Quantum process in probability representation of quantum mechanics

Abstract

In this work, the operator-sum representation of a quantum process is extended to the probability representation of quantum mechanics. It is shown that each process admitting the operator-sum representation is assigned a kernel, convolving of which with the initial tomogram set characterizing the system state gives the tomographic state of the transformed system. This kernel, in turn, is broken into the kernels of partial operations, each of them incorporating the symbol of the evolution operator related to the joint evolution of the system and an ancillary environment. Such a kernel decomposition for the projection to a certain basis state and a Gaussian-type projection is demonstrated as well as qubit flipping and amplitude damping processes.

Probability Distributions Describing Qubit-State Superpositions

“…The construction of probability representations was developed using the concept of quantizer and dequantizer [17,18]. Some properties of the probability distributions used in quantum mechanics were discussed in [19][20][21][22][23].…”

Inverted Oscillator Quantum States in the Probability Representation

“…All these functions are functions of two variables q and p, but they cannot be analogs of position and momentum, as in classical mechanics, because the Heisenberg uncertainty relation forbids the existence of such probability distributions. The problem was solved [10] when the probability distribution function of one random position X was found (see also [11][12][13]). This probability distribution contains all the information about the quantum state of a particle and is related to the density matrix and all known quasidistrubition functions by invertible integral transforms.…”

Dynamics of System States in the Probability Representation of Quantum Mechanics

“…The entanglement phenomenon in quantum physics provides the possibility to apply this notion in classical probability theory [12]. The functions that define the states of a quantum system, and that are probability distribution functions, were introduced in [13]; they were named symplectic tomograms, and this representation was called probability representation of quantum mechanics (see also [14][15][16][17][18]). Some mathematical aspects of the probability representation of quantum and classical states were discussed in [19,20].…”