Holevo boundtheorem

In 1973, Soviet mathematician Alexander Holevo from the Steklov Mathematical Institute proved his famous theorem. It states that the amount of classical information accessible to an observer about a quantum state does not exceed the von Neumann entropy of that state (a measure of its uncertainty) minus the average entropy of the states composing the ensemble. This work became a cornerstone of quantum information theory.

If messages are transmitted using quantum particles (e.g., photons), the amount of classical information (ordinary bits) that the receiver can extract from measurements is limited. The Holevo bound says: n quantum bits (qubits) cannot carry more than n classical bits of information. This is like a USB flash drive that physically cannot hold more data than its capacity, no matter how cunningly one tries to encode it.

The Holevo theorem (or Holevo bound) is a fundamental inequality in quantum information theory that limits the mutual information I(X;Y) between a classical random variable X and the result of a measurement Y on a quantum state. If the sender sends states with probabilities p_i and density matrices ρ_i, then the receiver, by performing any measurement, can extract at most χ = S(∑ p_i ρ_i) — ∑ p_i S(ρ_i) bits of information. Here S(ρ) = –Tr(ρ log₂ ρ) is the von Neumann entropy (the quantum analogue of Shannon entropy, quantitatively characterizing the uncertainty of the state). An important corollary: one qubit cannot transmit more than one classical bit.