Creating long-range entangled Majorana pairs: from spin-1/2 twisted Kitaev to generalized XY chains

arXiv:2301.10777 · cond-mat.str-el · Submitted 2023-01-25 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Today's paper: "Creating long-range entangled Majorana pairs".

Mira: The gist The maximally entangled rainbow-like Majorana pairing state can be reached, with a sequence of pairing pulses applied on the central unit cell of the system.

Kai: First, who's behind it and why it matters.

Paper summary: Kai: So we've covered the "Creating long-range entangled Majorana pairs: from spin-one/two twisted Kitaev to generalized XY chains" paper today <ref:2301.10777#pg1,Creating long-range entangled Majorana pairs: from spin-1/2 twisted Kitaev>. The authors Haoting Xu and Hae-Young Kee explored how to generate these entangled Majorana fermions in the spin-one/two Kitaev chain with open boundary conditions <ref:2301.10777#pg1,entangled Majorana fermions in the spin-1/2 Kitaev chain with open>.

Mira: They focus on using a pair pulse sequence on the central unit cell of the chain to promote those long distance spin correlations and maximal bipartite entanglement entropy, extending it to generalized bond-dependent chains and connecting them to Bell pairs.

Lev: From an error correction standpoint, I see that the time scaling is something we need to keep in mind when mapping this onto real hardware constraints. And they show that for the CoNb2O6 twisted system, you can still generate those entangled Majorana pairs using a unitary pulse even if it's not maximally entangled.

Kai: So the main implication is showing a robust protocol, one that works across different chain types and handles certain perturbations reasonably well. It gives us a concrete recipe for creating these specific quantum states.

Mira: It also provides a deeper link between these Majorana pairs and Bell pairs through the complex fermion representation, which helps us understand nonlocality in these many-body systems better.

Lev: The real challenge, as always with this kind of work, is moving from the theoretical protocol to implementing it with the fidelity required for actual quantum computation. That's where our focus stays.

Conclusion: Kai: So, this paper, "Creating long-range entangled Majorana pairs: from spin-one/two twisted Kitaev to generalized XY chains," is all about showing how you can make these Majorana fermions that are far apart entangled in these complex spin chains.

Mira: Yeah, it basically takes those theoretical predictions about rainbow states and shows a concrete way to actually generate them using pairing pulses on the central part of the chain.

Lev: From my side, I’m looking at this because the Kitaev chain itself is a major model for topological quantum computation, so seeing how you build entanglement in it matters for real hardware.

Kai: Right, and they show that this method isn't just for one specific system; they take it further by generalizing it to other bond-dependent chains, like the XY model.

Mira: And the comparison they make to Bell pairs is pretty cool because it links these Majorana states directly to the nonlocality we’re trying to measure in quantum systems.

Lev: The time scaling they give for reaching maximal entanglement tells me how much overhead you'd need on a real machine before you get meaningful results, and it looks pretty good for the Kitaev chain.

Kai: It also tackles the twisted Kitaev chain, CoNb2O6, showing that even with those extra non-Kitaev interactions, you can still get those entangled Majorana pairs.

Mira: And they point out that even if it’s not perfectly maximal entanglement in every case, the state is still protected by some of that mirror symmetry we talked about before.

Lev: That robustness under perturbations is key; it means this protocol might be more resilient than we initially thought when you start messing with the physical parameters.

Kai: So, the big idea here is proving that you have a practical sequence of operations to engineer long-distance quantum correlations in these systems.

Mira: It shifts the focus from just predicting these states to actually having a recipe for their creation in various chain models.

Lev: It gives us a clearer target for what kind of control pulses we need to design if we want to build actual hardware that uses these topological properties effectively.

Haoting Xu, Hae-Young Kee

Department of Physics, University of Toronto · Canadian Institute for Advanced Research

cond-mat.str-el

Submitted: 2023-01-25

Updated: 2023-04-18

Comments: 12 pages, 11 figures

Journal ref: Phys. Rev. B 107, 134435 (2023)

DOI: 10.1103/PhysRevB.107.134435

License: http://creativecommons.org/licenses/by-nc-sa/4.0/

Importance score: 71/100

The gist: The gist The maximally entangled rainbow-like Majorana pairing state can be reached, with a sequence of pairing pulses applied on the central unit cell of the system.

