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Minimum uncertainty state and non-negative quasi probability of qudit

T. Hashimoto, A. Hayashi, M. Horibe

Published Apr 11, 2017• 0 citations
OpenAlex

Abstract

For a qudit, a quantum system on a finite dimensional Hilbert space, we construct minimum uncertainty states and a non-negative quasi probability distribution. They are the discrete version of the coherent states and the Husimi function in the continuous quantum mechanics. The two bases related by the discrete Fourier transformation are identified with the position and momentum bases. We prove an inequality concerning uncertainties of the position and momentum of a qudit. The minimum uncertainty states are the ones which saturate the inequality. Like the coherent states, they are not mutually orthogonal but comprise an overcomplete set in the state space. As the dimension of the Hilbert space goes to infinity, these states approach the coherent states. Using the minimum uncertain states, we construct a non-negative quasi probability distribution for a qudit. Its marginal distributions are smeared out, but we show that there does not exist a non-negative quasi probability distribution with sharper marginal properties if the translational covariance is assumed.

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For a qudit, a quantum system on a finite dimensional Hilbert space, we construct minimum uncertainty states and a non-negative quasi probability distribution. They are the discrete version of the coherent states and the Husimi function in the continuous quantum mechanics. The two bases related by the discrete Fourier transformation are identified with the position and momentum bases. We prove an inequality concerning uncertainties of the position and momentum of a qudit. The minimum uncertainty states are the ones which saturate the inequality. Like the coherent states, they are not mutually orthogonal but comprise an overcomplete set in the state space. As the dimension of the Hilbert space goes to infinity, these states approach the coherent states. Using the minimum uncertain states, we construct a non-negative quasi probability distribution for a qudit. Its marginal distributions are smeared out, but we show that there does not exist a non-negative quasi probability distribution with sharper marginal properties if the translational covariance is assumed.
We construct minimum-uncertainty states and a non-negative quasiprobability distribution for quantum systems on a finite-dimensional space. We reexamine the theorem of Massar and Spindel for the uncertainty relation of 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 quasiprobability distribution. Its marginal distributions are smeared out. However, we show that this quasiprobability is optimal in the sense that there does not exist a non-negative quasiprobability distribution with sharper marginal properties if the translational covariance in the phase space is assumed. Generally, it is desirable that the quasiprobability is complete, i.e., it contains full information of the state. We show that the obtained quasiprobability is indeed complete if the dimension of the state space is odd, whereas it is not if the dimension is even.
Uncertainty relations are usually stated as bounds on selected combinations of variances, but the full covariance matrix contains substantially richer information about the geometry of quantum state space and about the operational capabilities of quantum systems. Here we characterize the covariance matrices attainable by a finite-dimensional canonical pair of observables related by the discrete Fourier transform, the natural analogue of position and momentum in a finite Hilbert space. We combine analytic arguments with convex-geometric and semidefinite-programming methods based on joint numerical ranges to describe the admissible region through unitary invariants, in particular the trace and determinant of the covariance matrix. This provides a systematic way to identify extremal states, generalizing the notion of minimum-uncertainty states, and to quantify how the discrete uncertainty geometry approaches its continuous counterpart with increasing dimension. We further show that the resulting covariance-matrix characterization has direct consequences for applications: it yields accuracy bounds for multi parameter estimation protocols and separability criteria for finite-dimensional bipartite systems, including discrete analogues of continuous-variable EPR-type witnesses. Our results establish a systematic and versatile platform for connecting uncertainty relations, convex quantum geometry, metrology, and entanglement detection in finite-dimensional systems.
We derive an uncertainty relation for two unitary operators which obey a commutation relation of the form UV=e(i phi) VU. Its most important application is to constrain how much a quantum state can be localized simultaneously in two mutually unbiased bases related by a discrete fourier transform. It provides an uncertainty relation which smoothly interpolates between the well-known cases of the Pauli operators in two dimensions and the continuous variables position and momentum. This work also provides an uncertainty relation for modular variables, and could find applications in signal processing. In the finite dimensional case the minimum uncertainty states, discrete analogues of coherent and squeezed states, are minimum energy solutions of Harper's equation, a discrete version of the harmonic oscillator equation.
Classical mechanics, in the Koopman-von Neumann formulation, is described in Hilbert space. It is shown here that classical canonical transformations are generated by Hermitian operators that are in general noncommutative. This naturally brings about uncertainty relations inherent in classical mechanics, for example between position and the generator of space translations, between momentum and the generator of momentum translations, and between dynamical time and the Liouvillian, to name a few. Further, it is shown that the Wigner representation produces a quasi-probability distribution that can take on negative values. Thus, two of the hallmark features of quantum mechanics are reproduced, and become apparent, in a Hilbert-space formulation of classical mechanics.
Negativity in a quasiprobability representation is typically interpreted as an indication of nonclassical behavior. However, this does not preclude states that are non-negative from exhibiting phenomena typically associated with quantum mechanics---the single qubit stabilizer states have non-negative Wigner functions and yet play a fundamental role in many quantum information tasks. We seek to determine what other sets of quantum states and measurements of a qubit can be non-negative in a quasiprobability distribution, and to identify nontrivial groups of unitary transformations that permute the states in such a set. These sets of states and measurements are analogous to the single qubit stabilizer states. We show that no quasiprobability representation of a qubit can be non-negative for more than two bases in any plane of the Bloch sphere. Furthermore, there is a unique set of four bases that can be non-negative in an arbitrary quasiprobability representation of a qubit. We provide an exhaustive list of the sets of single qubit bases that are non-negative in some quasiprobability distribution and are also closed under a group of unitary transformations. This list includes two nontrivial families of three bases that both include the single qubit stabilizer states as a special case. For qudits, we prove that there can be no more than 2d2{2}^{{d}^{2}}2d2 states in non-negative bases of a ddd-dimensional Hilbert space in any quasiprobability representation. Furthermore, these bases must satisfy certain symmetry constraints, corresponding to requiring the bases to be sufficiently different from each other.
The Born rule provides a probability vector (distribution) with a quantum state for a measurement setting. For two settings, we have a pair of vectors from the same quantum state. Each pair forms a combined-probability vector that obeys certain quantum constraints, which are triangle inequalities in our case. Such a restricted set of combined vectors, called the combined-probability space, is presented here for a ddd-level quantum system (qudit). The combined space is a compact convex subset of a Euclidean space, and all its extreme points come from a family of parametric curves. Considering a suitable concave function on the combined space to estimate the uncertainty, we deliver an uncertainty relation by finding its global minimum on the curves for a qudit. If one chooses an appropriate concave (or convex) function, then there is no need to search for the absolute minimum (maximum) over the whole space; it will be on the parametric curves. So these curves are quite useful for establishing an uncertainty (or a certainty) relation for a general pair of settings. We also demonstrate that many known tight certainty or uncertainty relations for a qubit can be obtained with the triangle inequalities.