Holevo's theorem

In physics, in the area of quantum information theory, Holevo's theorem (sometimes called Holevo's bound, since it establishes an upper bound) is an important limitative theorem in quantum computing which was published by Alexander Holevo in 1973. According to the theorem, the amount of information accessible given a quantum state $ho$ is limited by its "Holevo information":$S(\; ho)\; -sum\_\{i\}^\{\}\; p(i)\; S(\; ho\_i)$where $S(\; ho)=-operatorname\{tr\}\; holog\_2\; ho$ is the von Neumann entropy, $ho=sum\_\{i\}\; p(i)\; ho\_i$ and $ho\_i$ are the states used to encode the information under the prior distribution $p(i)$.

In essence, it proves that "n" qubits can represent only up to "n" classical (non-quantum encoded) bits. This is surprising, for two reasons: quantum computing is so often more powerful than classical computing, that results which show it to be only as good or inferior to conventional techniques are unusual, and because it takes $2^n-1$ complex numbers to encode the qubits which represent a mere "n" bits.

References

*Mathematical Sciences Research Institute Holevo's theorem and its implications for quantum communication and computation [http://www.msri.org/publications/ln/msri/2000/qcomputing/nayak/1/index.html]