In physics

Physics

Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...

, in the area of quantum information theory, Holevo's theorem (sometimes called Holevo's bound, since it establishes an upper bound

Upper bound

In mathematics, especially in order theory, an upper bound of a subset S of some partially ordered set is an element of P which is greater than or equal to every element of S. The term lower bound is defined dually as an element of P which is lesser than or equal to every element of S...

) 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 is limited by its Holevo information,
where is the von Neumann entropy

Von Neumann entropy

In quantum statistical mechanics, von Neumann entropy, named after John von Neumann, is the extension of classical entropy concepts to the field of quantum mechanics....

 and the quantum state is defined in terms of the encoded state with a prior distribution . When the state is interrogated using measurements described by a set of POVM

POVM

In functional analysis and quantum measurement theory, a POVM is a measure whose values are non-negative self-adjoint operators on a Hilbert space. It is the most general formulation of a measurement in the theory of quantum physics...

 elements , then the mutual information

Mutual information

In probability theory and information theory, the mutual information of two random variables is a quantity that measures the mutual dependence of the two random variables...

  between the encoded states (indexed by ) and the measurement outcomes (index by ) is bounded from above by the Holevo quantity as:



In essence, the Holevo bound proves that n qubit

Qubit

In quantum computing, a qubit or quantum bit is a unit of quantum information—the quantum analogue of the classical bit—with additional dimensions associated to the quantum properties of a physical atom....

s can represent only up to n classical (non-quantum encoded) bit

Bit

A bit is the basic unit of information in computing and telecommunications; it is the amount of information stored by a digital device or other physical system that exists in one of two possible distinct states...

s. 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 complex number

Complex number

A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

s to encode the qubits which represent a mere n bits.