⚛️ quantum physics

Teleportation of unknown qubit via Star type tripartite states

This paper demonstrates that perfect standard teleportation of an unknown qubit is achievable using Star-type tripartite states, including specific superpositions of non-prototype W states, thereby refuting Jung's conjecture that a Groverian entanglement measure of $1/\sqrt{2}$ is a necessary condition for such teleportation.

Original authors: Anushree Pandey, Abhijit Mandal, Sovik Roy

Published 2026-03-16

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Anushree Pandey, Abhijit Mandal, Sovik Roy

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to send a secret, fragile message (a "qubit") from your house (Alice) to your friend's house (Bob) across the city. In the quantum world, you can't just copy the message or look at it, because doing so would break it. You need a special "quantum bridge" to transport it.

This paper is about finding new, better, and more surprising types of bridges to build that quantum teleportation system.

Here is the story of their discovery, broken down into simple concepts:

1. The Old Rules vs. The New Discovery

For a long time, scientists thought there was a strict rule for building a perfect teleportation bridge.

The Old Rule (The "Jung Conjecture"): A famous scientist named Jung suggested that to teleport a message perfectly, the bridge (the entangled state) had to be "perfectly balanced." Specifically, it had to have a specific "strength" of connection (called Groverian entanglement) equal to 0.71 (or $1/\sqrt{2}$ ).
The Analogy: Imagine you think you need a bridge made of 100% steel to cross a river. If the bridge is 90% steel, you believe it will collapse.

The Paper's Big Reveal: The authors found bridges that are not made of 100% steel (they are only about 86% "strong" in the old sense), yet they still cross the river perfectly without collapsing! They proved that the old rule was too strict. You don't need that specific "perfect balance" to teleport a quantum state successfully.

2. The Characters: The "Star" and the "W"

To understand their experiment, we need to know the players:

The GHZ State: The classic, reliable bridge. Everyone knew this worked.
The Prototype W State: A popular bridge that everyone thought was useless for teleportation. It was like a wobbly wooden plank that seemed too weak.
The Non-Prototype W State: The authors took that "useless" plank and tweaked it slightly. Suddenly, it worked!
The "Star" State: This is the main character of the paper. Imagine a star shape with a center and points.
- In their setup, Alice holds two "points" (peripheral qubits), and Bob holds the "center" (central qubit).
- The Magic Trick: If you remove the center (Bob's part), the two points Alice holds become completely unconnected (separable). But if you remove the points, the center stays connected to the rest. It's an asymmetric relationship, unlike a circle where everyone is equal.

3. The Experiment: Building the "Sw~w" Bridge

The authors did something clever. They took a "Non-Prototype W" state and mixed it with its "spin-flipped" version (imagine flipping a coin and mixing heads and tails).

The Result: This mixture created a new state they called $S_{w\tilde{w}}$ .
The Surprise: Even though this new state didn't fit the "perfect balance" rule (it wasn't a $P_{max} = 1/2$ state), they proved mathematically that it works perfectly for teleportation.
The Metaphor: It's like discovering that a bridge made of a specific mix of wood and plastic works better than a bridge made of pure steel, even though everyone told you "steel is the only way."

4. Why Does This Matter? (The "Tangle" vs. The "Groverian")

The paper dives into two ways of measuring how "knotted" or connected these quantum bridges are:

Tangle (Genuine Tripartite Entanglement): This measures if all three parts are deeply connected together.
- Finding: Some bridges that worked perfectly (like the $W_1$ state) had zero "Tangle." This means you don't need all three parts to be deeply knotted together to teleport a message. You can do it with a simpler connection.
Groverian Entanglement: This is the "strength" measure from the old rule.
- Finding: The new "Star" bridges had a strength of $\sqrt{3}/2$ (approx 0.866), which is different from the old rule's 0.71.

5. The Conclusion: Breaking the Rules

The authors conclude with a powerful message:

Genuine tripartite entanglement (the deep 3-way knot) is NOT required to teleport a quantum state.
The old rule that "Groverian entanglement must be exactly $1/\sqrt{2}$ " is NOT necessary.

In Everyday Language:
Imagine you were told, "To bake a perfect cake, you must use exactly 200 grams of sugar."
This paper is like a baker who says, "Actually, I just baked a perfect cake using 173 grams of sugar and a secret pinch of spice. The old rule was wrong. You have more freedom in how you build your quantum bridges than we thought!"

This opens the door for scientists to use a much wider variety of quantum states to build future quantum computers and communication networks, making the technology more flexible and potentially easier to create.

1. Problem Statement

The paper addresses two primary challenges in quantum information theory:

Limitations of Standard Entanglement Classes: While the GHZ state is a known resource for perfect quantum teleportation, the standard "prototype" W state is generally considered unsuitable for standard (perfect) teleportation protocols.
Validity of the Jung Conjecture: Eylee Jung et al. conjectured that for perfect two-party quantum teleportation to occur, the entanglement resource must satisfy specific conditions: the maximum probability of success in Grover's search algorithm ( $P_{max}$ ) must be exactly $1/2$ , and the Groverian measure of entanglement must be $1/\sqrt{2}$ . The authors aim to test the necessity of this conjecture by investigating whether states violating these conditions can still serve as perfect teleportation channels.

