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arXiv:1906.06223

The Limits of Quantum State Transfer for Field-Free Heisenberg Chains

Alastair Kay

Published Jun 14, 2019• 1 citation
arXivHTMLPDF

Abstract

In a one-dimensional Heisenberg chain, we show that there are no sets of coupling strengths such that the evolution perfectly transfers a quantum state between the two ends of the chain without the addition of magnetic fields. In lieu of perfect transfer, we consider a range of options for achieving high quality transfer, whether in finite time, or via "pretty good" transfer where one waits long times in the hope of getting arbitrarily close to perfect transfer. In attempting to engineer arbitrarily accurate transfer, we explore a new paradigm that facilitates time estimates for achieving any target accuracy $ε$ for the transfer.

DOI
10.48550/arxiv.1906.06223

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In a one-dimensional Heisenberg chain, we show that there are no sets of coupling strengths such that the evolution perfectly transfers a quantum state between the two ends of the chain without the addition of magnetic fields. In lieu of perfect transfer, we consider a range of options for achieving high quality transfer, whether in finite time, or via "pretty good" transfer where one waits long times in the hope of getting arbitrarily close to perfect transfer. In attempting to engineer arbitrarily accurate transfer, we explore a new paradigm that facilitates time estimates for achieving any target accuracy εεε for the transfer.
In a one-dimensional Heisenberg chain, we show that there are no sets of coupling strengths such that the evolution perfectly transfers a quantum state between the two ends of the chain without the addition of magnetic fields. In attempting to engineer arbitrarily accurate transfer, where explore a new paradigm that facilitates time estimates for achieving any target accuracy ϵ\epsilonϵ for the transfer.
Pretty good state transfer in networks of qubits occurs when a continuous-time quantum walk allows the transmission of a qubit state from one node of the network to another, with fidelity arbitrarily close to 1. We prove that in a Heisenberg chain with n qubits, there is pretty good state transfer between the nodes at the jth and (n − j + 1)th positions if n is a power of 2. Moreover, this condition is also necessary for j = 1. We obtain this result by applying a theorem due to Kronecker about Diophantine approximations, together with techniques from algebraic graph theory.
Pretty good state transfer in networks of qubits occurs when a continuous-time quantum walk allows the transmission of a qubit state from one node of the network to another, with fidelity arbitrarily close to 1. We prove that in a Heisenberg chain with n qubits, there is pretty good state transfer between the nodes at the jth and (n − j + 1)th positions if n is a power of 2. Moreover, this condition is also necessary for j = 1. We obtain this result by applying a theorem due to Kronecker about Diophantine approximations, together with techniques from algebraic graph theory.
The basic idea of spin-chain engineering for perfect quantum state transfer (QST) is to find a set of coupling constants in the Hamiltonian such that a particular state initially encoded on one site will evolve freely to the opposite site without any dynamical controls. The minimal possible evolution time represents a speed limit for QST. We prove that the optimal solution is the one simulating the precession of a spin in a static magnetic field. We also argue that, at least for solid-state systems where interactions are local, it is more realistic to characterize the computation power by the couplings than the initial energy.
Efficient quantum-state transfer is achieved in a uniformly coupled spin-1/21/21/2 chain, with open boundaries, by application of local magnetic fields on the second and last-but-one spins, respectively. These effective barriers induce the appearance of two eigenstates, bilocalized at the edges of the chain, which allow a high-quality transfer also at relatively long distances. The same mechanism may be used to send an entire e-bit (e.g., an entangled qubit pair) from one to the other end of the chain.
We suggest a protocol for perfect quantum communication through spin-chain channels. By combining a dual-rail encoding with measurements only at the receiving end, we can get conclusively perfect state transfer, whose probability of success can be made arbitrarily close to unity. As an example of such an amplitude-delaying channel, we show how two parallel Heisenberg spin chains can be used as quantum wires. Perfect state transfer with a probability of failure lower than PPP in a Heisenberg chain of NNN spin-12\frac{1}{2}21​ particles can be achieved in a timescale of the order of $(0.33\ensuremath{\hbar}∕J){N}^{1.7}\ensuremath{\mid}\mathrm{ln}\phantom{\rule{0.2em}{0ex}}P\ensuremath{\mid}$. We demonstrate that our scheme is more robust to decoherence and nonoptimal timing than any scheme using single spin chains.