Generalised simultaneous transmission of arbitrary quantum states and classical information

This paper proposes a protocol that enables the simultaneous transmission of arbitrary optical quantum states and classical data without compromising the integrity of either, by encoding classical information via phase-space displacements and retrieving both signals through Gaussian continuous-variable teleportation and inverse displacement operations.

The Gist

The Big Idea: The "Double-Decker" Message

Imagine you want to send a very fragile, delicate glass sculpture (representing quantum information) to a friend. At the same time, you want to send them a standard text message (representing classical information).

Usually, in the world of quantum physics, trying to send the text message along with the sculpture is risky. If you try to read the text message, you might accidentally break the glass sculpture, or the noise from the text message might distort the sculpture's shape. Previous methods forced you to choose: either send the sculpture perfectly and the text poorly, or send the text perfectly and ruin the sculpture.

This paper proposes a new way to do both at the same time without ruining either. The authors call this Classically-Modulated Quantum Communication (CMQC).

How It Works: The "Moving Box" Analogy

Here is the step-by-step process of their protocol, explained simply:

1. Packing the Box (Encoding)
Alice (the sender) has her fragile glass sculpture. She also has a text message she wants to send.
Instead of putting the text inside the box with the sculpture (which would clutter it), she decides to move the box itself.

• If the text says "A," she pushes the box slightly to the left.
• If the text says "B," she pushes the box slightly to the right.
• The sculpture inside remains untouched; only the position of the box in the room changes.

2. The Journey (Transmission)
Alice sends the box down a bumpy road (the communication channel). The road might be a bit shaky (noise) or the box might get slightly smaller (loss), but the sculpture inside is still safe.

3. The Magic Trick (Teleportation)
When the box arrives at Bob's (the receiver) end, he doesn't just open it. Instead, he performs a "quantum magic trick" called Continuous-Variable Teleportation.

• Think of this as having a special scanner that can look at the box and instantly recreate a perfect copy of the sculpture on a new table in his lab.
• Crucially, this scanner also tells Bob exactly where the box was pushed (the text message). Because the push was strong enough, the scanner can easily tell, "Ah, this box was pushed to the left, so the message was 'A'."

4. The Cleanup (Restoration)
Now Bob has two things:

• He knows the text message ("A").
• He has a copy of the sculpture, but it's still sitting in the "pushed" position (because the original box was pushed).

To fix this, Bob uses his knowledge of the text message to push the new sculpture back to its original, neutral position.

• If he guessed the message correctly, the sculpture is now perfectly restored to its original state, as if it never moved.
• If he guessed wrong (because the road was too bumpy), the sculpture might be slightly off-center, but this happens very rarely if the "push" (the signal) was strong enough.

The Key Findings

The paper proves two main things about this method:

1. Perfect Recovery is Possible: If the text message signal is loud enough (strong) and the "magic scanner" (teleportation) is high-quality, Bob can recover the text message perfectly and restore the quantum sculpture perfectly. The quantum state keeps all its delicate "coherence" (its quantum magic) intact.
2. The Trade-off: There is a catch. To make the text message readable, you need to push the box hard. However, if you push it too hard while the "magic scanner" is also getting more sensitive (a concept called "squeezing"), the noise from the scanner can actually make it harder to read the text.
   • The Solution: You don't need infinite power. You just need the text signal to be strong enough to overcome the noise. If you do this, you get the best of both worlds.

A Real-World Example: The "Entangled Twins"

To prove this works, the authors tested it with a specific type of quantum object called a Bell State (think of it as two "entangled twins" that are magically linked, no matter how far apart they are).

They showed that even while sending a text message, they could still verify that the twins remained perfectly linked.

• The Result: Even with some signal loss (like a long, bumpy road), as long as the text message was strong enough, the "twins" remained perfectly entangled.
• The "Post-Selection" Trick: They also noted that if Bob only counts the times when he is sure he got the message right, the entanglement looks perfect, even over long distances. This is like only counting the photos where the focus was sharp and ignoring the blurry ones.

Summary

This paper introduces a protocol that acts like a dual-purpose delivery service. It allows you to send a standard text message by slightly shifting the position of a quantum package. By using a special teleportation technique, the receiver can read the text and then "un-shift" the package to retrieve the original quantum data perfectly.

This is a big deal because it means future quantum networks (like a "Quantum Internet") won't have to choose between sending quantum data or classical data. They can send both simultaneously, using the same channel, without one ruining the other.

Technical Summary

Title: Generalised Simultaneous Transmission of Arbitrary Quantum States and Classical Information

Problem Statement
Quantum communication using optical-frequency states offers capabilities beyond classical information theory, yet a fundamental question remains regarding the coexistence of quantum and classical information within a single channel. While previous work by Devetak and Shor established theoretical limits, and Wilde et al. proposed trade-off coding schemes, practical protocols for simultaneous transmission often face significant limitations. Specifically, earlier schemes (e.g., Qi [8, 9]) that modulate the phase space of quantum states to encode classical data suffer from a critical flaw: the measurement required to extract the classical information destroys the quantum state. Consequently, these schemes are restricted to point-to-point networks and are incompatible with protocols requiring the preservation of the quantum state, such as quantum repeaters or entanglement distribution. Furthermore, existing methods often incur a reduction in effective classical signal-to-noise ratio (SNR) and quantum coherence.