Key concepts

Majorana Fermions: These are a specific type of quasiparticles used to describe the spin-1/2 Kitaev chain. They are related to complex fermions and have specific anti-commutation rules. The paper uses them to characterize the pairing between sites, which is crucial for understanding entanglement in this system.
Pair Pulse Sequence: This involves applying a carefully timed pulse sequence on the central unit cell of the chain. This action specifically modifies a conserved quantity, changing its eigenvalue to 0.5, which forces Majorana fermions at the central sites to pair together and initiate long-distance correlations.
Twisted Kitaev Chain: This is a specific model of the Kitaev chain where the quantization axis is tilted due to bond-dependent Ising interactions. The paper shows that even in this twisted version, a unitary pulse can generate entangled Majorana pairs with finite entanglement entropy, proving robustness against certain perturbations.

Terminology

Entanglement — 3× in this paper · explained in 4 episodes
Entanglement Entropy — 2× in this paper · explained in 2 episodes
Degeneracy — 1× in this paper · explained in 4 episodes

Summary

The gist The maximally entangled rainbow-like Majorana pairing state can be reached, with a sequence of pairing pulses applied on the central unit cell of the system.

Investigating Entanglement Generation in Spin-1/2 Chains

The research investigates a way to create entangled Majorana fermions in the spin-1/2 Kitaev chain with open boundary conditions. The method involves applying a pair pulse sequence on the central unit cell of the chain to promote long distance spin correlations and maximal bipartite entanglement entropy. This approach is extended to the generalized bond-dependent spin-1/2 chain by introducing another set of Majorana fermions and comparing it to entangled Bell pairs. The time to reach maximal bipartite entanglement entropy is shorter in the Kitaev chain as the zero modes do not participate in the entangled pairs.

Majorana Fermion Description and Conserved Quantities

The spin-1/2 bond-dependent Kitaev chain Hamiltonian is represented using Majorana fermions defined by complex fermions, leading to specific anti-commutation relations for the Majorana operators. A conserved quantity, denoted as Cα, characterizes the pairing of the Majorana fermions between left-right symmetric sites such as (jB, −jA) of the Kitaev chain. The eigenvalue of Cα can range from −l − 1/2 to l + 1/2, with maximum and minimum values corresponding to the maximum pairing of Majorana fermions between left-right mirror symmetric sites.

Generating Long-Range Pairs via Pairing Pulses

The process of generating longrange Majorana pairs starts from a paramagnetic ordered state along the z-direction, and then a pairing pulse (a0B − ia0A)/2 is applied on the central bond. This pulse changes hCα,0i to 0.5, making the Majorana fermions at the central sites pair together. The time evolution governed by the Kitaev Hamiltonian then creates an entangled state involving one more site each on the left and the right of the chain. Monitoring hCα,0i until it drops by a tiny value, such as 0.001, signals when to apply the pairing pulse.

Generalization to the Generalized XY Chain

The method is extended to the generalized XY spin chain where another set of Majorana fermions is needed to describe the full Hamiltonian. A second conserved quantity, Cβ, connects biµ Majorana fermions between left and right sites, and its pairing can also be generated by a pairing pulse switched to β†0 = (b0A − ib0B)/2. The results show that the Majorana pairs of the aiµ state generated in Sec. II also correspond to Majorana pairs of biµ.

Application to Twisted Kitaev Chains and Robustness

The theory is applied to the twisted Kitaev chain, CoNb2O6, where the entangled Majorana pairs with finite entanglement entropy can also be generated by the pairing pulse. Even with cross terms (non-Kitaev interactions), a sequence of l pulses results in a state that still reaches the rainbow-like state of Majorana pairs, which is the only state in the eigensector with the highest hCβi and hCαi eigenvalue. The rainbow state is still protected by the mirror symmetry of the 1D chain in the twisted Kitaev chain, despite Cβ no longer being conserved. The protocol demonstrates robustness under a wide range of perturbations.

Connection to Bell Pairs and Time Scaling

The entangled Majorana pairs are related to the Bell pairs through the complex fermion number described by specific operators. For all different ratios of (J1x, J1y, J2x, J2y), the state can reach a direct product state of Bell states with hCαi = hCβi = l + 1/2. The time taken to reach the maximally entangled rainbow-like state is approximately (N/2) (1/J1x + 1/J2y). The Kitaev chain has the minimal time to reach the maximum entanglement entropy.

Effect of Spin-Flip Pulses

A spin-flip pulse, which is a linear combination of two spin-flip pulses, can increase hCαi when hCα,0i < 0. This pulse promotes the ajµ Majorana fermions pairing on the central unit cell, while for other parameter settings the Hamiltonian involves dynamics of the bjµ Majorana fermions. The spin-flip operator α†0 + α0 changes hCα,0i to −hCα,0i. This pulse results in the decrease of hCα,0i when the system reaches a possibly maximally entangled state.

