# 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 $\rho$. Then Ben Schumacher showed that the number of qubits needed to faithfully transmit these states is given by von Neumann entropy $S\left(\rho\right)=-Tr\rho\log\rho$. 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 $2^\left\{nS\left(\rho\right)\right\}$. 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 $\rho$. An elegent way to think of this is to think of $\rho$ to be on system $A$, and consider its purification, on a reference system $R$, such that one has the pure state $|\psi\rangle_\left\{AR\right\}$. Faithful compression of $\rho$ will then produce a state which has high fidelity with $|\psi\rangle_\left\{AR\right\}$. The protocol for Schumacher's compression scheme is as follows: ....someone to insert.... -Jono 23:18, 9 Dec 2005 (GMT) ==References== Schumacher, B. [http://prola.aps.org/abstract/PRA/v51/i4/p2738_1 Quantum coding]. Phys. Rev. A 51, 2738–-2747 (1995). ==See also== Partial quantum information Category:Quantum Information Theory Category:Handbook of Quantum Information