Methodology: Classically-Modulated Quantum Communication (CMQC)
The authors propose a protocol termed Classically-Modulated Quantum Communication (CMQC) to enable the simultaneous transmission of arbitrary optical quantum states and classical data without sacrificing the integrity of either. The protocol operates as follows:

1. Encoding: Alice prepares an arbitrary $N$-mode quantum state $\hat{\rho}$. She encodes classical information by applying a unitary phase-space displacement $\hat{D}(\alpha)$ to the state. Each mode is displaced by a vector $\alpha_i$ corresponding to a classical symbol (e.g., from a Phase-Shift Keying alphabet).
2. Transmission: The displaced state $\hat{\rho}' = \hat{D}(\alpha)\hat{\rho}\hat{D}^\dagger(\alpha)$ is transmitted over an arbitrary $N$-mode channel $E_N$.
3. Reception and Teleportation: Upon receipt, Bob does not perform a direct measurement that would collapse the state. Instead, he passes the displaced state through a continuous-variable (CV) teleportation circuit.
   • He mixes the incoming mode with one half of an Einstein-Podolsky-Rosen (EPR) entangled state (squeezed vacuum) on a balanced beamsplitter.
   • He performs a dual homodyne measurement on the output ports to obtain measurement results $x_i = (x_i, y_i)^T$.
4. Classical Decoding: Because dual homodyne measurement is a tomographic measurement of the phase-space distribution, the measurement outcomes are distributed around the central classical displacement $\alpha_i$. Bob uses these outcomes to probabilistically infer the classical symbol $\tilde{\alpha}_i$ sent by Alice.
5. State Restoration: Bob applies a feed-forward unitary displacement to the teleported mode based on the measurement results to reconstruct the quantum state. Crucially, he then applies an inverse displacement based on his inferred classical symbol to remove the encoding.
   • If the classical symbol is inferred correctly, the inverse displacement perfectly restores the original non-displaced state $\hat{\rho}$ (subject to channel loss and teleportation noise).
   • If a symbol error occurs, the inverse displacement is incorrect, leaving a residual displacement that acts as noise on the quantum state.

Key Contributions and Theoretical Results
The paper provides a rigorous theoretical analysis of the CMQC protocol, modeling the entire process as a Completely Positive Trace-Preserving (CPTP) map.

• Channel Modeling: The authors derive that the effective channel for the quantum signal is equivalent to an initial attenuation followed by a random phase-space displacement. The magnitude of this random displacement depends on the accuracy of the classical symbol inference.
• Limiting Cases:
  • High Classical SNR: In the limit of high classical signal strength, the probability of symbol error vanishes. Bob can perfectly infer the displacement and remove it, allowing the quantum state to be reconstructed with coherence limited only by the channel characteristics and the teleportation fidelity.
  • High Squeezing: In the limit of infinite EPR squeezing, CV teleportation can perfectly reconstruct the input state. However, the authors identify a critical asymmetry: increasing squeezing without a corresponding increase in classical signal strength degrades performance. As squeezing increases, the effective noise on the classical signal grows (scaling with $\cosh 2r$). If the classical signal is not sufficiently strong to overcome this noise, symbol errors increase, leading to incorrect inverse displacements that destroy quantum coherence.
• Dual-Rail Bell State Example: The protocol is applied to the transmission of a dual-rail Bell state ($|\phi^-\rangle$) encoded with Binary Phase-Shift Keying (BPSK). The authors calculate the CHSH Bell criterion ($S$) for the output ensemble.
  • They demonstrate that for sufficiently high classical signal strength, the protocol can transmit a logical Bell state that violates the Bell inequality ($S > 2$) even in the presence of significant channel loss, provided the post-selected regime is considered.
  • The analysis shows that the protocol achieves near-ideal post-selected Bell statistics for modest classical signal strengths and squeezing resources.

Significance and Claims
The authors claim that the CMQC protocol offers greater generality than previous trade-off encoding-based protocols while retaining the simplicity of continuous-variable SQCC.

• Simultaneity without Destruction: Unlike previous schemes where extracting classical data destroys the quantum state, CMQC allows for the retrieval of classical data while preserving the quantum state for further processing or entanglement distribution.
• Network Compatibility: The scheme supports arbitrary network configurations, enabling a classical network to simultaneously host a parallel quantum network (or vice versa) with minimal loss of quantum coherence or increase in classical error rates.
• Practical Feasibility: The paper argues that the protocol does not require infinite squeezing to function effectively. Instead, it highlights that a sufficiently high classical signal strength is the primary requirement for maintaining quantum coherence, making the scheme experimentally feasible with current or near-future resources.
• Application to Quantum Internet: The authors suggest immediate applicability to the future quantum internet, envisioning scenarios where entanglement distribution occurs simultaneously with high-volume data transfer, with distributed entangled bits (e-bits) stored in quantum memories at network nodes.

The paper concludes that in the limit of sufficiently high classical signal and appropriate squeezing, the protocol can perfectly reconstruct both the input classical signal and the input quantum state without loss of coherence, effectively decoupling the two information streams within a single channel use.