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arXiv:1105.6058 · New Journal of Physics

Long quantum channels for high-quality entanglement transfer

Leonardo Banchi, Tony J. G. Apollaro, Alessandro Cuccoli, Ruggero Vaia, Paola Verrucchi

Published Dec 2, 2011• 89 citations
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Abstract

High-quality quantum-state and entanglement transfer can be achieved in an unmodulated spin bus operating in the ballistic regime, which occurs when the endpoint qubits A and B are nonperturbatively coupled to the chain by a suitable exchange interaction j 0 . Indeed, the transition amplitude characterizing the transfer quality exhibits a maximum for a finite optimal value j opt 0 ( N ), where N is the channel length. We show that j opt 0 ( N ) scales as N −1/6 for large N and that it ensures a high-quality entanglement transfer even in the limit of arbitrarily long channels, almost independently of the channel initialization. For instance, for any chain length the average quantum-state transmission fidelity exceeds 90% and decreases very little in a broad neighbourhood of j opt 0 ( N ). We emphasize that, taking the reverse point of view, should j 0 be experimentally constrained, high-quality transfer can still be obtained by adjusting the channel length to its optimal value.

DOI
10.1088/1367-2630/13/12/123006

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High-quality quantum-state and entanglement transfer can be achieved in an unmodulated spin bus operating in the ballistic regime, which occurs when the endpoint qubits A and B are nonperturbatively coupled to the chain by a suitable exchange interaction j 0 . Indeed, the transition amplitude characterizing the transfer quality exhibits a maximum for a finite optimal value j opt 0 ( N ), where N is the channel length. We show that j opt 0 ( N ) scales as N −1/6 for large N and that it ensures a high-quality entanglement transfer even in the limit of arbitrarily long channels, almost independently of the channel initialization. For instance, for any chain length the average quantum-state transmission fidelity exceeds 90% and decreases very little in a broad neighbourhood of j opt 0 ( N ). We emphasize that, taking the reverse point of view, should j 0 be experimentally constrained, high-quality transfer can still be obtained by adjusting the channel length to its optimal value.
The capability of faithfully transmit quantum states and entanglement through quantum channels is one of the key requirements for the development of quantum devices. Different solutions have been proposed to accomplish such a challenging task, which, however, require either an ad hoc engineering of the internal interactions of the physical system acting as the channel or specific initialization procedures. Here we show that optimal dynamics for efficient quantum-state and entanglement transfer can be attained in generic quantum systems with homogeneous interactions by tuning the coupling between the system and the two attached qubits. We devise a general procedure to determine the optimal coupling, and we explicitly implement it in the case of a channel consisting of a spin-12\frac{1}{2}21​XY\mathit{XY}XY chain. The quality of quantum-state and entanglement transfer is found to be very good and, remarkably, almost independent of the channel length.
Quantum-state transfer with fidelity higher than 0.990.990.99 can be achieved in the ballistic regime of an arbitrarily long one-dimensional chain with uniform nearest-neighbor interaction, except for the two pairs of mirror-symmetric extremal bonds, say xxx (first and last) and yyy (second and penultimate). These have to be roughly tuned to suitable values $x\ensuremath{\sim}2{N}^{\ensuremath{-}1/3}$ and $y\ensuremath{\sim}{2}^{3/4}{N}^{\ensuremath{-}1/6}$, where NNN is the chain length. The general framework can describe the end-to-end response in different models, such as fermion or boson hopping models and XXXXXX spin chains.
In this paper, we provide a simple strategy to optimize the performances of a spin chain as a quantum channel for state transfer.We investigate the most general conditions under which a long range interacting chain achieves the highest fidelity and the shortest transfer time. In this respect, we show that the symmetry of the spin chain plys a key role to identify the ideal system, which involves sender and receiver only, and provide numerical evidence. However, the two-spin ideal chain is unpractical due to the decrease of the coupling strength with distance. Therefore we show how to design a spin chain, which nearly approaches the optimal behaviour, while keeping the interaction strength still reasonably high.
We study the quality of state and entanglement transmission through quantum channels described by spin chains varying both the system parameters and the initial state of the channel. We consider a vast class of one-dimensional many-body models which contains some of the most relevant experimental realizations of quantum data buses. In particular, we consider spin-1/21/21/2 XY\mathit{XY}XY and XXZ\mathit{XXZ}XXZ models with open boundary conditions. Our results show a significant difference between free-fermionic (noninteracting) systems (XY\mathit{XY}XY) and interacting ones (XXZ\mathit{XXZ}XXZ), where in the former case initialization can be exploited for improving the entanglement distribution, while in the latter case it also determines the quality of state transmission. In fact, we find that in noninteracting systems the exchange with fermions in the initial state of the chain always has a destructive effect, and we prove that it can be completely removed in the isotropic XX\mathit{XX}XX model by initializing the chain in a ferromagnetic state. On the other hand, in interacting systems constructive effects can arise by scattering between hopping fermions and a proper initialization procedure. Our results are an example in which state and entanglement transmission show maxima at different points as the interactions and initializations of spin chain channels are varied.
We calculate the fidelity of transmission of a single qubit between distant sites on semi-infinite and finite chains of spins coupled via the magnetic dipole interaction. We show that such systems often perform better than their Heisenberg nearest-neighbor coupled counterparts, and that fidelities closely approaching unity can be attained between the ends of finite chains without any special engineering of the system, although state transfer becomes slow in long chains. We discuss possible optimization methods, and find that, for any length, the best compromise between the quality and the speed of the communication is obtained in a nearly uniform chain of four spins.
The XX model with uniform couplings represents the most natural choice for quantum state transfer through spin chains. Given that it has long been established that single-qubit states cannot be transferred with perfect fidelity in this model, the notion of pretty good state transfer has been recently introduced as a relaxation of the constraints on fidelity. In this paper, we study the transfer of multi-qubit entangled and unentangled states through unmodulated spin chains, and we prove that it is possible to have pretty good state transfer of any multi-particle state. This significantly generalizes the previous results on single-qubit state transfer and opens the way to using uniformly coupled spin chains as short-distance quantum channels for the transfer of arbitrary states of any dimension. Our results could be tested with current technology.