No-Broadcasting as Principle

Broadcasting produces two output systems whose individual states each equal the input state. It allows correlations between the outputs, so it is weaker than cloning. Its foundational importance is that finite-dimensional quantum states admit a common broadcasting channel exactly when they commute. The mathematical theorem and proof belong to No-Broadcasting Theorem.

Marginal copying rather than independent copies

For a set of states $\mathcal S$, a broadcasting channel $\mathcal B$ must satisfy

$$\operatorname{Tr}_B\mathcal B(\rho)=\rho, \qquad \operatorname{Tr}_A\mathcal B(\rho)=\rho \quad(\rho\in\mathcal S).$$

The same channel must work for every member of the specified set. If a different channel could be chosen after learning the input state, state preparation would trivialize the question.

Cloning imposes the stronger condition $\mathcal B(\rho)=\rho\otimes\rho$. A correlated output can broadcast without cloning.

A commuting example

Suppose all states are diagonal in a common basis: $\rho=\sum_i p_i\lvert i\rangle\langle i\rvert$. Measuring that basis and preparing the same classical label at both outputs gives

$$\mathcal B(\rho)= \sum_i p_i \lvert i\rangle\langle i\rvert \otimes \lvert i\rangle\langle i\rvert.$$

Each marginal is $\rho$, but the joint distribution correlates the labels perfectly. For a fair bit, it assigns probability $1/2$ each to $00$ and $11$, whereas $\rho\otimes\rho$ assigns $1/4$ to all four pairs.

The commuting family therefore behaves like classical probability for this task. The theorem says that introducing genuinely noncommuting states obstructs a single exact broadcasting channel.

The reconstruction lesson

“No broadcasting” can express nonclassicality without assuming pure-state language or an attempted unitary copier. In an algebraic setting, universal broadcastability is closely tied to commutativity. Clifton, Bub, and Halvorson use information-theoretic restrictions within a C*-algebraic framework to constrain physical theory.

The starting framework must remain visible. Many generalized probabilistic theories also forbid universal broadcasting. The prohibition alone does not derive complex Hilbert space, a particular tensor product, or the Born rule. It supplies one structural discriminator.

Approximate broadcasting is a quantitative optimization problem and is not excluded by the exact theorem. Similarly, broadcasting a restricted commuting subset does not imply universal broadcastability of the whole system.

Exercises

A machine measures $Z$ and prepares two copies of its recorded eigenstate. What happens to an input $\lvert+\rangle$?

Solution

Each output marginal becomes $I/2$, not $\lvert+\rangle\langle+\rvert$. The machine broadcasts the $Z$-diagonal family but destroys the coherence needed for the $X$-eigenstate. This connects the theorem to the distinction between effects and measurement disturbance.

References

• Barnum, Howard, Carlton M. Caves, Christopher A. Fuchs, Richard Jozsa, and Benjamin Schumacher, “Noncommuting mixed states cannot be broadcast,” Physical Review Letters 76, 2818–2821 (1996). Paper.
• Clifton, Rob, Jeffrey Bub, and Hans Halvorson, “Characterizing quantum theory in terms of information-theoretic constraints,” Foundations of Physics 33, 1561–1591 (2003). Paper.