Just as classical information is the amount of classical communication needed to convey a classical message, **quantum information** can be defined as the amount of quantum communication needed to convey unknown quantum states.

Imagine that a party is given a sequence of unknown quantum states chosen randomly from some distribution denoted by the density matrix *ρ*. Then Ben Schumacher showed that the number of qubits needed to faithfully transmit these states is given by von Neumann entropy

*S*(*ρ*) = − *T**r**ρ*log*ρ*.

Essentially, given a sample of *n* such states, one can compress the states, so that they reside on a smaller Hilbert space of dimension just over 2^{nS(ρ)}.

Note that the compression works even though one doesn't know what the states are, nor from what ensemble they are chosen from. One only knows the density matrix *ρ*. An elegent way to think of this is to think of *ρ* to be on system *A*, and consider its purification, on a reference system *R*, such that one has the pure state ∣*ψ*⟩_{AR}. Faithful compression of *ρ* will then produce a state which has high fidelity with ∣*ψ*⟩_{AR}.

The protocol for Schumacher's compression scheme is as follows: ....someone to insert....

-Jono 23:18, 9 Dec 2005 (GMT)

### References

Schumacher, B. Quantum coding. Phys. Rev. A 51, 2738–-2747 (1995).

### See also

Category:Quantum Information Theory Category:Handbook of Quantum Information