Although the existence of unitary evolution for probabilistic identification and clone of quantum states has been demonstrated, the concrete form is undeveloped. In this paper, we first derived unitary evolution matrixes with the given probability for $n$ -state identification. On the base of the work, the corresponding Hamiltonian was illustrated in detail. To some generalization, we found unitary representations and Hamiltonians of $N$ probabilistic clones given $M$ initial exact copies using the similar method. And the probabilistic identification given $M$ initial copies come out the result in the similar way. For the specialization of quantum computer, the evolution of probabilistic identification and clone in $2^n$ -dimension quantum system is successfully derived and we realize it with the universal quantum logic gates.Although the existence of unitary evolution for probabilistic identification and clone of quantum states has been demonstrated, the concrete form is undeveloped. In this paper, we first derived unitary evolution matrixes with the given probability for $n$-state identification. On the base of the work, the corresponding Hamiltonian was illustrated in detail. To some generalization, we found unitary representations and Hamiltonians of $N$ probabilistic clones given $M$ initial exact copies using the similar method. And the probabilistic identification given $M$ initial copies come out the result in the similar way. For the specialization of quantum computer, the evolution of probabilistic identification and clone in $2^n$-dimension quantum system is successfully derived and we realize it with the universal quantum logic gates.

Jul 30, 19990 citations

Evolution of a Qubit Density Matrix in a Probabilistic Representation of Quantum States

Olga V. Man’ko, Vladimir N. Chernega

Abstract A system of equations is obtained for a generalized qubit density matrix that depends on two moments in time. It determines the evolution of the quantum state of a qubit and transforms into the von Neumann equation for a qubit density matrix at coinciding times. The states of a harmonic oscillator in a probabilistic representation of quantum states are considered. Kinetic equations for the probabilities of projections of a spin +1/2 onto three perpendicular directions in space that determine the state of a qubit are obtained.Abstract A system of equations is obtained for a generalized qubit density matrix that depends on two moments in time. It determines the evolution of the quantum state of a qubit and transforms into the von Neumann equation for a qubit density matrix at coinciding times. The states of a harmonic oscillator in a probabilistic representation of quantum states are considered. Kinetic equations for the probabilities of projections of a spin +1/2 onto three perpendicular directions in space that determine the state of a qubit are obtained.

Bulletin of the Russian Academy of Sciences PhysicsMar 1, 20200 citations

Probabilistic quantum phase-space simulation of Bell violations and their dynamical evolution

L. Rosales-Zárate, Bogdan Opanchuk, P. D. Drummond +1 more

Quantum simulations of Bell inequality violations are numerically obtained using probabilistic phase-space methods, namely, the positive- $P$ representation. In this approach the moments of quantum observables are evaluated as moments of variables that have values outside the normal eigenvalue range. There is thus a parallel with quantum weak measurements and weak values. Nevertheless, the representation is exactly equivalent to quantum mechanics. A number of states violating Bell inequalities are sampled, demonstrating that these quantum paradoxes can be treated with probabilistic methods. We treat quantum dynamics by simulating the time evolution of the Bell state formed via parametric down-conversion and discuss multimode generalizations.Quantum simulations of Bell inequality violations are numerically obtained using probabilistic phase-space methods, namely, the positive-$P$ representation. In this approach the moments of quantum observables are evaluated as moments of variables that have values outside the normal eigenvalue range. There is thus a parallel with quantum weak measurements and weak values. Nevertheless, the representation is exactly equivalent to quantum mechanics. A number of states violating Bell inequalities are sampled, demonstrating that these quantum paradoxes can be treated with probabilistic methods. We treat quantum dynamics by simulating the time evolution of the Bell state formed via parametric down-conversion and discuss multimode generalizations.

arXiv:1405.1168Physical Review AAug 13, 201420 citations

Molchanov's Formula and Quantum Walks: A Probabilistic Approach

Hoang Vu

This paper establishes a robust link between quantum dynamics and classical ones by deriving probabilistic representation for both continuous time and discrete time quantum walks. We first adapt Molchanov formula, originally employed in the study of Schrodinger operators on multidimensional integer lattice, to characterize the evolution of continuous time quantum walks. Extending this framework, we develop a probabilistic method to represent discrete time quantum walks on an infinite integer line, bypassing the locality constraints that typically inhibit direct application of Molchanov formula. The validity of our representation is empirically confirmed through a benchmark analysis of the Hadamard walk, demonstrating high fidelity with traditional unitary evolution. Our results suggest that this probabilistic lens offer a powerful alternative for learning multidimensional quantum walks and provides new analytical pathways for investigating quantum systems via classical stochastic processes.This paper establishes a robust link between quantum dynamics and classical ones by deriving probabilistic representation for both continuous time and discrete time quantum walks. We first adapt Molchanov formula, originally employed in the study of Schrodinger operators on multidimensional integer lattice, to characterize the evolution of continuous time quantum walks. Extending this framework, we develop a probabilistic method to represent discrete time quantum walks on an infinite integer line, bypassing the locality constraints that typically inhibit direct application of Molchanov formula. The validity of our representation is empirically confirmed through a benchmark analysis of the Hadamard walk, demonstrating high fidelity with traditional unitary evolution. Our results suggest that this probabilistic lens offer a powerful alternative for learning multidimensional quantum walks and provides new analytical pathways for investigating quantum systems via classical stochastic processes.

quant-pharXiv:2601.01071Jan 6, 20260 citations

n

-state identification. On the base of the work, the corresponding Hamiltonian was illustrated in detail. To some generalization, we found unitary representations and Hamiltonians of

N

probabilistic clones given

M

initial exact copies using the similar method. And the probabilistic identification given

M

initial copies come out the result in the similar way. For the specialization of quantum computer, the evolution of probabilistic identification and clone in

2^n

-dimension quantum system is successfully derived and we realize it with the universal quantum logic gates.

Quantum simulations of Bell inequality violations are numerically obtained using probabilistic phase-space methods, namely, the positive-

P

representation. In this approach the moments of quantum observables are evaluated as moments of variables that have values outside the normal eigenvalue range. There is thus a parallel with quantum weak measurements and weak values. Nevertheless, the representation is exactly equivalent to quantum mechanics. A number of states violating Bell inequalities are sampled, demonstrating that these quantum paradoxes can be treated with probabilistic methods. We treat quantum dynamics by simulating the time evolution of the Bell state formed via parametric down-conversion and discuss multimode generalizations.