Perturbation Effects on Entanglement Entropy

The protocol demonstrates robustness under various perturbations including nonuniform bond interaction, longer range interactions, external magnetic field, and ZZ spin interaction. For nonuniform interactions, maximum hCαi ∼ 1.5 for p = 0.1 in a 10-site chain. The homogeneous magnetic field along the z-direction does not affect maximum entanglement at all because the final state is a simultaneous eigenstate of Htotal and Cα operator. The protocol demonstrates robustness under a wide range of perturbations albeit the entanglement is smaller in some limits.

Conclusion on Implementation

The paper concludes that for CoNb2O6, one can use the unitary pulse which also results in the entangled state, even though it is not maximally entangled. The emergence of spin-spin correlations in this case under the spin-flip pulse signals the entangled Majorana pairs. A challenge remains in implementing the non-unitary pulse which often requires post-selection process and takes considerably longer time than applying a unitary pulse. The observation of the constant hCαi = 2.5 under the z-direction magnetic field confirms that we have reached the rainbow state independent of the field strength.

Summary for CoNb2O6

The theory can be applied to a twisted Kitaev chain, CoNb2O6, with an effective moment of Jeff = 1/2. The quasi-one dimensional chain has been known as one of the best examples of Ising ferromagnet. The bond-dependent Ising interaction, i.e., the Kitaev interaction with a tilted quantization axis, dubbed a twisted Kitaev chain model, describes its dynamics better. The results show that the entangled Majorana pairs with finite entanglement entropy can also be generated by the pairing pulse. One can use the unitary pulse, which also results in the entangled state, even though it is not maximally entangled. The emergence of spin-spin correlations in this case under the spin-flip pulse signals the entangled Majorana pairs. The final state also has hCαi = hCβi = l + 1/2, and the wave function is the same as the state generated in the untwisted Kiteav chain.

Improvements for AI systems

Bold Header: Robust Long-Range Entanglement Generation in Many-Body Systems

The improved AI system can generate long-range entangled Majorana pairs in spin systems like the Kitaev chain by applying a pairing pulse sequence on the central unit cell of the chain, which promotes long distance spin correlations and maximal bipartite entanglement entropy.

Bold Header: Entanglement Maximization via Conserved Quantities

The system can utilize conserved quantities, such as [Cα, H] = 0, to drive the state toward maximal entanglement; specifically, applying a pairing pulse increases the expectation value hCαi, and the protocol proceeds until hCαi reaches its maximum value of l + 1/2.

Bold Header: Application to Realistic Materials (CoNb2O6)

The improved system can be applied to materials like CoNb2O6, demonstrating that maximal entanglement entropy can be reached despite its deviation from the ideal Kitaev chain, allowing for the creation of entangled Majorana pairs with finite entanglement entropy even under non-ideal conditions.

Bold Header: Generalized Chain Modeling and Parameter Space Exploration

The system can handle generalized XY chains by treating them as two independent Kitaev chains, allowing for parameterization via ratios like (J1x, J1y, J2x, J2y) to find the shortest time to reach maximal entanglement based on the ratio of (θ, φ).

Bold Header: Spin-Flip Pulse Implementation

The system can implement a spin-flip pulse that acts as a linear combination of two spin-flip pulses, which can be used to increase the conserved quantity when it is negative, enabling the system to reach the possibly maximally entangled state by applying the pulse when hCα,0i reaches its negative minimum.

Abstract

Entangling a pair of far-distant qubits in many-body systems has been a challenging task in quantum computing. A robust entanglement was predicted in the rainbow states and generating nonlocal Bell pairs protected by a mirror symmetry was recently proposed. We investigate a way to create entangled Majorana fermions in the spin-1/2 Kitaev chain with open boundary conditions. The spin-1/2 Kitaev chain, a one-dimensional version of the honeycomb lattice with bond-dependent Ising interactions, has a macroscopic degeneracy related to the zero modes containing non-local spin strings. We show that applying a pair pulse sequence on the central unit cell of the chain promotes long distance spin correlations and maximal bipartite entanglement entropy. We extend this method to the generalized bond-dependent spin-1/2 chain by introducing another set of Majorana fermions and make a comparison to the entangled Bell pairs. The time to reach maximal bipartite entanglement entropy is shorter in the Kitaev chain as the zero modes do not participate in the entangled pairs. An application of our results to a recently proposed twisted Kitaev chain, CoNb2O6, is presented and future directions are also discussed.