Stéphane Attal

Open Quantum Random Walks

A new model of quantum random walks is introduced, on lattices as well as on finite graphs. These quantum random walks take into account the behavior of open quantum systems. They are the exact quantum analogues of classical Markov chains. We explore the “quantum trajectory” point of view on these quantum random walks, that is, we show that measuring the position of the particle after each time-step gives rise to a classical Markov chain, on the lattice times the state space of the particle. This quantum trajectory is a simulation of the master equation of the quantum random walk. The physical pertinence of such quantum random walks and the way they can be concretely realized is discussed. Differences and connections with the already well-known quantum random walks, such as the Hadamard random walk, are established.

Open Quantum Walks on Graphs

Open quantum walks (OQW) are formulated as quantum Markov chains on graphs. It is shown that OQWs are a very useful tool for the formulation of dissipative quantum computing algorithms and for dissipative quantum state preparation. In particular, single qubit gates and the CNOT-gate are implemented as OQWs on fully connected graphs. Also, dissipative quantum state preparation of arbitrary single qubit states and of all two-qubit Bell-states is demonstrated. Finally, the discrete time version of dissipative quantum computing is shown to be more efficient if formulated in the language of OQWs.

Approximating the Fock space with the toy Fock space

I. The toy Fock space II. The Fock space III. Embedding the toy Fock space into the Fock space IV. Projections on the toy Fock space V. Approximations VI. Probabilistic interpretations VII. The Ito tables Bibliography

From

Abstract In quantum physics, the state space of a countable chain of ( n + 1 ) -level atoms becomes, in the continuous field limit, a Fock space with multiplicity n. In a more functional analytic language, the continuous tensor product space over R + of copies of the space C n + 1 is the symmetric Fock space Γ s ( L 2 ( R + ; C n ) ) . In this article we focus on the probabilistic interpretations of these facts. We show that they correspond to the approximation of the n-dimensional normal martingales by means of obtuse random walks, that is, extremal random walks in R n whose jumps take exactly n + 1 different values. We show that these probabilistic approximations are carried by the convergence of the matrix basis a j i ( p ) of ⊗ N C n + 1 to the usual creation, annihilation and gauge processes on the Fock space.

(n+1)

It is known, from a simple algebraic computation, that every Hilbert-Schmidt operator on the Fock space admits a Maassen-Meyer kernel. Maassen-Meyer kernels are a non-commutative extension of the usual notion of chaotic expansion of random variables. Using an extension of the non-commutative stochastic integrals which allows to define these integrals on the whole Fock space, we prove that a Hilbert-Schmidt operator on Fock space is the sum of a series of iterated non-commutative stochastic integrals with respect to the basic theree quantum noises. In this way we recover its Maassen-Meyer kernel which can be completely described from the operator itself.

-level atom chains to n -dimensional noises

2.1 First definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 2.2 Spectral analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 2.3 Representations and states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 2.4 Commutative C∗-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 2.5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Non-commutative chaotic expansion of Hilbert-Schmidt operators on Fock space

We derive stochastic master equations which describe the evolution of open quantum systems in contact with a heat bath and undergoing indirect measurements. These equations are obtained as a limit of a quantum repeated measurement model where we consider a small system in contact with an infinite chain at positive temperature. It is well-known that at zero temperature one obtains stochastic differential equations of jump-diffusion type. We show that only pure diffusion type equations are relevant at strictly positive temperatures.

Elements of Operator Algebras and Modular Theory

We consider a non-interacting bipartite quantum system

Entanglement of Bipartite Quantum Systems Driven by Repeated Interactions

\mathcal H_S^A\otimes \mathcal H_S^B