# 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

ρ = ∑ipiρi

The Holevo quantity χ is subsequently defined as

χ = S(ρ) − ∑ipiS(ρi)

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 Holevo1973.

Holevo Holevo1998 and, independently, Schumacher and Westmorland SchumacherWestmorland1997 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.