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arXiv:0811.3581 · Physical Review A

Relaxation due to random collisions with a many-qudit environment

Giuseppe Gennaro, Giuliano Benenti, G. Massimo Palma

Published Feb 5, 2009• 34 citations
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Abstract

We analyze the dynamics of a system qudit of dimension $\ensuremath{\mu}$ sequentially interacting with the $\ensuremath{\nu}$-dimensional qudits of a chain playing the role of an environment. Each pairwise collision has been modeled as a random unitary transformation. The relaxation to equilibrium of the purity of the system qudit, averaged over random collisions, is analytically computed by means of a Markov chain approach. In particular, we show that the steady state is the one corresponding to the steady state for random collisions with a single environment qudit of effective dimension ${\ensuremath{\nu}}_{e}=\ensuremath{\nu}\ensuremath{\mu}$. Finally, we numerically investigate aspects of the entanglement dynamics for qubits $(\ensuremath{\mu}=\ensuremath{\nu}=2)$ and show that random unitary collisions can create multipartite entanglement between the system qudit and the qudits of the chain.

DOI
10.1103/physreva.79.022105

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We analyze the dynamics of a system qudit of dimension \ensuremathμ\ensuremath{\mu}\ensuremathμ sequentially interacting with the \ensuremathν\ensuremath{\nu}\ensuremathν-dimensional qudits of a chain playing the role of an environment. Each pairwise collision has been modeled as a random unitary transformation. The relaxation to equilibrium of the purity of the system qudit, averaged over random collisions, is analytically computed by means of a Markov chain approach. In particular, we show that the steady state is the one corresponding to the steady state for random collisions with a single environment qudit of effective dimension \ensuremathνe=\ensuremathν\ensuremathμ{\ensuremath{\nu}}_{e}=\ensuremath{\nu}\ensuremath{\mu}\ensuremathνe​=\ensuremathν\ensuremathμ. Finally, we numerically investigate aspects of the entanglement dynamics for qubits (\ensuremathμ=\ensuremathν=2)(\ensuremath{\mu}=\ensuremath{\nu}=2)(\ensuremathμ=\ensuremathν=2) and show that random unitary collisions can create multipartite entanglement between the system qudit and the qudits of the chain.
We analyze the dynamics of the system consisting of a qubit sequentially interacting with a chain of qubits that are initially individually in a pure random state. Each pairwise collision has been modeled as a partial swap transformation. The relaxation to equilibrium of the purity of the system qubit, averaged over all the initial states of the environment, is analytically computed. In particular, we show that the steady state depends on the parameter η of the partial swap transformation. Finally, we investigate aspects of the entanglement dynamics for qubits and show that such process can create typical multipartite entanglement between the system qubit and the qubits of the chain.
The dynamics of a single qubit interacting by a sequence of pairwise collisions with an environment consisting of just two more qubits is analyzed. Each collision is modeled in terms of a random unitary operator with a uniform probability distribution described by the uniform Haar measure. We show that the purity of the system qubit as well as the bipartite and the tripartite entanglement reach time-averaged equilibrium values characterized by large instantaneous fluctuations. These equilibrium values are independent of the order of collision among the qubits. The relaxation to equilibrium is analyzed also in terms of an ensemble average of random collision histories. Such average allows for a quantitative evaluation and interpretation of the decay constants. Furthermore a dependence of the transient dynamics on the initial degree of entanglement between the environment qubits is shown to exist. Finally the statistical properties of bipartite and tripartite entanglement are analyzed.
We analyze the dynamics of a system qubit interacting by means of a sequence of pairwise collisions with an environment consisting of just two qubits. We show that the density operator of the qubits approaches a common time-averaged equilibrium state, characterized by large fluctuations, only for a random sequence of collisions. For a regular sequence of collisions the qubit states of the system and of the reservoir undergo instantaneous periodic oscillations and do not relax to a common state. Furthermore we show that pure bipartite entanglement is developed only when at least two qubits are initially in the same pure state while otherwise also genuine multipartite entanglement builds up.
We investigate decoherence channels that are modeled as a sequence of collisions of a quantum system (e.g., a qubit) with particles (e.g., qubits) of the environment. We show that collisions induce decoherence when a bipartite interaction between the system qubit and an environment (reservoir) qubit is described by the controlled-UUU unitary transformation (gate). We characterize decoherence channels and in the case of a qubit we specify the most general decoherence channel and derive a corresponding master equation. Finally, we analyze entanglement that is generated during the process of decoherence between the system and its environment.
We study decoherence of one, two, and nnn non-interacting qubits. Decoherence, measured in terms of purity, is calculated in linear response approximation, making use of the spectator configuration. The environment and its interaction with the qubits are modelled by random matrices. For two qubits, numerical studies reveal a simple one to one correspondence between its decoherence and its internal entanglement decay. Using this relation we are able to give a formula for concurrence decay. For large environments the evolution induces a unital channel in the two qubits, providing a partial explanation for the relation above. Using a kicked Ising spin network, we study the exact evolution of two non-interacting qubits in the presence of a spin bath. We find that the entanglement (as measured by concurrence) of the two qubits has a close relation to the purity of the pair, and closely follows an analytic relation derived for Werner states. As a collateral result we find that an integrable environment causes quadratic decay of concurrence as well as of purity, while a chaotic environment causes linear decay. Both quantities display recurrences in some integrable environments. Good agreement with the results found using random matrix theory is obtained. Finally, we analyze decoherence of a quantum register in the absence of non-local operations. The problem is solved in terms of a sum rule which implies linear scaling in the number of qubits. Each term involves a single qubit and its entanglement with the remaining ones. Two conditions are essential: decoherence must be small and the coupling of different qubits must be uncorrelated in the interaction picture. We apply the result to the random matrix model, and illustrate its reach considering a GHZ state coupled to a spin bath.
The asymptotic dynamics of many-qubit quantum systems is investigated under iteratively and randomly applied unitary transformations. For a one-parameter family of unitary transformations, which entangle pairs of qubits, two main theorems are proved. They characterize completely the dependence of the resulting asymptotic dynamics on the topology of the interaction graph that encodes all possible qubit couplings. These theorems exhibit clearly which aspects of an interaction graph are relevant and which ones are irrelevant to the asymptotic dynamics. On the basis of these theorems, the local entropy transport between an open quantum system and its environment are explored for strong non-Markovian couplings and for different sizes of the environment and different interaction topologies. It is shown that although the randomly applied unitary entanglement operations cannot decrease the overall entropy of such a qubit network, a local entropy decrease or 'cooling' of subsystems is possible for special classes of interaction topologies.