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arXiv:2206.14230 · Journal of Physics A Mathematical and Theoretical

Exact solution of a family of staggered Heisenberg chains with conclusive pretty good quantum state transfer

Pablo J. Serra, Alejandro Ferrón, Omar Osenda

Published Sep 7, 2022• 10 citations

Abstract

Abstract We construct the exact solution for a family of one-half spin chains explicitly. The spin chains Hamiltonian corresponds to an isotropic Heisenberg Hamiltonian, with staggered exchange couplings (ECs) that take only two different values. We work out the exact solutions in the one-excitation subspace. Regarding the problem of quantum state transfer, we use the solution and some theorems concerning the approximation of irrational numbers, to show the appearance of conclusive pretty good transmission for chains with particular lengths. We present numerical evidence that pretty good transmission is achieved by chains whose length is not a power of two. The set of spin chains that shows pretty good transmission is a subset of the family with an exact solution. Using perturbation theory, we thoroughly analyze the case when one of the EC strengths is orders of magnitude larger than the other. This strong coupling limit (SCL) allows us to study, in a simple way, the appearance of pretty good transmission. The use of analytical closed expressions for the eigenvalues, eigenvectors, and transmission probabilities allows us to obtain the precise asymptotic behavior of the time where the pretty good transmission is observed. Moreover, we show that this time scales as a power law whose exponent is an increasing function of the chain length. We also discuss the crossover behavior obtained for the pretty good transmission time between the regimes of SCL and the one observed when the ECs are of the same order of magnitude.

DOI: 10.1088/1751-8121/ac901d

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

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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 $ε$ for the transfer.

arXiv:1906.06223•Jun 14, 2019•1 citation

Abstract We construct the exact solution for a family of one-half spin chains explicitly. The spin chains Hamiltonian corresponds to an isotropic Heisenberg Hamiltonian, with staggered exchange couplings (ECs) that take only two different values. We work out the exact solutions in the one-excitation subspace. Regarding the problem of quantum state transfer, we use the solution and some theorems concerning the approximation of irrational numbers, to show the appearance of conclusive pretty good transmission for chains with particular lengths. We present numerical evidence that pretty good transmission is achieved by chains whose length is not a power of two. The set of spin chains that shows pretty good transmission is a subset of the family with an exact solution. Using perturbation theory, we thoroughly analyze the case when one of the EC strengths is orders of magnitude larger than the other. This strong coupling limit (SCL) allows us to study, in a simple way, the appearance of pretty good transmission. The use of analytical closed expressions for the eigenvalues, eigenvectors, and transmission probabilities allows us to obtain the precise asymptotic behavior of the time where the pretty good transmission is observed. Moreover, we show that this time scales as a power law whose exponent is an increasing function of the chain length. We also discuss the crossover behavior obtained for the pretty good transmission time between the regimes of SCL and the one observed when the ECs are of the same order of magnitude.

Abstract The pretty good transmission (PGT) scenario implies that the probability of sending one excitation from one extreme of a spin chain to the other can reach values arbitrarily close to the unity just by waiting a time long enough. The conditions that ensure the appearance of this scenario are known for chains with different interactions and lengths. Sufficient conditions for the presence of PGT depend on the spectrum of the Hamiltonian of the spin chain. Some works suggest that the time t ɛ at which the PGT takes place scales as 1 / ( | ε | ) f ( N ) , where ɛ is the difference between the probability that a single excitation propagates from one extreme of the chain to the other and the unity, while f(N) is an unknown function of the chain length. In this paper, we show that the exponent is not a simple function of the chain length but a power law of the number of linearly independent irrational eigenvalues of the one-excitation block of the Hamiltonian that enter into the expression of the probability of transmission of one excitation. We explicitly provide examples of a chain showing that the exponent changes when the couplings between the spins change while the length remains fixed. For centrosymmetric spin chains the exponent is at most N / 2 .

Using a global optimization algorithm, interchange coupling distributions are obtained for chains of quantum spins. The couplings give the magnitude of the interaction between the spins of the chain, both chains with Heisenberg Hamiltonian and XXZ type anisotropic Hamiltonian are considered. These couplings allow the chains to be used as channels to transmit quantum states according to the simplest transmission protocol. It is analyzed under which conditions it is possible to obtain very high transmission fidelities, both as a function of the arrival time and the maximum intensity allowed for the couplings. Special emphasis is placed on finding coupling sets that allow fast transmission, with short arrival times on the order of

N

, where

N

is the number of spins in the chain. The stability of the chains as transmission channels is analyzed also when the couplings are subject to static disorder. Finally it is shown that the transmission fidelity at different arrival times shows scaling properties.

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.

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.

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.