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 $\backslash rho\_i$. 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 $\backslash rho\_i$. Then, if each message occurs with probability $p\_i$, the receiver of the message will get the quantum state $\backslash rho=\backslash sum\_i\; p\_i\; \backslash rho\_i$ The [[Holevo quantity]] $\backslash chi$ is subsequently defined as $\backslash chi=S(\backslash rho)-\backslash sum\_i\; p\_i\; S(\backslash rho\_i)$ where $S$ is the [[von Neumann entropy]]. By [[convexity]] of the von Neumann entropy, the Holevo quantity $\backslash chi$ is always positive. Moreover, Holevo showed that $\backslash chi$ 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 $\backslash chi$ 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. {{stub}} [[Category:Quantum Information Theory]] [[Category:Handbook of Quantum Information]]