Remote state preparation of single-partite high-dimensional states in complex Hilbert spaces

1. Problem Statement

Remote State Preparation (RSP) is a protocol allowing a sender (Alice) to prepare a specific quantum state at a receiver's (Bob) location using pre-shared entanglement and classical communication, without physically transmitting the quantum system. While RSP has been extensively studied for qubits (2-level systems) and in real Hilbert subspaces, significant gaps remain:

• Dimensionality: Most existing high-dimensional RSP schemes are restricted to real Hilbert subspaces. Schemes for complex high-dimensional Hilbert spaces (specifically 4-level and 8-level systems) are largely absent.
• Resource Efficiency: Many existing schemes rely on Positive Operator-Valued Measures (POVMs), which are experimentally difficult to implement, or require complex multi-particle entangled resources (like cluster states).
• Channel Noise: Practical quantum channels often degrade from maximally entangled states to non-maximally (partially) entangled states due to decoherence. Efficient RSP schemes for these noisy channels in high dimensions are needed.

2. Methodology

The authors propose deterministic and probabilistic RSP protocols for four-level (qudit) and eight-level (qunit) equatorial states in complex Hilbert spaces. The methodology involves three main components:

A. Protocol Design

1. State Definition: The target states are equatorial states defined as superpositions of basis states with real coefficients and specific phase parameters (e.g., $|\psi\rangle = \sum c_i e^{i\theta_i}|i\rangle$).
2. Maximally Entangled Channel:
   • Alice and Bob share a maximally entangled state (e.g., $|\Psi\rangle = \frac{1}{2}\sum |ii\rangle$ for 4-level).
   • Alice performs a local unitary transformation on her particle to encode the phase information.
   • Alice performs a Von Neumann projective measurement in a newly constructed set of orthogonal bases. These bases are derived by identifying a specific orthogonal matrix ($T_4$ or $T_8$) that relates the computational basis to the target state basis.
   • Alice sends the measurement result to Bob via classical communication.
   • If the measurement outcome corresponds to the target state, Bob obtains the state directly. If not, Bob applies specific feed-forward unitary operations (if the phases are restricted to zero/real subspace) to recover the state.
3. Non-Maximally Entangled Channel:
   • To handle noisy channels where the shared state is non-maximally entangled ($\sum a_i |ii\rangle$), the protocol introduces an auxiliary particle.
   • Bob performs a collective unitary transformation on his particle and the auxiliary particle to "concentrate" the non-maximally entangled state into a maximally entangled one.
   • A subsequent measurement on the auxiliary particle determines success. If successful, the desired state is recovered; otherwise, the scheme fails (or yields a lower-dimensional state).

B. Physical Implementation Strategy

• Encoding: The authors propose encoding the high-dimensional computational basis ($|0\rangle$ to $|N-1\rangle$) into the spatial modes (path) of single-photon systems.
• Avoiding Collective Operations: A key innovation is avoiding the difficult implementation of high-dimensional collective operations (like the $W$ and $Z$ matrices required for non-maximally entangled channels). Instead, they propose using Variable Reflectivity Beam Splitters (VBSs) on the spatial modes.
  • For a 4-level system, 3 VBSs are used.
  • For an 8-level system, 7 VBSs are used.
  • These VBSs effectively perform the necessary concentration of the state without requiring complex multi-particle gates.
• Linear Optics: The single-qudit operations ($T_4$ and $T_8$) required for the measurement basis are decomposed into sequences of Mach-Zehnder Interferometers (MZIs) composed of 50:50 Beam Splitters (BS) and Phase Shifters (PS).

3. Key Contributions

1. Complex Hilbert Space RSP: The paper presents the first protocols for remotely preparing 4-level and 8-level equatorial states in complex Hilbert spaces, extending beyond previous work limited to real subspaces.
2. Orthogonal Measurement Bases: The authors identify a specific set of orthogonal measurement bases (derived from matrices $T_4$ and $T_8$). This allows the use of standard Von Neumann projective measurements, which are experimentally more feasible than POVMs.
3. Scalability and Flexibility:
   • The 4-level and 8-level schemes are general. By setting specific coefficients to zero, the protocols naturally reduce to 3-level, 5-level, 6-level, and 7-level RSP schemes.
   • The schemes work for both maximally and non-maximally entangled channels.
4. Experimental Feasibility: The paper demonstrates that the necessary high-dimensional collective operations can be replaced by linear optical elements (BS, PS, VBS) acting on spatial modes of single photons, making the scheme viable with current technology.

4. Results and Evaluations

• Success Probabilities:
  • Maximally Entangled Channel: In the general complex case, the success probability is $1/N$ (where $N=4$ or $8$). However, if the state is restricted to the real subspace (phases are zero), the success probability can be boosted to 100% using feed-forward unitary corrections.
  • Non-Maximally Entangled Channel: The success probability is determined by the smallest coefficient of the entangled state (e.g., $|a_3|^2$ for 4-level). In the real subspace limit, this can be boosted to $N|a_{min}|^2$.
• Fidelity Analysis: The authors evaluated the impact of experimental imperfections (non-ideal beam splitters and phase shifters).
  • Simulations show that with realistic imperfections (e.g., transmission ratio deviation $\epsilon$), the average fidelity remains very high (approx. 0.996 to 0.998).
  • Dark counts in detectors are estimated to reduce performance by only 5-10%.
• Resource Efficiency: The proposed schemes minimize the number of entangled pairs required compared to cluster-state-based approaches and avoid the need for full Bell-state measurements.

5. Significance

• Advancement in Quantum Communication: This work bridges the gap between theoretical high-dimensional RSP and experimental realization, specifically addressing the complex phase space which is crucial for high-capacity quantum communication.
• Noise Resilience: By providing a method to handle non-maximally entangled states (common in real-world noisy channels) without complex collective gates, the protocol enhances the robustness of quantum networks.
• Experimental Roadmap: The proposal to use spatial modes and linear optics (VBSs) provides a concrete, implementable roadmap for building high-dimensional quantum networks using existing photonic technology.
• Scalability: The ability to easily scale from 4/8 levels down to 3/5/6/7 levels by parameter adjustment makes these protocols versatile for various quantum information processing tasks, including Quantum Key Distribution (QKD) and quantum computing.

In summary, Zhao et al. provide a theoretically rigorous and experimentally practical framework for high-dimensional Remote State Preparation in complex Hilbert spaces, overcoming previous limitations regarding dimensionality, channel noise, and experimental complexity.