Probability representation of quantum dynamics using pseudostochastic maps

In this work, we consider a probability representation of quantum dynamics for finite-dimensional quantum systems with the use of pseudostochastic maps acting on true probability distributions. These probability distributions are obtained via symmetric informationally complete positive operator-valued measure (SIC-POVM) and can be directly accessible in an experiment. We provide SIC-POVM probability representations both for unitary evolution of the density matrix governed by the von Neumann equation and dissipative evolution governed by Markovian master equation. In particular, we discuss whereas the quantum dynamics can simulated via classical random processes in terms of the conditions for the master equation generator in the SIC-POVM probability representation. We construct practical measures of nonclassicality non-Markovianity of quantum processes and apply them for studying experimental realization of quantum circuits realized with the IBM cloud quantum processor.

Figure 1

General relation among the set of SIC-POVM probability distributions, general classical probability distributions of the same dimension, and maps which keep distributions in a particular set. The set of SIC-POVM probability distributions, so-called Hilbert qplex [38], is smaller than the full probability distribution set, while the set of possible maps turning SIC-POVM probabilities into SIC-POVM probabilities (known to be a pseudostochastic) is wider than the set of classical stochastic maps. Points and arrows demonstrate actions of pseudostochastic (stochastic) maps on SIC-POVM (classical) probability distributions.

Figure 2

Scheme of a quantum-mechanical experiment.

Figure 3

The experimental setup. For the tomography of the SIC-POVM measurement operation, applying $U$ (${\mathbf{S}}_U$) is skipped. In panel (a), standard representation as a circuit is presented (standard notations for cnot and Hadamard gates are used). In panel (b), SIC-POVM probability representation is shown.

Figure 4

The Bloch vectors corresponded to SIC-POVM projectors $\left\{\right|{\psi }_i\rangle \langle |{\psi }_i\rangle {\left|\right\}}_{i=1}^4$ and the state $|\varphi \rangle \langle \varphi |$.

Figure 5

Experimentally reconstructed pseudostochastic matrices of the single-qubit gate $U$ (a) and the imperfect SIC-POVM measurement (b).