Optical quantum teleportation with known amplitude distorting factors of teleported qubits

This paper proposes a hybrid continuous-discrete variable quantum teleportation protocol for unknown optical single-rail qubits that, while not always perfect, enables Bob to either recover the original state or utilize it with known amplitude distortions by leveraging increased classical communication to identify the specific unitary transformations applied.

The Gist

Imagine you are trying to send a very delicate, secret message written on a piece of glass to a friend across the ocean. But there's a catch: you can't ship the glass itself, and if you try to look at it too closely, it shatters. This is the challenge of Quantum Teleportation.

In this paper, the authors propose a new, clever way to send these "quantum messages" (specifically, single-rail optical qubits) using a mix of two different types of "shipping containers."

Here is the breakdown of their idea using simple analogies:

1. The Problem: The "Perfect" vs. The "Real"

Usually, quantum teleportation is like trying to hit a bullseye with a blindfold. You have a 50% chance of getting it perfect, and the other 50% of the time, you get nothing or a broken message.

The authors wanted to improve this. They asked: What if, instead of just getting a "perfect" message or a "broken" one, we could get a message that is slightly warped, but we know exactly how it is warped?

2. The Solution: The Hybrid Truck

To solve this, they built a Hybrid Quantum Channel. Think of this as a delivery truck that carries two types of cargo at once:

• The "Smooth" Cargo (Continuous Variable): This is like a giant, continuous wave of water. It's smooth and hard to count, but very powerful.
• The "Countable" Cargo (Discrete Variable): This is like a bag of marbles. You can count them: 0 marbles, 1 marble, 2 marbles.

By mixing these two types of cargo together, they created a special "entanglement" link between the sender (Alice) and the receiver (Bob).

3. The Process: The Magic Beam Splitter

Here is how the teleportation happens, step-by-step:

1. The Setup: Alice has the secret message (the unknown qubit) and half of the Hybrid Truck. Bob has the other half.
2. The Mix: Alice takes her secret message and mixes it with her half of the truck on a special mirror called a Beam Splitter. Imagine pouring two different colored liquids into a funnel; they swirl together.
3. The Peek (Measurement): Alice looks at the result of the mix using super-sensitive detectors (like counting exactly how many drops of water and marbles came out).
4. The Text Message: Alice sends the count to Bob via a regular text message (classical information).

4. The Result: The "Distorted" Gift

This is where the paper gets interesting. Depending on what Alice counts, two things can happen:

• Scenario A (The Perfect Win - 50% chance): Alice counts a specific combination (like "0 drops, 1 marble"). Bob receives the message exactly as it was. No changes needed.
• Scenario B (The "Warped" Win - ~50% chance): Alice counts something else (like "2 drops, 0 marbles"). Bob receives the message, but the "volume" or "brightness" of the message is distorted. It's like receiving a photo that is slightly too bright or too dark.

Here is the genius part: Because Alice and Bob know the settings of their machine, they both know exactly how the photo is distorted. They know, "Oh, this photo is 20% too bright."

5. Why is this better?

In old methods, if you got a distorted message, you threw it away. In this new method, you don't throw it away.

• Option 1: Bob can use the distorted message anyway. Since he knows the distortion factor, he can adjust his equipment to compensate for it later.
• Option 2 (The Repair Shop): Bob can try to "fix" the message. He has a special tool (an auxiliary photon state) that he can mix with the distorted message. If he gets lucky with a second measurement, he can "un-warp" the message and get the perfect original back.

The Big Picture

Think of this protocol like a smart delivery service:

• Old Way: 50% chance of a perfect package, 50% chance of a lost package.
• New Way: 50% chance of a perfect package, and 50% chance of a package that is slightly dented. But, the shipping label tells you exactly how to fix the dent.

The authors show that by using this "Hybrid" approach, they can make the process nearly deterministic (meaning it almost always works). Even when the message isn't perfect, it's not useless; it's just "known to be imperfect," which allows Bob to either use it as-is or try to repair it.

In summary: They found a way to teleport quantum information where, even if the signal gets a little "scrambled," both the sender and receiver know the code to unscramble it, making the whole process much more reliable and useful for future quantum computers.

Technical Summary

Here is a detailed technical summary of the paper "Optical quantum teleportation with known amplitude distorting factors of teleported qubits" by Mikhail S. Podoshvedov and Sergey A. Podoshvedov.

1. Problem Statement

Standard quantum teleportation protocols often face a trade-off between fidelity and success probability.

• Discrete Variable (DV) systems: Typically offer unit fidelity but are often probabilistic (success probability < 1) or require complex resource states.
• Continuous Variable (CV) systems: Can be deterministic but suffer from finite squeezing, resulting in non-unit fidelity (noise/distortion).
• Hybrid Approaches: While hybrid entanglement resources exist, previous protocols for single-rail optical qubits (superpositions of vacuum $|0\rangle$ and single photon $|1\rangle$) often resulted in either perfect teleportation with low probability or total failure (vacuum state) in other cases.

The authors address the need for a protocol that is nearly deterministic (high success probability) while providing a mechanism to handle cases where the teleported state is not perfect. Specifically, they aim to develop a protocol where, even when the output qubit is distorted, the amplitude distortion factors are known to both parties, allowing for potential recovery or continued utility of the state.

