No-Broadcasting Theorem

A finite-dimensional family of density operators is exactly broadcastable by one quantum channel if and only if its members commute pairwise. The canonical No-Broadcasting Theorem page owns the constructive channel, necessity proof architecture, examples, and edge cases.

Helpful background. For the constructive proof and noncommuting obstruction behind this lookup card, consult the canonical No-Broadcasting Theorem page.

Channel statement

A CPTP map $\mathcal E$ broadcasts a family $\{\rho_i\}$ when

$$\widetilde\rho_i^{AB}=\mathcal E(\rho_i)$$

has both marginals equal to the input:

$$\operatorname{Tr}_B\widetilde\rho_i^{AB}=\rho_i, \qquad \operatorname{Tr}_A\widetilde\rho_i^{AB}=\rho_i$$

for every $i$. Such a common channel exists exactly when

$$[\rho_i,\rho_j]=0$$

for every pair in the family.

Meaning

Commuting states share an eigenbasis and behave as classical probability distributions over that basis. A channel can read the basis label and create two correlated records whose marginals reproduce each input state. Noncommuting states admit no exact deterministic common broadcasting channel.

Broadcasting is weaker than cloning. Cloning requires the product output $\rho_i\otimes\rho_i$; broadcasting permits correlations between the outputs and constrains only their marginals. For pure states, a pure marginal cannot be correlated with another system, so broadcasting reduces to cloning. Distinct pure states are jointly broadcastable only when they are orthogonal.

Scope

• One common channel must work for the entire specified family.
• A singleton family is trivially broadcastable.
• The theorem is exact and deterministic.
• Commuting mixed-state families can be broadcast.
• Approximate, probabilistic, and asymptotic variants are different tasks.

Quick check

Can two diagonal qubit states with different eigenvalues be broadcast by one channel?

Solution

Yes. They commute because they are diagonal in the same basis. A channel can measure that basis and write the classical outcome into two correlated output registers; each marginal then reproduces the original diagonal distribution.

References

• H. Barnum, C. M. Caves, C. A. Fuchs, R. Jozsa, and B. Schumacher, “Noncommuting mixed states cannot be broadcast,” Physical Review Letters 76, 2818–2821 (1996).
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th anniversary ed., Cambridge University Press (2010).
• J. Watrous, The Theory of Quantum Information, Cambridge University Press (2018).