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Entanglement-screening by nonlinear resonances

A member of the Laboratory, in collaboration with researchers from Australia, United States and Germany, has recently shown that nonlinear resonances in a classically mixed phase space allow to define generic, strongly entangled multi-partite quantum states. The robustness of their multipartite entanglement was shown to increase with the particle number, i.e. in the semiclassical limit, for those classes of diffusive noise which assist the quantum-classical transition.

The robustness of the entanglement produced by the implementation of the quantum version of the Harper’s map is examined under the action of noise, possibly due to interaction with some environment.

The system consists of k qubits and the computational basis of the k-partite system

|i\rangle,\ \ {\rm where}\ \ i=\sum_{n=0}^{k-1}i_n 2^n

is identified with the position basis of the quantum map on the torus (dim({\cal H})\equiv N=2^k or \hbar=1/(2\pi 2^k)) :

|i\rangle \leftrightarrow |q\rangle = |i/N\rangle.

Thus the action of the map, be it chaotic or regular, will naturally generate entanglement. It is well known that chaotic dynamics seems to produce entanglement faster that regular dynamics. However, the focus of this work is on how robust the entanglement produced is depending on the underlying classical phase space structure.

As a measure of entanglement, the k-partite concurrence C_{k} concurrence as defined by Mintert, et al. is used. On pure states, this quantity is given by the square root of a balanced average over the squared concurrence of all nontrivial bipartitions of the k-set under scrutiny, and vanishes exclusively for k-separable states. Moreover, it has the particularly advantageous property C_k(\ket{\phi}\otimes\ket{\psi}_{k-1})  =  C_{k-1}(\ket{\psi}_{k-1}), \forall \ket{\phi}\in {\cal H}_1, \ket{\psi}_{k-1}\in {\cal H}_2\otimes\ldots\otimes{\cal H}_k. The latter allows to compare the entanglement inscribed into quantum states composed of an increasing number of subsystems. Furthermore, C_k has a generalization for mixed states (through the convex roof construction), which is the one used.

Results : The initial state is a minimum uncertainty Gaussian wave packet (size \sim\sqrt{\hbar}=\sqrt{1/2\pi N}). The initial value of C_k for this types of states is non-zero and depends on the position of the center of the packet. Unitary (open symbols) and noisy (filled symbols) evolution of initial conditions is studied in the regular island (squares) and in the chaotic sea (triangles). Under purely unitary dynamics (open symbols), chaotic dynamics induces strong coupling between all position basis states and C_k will increase rapidly for the initial condition placed in the chaotic domain, and saturate once equilibrated over the chaotic eigenstates of the quantized Harper map. Saturation to maximal value of C_k for pure states would be reached in the limit k\to \infty. The qualitative behavior does not depend on the system size. On the other hand, for the state initially placed within the elliptic island, size does matter : for k=5, the initial coherent state cannot be well accomodated within the elliptic island in phase space (due to the finite size of \hbar_{\rm eff}), and exhibits non-negligible tunneling coupling to its chaotic environment. Consequently, as time proceeds, the coherently evolved state spreads more and more over the chaotic phase space component, and its entanglement finally reaches essentially the same value as for the initial condition within the chaotic domain, just after considerably longer time — essentially determined by the relevant tunneling matrix elements (which, in general, will be strongly fluctuating under small parameter changes. In contrast, for k=8, tunneling from the island into the chaotic sea occurs on a much longer time scale (which, on average, increases exponentially with \hbar_{\rm eff}), and remains invisible on the time scale covered in the figure. The small oscillations of C_{k=8}(t) are due to the spreading of the initial wave packet along the regular island’s tori.

This screening of the initial state from the chaotic sea when initially placed within the elliptic island, more and more efficient with increasing particle number, has an immediate consequence for he robustness of the state’s multipartite entanglement under the influence of decoherence, as illustrated by the filled symbols in the first figure : For chaotic initial conditions, an initial rise of C_k is rather quickly overruled by the loss of multiparticle coherence and hence of entanglement, and this is once again largely independent of k. However, for initial conditions within the island, k=5 again leads to asymptotically the same behaviour as for the chaotic initial condition, while k=8 induces entanglement dynamics almost completely unaffected by the noise. Thus, for sufficiently large k, equivalent to sufficiently small \hbar_{\rm eff}, and correspondingly suppressed tunneling rates, the classical nonlinear resonance creates strongly entangled multipartite states which, in addition, are robust against noise. This is further illustrated in the second figure where C_k=8(t=16)$ is plotted for different initial conditions, in the absence and in the presence of noise. Clearly, entanglement is robust when shielded by the resonance island. While chaotic dynamics produce slighty stronger Entanglement, this is significantly more fragile under decoherence.

Details on the type of decoherence considered and analysis of other (non-local) noises are presented in the corresponding paper, Entanglement-screening by nonlinear resonances., which has been published in Phys. Rev. Lett. 98, 120504 (2007). The corresponding author in the laboratory is García-Mata Ignacio . Hospitality at MPI-PKS is gratefully acknowledged.