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Perfect and Pretty Good State Transfer for Field-Free Heisenberg Chains

Alastair Kay

Published Jun 14, 2019• 1 citation
OpenAlex

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 attempting to engineer arbitrarily accurate transfer, where explore a new paradigm that facilitates time estimates for achieving any target accuracy $\epsilon$ for the transfer.

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

Alastair Kay

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.

arXiv:1906.06223•Jun 14, 2019•1 citation

Pretty good state transfer in qubit chains-The Heisenberg Hamiltonian

Leonardo Banchi, Coutinho, G, Godsil, C +1 more

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.

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Pretty good state transfer in qubit chains—The Heisenberg Hamiltonian

Leonardo Banchi, Gabriel Coutinho, Chris D. Godsil +1 more

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.

arXiv:1608.04722•Journal of Mathematical Physics•Mar 1, 2017•46 citations

Engineered spin chains for perfect quantum state transfer

Bich Ha Nguyen, Văn Hiếu Nguyễn

Abstract The transfer of quantum states along spin chains with the XY Heisenberg exchange interaction between nearest-neighbour spins and without decoherence was studied. It was shown that there exists a wide class of engineered mirror-symmetric N-spin-chains that allow the perfect quantum state transfer from one end of each chain to the other. The sets of values of physical parameters of these N-spin-chains were established.

Advances in Natural Sciences Nanoscience and Nanotechnology•Jun 1, 2010•8 citations

Conclusive and arbitrarily perfect quantum-state transfer using parallel spin-chain channels

Daniel Klaus Burgarth, Sougato Bose

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 $P$ in a Heisenberg chain of $N$ spin-$\frac{1}{2}$ 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.

arXiv:quant-ph/0406112•Physical Review A•May 13, 2005•247 citations

Pretty good quantum state transfer on isotropic and anisotropic Heisenberg spin chains with tailored site dependent exchange couplings

Pablo Serra, Alejandro Ferrón, Omar Osenda

Using a global optimization algorithm we obtain spin chains with site-dependent exchange coefficients which allow almost perfect quantum state transfer between the extremes of the chains without any further time-dependent external control. We consider chains with isotropic and anisotropic Heisenberg Hamiltonian with up to 100 spins. The method allow us to choose the arrival of the transferred state by changing the range available to the exchange coupling strengths. We consider short transferred times, in particular shorter than those achievable with known time-dependent control schemes. The chains obtained with the optimization method show some remarkable and interesting traits as, for instance, the scaling of the magnitude of the exchange couplings needed to achieve near perfect state transfer with the length of the chain and the arrival time. This scaling makes it possible to decide if the chain with site-dependent coefficients can be implemented in an actual system according to the range of interactions allowed in it. We compare the robustness of the transmission protocol against static disorder on the exchange coefficients using different figures of merit, which allow us to conclude that the isotropic Heisenberg chain is the best option when compared with anisotropic chains. This comparison is relevant since the method allow us to start with chains that all achieve near perfect quantum state transfer, regime that was not available previously in these chains without time-dependent external control.

arXiv:2101.03194•Jan 8, 2021•0 citations
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.
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.
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.
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.
Using a global optimization algorithm we obtain spin chains with site-dependent exchange coefficients which allow almost perfect quantum state transfer between the extremes of the chains without any further time-dependent external control. We consider chains with isotropic and anisotropic Heisenberg Hamiltonian with up to 100 spins. The method allow us to choose the arrival of the transferred state by changing the range available to the exchange coupling strengths. We consider short transferred times, in particular shorter than those achievable with known time-dependent control schemes. The chains obtained with the optimization method show some remarkable and interesting traits as, for instance, the scaling of the magnitude of the exchange couplings needed to achieve near perfect state transfer with the length of the chain and the arrival time. This scaling makes it possible to decide if the chain with site-dependent coefficients can be implemented in an actual system according to the range of interactions allowed in it. We compare the robustness of the transmission protocol against static disorder on the exchange coefficients using different figures of merit, which allow us to conclude that the isotropic Heisenberg chain is the best option when compared with anisotropic chains. This comparison is relevant since the method allow us to start with chains that all achieve near perfect quantum state transfer, regime that was not available previously in these chains without time-dependent external control.