Entanglement concentration of high-dimensional unknown partially entangled state

1. Problem Statement

In quantum communication and information processing (QIP), maximally entangled states are essential resources for tasks like teleportation, dense coding, and quantum key distribution. However, during practical transmission and storage, environmental noise inevitably degrades these states into partially entangled pure states or mixed states.

• The Gap: While Entanglement Concentration Protocols (ECPs) exist for two-level qubit systems, and some for high-dimensional systems with known parameters, there is a lack of universal schemes for high-dimensional (specifically qutrit) partially entangled states with unknown parameters.
• The Challenge: Existing high-dimensional protocols often require complex multi-round local unitary transformations, permutation operations, or prior knowledge of the Schmidt coefficients. Furthermore, many rely on hyper-entanglement (multiple degrees of freedom), which is difficult to manipulate simultaneously.

2. Methodology

The authors propose a universal scheme to distill a two-qutrit maximally entangled generalized Bell state from two identical copies of an unknown, less-entangled qutrit state. The protocol relies on cross-Kerr nonlinearities and linear optical elements.

Core Components:

1. Input State: Two identical copies of a partially entangled qutrit state:
   $$|\psi\rangle = \alpha|0_a0_b\rangle + \beta|1_a1_b\rangle + \gamma|2_a2_b\rangle$$
   where $\alpha, \beta, \gamma$ are unknown to the communicating parties (Alice and Bob).
2. Cross-Kerr Nonlinearity:
   • Bob interacts the signal photons (modes $b, d, f$) with coherent probe states ($|\alpha\rangle$ and $|\alpha'\rangle$).
   • The interaction induces a phase shift on the coherent state dependent on the photon number in the signal mode (e.g., $|0\rangle \to -\theta$, $|1\rangle \to 0$, $|2\rangle \to 2\theta$).
   • This creates entanglement between the signal photons and the coherent probe states.
3. Homodyne Measurements:
   • Bob performs X-quadrature homodyne measurements on the two coherent probe beams.
   • Specific measurement outcomes (phase shifts) project the three photon pairs into specific non-normalized states.
4. Single-Partite Projection Measurements:
   • Alice and Bob perform measurements on the auxiliary photons ($c, d, e, f$) in the three-dimensional Fourier basis ($\{|\phi_0\rangle, |\phi_1\rangle, |\phi_2\rangle\}$).
   • This basis is implemented using linear optical elements (variable beam splitters and phase shifters) to realize a spatial-based single-qutrit Fourier transform ($U_F$).
5. Feed-Forward Operations:
   • Based on the measurement results, Alice or Bob applies specific single-qutrit unitary operations ($U_0$ to $U_4$) to the remaining photons ($a$ and $b$) to correct phases and obtain the target state.

3. Key Contributions

• Universality for Unknown Parameters: Unlike previous high-dimensional ECPs that require knowledge of the Schmidt coefficients, this scheme works for completely unknown parameters $\alpha, \beta, \gamma$.
• Local Operations: All complex operations (Kerr interactions, homodyne detection, and projection measurements) are performed only at Bob's site, simplifying the protocol for Alice.
• Linear Optical Implementation: The crucial step of single-qutrit projection measurement is achieved using only linear optics (beam splitters and phase shifters), making the scheme experimentally feasible compared to schemes requiring strong nonlinearities for every step.
• Resource Efficiency: The protocol utilizes two identical copies of the input state. While this is higher than the single-copy requirement for some qubit ECPs, it is necessary for the higher information capacity of qutrits.
• By-Product Utility: The protocol does not discard "failed" outcomes. It generates concentrated partially entangled qubit states (e.g., states involving only $|0\rangle$ and $|1\rangle$) as by-products. These are valuable resources for other QIP tasks like circuit cutting and hierarchical sharing.

4. Results and Performance

• Success Probability:
  • The probability of obtaining the desired maximally entangled state $|\Psi_{40}\rangle = \frac{1}{\sqrt{3}}(|00\rangle + |11\rangle + |22\rangle)$ is $P = 6|\alpha\beta\gamma|^2$.
  • The paper notes that, unlike qubit ECPs, the success probability of this specific qutrit scheme cannot be improved by iteration because the remaining states after a failed attempt do not retain the necessary structure for further concentration in the same manner.
• Feasibility Analysis:
  • Linear Optics: The fidelity of the Fourier transformation block ($T_{mn}$) was analyzed against imperfections in beam splitters (deviation from 50:50) and phase shifters. The results show high fidelity is achievable with current technology.
  • Homodyne Detection: The success probability of the X-homodyne measurement was evaluated considering photon dissipation. With large coherent state amplitudes ($\alpha \approx 50-60$) and low dissipation, the success probability exceeds 98%.
• Known Parameter Variant: The authors also designed a simplified linear-optics-only ECP for the case where parameters are known, achieving a success probability of $|\gamma|^2/3$ (assuming $|\alpha| > |\beta| > |\gamma|$).

5. Significance

• Advancement in High-Dimensional QIP: This work bridges the gap between theoretical high-dimensional entanglement and practical implementation, specifically addressing the "unknown parameter" scenario which is the most realistic for long-distance quantum communication.
• Scalability: By demonstrating that high-dimensional states can be concentrated using linear optics and weak nonlinearities, the paper supports the scalability of quantum networks beyond simple qubit systems.
• Noise Resilience: High-dimensional systems inherently offer better noise resistance and larger channel capacity. Providing a method to recover these states from noise makes high-dimensional quantum communication more robust.
• Experimental Realism: The reliance on cross-Kerr nonlinearities (enhanced by EIT techniques) and standard linear optics aligns the proposal with current experimental capabilities, moving beyond purely theoretical constructs.

In summary, Du et al. present a robust, universal protocol for distilling high-dimensional entanglement from unknown noisy states, leveraging cross-Kerr effects for state selection and linear optics for measurement, thereby enhancing the feasibility of high-dimensional quantum communication networks.