⚛️ quantum physics

No-local-broadcasting theorem for non-signalling behaviours and assemblages

This paper proves the conjecture that nonlocal behaviours and steerable assemblages cannot be locally broadcast by demonstrating the monotonicity of relative entropy within correlation scenarios, thereby extending the no-broadcasting theorem to broader non-classical theories.

Original authors: Adrian Solymos, Carlos Vieira, Cristhiano Duarte, Zoltán Zimborás

Published 2026-03-20

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Adrian Solymos, Carlos Vieira, Cristhiano Duarte, Zoltán Zimborás

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 have a very special, secret recipe for a cake that tastes unlike anything else in the world. This recipe relies on a mysterious, "spooky" connection between two ingredients that can't be explained by normal physics. Let's call this a "Quantum Secret."

In the world of quantum information, scientists have long known a rule called the "No-Cloning Theorem." It basically says: You cannot make a perfect copy of a secret quantum recipe. If you try to photocopy it, you ruin the original or the copy is wrong. This is great for security (like in quantum encryption) because it means spies can't steal your secret without getting caught.

But there's a slightly different trick called "Broadcasting." Instead of making two exact copies of the same cake, imagine you want to take one cake and split it into two new cakes, such that if you taste either new cake, it tastes exactly like the original.

For a long time, scientists knew you couldn't do this with quantum secrets. But a big question remained: Does this rule apply to all kinds of weird, non-classical secrets, even ones that might exist in theories beyond our current understanding of physics?

This paper says: Yes. The rule holds everywhere. You cannot "locally broadcast" these secrets.

Here is the breakdown of their discovery using simple analogies:

1. The Two Types of "Secrets"

The paper looks at two specific types of quantum mysteries:

Non-local Behaviors (The "Spooky Correlations"): Imagine Alice and Bob are in different rooms. They press buttons on machines, and the lights flash in a pattern that seems impossible to predict unless they are secretly communicating faster than light. This is a "non-local behavior."
Steerable Assemblages (The "Remote Control"): Imagine Alice can press a button that instantly changes the state of a quantum object Bob is holding, even though she is far away. She can "steer" his object into different shapes. This is "steering."

2. The "Local Broadcast" Attempt

The researchers asked: Can Alice and Bob take their shared secret and use only "local" tools (tools they have in their own rooms, plus some shared random notes) to create two new pairs of machines (Alice-0, Bob-0 and Alice-1, Bob-1) that both perfectly reproduce the original secret?

Think of it like this:

You have a Magic Box that produces a specific, weird pattern of lights.
You want to build two new Magic Boxes using only local wiring and a shared instruction manual.
You want both new boxes to produce the exact same weird pattern as the original.

3. The "Energy" of Secrets (Relative Entropy)

To prove this is impossible, the authors used a mathematical tool called Relative Entropy.

The Metaphor: Think of "Non-locality" or "Steering" as a form of fuel or energy.
The Rule: In the universe of these secrets, this "fuel" is monotonic. This means you can never create more of it using local tools. You can only use it up or keep it the same. It's like trying to make more gold out of lead using only a hammer; you can't create the value, you can only move it around.

4. The Proof: The Impossible Math

The authors proved their point with a logical trap (a contradiction):

Assumption: Let's pretend you can broadcast the secret. You take one "Magic Box" and make two new ones.
The Cost: Because the new boxes are created locally, the "fuel" (non-locality) inside them cannot increase. In fact, the math shows it should stay the same or go down.
The Twist: However, the act of broadcasting itself creates a situation where the "fuel" must increase. Why? Because to have two independent pairs that both perfectly mimic the original, you are essentially doubling the complexity of the "spooky connection."
The Contradiction: You cannot have a process that doesn't create fuel, yet the result requires more fuel. It's like saying, "I can build a car that runs on water, but the engine requires more water than I put in."

Since this is a logical impossibility, the assumption must be wrong. You cannot broadcast the secret.

