# 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 ρi. 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 ρi. Then, if each message occurs with probability pi, 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 2n − 1 complex numbers, in fact, such a state can communicate at most n bits of decodable information.

1. Holevo1973
2. Holevo1998
3. SchumacherWestmorland1997