Entanglement between quantum dots transmitted via Majorana wire: Insights from the fermionic negativity, concurrence and quantum mutual information

The study investigates quantum entanglement in a system where two quantum dots are interconnected through a short topological superconducting nanowire hosting overlapping boundary Majorana modes, providing insights into how entanglement behaves under varying energy levels and hybridization.

Key Entanglement Measures

The research focuses on quantifying entanglement using three key measures: the fermionic negativity, thermal concurrence, and quantum mutual information. The authors analyze these quantities to understand the properties of superimposed quantum states dwelling in a topological superconducting nanowire, specifically focusing on the regime where Majorana modes emerge. They investigate how entanglement varies against the position of energy levels of quantum dots and their hybridization with the nanowire.

Model and Effective Hamiltonian

The low-energy physics is captured by an effective Hamiltonian:

Hˆ = X i=1,2 Hˆ QD i + V. ˆ (1)

The quantum dots are treated as single level spin-less impurities with energies defined by:

Hˆ QD i = εi ˆd†i ˆdi, (2)

The hybridization between the quantum dots and the Majorana quasiparticles is described by:

Vˆ = iεMγˆ1γˆ2 + λ1 ˆd†1 − ˆd1 γ 1 + iλ2γ squared ˆd†2 + ˆd2, (3)

The system is analyzed using a basis of occupation number states, leading to a Hamiltonian matrix (7) with a block-diagonal structure based on charge parity. The effective parameters used in the analysis are defined as:

H33 = ε1 − εM squared, H44 = ε1 + εM squared, H55 = ε2 − εM squared, H66 = ε2 + εM squared, H77 = ε1 + ε2 − E M squared, and H88 = E max.

Entanglement Negativity Analysis

The logarithmic negativity is employed as a practical method to capture entanglement in this composite system. The authors utilize the partial transpose of the reduced density matrix of the quantum dots, denoted as T f nd1, which is defined by:

(13) ρˆ Tf nd1 R = [matrix shown]

The logarithmic entanglement negativity (N) is calculated using:

(14) N = ln(Trs(ρˆ Tf nd1 R / ρˆ Tf nd1 R†)).

The analysis reveals that in the absence of electron correlations, optimal entanglement occurs when the energy levels coincide with the zero-energy Majorana modes. However, when quantum dot levels are detuned from zero-energy, they become maximally entangled for a certain optimal hybridization.

Optimal Entanglement Conditions

The study identifies several conditions for maximal negativity:

1. The optimal entanglement occurs when quantum dot energy levels are around the overlap energies of Majorana modes leaking onto these quantum dots.

2. The optimal coupling parameters increase with QDs energies, indicating that a stronger coupling is needed to reach sensible entanglement, though the entanglement degree does not depend on the QDs energies separately but rather on their sum ε1 + ε2.

3. For symmetric couplings between quantum dots and Majorana modes (λ1 = λ2), considerable maximal negativity develops for small detunings of the quantum dot energy levels from resonance, with the limit of small energies corresponding to maximal negativity.

Finite Temperature Entanglement

To explore entanglement at finite temperatures, the authors compute thermal concurrence and quantum mutual information.

The thermal concurrence is calculated using a formula involving eigenvalues (23) of a matrix R derived from the reduced density matrix ρd.

In the low-temperature and strong coupling limit (T < ω, λ1, λ2), the expression for the density matrix simplifies to:

(25) ρˆ d = 1/2 [η 2++ η 2- Φ−(T)⟩⟨Φ−(T) + (ζ 2++ ζ 2-)Ψ+(T)⟩⟨Ψ+(T).

In this limit, the analytic result for the concurrence is deduced as:

(26) C = 1 − η 2+ − η 2-..

The concurrence is maximal for ω > λ > T, approaching zero if λ > ω.

Quantum Mutual Information

The quantum mutual information between the dots, I(ρR), is calculated using the definition:

(28) I(ρR) = S(ρd1 R) + S(ρd2 R) + X 4 n=1 ρR,nn log(ρR,nn)..

The quantum mutual information between QDs increases monotonically with growing the overlap between Majorana modes ω, but simultaneously decreases with the growing ratio of λ/ω.

As a fastidious and diligent researcher, I have analyzed the provided paper, Entanglement between quantum dots transmitted via Majorana wire: Insights from the fermionic negativity, concurrence and quantum mutual information.