2. Methodology

The authors employ a theoretical framework involving quantum teleportation protocols, entanglement measures, and state classification:

Teleportation Protocol: They utilize the standard Bennett et al. protocol extended to tripartite systems. Alice (sender) holds two qubits, and Bob (receiver) holds one. Alice performs a Bell-like measurement on her two qubits (the unknown state and her share of the channel) and communicates the result classically to Bob, who applies a unitary correction.
State Construction:
- Non-prototype W States: They utilize the $|W_n\rangle$ class defined by Agrawal and Pati, specifically the case $|W_1\rangle$ , which differs from the symmetric prototype W state.
- Spin-Flipped States: They construct spin-flipped versions of these states (denoted $\tilde{W}$ ).
- Star Class States: They define a specific "Star" state ( $|Star_1\rangle$ ) and construct a new state, $|S_{w\tilde{w}}\rangle$ , by taking the linear superposition of the non-prototype W state ( $|W_1\rangle$ ) and its spin-flipped version ( $|\tilde{W}_1\rangle$ ).
Analysis Metrics:
- Coherence: Calculated using the $l_1$ -norm.
- Bipartite Correlations: Quantified using Concurrence ( $C$ ) and Negativity ( $N$ ).
- Multipartite Entanglement: Measured using Tangle ( $\tau$ ) for genuine tripartite entanglement and the Groverian measure ( $G$ ) derived from $P_{max}$ .

3. Key Contributions

Validation of Non-Prototype W States: The paper confirms that the non-prototype W state ( $|W_1\rangle$ ) and its spin-flipped version ( $|\tilde{W}_1\rangle$ ) can serve as channels for perfect teleportation, extending previous work by Agrawal and Pati.
Discovery of Star Class Utility: The authors demonstrate that the constructed state $|S_{w\tilde{w}}\rangle$ (a superposition of $|W_1\rangle$ and $|\tilde{W}_1\rangle$ ) belongs to the Star class of states. They prove that this specific Star class state, along with a modified Star state ( $|Star_1\rangle$ ), successfully facilitates perfect teleportation.
Refutation of the Jung Conjecture: The most significant contribution is the demonstration that the Jung conjecture is not a necessary condition. The authors show that states with $P_{max} = 1/4$ (and consequently $G = \sqrt{3}/2$ ) can still achieve perfect teleportation, contradicting the claim that $P_{max} = 1/2$ is required.
Insight into Genuine Tripartite Entanglement: The study reveals that genuine tripartite entanglement (measured by Tangle) is not a prerequisite for perfect teleportation. States with zero Tangle (like $|W_1\rangle$ ) and states with non-zero Tangle (like $|Star_1\rangle$ ) both work effectively.

4. Key Results

A. Teleportation Success

The authors derived the measurement bases and corresponding unitary corrections for Bob for several states:

GHZ State: Confirmed as a perfect channel.
$|W_1\rangle$ and $|\tilde{W}_1\rangle$ : Successfully used as channels.
$|Star_1\rangle$ and $|S_{w\tilde{w}}\rangle$ : Successfully used as channels.
General $|W_n\rangle$ and $|W_n\tilde{W}_n\rangle$ : Shown to be useful for all $n \geq 1$ .

B. Entanglement and Correlation Properties

The paper provides a comparative analysis of the states (Table IX and Table X in the original text):

State	Tangle ( $\tau$ )	$P_{max}$	Groverian ( $G$ )	Concurrence (Bipartite)	Teleportation Success?
GHZ	1	$1/2$	$1/\sqrt{2}$	High	Yes
$\|W_1\rangle$	0	$1/2$	$1/\sqrt{2}$	Non-zero	Yes
$\|Star_1\rangle$	$1/4$	$1/4$	$\sqrt{3}/2$	Non-zero	Yes
$\|S_{w\tilde{w}}\rangle$	$1/4$	$1/4$	$\sqrt{3}/2$	Non-zero	Yes

Coherence: The Star class states ( $|Star_1\rangle$ and $|S_{w\tilde{w}}\rangle$ ) exhibit coherence distributed at all levels (1-qubit, 2-qubit, and 3-qubit), though asymmetrically.
Negativity: For Star states, the concurrence and negativity vanish when the central qubit is traced out, confirming their Star topology (central qubit is essential for entanglement).

5. Significance

Theoretical Advancement: The work challenges the established "Jung Conjecture," broadening the understanding of what constitutes a valid quantum resource for teleportation. It proves that high Groverian entanglement or specific $P_{max}$ values are not strict bottlenecks for perfect teleportation.
Resource Optimization: It identifies new classes of states (Star type and superpositions of W states) that are viable for quantum communication, potentially offering more robust or flexible resources than standard GHZ states.
Structural Insight: The paper highlights that genuine tripartite entanglement (Tangle) is not a necessary condition for perfect teleportation, as states with zero Tangle ( $|W_1\rangle$ ) work just as well as those with non-zero Tangle.
Future Directions: The authors suggest that Star states could be further explored for superdense coding and experimental realization, opening new avenues for multipartite quantum information processing.

In conclusion, the paper successfully demonstrates that Star-type tripartite states and superpositions of non-prototype W states are effective channels for perfect quantum teleportation, thereby disproving the necessity of the Jung conjecture's specific entanglement thresholds.