Planck oscillator is solved exactly in this framework. Discrete concircular orbits of constant energy are shown to be circles $S^{1}_{n}$ of radii $2E_n =\sqrt{2n+1}$ within the discrete (1 + 1)-dimensional phase plane. Moreover, the time evolution of these orbits sweep out world-sheet like geometrical entitie…

Discrete-phase-randomized coherent state source and its application in quantum key distribution

Coherent state photon sources are widely used in quantum information processing. In many applications, such as quantum key distribution (QKD), a coherent state is functioned as a mixture of Fock states by assuming its phase is continuously randomized. In practice, such a crucial assumption is often not satisfied and, therefore, the security of existing QKD experiments is not guaranteed. To bridge this gap, we provide a rigorous security proof of QKD with discrete-phase-randomized coherent state sources. Our results show that the performance of the discrete-phase randomization case is close to …

离散Wigner函数与量子计算机的相空间表示

Discrete Wigner functions and the phase space representation of quantum computers

我们展示了如何在相空间中表示量子计算机（或任何具有$N$ -维希尔伯特空间的系统）的状态和演化。为此目的，我们使用离散版本的维格纳函数，对于任意的$N$，定义在一个相位空间网格的$2N\times 2N$点。我们计算这样的维格纳函数的状态是相关的量子计算。最后讨论了相空间中量子算法的性质，给出了Grover量子搜索算法的相空间表示。

Discrete-to-continuous transition in quantum phase estimation

We analyze the problem of quantum phase estimation where the set of allowed phases forms a discrete $N$ element subset of the whole $[0,2π]$ interval, $φ_n = 2πn/N$, $n=0,\dots N-1$ and study the discrete-to-continuous transition $N\rightarrow\infty$ for various cost functions as well as the mutual information. We also analyze the relation between the problems of phase discrimination and estimation by considering a step cost functions of a given width $σ$ around the true estimated value. We show that in general a direct application of the theory of covariant measurements for a discrete subgrou…

Transition probabilities and transition rates in discrete phase space

The evolution of the discrete Wigner function is formally similar to a probabilistic process, but the transition probabilities, like the discrete Wigner function itself, can be negative. We investigate these transition probabilities, as well as the transition rates for a continuous process, aiming particularly to give simple criteria for deciding when a set of such quantities corresponds to a legitimate quantum process. We also show how the transition rates for any Hamiltonian evolution can be worked out by expanding the Hamiltonian as a linear combination of displacement operators in the disc…

Continuous phase-space methods on discrete phase spaces

We show that discrete quasiprobability distributions defined via the discrete Heisenberg-Weyl group can be obtained as discretizations of the continuous $SU(N)$ quasiprobability distributions. This is done by identifying the phase-point operators with the continuous quantisation kernels evaluated at special points of the phase space. As an application we discuss the positive-P function and show that its discretization can be used to treat the problem of diverging trajectories. We study the dissipative long-range transverse-field Ising chain and show that the long-time dynamics of local observa…

离散量子相空间中的波包

Wave Packets in Discrete Quantum Phase Space

研究了离散相空间的量子力学的性质。在连续极限下，找到了最小测不准态，这些态成为高斯波包。适当选择的哈密顿量，使自由粒子运动的连续极限，它被发现，完整或近似的周期性时间演化可以导致。这代表了波包复活的一个例子，在连续极限中，波包是人们熟悉的直线上的自由粒子运动。最后，我们研究了离散相空间的不确定原理，得到了连续相空间的修正项。

Zak Phase in Discrete-Time Quantum Walks

We report on a simple scheme that may present a non-trivial geometric Zak phase ($Φ_{Zak}$) structure, which is based on a discrete-time quantum walk architecture. By detecting the Zak phase difference between two trajectories connecting adjacent Dirac points where the quasi-energy gap closes for opposite values of quasi-momentum ($k$), it is possible to identify geometric invariants. These geometric invariants correspond to $|Φ_{Zak}^{+(-)}-Φ_{Zak}^{-(+)}|=π$ and $|Φ_{Zak}^{+(-)}-Φ_{Zak}^{+(-)}|=0$, we argue that this effect can be directly measured.

On discrete structures in finite Hilbert spaces

We present a brief review of discrete structures in a finite Hilbert space, relevant for the theory of quantum information. Unitary operator bases, mutually unbiased bases, Clifford group and stabilizer states, discrete Wigner function, symmetric informationally complete measurements, projective and unitary t--designs are discussed. Some recent results in the field are covered and several important open questions are formulated. We advocate a geometric approach to the subject and emphasize numerous links to various mathematical problems.

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…

A Family of Weyl-Wigner Transforms for Discrete Variables Defined in a Finite-Dimensional Hilbert Space

We study the Weyl-Wigner transform in the case of discrete variables defined in a Hilbert space of finite prime-number dimensionality $N$. We define a family of Weyl-Wigner transforms as function of a phase parameter. We show that it is only for a specific value of the parameter that all the properties we have examined have a parallel with the case of continuous variables defined in an infinite-dimensional Hilbert space. A geometrical interpretation is briefly discussed.

First considerations on the generalized uncertainty principle for finite-dimensional discrete phase spaces

Generalized uncertainty principle and breakdown of the spacetime continuum certainly represent two important results derived of various approaches related to quantum gravity and black hole physics near the well-known Planck scale. The discreteness of space suggests, in particular, that all measurable lengths are quantized in units of a fundamental scale (in this case, the Planck length). Here, we propose a self-consistent theoretical framework for an important class of physical systems characterized by a finite space of states, and show that such a framework enlarges previous knowledge about g…