# The Holevo bound

The **Holevo bound** puts an upper limit on how much information can be contained in a quantum system, using a particular ensemble. Essentially it says that one qubit can contain at most one bit of information.

For example, consider a classical message, labelled by the index *i*, to be encoded in a quantum state represented by density matrix *ρ**i*. Let us further assume that a serise of such classical messages is transmitted through a channel with each output state the same as the input state *ρ**i*. Then, if each message occurs with probability *p**i*, the receiver of the message will get the quantum state

\rho=\sum_i p_i \rho_i

The Holevo quantity *χ* is subsequently defined as

\chi=S(\rho)-\sum_i p_i S(\rho_i)

where *S* is the von Neumann entropy. By convexity of the von Neumann entropy, the Holevo quantity *χ* is always positive. Moreover, Holevo showed that *χ* gives the upper bound on the classical capacity of the channel 1.

Holevo 2 and, independently, Schumacher and Westmorland 3 were able to show that the rate *χ* is asymptotically achievable and, therefore, gives the classical capacity of the quantum channel. This result is known as the HSW theorem. Consequently, although a quantum state of *n* qubits can be thought to represent a large amount of information, in the sense that the state is specified by 2*n* − 1 complex numbers, in fact, such a state can communicate at most *n* bits of decodable information.

Category:Quantum Information Theory Category:Handbook of Quantum Information