Optimal quench for distance-independent entanglement and maximal block entropy

We optimize a quantum walk of multiple fermions following a quench in a spin chain to generate near-ideal resources for quantum networking. We first prove a useful theorem mapping the correlations evolved from specific quenches to the apparently unrelated problem of quantum state transfer between distinct spins. This mapping is then exploited to optimize the dynamics and produce large amounts of entanglement distributed in very special ways. Two applications are considered: the simultaneous generation of many Bell states between pairs of distant spins (maximal block entropy) and high entanglement between the ends of an arbitrarily long chain (distance-independent entanglement). Thanks to the generality of the result, we study its implementation in different experimental setups using present technology: nuclear magnetic resonance, ion traps, and ultracold atoms in optical lattices.

Figure 1

Schematic picture for dynamical generation of entanglement using a fully engineered XX Hamiltonian and the Néel initial state.

Figure 2

Entropy dynamics for fully engineered, minimal engineered, and uniform couplings. $N=51$.

Figure 3

Fully entangled fraction $F_{n,N-n+1}\left(t^{\prime }\right)$ for pairs of mirror-symmetric spins at the time $t^{\prime }\simeq t^{*}/2$ which maximizes $F_{1,N}$, $N=5$. For $N=5$, fully and minimally engineered chains coincide. The pulse error $\epsilon$ is defined in the text. An average over 100 realizations of the imperfections is considered. Inset: $t^{\prime }$ (black line), time where $F_{1,5}\ge 0.9F_{1,5}\left(t^{\prime }\right)$ (red or light gray area), time where $F_{1,5}\ge 0.5F_{1,5}\left(t^{\prime }\right)$ (blue or dark gray area).

Figure 4

$F_{n,N-n+1}\left(t^{\prime }\right)$ as in Fig. 3. Fully engineered (left) and minimally engineered (right) chains for $N=10$. Decoherence is modeled by the master equation $\dot{\rho }=-i[{\mathcal{H}}_0,\rho ]+J\gamma {\sum }_i({\sigma }_i^z\rho {\sigma }_i^z-\rho )$ with dephasing rate $\gamma$ [29].

Figure 5

Maximum $F_{1,N}$ for the minimally engineered Hamiltonian vs the chain length.

Figure 6

(a) Time at which the first peak of $F_{1,N}$ occurs vs the chain length for the minimally engineered Hamiltonian and Néel initial state. (b) Reading time defined as the width of $F_{1,N}\left(t\right)$ at half the maximum.

Figure 7

Fully entangled fraction for $N=10$ for (a),(b) Néel initial state with the fully engineered and minimally engineered Hamiltonian, respectively. (c),(d) Series of Bell states initial state with the fully engineered and minimally engineered Hamiltonian, respectively.