Holevo's theorem

As for several concepts in quantum information theory, accessible information is best understood in terms of a 2-party communication. So we introduce two parties, Alice and Bob. Alice has a classical random variable X, which can take the values {1, 2, ..., n} with corresponding probabilities {p₁, p₂, ..., p_n}. Alice then prepares a quantum state, represented by the density matrix ρ_X chosen from a set {ρ₁, ρ₂, ... ρ_n}, and gives this state to Bob. Bob's goal is to find the value of X, and in order to do that, he performs a measurement on the state ρ_X, obtaining a classical outcome, which we denote with Y. In this context, the amount of accessible information, that is, the amount of information that Bob can get about the variable X, is the maximum value of the mutual information I(X : Y) between the random variables X and Y over all the possible measurements that Bob can do.[1]

There is currently no known formula to compute the accessible information. There are however several upper bounds, the best-known of which is the Holevo bound, which is specified in the following theorem.[1]

Let {ρ₁, ρ₂, ..., ρ_n} be a set of mixed states and let ρ_X be one of these states drawn according to the probability distribution P = {p₁, p₂, ..., p_n}.

Then, for any measurement described by POVM elements {E_Y} and performed on ${\displaystyle \rho =\sum _{X}p_{X}\rho _{X}}$, the amount of accessible information about the variable X knowing the outcome Y of the measurement is bounded from above as follows:

${\displaystyle I(X:Y)\leq S(\rho )-\sum _{i}p_{i}S(\rho _{i})}$

where ${\displaystyle \rho =\sum _{i}p_{i}\rho _{i}}$ and ${\displaystyle S(\cdot )}$ is the von Neumann entropy.

The quantity on the right hand side of this inequality is called the Holevo information or Holevo χ quantity:

${\displaystyle \chi$ :=S(\rho )-\sum _{i}p_{i}S(\rho _{i})} .

The proof can be given using three quantum systems, called ${\displaystyle P,Q,M}$. ${\displaystyle P}$ can be intuitively thought as the preparation, ${\displaystyle Q}$ can be thought as the quantum state prepared by Alice and given to Bob, and ${\displaystyle M}$ can be thought as Bob's measurement apparatus.

The compound system ${\displaystyle P\otimes Q\otimes M}$ at the beginning is in the state

${\displaystyle \rho ^{PQM}:=\sum _{x}p_{x}|x\rangle \langle x|\otimes \rho _{x}\otimes |0\rangle \langle 0|}$

Alice's state ${\displaystyle P}$ can be thought of as Alice having the value ${\displaystyle x}$ for the random variable ${\displaystyle X}$. Then the preparation state is the mixed state described by the density matrix ${\displaystyle \sum _{x}p_{x}|x\rangle \langle x|}$, and the quantum state given to Bob is ${\displaystyle \sum _{x}p_{x}\rho _{x}}$, and Bob's measurement apparatus is in its initial or rest state ${\displaystyle |0\rangle }$. Using known results of quantum information theory it can be shown that

${\displaystyle S(P';M')\leq S(P;Q)}$

which, after some algebraic manipulation, can be shown to be equivalent to the statement of the theorem.[1]

In essence, the Holevo bound proves that given n qubits, although they can "carry" a larger amount of (classical) information (thanks to quantum superposition), the amount of classical information that can be retrieved, i.e. accessed, can be only up to n classical (non-quantum encoded) bits. This is surprising, for two reasons: (1) 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 (2) because it takes ${\displaystyle 2^{n}}$ complex numbers to encode the qubits that represent a mere n bits.

• Superdense coding

• Holevo, Alexander S. (1973). "Bounds for the quantity of information transmitted by a quantum communication channel". Problems of Information Transmission. 9: 177–183.
• Nielsen, Michael A.; Chuang, Isaac L. (2000). Quantum Computation and Quantum Information. Cambridge, UK: Cambridge University Press. ISBN 978-0-521-63235-5. OCLC 43641333. (see page 531, subsection 12.1.1 - equation (12.6) )
• Wilde, Mark M. (2011). "From Classical to Quantum Shannon Theory". arXiv:1106.1445v2 [quant-ph].CS1 maint: ref=harv (link). See in particular Section 11.6 and following. Holevo's theorem is presented as exercise 11.9.1 on page 288.