2. Methodology

The proposed protocol utilizes a hybrid quantum channel composed of Continuous Variable (CV) and Discrete Variable (DV) states.

• The Quantum Channel:

  • The channel is a hybrid entangled state $|\triangle_1\rangle_{13}$ shared between Alice (mode 1, CV part) and Bob (mode 3, DV part).
  • The CV part consists of parity-dependent states ($|\Psi^{(0)}\rangle$ and $|\Psi^{(1)}\rangle$) derived from a Single-Mode Squeezed Vacuum (SMSV) state.
  • The DV part is a single-photon state.
  • The channel is characterized by two independent parameters: the squeezing parameter of the SMSV state ($S_{SMSV}$) and a beam splitter (BS) parameter ($B_0$) used during channel preparation.

• Teleportation Process:

  1. Mixing: Alice mixes the CV part of the channel with the unknown single-rail qubit ($|\varphi\rangle_2 = a_0|0\rangle + a_1|1\rangle$) on a second beam splitter with parameter $B = r^2/t^2$.
  2. Measurement: Alice performs Photon Number Resolving (PNR) detection on the two output modes of the beam splitter, obtaining results $(k_1, k_2)$.
  3. Classical Communication: Alice sends the measurement results to Bob.
  4. State Reconstruction: Bob's state collapses into a conditional state depending on the parity of $k_1 + k_2$.

• Key Control Mechanism:
  The protocol relies on three independent parameters: $S_{SMSV}$, $B_0$ (channel creation), and $B$ (mixing). By tuning these, the authors analyze the conditions under which the amplitude distortion factor ($b$) equals 1 (perfect teleportation) or takes other known values.

3. Key Contributions

• Generalization of Nonlocal Photon Teleportation: The protocol generalizes the teleportation of single-rail qubits using a nonlocal photon (which corresponds to the limit where $S_{SMSV} = 0$). By introducing non-zero squeezing, the protocol extends the capabilities to include cases where the output is distorted but recoverable.
• Known Amplitude Distortion: Unlike standard probabilistic teleportation where failure results in a lost state, this protocol ensures that in "failed" cases (where fidelity is not 1), the distortion factor is known to both Alice and Bob. This allows Bob to apply specific unitary corrections or use the state for other purposes.
• Nearly Deterministic Operation: The protocol achieves a success probability of 0.5 for perfect teleportation (without distortion) and an additional probability of receiving a distorted but usable qubit in the remaining cases (excluding rare failure events like $k_1=k_2=0$).
• Probabilistic Recovery Scheme: The authors propose a method for Bob to probabilistically reconstruct the original qubit from the distorted state using pre-prepared auxiliary photon states and an additional beam splitter mixing step.

4. Results

• Perfect Teleportation Probability:

  • When the measurement outcomes are $(0,1)$ or $(1,0)$ (sum is odd), and the beam splitter is balanced ($B=1$) with specific channel parameters, Bob receives the original qubit with probability 0.5.
  • In these cases, Bob only needs to apply a Pauli-Z gate (for outcome 01) or do nothing (for outcome 10) to recover the state.

• Distorted Teleportation:

  • For other measurement outcomes (e.g., $(2,0), (0,2), (1,1)$), Bob receives a qubit with an amplitude distortion factor $b \neq 1$.
  • Crucially, the value of $b$ is calculable by both parties based on the known parameters ($S_{SMSV}, B_0, B$) and the measurement result.
  • The probability of total failure (receiving a vacuum state or losing the superposition, e.g., outcome 00 or 11 with $B=1$) is negligible and decreases rapidly as squeezing increases.

• Recovery Efficiency:

  • By employing the proposed recovery scheme (mixing the distorted qubit with an auxiliary state), the total probability of obtaining the original qubit (either directly or via recovery) can be increased from 0.5 to approximately 0.56.
  • This recovery is probabilistic and depends on the magnitude of the distortion factor; larger distortion factors allow for higher recovery probabilities.

• Parameter Dependencies:

  • The paper provides contour plots showing that perfect teleportation ($b=1$) for specific outcomes (like 01 and 10) is achievable over a wide range of $S_{SMSV}$ and $B_0$ values when $B=1$.
  • However, achieving $b=1$ for all possible measurement outcomes simultaneously is mathematically impossible with a single set of parameters.

5. Significance

• Practical Feasibility: The protocol is experimentally feasible using current technology, specifically superconducting transition edge sensors (TES) for photon number resolution and standard optical components for generating SMSV states and beam splitters.
• Resource Efficiency: It offers a "nearly deterministic" alternative to standard probabilistic teleportation, maximizing the utility of the quantum channel. Even when the state is not perfectly teleported, the information is not lost; the known distortion allows for partial information usage or active correction.
• Scalability and Generalization: The approach serves as a generalization of existing single-rail teleportation schemes. It bridges the gap between DV and CV teleportation, demonstrating how hybrid resources can mitigate the limitations of finite squeezing in CV systems while maintaining the discrete nature of the qubit.
• Application in Quantum Networks: The ability to know the distortion factor makes this protocol suitable for quantum repeaters and quantum computing networks where intermediate states might be imperfect but still useful for subsequent operations (e.g., further teleportation or gate operations).

In summary, the paper presents a robust optical quantum teleportation protocol that trades the strict requirement of unit fidelity for a high-probability, nearly deterministic process where the nature of any fidelity loss is known and potentially correctable.