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Fortschritte der Physik

Fidelity for qubits and photon states in tomographic probability representation

Olga V. Man’ko, Vladimir I. Man’ko

Published Sep 1, 2009• 8 citations
Publisher

Abstract

Abstract Review of the probability representation of quantum states based on the tomographic‐probability notion is presented. Classical and quantum states are studied and the difference of tomographic probabilities describing the states in both domains is elucidated. New expressions for fidelity in quantum domain in terms of the tomographic probabilities for both discrete qubit states and continuous quadrature components of photon are obtained.

DOI
10.1002/prop.200900086

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Abstract Review of the probability representation of quantum states based on the tomographic‐probability notion is presented. Classical and quantum states are studied and the difference of tomographic probabilities describing the states in both domains is elucidated. New expressions for fidelity in quantum domain in terms of the tomographic probabilities for both discrete qubit states and continuous quadrature components of photon are obtained.
Starting from the famous Pauli problem on the possibility of associating quantum states with probabilities, the formulation of quantum mechanics in which quantum states are described by fair probability distributions (tomograms, i.e. tomographic probabilities) is reviewed in a pedagogical style. The relation between the quantum state description and the classical state description is elucidated. The difference between those sets of tomograms is described by inequalities equivalent to a complete set of uncertainty relations for the quantum domain and to non-negativity of probability density on phase space in the classical domain. The intersection of such sets is studied. The mathematical mechanism that allows us to construct different kinds of tomographic probabilities like symplectic tomograms, spin tomograms, photon number tomograms, etc is clarified and a connection with abstract Hilbert space properties is established. The superposition rule and uncertainty relations in terms of probabilities as well as quantum basic equations like quantum evolution and energy spectra equations are given in an explicit form. A method to check experimentally the uncertainty relations is suggested using optical tomograms. Entanglement phenomena and the connection with semigroups acting on simplexes are studied in detail for spin states in the case of two-qubits. The star-product formalism is associated with the tomographic probability formulation of quantum mechanics.
We present a new formulation of conventional quantum mechanics, in which the notion of a quantum state is identified via a fair probability distribution of the position measured in a reference frame of the phase space with rotated axes. In this formulation, the quantum evolution equation as well as the equation for finding energy levels are expressed as linear equations for the probability distributions that determine the quantum states. We also give the integral transforms relating the probability distribution (called the tomographic-probability distribution or the state tomogram) to the density matrix and the Wigner function and discuss their connection with the Radon transform. Qudit states are considered and the invertible map of the state density operators onto the probability vectors is discussed. The tomographic entropies and entropic uncertainty relations are reviewed. We demonstrate the uncertainty relations for the position and momentum and the entropic uncertainty relations in the tomographic-probability representation, which is suitable for an experimental check of the uncertainty relations.
A review of the tomographic-probability representation of classical and quantum states is presented. The tomographic entropies and entropic uncertainty relations are discussed in connection with ambiguities in the interpretation of the state tomograms which are considered either as a set of the probability distributions of random variables depending on extra parameters or as a single joint probability distribution of these random variables and random parameters with specific properties of the marginals. Examples of optical tomograms of photon states, symplectic tomograms, and unitary spin tomograms of qudits are given. A new universal integral inequality for generic wave function is obtained on the base of tomographic entropic uncertainty relations.
The review of new formulation of conventional quantum mechanics where the quantum states are identified with probability distributions is presented. The invertible map of density operators and wave functions onto the probability distributions describing the quantum states in quantum mechanics is constructed both for systems with continuous variables and systems with discrete variables by using the Born's rule and recently suggested method of dequantizer-quantizer operators. Examples of discussed probability representations of qubits (spin-1/2, two-level atoms), harmonic oscillator and free particle are studied in detail. Schrödinger and von Neumann equations, as well as equations for the evolution of open systems, are written in the form of linear classical-like equations for the probability distributions determining the quantum system states. Relations to phase-space representation of quantum states (Wigner functions) with quantum tomography and classical mechanics are elucidated.
Different kinds of entropy associated with probability distribution functions characterizing the system state in classical and quantum domains are reviewed. Shannon entropy and R?nyi entropy are discussed. The notion of tomographic entropy determined by the probability distribution in the phase space of the classical system and by the density operator of the quantum system is considered. Inequalities for the tomographic entropies in classical and quantum domains are studied, and a difference in the form of these inequalities in corresponding domains is suggested as a test to clarify the classicality and quantumness of the system state in quantum optics experiments. A new bound for tomographic entropy (ln??e)?(?) depending on the local oscillator phase difference in homodyne photon detection experiments is discussed.