# Quantum information

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) =  − Trρlogρ.

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 2nS(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 ∣ψAR. Faithful compression of ρ will then produce a state which has high fidelity with ∣ψAR.

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).