5. Why This Matters

Security: This confirms that the "un-copyable" nature of quantum secrets is a fundamental law of the universe, not just a quirk of our current physics. Even if we discover "Post-Quantum" physics (physics beyond what we know now), spies still can't copy these secrets.
Resource Management: It tells us that "spooky connections" are a finite resource. You can't amplify them or distribute them freely. If you have a little bit of non-locality, you have to be very careful with it; you can't just clone it to get more.

The Bottom Line

Just as you cannot photocopy a unique, one-of-a-kind painting without destroying the original or making a fake, you cannot "broadcast" a quantum secret to create two perfect copies using only local tools. The universe has a hard limit on how much "spooky connection" can be shared, and this paper proves that limit applies to the broadest possible class of theories.

1. Problem Statement and Motivation

The paper addresses a fundamental question in quantum information theory: Is the impossibility of broadcasting quantum information an intrinsic property of quantum mechanics, or does it hold for broader classes of non-classical theories?

Background: The standard No-Broadcasting Theorem states that a quantum channel cannot create a bipartite state where both marginals are identical to an unknown input state, unless the input states commute. This is a generalization of the No-Cloning Theorem.
Context: Previous work established that no-broadcasting holds for general non-classical probabilistic models satisfying the no-signalling criterion. However, a specific conjecture by Joshi, Grudka, and the Horodeckis remained open: Can non-local behaviours be locally broadcast?
The Specific Question: In a "local broadcasting" scenario, two parties (Alice and Bob) share a known correlation (behaviour or assemblage). They apply local operations (potentially with shared randomness) to produce two copies of the correlation (Alice $_0$ , Bob $_0$ and Alice $_1$ , Bob $_1$ ). The conjecture posits that if the original correlation is non-local (or steerable), it is impossible to create two local copies using only local transformations.
Gap: Previous proofs were restricted to specific scenarios (e.g., the $(2,2,2)$ Bell scenario) or relied on properties specific to quantum states. This paper aims to prove the conjecture for general non-signalling behaviours and steerable assemblages in arbitrary dimensions and input/output configurations.

2. Methodology

The authors employ an information-theoretic approach based on the monotonicity of relative entropy under specific classes of free transformations.

A. Frameworks Defined

Correlation Scenarios (Behaviours):
- Described by joint probability distributions $P(a,b|x,y)$ where inputs are buttons and outputs are lights.
- Local Behaviours: Admit a Local Hidden Variable (LHV) model.
- Non-signalling Behaviours: Marginal probabilities are independent of other parties' inputs.
- Transformations: The authors utilize LOSR (Local Operations and Shared Randomness) transformations. Crucially, they define a specific subset called LR $_{ns}$ -LOSR, which maps local behaviours to Locally Realistic Non-Signalling (LR $_{ns}$ ) behaviours in the multipartite broadcast scenario.
Steering Scenarios (Assemblages):
- Hybrid objects combining classical probabilities (Alice's outcomes) and quantum states (Bob's conditional states): $\{P(a|x)\rho_{a|x}\}$ .
- Unsteerable Assemblages: Admit a Local Hidden State (LHS) model.
- Transformations: UR $_{ns}$ -LOSR transformations, which map unsteerable assemblages to Unsteerable Realistic Non-Signalling (UR $_{ns}$ ) assemblages.

B. Key Mathematical Tools

Kullback-Leibler (KL) Divergence:
- For behaviours: $S_b(P||Q) = \sup_{\pi} S(\pi P || \pi Q)$ , where the supremum is over input distributions.
- For assemblages: Defined via the quantum relative entropy of associated Classical-Quantum (CQ) states.
Relative Entropy of Nonlocality ( $E_{LR}$ ) and Steering ( $E_{UR}$ ):
- Defined as the infimum of the KL-divergence between a given object ( $P$ or $R$ ) and the set of "free" objects (local or unsteerable):
  $E(P) = \inf_{Q \in \text{Free}} S(P || Q)$
- These quantities serve as monotones (measures of non-classicality).