This paper focuses on quantifying entanglement in a specific condensed matter system—two quantum dots coupled through a topological superconducting nanowire hosting Majorana zero modes. The findings provide theoretical frameworks for understanding and potentially harnessing non-local correlations in complex fermionic systems.

Here are the specific improvements to AI systems that can be derived from this research, along with the capabilities of those improved systems:

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Improved AI Systems and Capabilities:

1. (System: Quantum State Characterization & Entanglement Verification System - QSCEVS)

2. (System: Topological Correlation Predictor & Optimization Engine - TCOE)

3. (System: Finite-Temperature Entanglement Simulator - FTES)

4. (System: Quantum State Characterization & Entanglement Verification System - QSCEVS)

5. The QSCEVS will be a specialized AI module designed to analyze complex, multi-partite quantum states (like the tripartite system described in the paper involving two quantum dots and Majorana modes).

6. Specific Capabilities:

7. a) Non-Local Entanglement Quantification: It can calculate and report key entanglement measures (Logarithmic Negativity, Concurrence, Quantum Mutual Information) directly from system parameters (QD energies ε1, ε2; coupling strengths λ1, λ2; overlap energy ω).

8. b) Topological Feature Extraction: It can identify the specific regimes of parameter space where maximal entanglement occurs (e.g., when QD levels coincide with zero-energy Majorana modes or at detuned energies requiring optimal hybridization), effectively mapping the entanglement landscape of the topological system.

9. c) Separability Testing: It can rigorously test for separability in complex fermionic density matrices, leveraging the partial transpose method adapted for fermionic statistics to determine if a state is entangled or separable under specific physical constraints.

10. (System: Topological Correlation Predictor & Optimization Engine - TCOE)

11. The TCOE will be an optimization AI trained on the model's Hamiltonian and entanglement landscape, designed to guide experimental design or material engineering efforts in topological quantum circuits.

12. Specific Capabilities:

13. a) Optimal Coupling Prediction: It can predict the optimal coupling strengths (λ1(opt), λ2(opt)) required between quantum dots and Majorana modes for achieving maximal entanglement under given energy detunings, directly informing experimental control protocols.

8. b) Robust State Design: It can suggest parameter sets (ε1, ε2, λ1, λ2) that lead to robust entanglement transmission across the nanowire by identifying regions where the state is maximally entangled even when subjected to small perturbations or finite temperatures.

9. c) Resonance Mapping: It can map out the dependence of entanglement on the sum of QD energies and predict which energy detuning configuration yields maximal negativity, streamlining experimental searches for optimal operating points.

5. (System: Finite-Temperature Entanglement Simulator - FTES)

6. The FTES will be a simulation tool specifically designed to model and predict entanglement behavior in realistic, finite-temperature environments relevant to superconducting circuits or quantum memories.

7. Specific Capabilities:

8. a) Thermal Concurrence Prediction: It can calculate the thermal concurrence of the reduced density matrix of the quantum dot subsystem under various temperature regimes (T), utilizing the derived energy eigenvalues and state vectors from the model (Eqs. 22a-d).

9. b) Temperature Threshold Analysis: It can determine the effective superconducting gap threshold where entanglement is most robust, advising experimentalists on safe operating temperature ranges to avoid hybridization effects that destroy the topological state.

10. c) Mutual Information Forecasting: It can forecast the behavior of Quantum Mutual Information under varying temperature and coupling strengths, indicating how quickly correlations decay as thermal noise increases or as coupling ratios deviate from optimal values.

We study quantum entanglement in a system comprising two quantum dots interconnected through the short topological superconducting nanowire, which hosts overlapping boundary Majorana modes. Inspecting the fermionic negativity, we analyze the variation of entanglement against the position of the energy levels of quantum dots and their hybridization with the topological superconducting nanowire. In the absence of electron correlations, the optimal entanglement occurs when the energy levels coincide with the zero-energy Majorana modes, whereas upon increasing the hybridizations, the entanglement is gradually suppressed. Such monotonous behavior is no longer valid when the quantum dot levels are detuned from the zero-energy. Under these circumstances, the quantum dots become maximally entangled for a certain optimal hybridization. Moreover, we study the thermal concurrence to explore the entanglement properties at finite temperatures. We also compute the quantum mutual information and propose recipes for robust finite-temperature entanglement transmission via Majorana modes.