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arXiv:0909.2746 · Journal of Russian Laser Research

Probability representation and quantumness tests for qudits and two-mode light states

Sergey N. Filippov, Vladimir I. Man’ko

Published Sep 1, 2009• 4 citations
arXivHTMLPDF
DOI
10.1007/s10946-009-9099-5

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The question of the discrimination of the Bell states of two qudits (i.e., d-dimensional quantum systems) by means of passive linear optical elements and conditional measurements is discussed. A qudit is supposed to be represented by d optical modes containing exactly one photon altogether. From recent results of Calsamiglia it follows that there is no way how to distinguish the Bell states of two qudits for d>2 - not even with the probability of success lower than one - without any auxiliary photons in ancillary modes. Following the results of Carollo and Palma it is proved that it is impossible to distinguish even only one such a Bell state with certainty (i.e., with the probability of success equal to one), irrespective of how many auxiliary photons are involved. However, it is shown that auxiliary photons can help to discriminate the Bell states of qudits with the high probability of success: A Bell-state analyzer based on the idea of linear optics quantum computation that can achieve the probability of success arbitrarily close to one is described. It requires many auxiliary photons that must be first "combined" into entangled states.
This paper proposes an alternative geometric representation of single qudit states based on probability simplexes to describe the quantum properties of noncomposite systems. In contrast to the known high dimension pictures, we present the planar picture of quantum states, using the elementary geometry. The approach is based on, so called, Malevich square representation of the single qubit state. It is shown that the quantum statistics of the single qudit with some spin j and observables are formally equivalent to statistics of the classical system with N 2 − 1 random vector variables and N 2 − 1 classical probability distributions, obeying special constrains, found in this study. We present a universal inequality, that describes the single qudits state quantumness. The inequality provides a possibility to experimentally check up entanglement of the system in terms of the classical probabilities. The simulation study for the single qutrit and ququad systems, using the Metropolis Monte-Carlo method, is obtained. The geometrical representation of the single qudit states, presented in the paper, is useful in providing a visualization of quantum states and illustrating their difference from the classical ones.