No-broadcast theorem

Short description: Theorem of quantum information processing

In physics, the no-broadcast theorem is a result of quantum information theory. In the case of pure quantum states, it is a corollary of the no-cloning theorem: since quantum states cannot be copied in general, they cannot be broadcast. Here, the word "broadcast" is used in the sense of conveying the state to two or more recipients. For multiple recipients to each receive the state, there must be, in some sense, a way of duplicating the state. The no-broadcast theorem generalizes the no-cloning theorem for mixed states. The no-cloning theorem says that it is impossible to create two copies of an unknown state given a single copy of the state.

The no-broadcast theorem says that, given a single copy of a state drawn from a restricted non-commuting set, it is impossible to create a state such that one part of it is the same as the original state and the other part is also the same as the original state. That is, given an initial state [math]\displaystyle{ \rho_1, }[/math] it is impossible to create a state [math]\displaystyle{ \rho_{AB} }[/math] in a Hilbert space [math]\displaystyle{ H_A \otimes H_B }[/math] such that the partial trace [math]\displaystyle{ Tr_A\rho_{AB} = \rho_1 }[/math] and [math]\displaystyle{ Tr_B\rho_{AB} = \rho_1 }[/math]. Remarkably, the theorem does not hold if more than one copy of the initial state are provided: for example, broadcasting six copies starting from four copies of the original state is allowed, even if the states are drawn from a non-commuting set. The purity of the state can even be increased in the process, a phenomenon known as superbroadcasting.

See also

• No-communication theorem
• No-hiding theorem
• Quantum teleportation
• Quantum entanglement
• Quantum information
• Uncertainty principle

References

• Barnum, Howard; Caves, Carlton M.; Fuchs, Christopher A.; Jozsa, Richard; Schumacher, Benjamin (1996-04-08). "Noncommuting Mixed States Cannot Be Broadcast". Physical Review Letters (American Physical Society (APS)) 76 (15): 2818–2821. doi:10.1103/physrevlett.76.2818. ISSN 0031-9007.