C. Proof Strategy (Proof by Contradiction)

The proof relies on two conflicting propositions regarding the behavior of these monotones under broadcasting:

Monotonicity (Contraction): Free transformations (LOSR) cannot increase the relative entropy of nonlocality/steering. If $M$ is a free transformation, $E(M(P)) \leq E(P)$ .
Super-additivity in Broadcasting: If $P'$ is a valid broadcast version of a non-local $P$ , then the relative entropy of the broadcast version must be strictly greater than the original: $E(P') > E(P)$ .

If a local broadcast were possible, both conditions would have to hold simultaneously, leading to the contradiction $E(P') \leq E(P) < E(P')$ .

3. Key Contributions and Results

Contribution 1: General No-Local-Broadcasting for Behaviours

Theorem 1: It is impossible to locally broadcast any known bipartite non-local non-signalling behaviour using LR $_{ns}$ -LOSR transformations.
Method:
- Proposition 1: Proved that $E_{LR}$ is contractive under LR $_{ns}$ -LOSR transformations.
- Proposition 2: Proved that for a broadcast version $P'$ of a non-local $P$ , $E_{LR}(P') > E_{LR}(P)$ . This was achieved by adapting the chain rule of probability distributions to behaviours. The proof showed that if the original behaviour is non-local, the conditional terms in the chain rule contribute a strictly positive amount to the divergence, ensuring the broadcast version is "more non-local" than the original.
Scope: Applies to general $(N, m, o)$ scenarios, including post-quantum correlations (e.g., PR boxes).

Contribution 2: General No-Local-Broadcasting for Assemblages

Theorem 2: It is impossible to locally broadcast any known steerable non-signalling assemblage using UR $_{ns}$ -LOSR transformations.
Method:
- Proposition 3: Established the contractivity of the relative entropy of steering ( $E_{UR}$ ) under LOSR transformations.
- Proposition 4: Proved that $E_{UR}(R') > E_{UR}(R)$ for a broadcast version $R'$ . Unlike the behaviour case, this proof utilized an inequality by Piani (related to the relative entropy of entanglement) rather than the chain rule. It demonstrated that the "steering" resource increases strictly upon broadcasting.
Significance: This bridges the gap between the no-broadcasting of quantum states and the no-broadcasting of classical correlations, treating assemblages as the intermediate object.

4. Significance and Implications

Foundational Insight: The results confirm that the impossibility of broadcasting is not unique to quantum mechanics but is a fundamental feature of any non-classical theory satisfying the no-signalling principle. It characterizes non-classicality (non-locality and steerability) as a resource that cannot be locally amplified.
Operational Resource Theory: By defining specific free transformations (LOSR) and monotones (Relative Entropy), the paper strengthens the resource-theoretic framework for non-locality and steering. It establishes that these resources are "consumable" but not "copyable" locally.
Security Implications: Since non-locality and steerability are crucial for Device-Independent Quantum Key Distribution (DI-QKD) and randomness generation, these results reinforce the security of such protocols. An eavesdropper cannot locally clone or broadcast the non-classical correlations to gain information without destroying the resource or being detected.
Generalization: The work resolves the conjecture by Joshi et al. for general scenarios, moving beyond the restrictive $(2,2,2)$ Bell scenario. It also unifies the treatment of behaviours (purely classical statistics) and assemblages (hybrid quantum-classical) under a single theoretical umbrella.

Conclusion

Solymos et al. provide a rigorous proof that non-local behaviours and steerable assemblages cannot be locally broadcast. By leveraging the monotonicity of relative entropy measures and demonstrating that broadcasting inherently increases the "distance" from the set of local/unsteerable objects, they establish a fundamental limit on the manipulation of non-classical correlations. This confirms that the no-broadcasting principle is a universal feature of non-signalling theories, extending the foundational understanding of quantum information limits.